Geometric Harmonic Analysis II: Function Spaces Measuring Size and Smoothness on Rough Sets [1 ed.] 3031137175, 9783031137174, 9783031137204, 9783031137181

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Developments in Mathematics Volume 73

Series Editors Krishnaswami Alladi, Department of Mathematics, University of Florida, Gainesville, FL, USA Pham Huu Tiep, Department of Mathematics, Rutgers University, Piscataway, NJ, USA Loring W. Tu, Department of Mathematics, Tufts University, Medford, MA, USA

Aims and Scope The Developments in Mathematics (DEVM) book series is devoted to publishing well-written monographs within the broad spectrum of pure and applied mathematics. Ideally, each book should be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High-quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVM series appeals to a variety of audiences including researchers, postdocs, and advanced graduate students.

Dorina Mitrea Irina Mitrea Marius Mitrea •

•

Geometric Harmonic Analysis II Function Spaces Measuring Size and Smoothness on Rough Sets

123

Dorina Mitrea Department of Mathematics Baylor University Waco, TX, USA

Irina Mitrea Department of Mathematics Temple University Philadelphia, PA, USA

Marius Mitrea Department of Mathematics Baylor University Waco, TX, USA

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-031-13717-4 ISBN 978-3-031-13718-1 (eBook) https://doi.org/10.1007/978-3-031-13718-1 Mathematics Subject Classification: 32A, 26B20, 31B, 35J, 42B © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated with love to our parents

Prefacing the Full Series

The current work is part of a series, comprised of five volumes, [133], [134], [135], [136], [137]. In broad terms, the principal aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. In Volume I ([133]) we establish a sharp version of Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense. Volume II ([134]) is concerned with function spaces measuring size and/or smoothness, such as Hardy spaces, Besov spaces, Triebel–Lizorkin spaces, Sobolev spaces, Morrey spaces, Morrey-Campanato spaces, spaces of functions of Bounded Mean Oscillations, etc., in general geometric settings. Work here also highlights the close interplay between differentiability properties of functions and singular integral operators. The topic of singular integral operators is properly considered in Volume III ([135]), where we develop a versatile Calderón-Zygmund theory for singular integral operators of convolution type (and with variable coefficient kernels) on uniformly rectifiable sets in the Euclidean ambient, and the setting of Riemannian manifolds. Applications to scattering by rough obstacles are also discussed in this volume. In Volume IV ([136]) we focus on singular integral operators of boundary layer type which enjoy more specialized properties (compared with generic, garden variety singular integral operators treated earlier in Volume III). Applications to Complex Analysis in several variables are subsequently presented, starting from the realizations that many natural integral operators in this setting, such as the Bochner-Martinelli operator, are actual particular cases of double layer potential operators associated with the complex Laplacian. In Volume V ([137]), where everything comes together, finer estimates for a certain class of singular integral operators (of chord-dot-normal type) are produced in a manner which indicates how their size is affected by the (infinitesimal and global) flatness of the “surfaces” on which they are defined. Among the library of double layer potential operators associated with a given second-order system, we then identify those double layers which fall under this category of singular integral vii

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operators. It is precisely for this subclass of double layer potentials that Fredholm theory may then be implemented assuming the underlying domain has a compact boundary, which is sufficiently flat at infinitesimal scales. For domains with unbounded boundaries, this very category of double layer potentials may be outright inverted, using a Neumann series argument, assuming the “surface” in question is sufficiently flat globally. In turn, this opens the door for solving a large variety of boundary value problems for second-order systems (involving boundary data from Muckenhoupt weighted Lebesgue spaces, Lorentz spaces, Hardy spaces, Sobolev spaces, BMO, VMO, Morrey spaces, Hölder spaces, etc.) in a large class of domains which, for example, are allowed to have spiral singularities (hence more general than domains locally described as upper-graphs of functions). In the opposite direction, we show that the boundary value problems formulated for systems lacking such special layer potentials may fail to be Fredholm solvable even for really tame domains, like the upper half-space, or the unit disk. Save for the announcement [132], all principal results appear here in print for the first time. We close with a short epilogue, attempting to place the work undertaken in this series into a broader picture. The main goal is to develop machinery of geometric harmonic analysis flavor capable of ultimately dealing with boundary value problems of a very general nature. One of the principal tools (indeed, the piecè de résistance) in this regard is a new and powerful version of the Divergence Theorem, devised in Volume I, whose very formulation has been motivated and shaped from the outset by its eventual applications to Harmonic Analysis, Partial Differential Equations, Potential Theory, and Complex Analysis. The fact that its footprints may be clearly recognized in the makeup of such a diverse body of results, as presented in Volumes II-V, serves as testament to the versatility and potency of our brand of Divergence Theorem. Alas, our enterprise is multifaceted, so its ssuccess is crucially dependent on many other factors. For one thing, it is necessary to develop a robust Calderón-Zygmund theory for singular integrals of boundary layer type (as we do in Volumes III-IV), associated with generic weakly elliptic systems, capable of accommodating a large variety of function spaces of interest considered in rather inclusive geometric settings (of the sort discussed in Volume II). This renders these (boundary-to-domain) layer potentials useful mechanisms for generating lots of null-solutions for the given system of partial differential operators, whose format is compatible with the demands in the very formulation of the boundary value problem we seek to solve. Next, in order to be able to solve the boundary integral equation to which matters are reduced in this fashion, the success of employing Fredholm theory hinges on the ability to suitably estimate the essential norms of the (boundary-to-boundary) layer potentials. In this vein, we succeed in relating the distance from such layer potentials to the space of compact operators to the flatness of the boundary of the domain in question (measured in terms of infinitesimal mean oscillations of the unit normal) in a desirable manner which shows that, in a precise quantitative fashion, the flatter the domain the smaller the proximity to compact operators. This subtle and powerful result, bridging between analysis and geometry, may be regarded as a far-reaching extension of the pioneering work of Radon and Carleman in the early 1900’s.

Prefacing the Full Series

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Ultimately, our work aligns itself with the program stemming from A.P. Calderón’s 1978 ICM plenary address in which he advocates the use of layer potentials “for much more general elliptic systems [than the Laplacian]” – see [24, p.90], and may be regarded as an optimal extension of the pioneering work of E.B. Fabes, M. Jodeit, and N.M. Rivière in [50] (where layer potential methods have been first used to solve boundary value problems for the Laplacian in bounded C1 domains). In this endeavor, we have been also motivated by the problem1 posed by A.P. Calderón on [24, p. 95], asking to identify the function spaces on which singular integral operators (of boundary layer type) are well defined and continuous. This is relevant since, as Calderón mentions, “A clarification of this question would be very important in the study of boundary value problems for elliptic equations [in rough domains]. The methods employed so far seem to be insufficient for the treatment of these problems.” We also wish to mention that our work is also in line with the issue raised as an open problem by C. Kenig in [113, Problem 3.2.2, pp. 116–117], where he asked whether operators of layer potential type may be inverted on appropriate Lebesgue and Sobolev spaces in suitable subclasses on NTA domains with compact Ahlfors regular boundaries. The task of making geometry and analysis work in unison is fraught with difficulties, and only seldom can a two-way street be built on which to move between these two worlds without loss of information. Given this, it is actually surprising that in many instances we come very close to having optimal hypotheses, almost an accurate embodiment of the slogan if it makes sense to write it, then it’s true. Acknowledgments: The authors gratefully acknowledge partial support from the Simons Foundation (through grants # 426669, # 958374, # 318658, # 616050, # 637481), as well as NSF (grant # 1900938). Portions of this work have been completed at Baylor University in Waco, Temple University in Philadelphia, the Institute for Advanced Study in Princeton, MSRI in Berkeley, and the American Institute of Mathematics in San Jose. We wish to thank these institutions for their generous hospitality. Last, but not least, we are grateful to Michael E. Taylor for gently yet persistently encouraging us over the years to complete this project. Waco, USA Philadelphia, USA Waco, USA

Dorina Mitrea Irina Mitrea Marius Mitrea

In the last section of [24], simply titled “Problems,” Calderón singles two directions for further study. The first one is the famous question whether the smallness condition on ka0 kL1 (the Lipschitz constant of the curve fðx; aðxÞÞ : x 2 Rg on which he proved the L2 -boundedness of the Cauchy operator) may be removed (as is well known, this has been solved in the affirmative by Coifman, McIntosh, and Meyer in [32]). We are referring here to the second (and final) problem formulated by Calderón on [24, p. 95]. 1

Description of Volume II

An enthralling aspect of modern mathematics is the study of how analysis and geometry affect one another. Webster’s dictionary defines the word ‘analysis’ as a breaking up of a whole into its parts as to find out their nature. This point of view is emblematic of a fundamental principle in Harmonic Analysis, in which one decomposes a mathematical entity (such as a function/distribution, or an operator) into simpler building blocks (enjoying certain specialized properties, typically having to do with localization, cancellation, and size), analyzing these smaller pieces individually, and then organizing this compartmentalized information in a global, coherent manner, in order to derive conclusions about the original object of study. Such a thesis goes at least as far back as the ground-breaking work of J. Fourier in the early 1800’s, who had the vision of superimposing sine and cosine curves with various amplitudes in an attempt to obtain the shape of the graph of a fairly arbitrary given function. In this endeavor, the goal is then to understand the quantitative correlation between the nature of said amplitudes (aka Fourier coefficients) on the one hand, and the functional-analytic properties of the original function (such as membership to L2 or Lp ), on the other hand. This technology, as a means of measuring size and smoothness for a given function, has received further impetus through the advent of the Littlewood–Paley theory (particularly in relation to the Lp -setting with p 6¼ 2), eventually culminating in the development of the modern theory of function spaces of Triebel–Lizorkin and Besov type. Yet another manifestation of Fourier’s pioneering ideas, that has fundamentally altered the landscape of contemporary Harmonic Analysis, is the theory of Hardy spaces viewed through the lenses of atomic and molecular techniques. In this scenario, the so-called atoms and molecules play the role of the sine and cosine building blocks (though, this times, they form an “overdetermined basis” as opposed to a genuine linear basis). Originally considered in the work of R.R. Coifman in [31] (on the real line), and R.H. Latter in [118] (in the higher-dimensional setting), then benefiting from advances due to many authors (see, e.g., [9], [18], [25], [36], [40], [56], [62], [71], [100], [88], [121], [138], [195], [176], [181], [182], [205], as well as

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the references therein) this body of work has matured into a complex, multifaceted theory, with deep and far-reaching implications in many areas of mathematics. To put matters into a broader perspective it is worth recalling that the theory of H p spaces has been originally developed as a bridge between complex function theory and Fourier Analysis, two branches of mathematics which tightly interfaced with one-another in the lower dimensional setting (where methods of complex function theory such as Blaschke products and conformal mappings played a decisive role). It was natural that higher-dimensional extensions of this theory would be sought involving systems of harmonic functions in the upper half-space satisfying natural generalizations of the Cauchy-Riemann equations (cf. [178]) and having their nontangential maximal functions in Lp . This is the route taken in [52], building on the earlier work in [177], [175], [179]. In this theory of harmonic H p spaces the focus shifts from the harmonic functions themselves to their boundary values, themselves tempered distributions in Rn , from which the harmonic functions can then be recovered via Poisson’s integral formula. In their pioneering work in [52], C. Fefferman and E. Stein have shown that the n-dimensional Hardy spaces in the Euclidean setting, developed in [177], have purely real-variable characterizations as the space of tempered distributions in Rn having some sort of maximal function belonging to Lp ðRn Þ. Such Ra maximal operator may be fashioned out of a fixed Schwartz function U with U 6¼ 0 (or even the classical Poisson kernel, as historically has been the case). Alternatively, one can take into account “all” such possible approximate identities, thus creating the so-called grand maximal function. More specifically, for each tempered distribution f 2 S0 ðRn Þ, its grand maximal function is defined at each x 2 Rn as f ] ðxÞ :¼ sup sup jðf Ut ÞðxÞj; U2A t [ 0

ð0:0:1Þ

where A is a family of suitably normalized Schwartz functions defined in Rn , and Ut :¼ tn Uð=tÞ, for each U 2 A and t 2 ð0; 1Þ. Concerning the role of geometry, one fundamental development (from the perspective of the work undertaken here) has been the consideration of environments much more general than the Euclidean ambient, through the introduction of the so-called spaces of homogeneous type.2 Indeed, by the late 1970’s it has become increasingly evident that a large portion of basic real analysis (such as covering lemmas, the Hardy-Littlewood maximal operator, functions of bounded mean oscillation, Lebesgue’s differentiation type theorem, Whitney decompositions, and even singular integral operators of Calderón-Zygmund type, among other things) requires relatively little structure from the geometric ambient. The emergence of the theory of Hardy spaces in spaces of homogeneous type perhaps best typifies these developments. For example, one may introduce Hardy spaces in such a setting by 2

Basic references in this regard are [35] and [36], which have retained their significance many decades after appearing in print.

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naturally adapting the definition of the Fefferman–Stein grand maximal function reviewed in (0.0.1), now considering for any “distribution” f the function defined pointwise as   f ] ðxÞ :¼ sup hU; f i for each x: U2T ðxÞ

ð0:0:2Þ

Above, h; i is interpreted as the duality paring between the space of Lipschitz functions with bounded support and its dual, while T ðxÞ (playing the role of the family A in the earlier discussion) is a collection of suitably normalized “test functions” concentrated near the point x (i.e., Hölder continuous functions, with a bounded support containing x, relative to which their size is correlated). This is the point of view adopted in Chapter 4 where, in keeping with the program initiated by R. Coifman and G. Weiss in the 1970’s, we develop a theory of (Lebesgue-based and Lorentz-based) Hardy spaces on Ahlfors regular subsets of Rn , by considering basic topics such as: the Fefferman–Stein grand maximal function, real interpolation of Lebesgue-based and Lorentz-based Hardy spaces, atoms and molecules, duality, weak- convergence, and the compatibility of various duality pairings, among other things. The starting point in Chapter 5 is the consideration of a more inclusive brand of Banach function space (than traditionally used in the literature; cf. e.g., [14]), which we dub Generalized Banach Function Space. This is done in x5.1. The relevance of this extension is that a variety of scales of spaces of interest, such as Morrey spaces, block spaces, as well as Beurling algebras and their pre-duals, now fit naturally into this more accommodating label. Most significantly, in x5.2 we develop powerful and versatile extrapolation results serving as portal, allowing us to pass from estimates on Muckenhoupt weighted Lebesgue spaces (for a fixed integrability exponent and arbitrary weights) to estimates on the brand of Generalized Banach Function Spaces introduced earlier, on which the Hardy-Littlewood maximal operator happens to be bounded. Finally, in x5.3 we focus on Orlicz spaces which, in particular, are natural examples of classical Banach function spaces for which the machinery developed so far applies. Next, in Chapter 6, we introduce a natural brand of Morrey-Campanato and Morrey spaces on Ahlfors regular subsets of Rn . The function theory developed in relation to these spaces (in x6.1 and x6.2) features: embeddings, density results, atomic decompositions, duality, extrapolation, as well as the boundedness of the HardyLittlewood maximal operator and of fractional integration operators on these spaces. Chapter 7 is reserved for a discussion pertaining to the scales of Besov and Triebel–Lizorkin spaces on Ahlfors regular sets in Rn . Modeled upon their Euclidean counterparts, these scales of spaces are considered through the perspective of Littlewood–Paley theory (suitably adapted to the context of spaces of homogeneous type). Topics covered here include: atomic and molecular theory, Calderón reproducing formula and frame theory, real and complex interpolation, duality and embeddings, quasi-Banach envelopes, and various intrinsic characterizations.

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In Chapter 8 the main focus is on boundary traces from weighted Sobolev spaces in open subsets of the Euclidean ambient (in which the weight is a suitable power of the distance to the boundary) into Besov spaces (discussed in Chapter 7) that are defined on the boundary of the domain in question. In this vein, the main issues of interest are: the well-definiteness and boundedness of such a boundary trace operator, the fact that said trace operator is surjective (which we establish via constructive methods, by showing the existence of a linear and bounded extension operator), and the explicit description of its null-space. Of basic importance for future applications is that this program is carried out in a very general geometric setting. A program of somewhat similar aims is executed in Chapter 9, this time starting from Besov and Triebel–Lizorkin spaces in open subsets of the Euclidean ambient (in lieu of the weighted Sobolev spaces employed earlier). Going further, in Chapter 10 we study the strong (i.e., pointwise) and weak (i.e., distributional, in the sense of our “bullet product”, introduced in [133, §4.2]) normal traces of vector fields, seeking conditions guaranteeing membership to (Lebesgue-based and Lorentz-based) Hardy spaces on the boundary of a given rough domain. This task is first accomplished working in the Euclidean ambient (in x10.1 and x10.2), then in the setting of Riemannian manifolds (in x10.3). Our brand of Divergence Theorem developed in Volume I ([133]) is a basic ingredient in proving such results. In turn, they play a crucial role in much of the subsequent work, from Volume III ([135]) where we discuss Fatou-type results in geometric settings of a desirable degree of generality, all the way to Volume V ([137]) where normal traces (more specifically, directional derivatives along the unit normal on the boundary of a given domain) feature prominently in the very formulation of the Neumann and Transmission boundary value problems. In Chapter 11, the final chapter in this volume, we expand on the work in [97], [139], [141] by considering a scale of “boundary” Sobolev spaces, which may be meaningfully defined on the geometric measure theoretic boundary of sets of locally finite perimeter (both in the Euclidean setting and that of manifolds). It is of interest to compare our brand of Sobolev spaces with other types of Sobolev spaces on generic measure metric spaces which have been introduced and studied elsewhere in the literature (see, e.g., [79], [81] and the references there). The notion of “weak gradient” employed in defining the latter is, by necessity (given the environment considered), of a rather algebraic nature (no differential calculus is involved). In sharp contrast, our Sobolev spaces make essential use of “weak derivatives” and integration by parts along the boundary (itself a consequence of the Divergence Theorem, suitably applied). This renders them particularly useful and natural in connection with problems in Partial Differential Equations. Moreover, we do show that the two scales are compatible, in the sense that they agree with one another when considered on the boundary of an Ahlfors regular domain satisfying a two-sided local John condition. In these endeavors we shall make crucial use of the Calderón–Zygmund theory for singular integrals of layer potential type, developed in [136] (independently of the present considerations). This is not entirely surprising: indeed, it has long been known that there is a close interplay between singular integrals and

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differentiability properties of functions (the 1970 monograph [175] of E.M. Stein, is famously titled as such). While such considerations have, so far, been largely restricted to the entire Euclidean space, here we take the first steps in the direction of employing a similar philosophy for more general geometries, i.e., when the ambient is a set of more general nature. The discussion in this volume is facilitated by the preparatory material presented in Chapter 1. Here we collect and develop basic results concerning vector spaces and operator theory (in x1.1), quasi-normed spaces (in x1.2), real interpolation of quasilinear operators (in x1.3), complex interpolation of quasi-Banach spaces (in x1.4), as well as Köthe function spaces and other categories of topological vector spaces (in x1.5). Chapter 2 is concerned with Fredholm theory, not only in the standard context of Banach spaces, but also of topological vector spaces of a much more general nature (which may lack local convexity). The latter aspect is relevant in view of the fact that many mainstream scales of function spaces (such as Hardy, Besov, and Triebel–Lizorkin) contain spaces which are not necessarily Banach. In this volume we also study functions whose oscillations vanish at infinitesimal scales (in Chapter 3). When said oscillations are measured in a mean (i.e., integral) sense, this gives rise to the Sarason space VMO, discussed in x3.1. When the oscillations in question are taken in a pointwise sense (via a local Hölder semi-norm), we come across a new brand of Hölder spaces, defined in x3.2. Lastly, we wish to note that while in this volume we study function spaces as an independent topic, something of interest in its own right, our work here has also been motivated by the problem posed by A.P. Calderón on [24, p. 95], asking to identify the function spaces on which singular integral operators (of boundary layer type) are well defined and continuous. We shall look into this latter aspect in subsequent volumes, bearing in mind that, as Calderón mentions, “A clarification of this question would be very important in the study of boundary value problems for elliptic equations [in rough domains]. The methods employed so far seem to be insufficient for the treatment of these problems.”

Contents

Prefacing the Full Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Description of Volume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

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1 1 5 25 36

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Abstract Fredholm Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fredholm Theory in Banach Spaces . . . . . . . . . . . . . . . . . . . . 2.2 Fredholm Theory in Topological Vector Spaces . . . . . . . . . . .

63 63 73

3

Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Functions of Vanishing Mean Oscillations on Measure Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A New Brand of Hölder Spaces . . . . . . . . . . . . . . . . . . . . . . .

4

Preliminary Functional Analytic Matters . . . . . . . . . 1.1 Algebraic and Operator Theoretic Rudiments . . 1.2 Quasi-Normed Spaces . . . . . . . . . . . . . . . . . . . 1.3 Real Interpolation of Quasilinear Operators . . . . 1.4 Complex Interpolation on Quasi-Banach Spaces 1.5 Köthe Function Spaces and Other Categories of Topological Vector Spaces . . . . . . . . . . . . .

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Hardy Spaces on Ahlfors Regular Sets . . . . . . . . . . . . . . . . 4.1 The Fefferman-Stein Grand Maximal Function . . . . . . 4.2 Lebesgue-Based and Lorentz-Based Hardy Spaces on Ahlfors Regular Sets . . . . . . . . . . . . . . . . . . . . . . 4.3 Real Interpolation of Hardy Spaces . . . . . . . . . . . . . . 4.4 Atomic Decompositions for Hardy Spaces . . . . . . . . . 4.5 Molecules in Hardy Spaces . . . . . . . . . . . . . . . . . . . . 4.6 Duality in Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . 4.7 Richness of the Duals of Lorentz-Based Hardy Spaces

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81 82 98

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113 119 143 174 182 191

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Contents

4.8 4.9

Weak- Convergence and More on the Compatibility of Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 More on H p Versus Lp : The Filtering Operator . . . . . . . . . . . . 212 217 217 233 252

5

Banach Function Spaces, Extrapolation, and Orlicz Spaces 5.1 Generalized Banach Function Spaces . . . . . . . . . . . . . 5.2 Extrapolation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Young Functions and Orlicz Spaces . . . . . . . . . . . . . .

6

Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals on Ahlfors Regular Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

7

Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets . 7.1 Definitions with Sharp Ranges of Indices and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Atomic and Molecular Theory . . . . . . . . . . . . . . . . . . . 7.3 Calderón’s Reproducing Formula and Frame Theory . . . 7.4 Interpolation of Besov and Triebel-Lizorkin Spaces via the Real Method . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Complex Interpolation of Besov and Triebel-Lizorkin Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Duality Results for Besov and Triebel-Lizorkin Spaces . 7.7 Loose and Tight Embeddings . . . . . . . . . . . . . . . . . . . . 7.8 Envelopes of Besov and Triebel-Lizorkin Spaces . . . . . 7.9 Intrinsic Characterizations of Besov Spaces . . . . . . . . .

8

9

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Boundary Traces from Weighted Sobolev Spaces into Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Traces from Weighted Sobolev Spaces Defined in Rn . . 8.2 Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Traces from Weighted Sobolev Spaces Defined in Open Subsets of Rn . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Extension Operators from Boundary Besov Spaces into Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . 8.5 Weighted Sobolev Spaces of Negative Order, Duality, and the Conormal Derivative Operator . . . . . . . . . . . . . 8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of Rn . . . . . . . . . . . . . . . . . . . . . . . .

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394 405 408 417 446

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Besov and Triebel-Lizorkin Spaces in Open Sets . . . . . . . . . . . . . . 517 9.1 Besov and Triebel-Lizorkin Spaces in Rn . . . . . . . . . . . . . . . . 517 9.2 Besov and Triebel-Lizorkin Spaces in Open Subsets of Rn . . . 527

Contents

9.3 9.4 9.5 9.6

xix

Envelopes of Besov and Triebel-Lizorkin Spaces in Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traces of Functions from Besov and Triebel-Lizorkin Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Conormal Derivative Operator on Besov and Triebel-Lizorkin Spaces . . . . . . . . . . . . . . . . . . . . . . Extension from Boundary Besov Spaces into Smoothness Spaces in Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Strong and Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Strong Normal Boundary Traces of Vector Fields in Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces . . . . . . . . . . . . . . . . . . . . . 10.3 The Manifold Setting . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 570 . . . . 572 . . . . 592 . . . . 602 . . . . 607 . . . . 607 . . . . 634 . . . . 679

11 Sobolev Spaces on the Geometric Measure Theoretic Boundary of Sets of Locally Finite Perimeter . . . . . . . . . . . . . . . . . . . . . . . 11.1 Weak Tangential Derivatives and Boundary Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Distributional Tangential Derivatives . . . . . . . . . . . . . . . . . 11.3 Distributional Derivatives Versus Weak Tangential Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 The Tangential Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Sobolev Spaces on Boundaries of Ahlfors Regular Domains Satisfying a Two-Sided Local John Condition . . . 11.6 Boundary Sobolev Spaces on Manifolds . . . . . . . . . . . . . . . 11.7 A General Recipe for Sobolev-Like Spaces . . . . . . . . . . . . 11.8 Boundary Sobolev Spaces of Negative Smoothness . . . . . . . 11.9 Principal-Value Distributions on Countably Rectifiable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10 Hardy-Based Boundary Sobolev Spaces . . . . . . . . . . . . . . . 11.11 Embedding Results Involving Boundary Sobolev Spaces . . . 11.12 Real Interpolation Involving Boundary Sobolev Spaces . . . . 11.13 Morrey-Based, and Block-Based, Homogeneous Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 681 . . 682 . . 702 . . 705 . . 728 . . . .

. . . .

736 784 791 796

. . . .

. . . .

804 811 824 831

. . 848

Appendix A: Terms and Notation used in Volume II. . . . . . . . . . . . . . . . 869 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915 Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921

Chapter 1

Preliminary Functional Analytic Matters

To set the stage for much of the subsequent discussion, in this chapter we elaborate on basic terminology and results pertaining to vector spaces and operator theory (in §1.1), quasi-normed spaces (in §1.2), real interpolation of quasilinear operators (in §1.3), complex interpolation of quasi-Banach spaces (in §1.4), and some useful categories of topological vector spaces (in §1.5).

1.1 Algebraic and Operator Theoretic Rudiments For any given vector space X we denote by dim X its dimension. Let X be a vector space, an suppose Y is a linear subspace of X. Denote by X/Y the quotient space, consisting of all classes of equivalences [x]X/Y = x + Y with x ∈ X, naturally regarded as a vector space. Lemma 1.1.1 Let X be a vector space and suppose V, W are subspaces of X such that V ⊆ W and       dim X V = dim X W < +∞. (1.1.1) Then necessarily V = W.   Proof Consider the mapping ι : X V → X W given by ι(x + V) := x + W for every x ∈ X. Then the fact that V ⊆ W implies that ι is a well-defined, linear mapping. Clearly, ι is surjective which, in concert with (1.1.1), actually forces ι to be a bijection. Writing out what it means for ι to be injective then yields the inclusion W ⊆ V. Thus, ultimately, V = W.  Remark 1.1.2 Assume X, Y, Z are three given vector spaces with the property that Z ⊆ Y ⊆ X. Then dim(X/Z) = dim(X/Y ) + dim(Y /Z). (1.1.2) As a consequence, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1_1

1

2

1 Preliminary Functional Analytic Matters

if dim(X/Z) < +∞ then Z = Y if and only if dim(X/Z) = dim(X/Y ).

(1.1.3)

A proof of the following result may be found in [142]. Theorem 1.1.3 Let X be a Hausdorff linear topological space. (1) If Y is a closed linear subspace of X and F is a finite-dimensional linear subspace of X, then Y + F is closed in X. (2) If Y is a closed linear subspace of X of finite codimension and F is any algebraic complement of Y , then X = Y ⊕ F (direct topological sum). (3) If Z is a closed subspace of finite codimension in X, then any linear subspace Y of X with the property that Z ⊆ Y is necessarily closed and of finite codimension in X. We next discuss a few generalities of functional analytic nature. For further reference, for each linear map T : X → Y where X, Y are vector spaces we shall let Im(T : X → Y ) := {T x : x ∈ X } and Ker(T : X → Y ) := {x ∈ X : T x = 0}

(1.1.4)

denote, respectively, the image (or range) and kernel (or null-space) of the operator T. Moreover, in order to lighten notation, we shall occasionally simply write Im T, Ker T when no confusion is likely to occur. Call T a finite-rank operator1 (or an operator of finite rank, or an operator having finite rank) if Im(T : X → Y ) is a finite dimensional subspace of Y . We shall often work in the category of topological vector spaces (or linear topological spaces). As far as this setting is concerned, we first wish to note (see, e.g., the discussion in [142]) that if X is a Hausdorff linear topological space with n := dim X < ∞, then X is linearly homeomorphic to Cn (with the standard topology).

(1.1.5)

Hence, on a finite dimensional vector space X over the field C there could only be one Hausdorff vector topology; specifically, if {x1, . . . , xn } is a basis in X, a set O ⊆ X is open if and only if there exist O1, . . . , On open sets in C such that O = O1 x1 + · · · + On xn .

(1.1.6)

Recall that, given a topological vector space X, a set A ⊆ X is said to be (topologically) bounded if for each zeroneighborhood U in X there exists εU > 0 such that ε A ⊆ U for each ε ∈ (0, εU ].

(1.1.7)

1 we emphasize that, even when the spaces involved are Banach, an operator of finite rank is not necessarily continuous

1.1 Algebraic and Operator Theoretic Rudiments

3

Hence, a set in a topological vector space is (topologically) bounded provided it is absorbed by any zero-neighborhood of said space. Clearly, any subset of a bounded set is itself bounded. Let us also recall that a linear topological space X is said to be locally bounded if it has a bounded zero-neighborhood.

(1.1.8)

If X, Y are two linear topological spaces, a linear operator T : X → Y is said to be bounded if there exists a zero-neighborhood U of X such that T(U) is a bounded subset of Y . Denote the set of bounded operators from X into Y by B(X → Y ), i.e.,   B(X → Y ) := T : X −→ Y : T is linear and bounded . (1.1.9) From (1.1.9) and (1.1.7) we see that any operator T ∈ B(X → Y ) maps bounded sets into bounded sets. Under the additional assumption that X is locally bounded, the converse implication is also true. If we also consider   L(X → Y ) := T : X −→ Y : T is linear and continuous , (1.1.10) then it is clear from definitions that every bounded operator is continuous, i.e., we have2 B(X → Y ) ⊆ L(X → Y ). (1.1.11) In general, continuous operators need not be bounded, though one may check without difficulty that if X, Y are two topological vector spaces, with X locally bounded, then any linear continuous operator T : X → Y is bounded, i.e., we actually have B(X → Y ) = L(X → Y ) in this case.

(1.1.12)

In addition to (1.1.9) and (1.1.10), for any two topological vector spaces X, Y let us also define3   Cp(X → Y ) := T : X → Y : T is a linear compact mapping (1.1.13) and abbreviate Cp(X) := Cp(X → X).

(1.1.14)

It is known (cf. [142]) that if X is a Hausdorff topological vector space, then every continuous finite-rank linear operator on X is compact.

(1.1.15)

2 A companion result states that if X is a metrizable locally convex topological vector space and Y is a locally convex topological vector space, and if T : X → Y is a linear operator mapping (topologically) bounded sets into (topologically) bounded sets, then T ∈ L(X → Y). 3 recall that a mapping is said to be compact if it maps (topologically) bounded sets into relatively compact sets

4

1 Preliminary Functional Analytic Matters

Straight from definitions it may be checked that every compact operator is bounded, i.e., for any two topological vector spaces X, Y we have Cp(X → Y ) ⊆ B(X → Y ).

(1.1.16)

Also, it is clear from definitions that if X is a topological vector space, T ∈ Cp(X), and Y is a closed subspace of X which is invariant under T, then T Y ∈ Cp(Y ). Given a topological vector space X, define its (topological) dual as   X ∗ := L(X → C) = Λ : X −→ C : Λ linear and continuous .

(1.1.17)

(1.1.18)

Clearly, X ∗ is a vector space, and, as may be seen from (1.1.12), if X is a locally bounded topological vector space then one has X ∗ = L(X → C) = B(X → C).

(1.1.19)

Suppose X, Y are two topological vector spaces. For each T ∈ L(X → Y ) define T ∗ : Y ∗ −→ X ∗,

T ∗ (Λ) := Λ ◦ T for each Λ ∈ Y ∗ .

(1.1.20)

Hence, T ∗ is a well-defined linear operator. In relation to this, we have the following useful result. Lemma 1.1.4 If X, Y are two topological vector spaces, with Y Hausdorff, and T ∈ L(X → Y ) is a finite-rank operator, then T ∗ : Y ∗ → X ∗ is also a finite-rank operator. Proof Let {y1, . . . , yn } be a basis in Im(T : X → Y ). Hence, for each x ∈ X there exist unique numbers Λi (x) ∈ C, for 1 ≤ i ≤ n, such that Tx =

n 

Λi (x)yi .

(1.1.21)

i=1

From the linearity and continuity of T : X → Im(T : X → Y ), the fact that the latter is a finite dimensional Hausdorff topological vector space, and (1.1.6) we then conclude that each assignment X x → Λi (x) ∈ C is linear and continuous. Hence, Λi ∈ X ∗ for each {1, . . . , n}. Next, for any given Λ ∈ Y ∗ we rely on (1.1.20) and (1.1.21) to compute (T ∗ Λ)(x) = (Λ ◦ T)(x) = Λ(T x) = =

n  i=1

Hence,

n 

Λi (x)Λ(yi )

i=1

 Λ(yi )Λi (x) for each x ∈ X.

(1.1.22)

1.2 Quasi-Normed Spaces

5

T ∗Λ =

n 

Λ(yi )Λi for each Λ ∈ Y ∗,

(1.1.23)

i=1

which goes to show that Im(T ∗ : Y ∗ → X ∗ ) has dimension at most n.



1.2 Quasi-Normed Spaces For a quasi-normed space (X, · X ), we denote by ρ = ρ(X) its modulus of concavity, i.e., the smallest positive constant for which x + y X ≤ ρ(X)( x X + y X ),

∀x, y ∈ X.

(1.2.1)

Note that always ρ(X) ≥ 1. Also, for any linear subspace Y of X, regarded as a quasi-normed space equipped with the quasi-norm · X , we have ρ(Y ) ≤ ρ(X). Call (X, · X ) a quasi-Banach space provided X is complete wit respect to · X . It follows that any closed linear subspace of a quasi-Banach space is itself a quasi-Banach. (1.2.2) Given p ∈ (0, 1], a p-norm on a vector space X is a quasi-norm · on X which satisfies (1.2.3) x + y p ≤ x p + y p for all x, y ∈ X. It follows that if · is a p-norm, for some p ∈ (0, 1], on a vector space X then x + y ≤ 2(1/p)−1 x + y for each x, y ∈ X.

(1.2.4)

Also, a quasi-normed space X is said to be a p-Banach space for some p ∈ (0, 1] provided there exists a p-norm which is equivalent to the original quasi-norm on X, and with respect to which X is complete. We recall the Aoki-Rolewicz theorem, which asserts (cf., e.g., [112], [138]) that for each quasi-normed space X there exists an equivalent p-norm where p ∈ (0, 1] is such that 2(1/p)−1 = ρ(X), i.e., when p := (1 + log2 ρ(X))−1 .

(1.2.5)

Note that by raising both sides of the inequality in (1.2.3) to an arbitrary power θ ∈ (0, 1) and then using the general fact that (a + b)θ < a θ + bθ for

each a, b ∈ [0, ∞) leads to the conclusion that for any quasi-normed space X, · we have  x + y r ≤ x r + y r for all r ∈ 0, (1 + log2 ρ(X))−1 and all x, y ∈ X. (1.2.6) In general a quasi-norm need not be continuous but any p-norm is. Indeed, using the elementary observation that

6

1 Preliminary Functional Analytic Matters



γ−1 |aγ − bγ | ≤ γ|a − b| · max{a, b} if a, b ∈ (0, ∞) and γ ≥ 1,

(1.2.7)

for each x, y ∈ X we may estimate, using (1.2.7) with γ := 1/p ≥ 1 and (1.2.3),      x − y  = ( x p )1/p − ( y p )1/p   

(1/p)−1 ≤ p−1  x p − y p  · max{ x p, y p }

1−p ≤ p−1 x − y p · max{ x , y } ,

(1.2.8)

from which we conclude that any p-norm is locally Hölder continuous, with Hölder exponent p ∈ (0, 1].

(1.2.9)

By replacing the original quasi-norm by some equivalent p-norm (cf. (1.2.5)), there is no loss of generality in assuming that (1.2.10) all quasi-norms considered are continuous.



Moving on, given two quasi-normed vector spaces X, · X and Y, · Y , consider a positively homogeneous mapping T : X → Y , i.e., a function T sending X into Y and which satisfies T(λu) = λT(u) for each u ∈ X and each λ ∈ (0, ∞) (note that taking u := 0 ∈ X and λ := 2 implies T(0) = 0 ∈ Y ). We shall denote by   T X→Y := sup Tu Y : u ∈ X, u X = 1 ∈ [0, +∞] (1.2.11) the operator “norm” of such a mapping T. In particular, Tu Y ≤ T X→Y u X for each u ∈ X.

(1.2.12)

It is also useful to observe that for any positively homogeneous mapping T : X → Y we have T X→Y =

sup

x ∈X\{0}

T x Y = x X

sup

x ∈X, x X =1

T x Y .

(1.2.13)

It is straightforward to check that a positively homogeneous mapping T : X → Y is continuous at 0 ∈ X if and only if T is bounded (i.e., it maps bounded subsets of X into bounded subsets of Y ) if and only if T X→Y < +∞.

(1.2.14)

Consider next the special case when (X, · X ) is an arbitrary quasi-normed space, the quasi-normed space (Y, · Y ) is a sub-lattice of the space of measurable functions associated with a generic measure space, and assume that T is some sublinear mapping of X into Y , i.e., T : X → Y satisfies T(λu) = |λ|T(u) for each u ∈ X and each scalar λ, as well as T(u + w) ≤ Tu +T w at a.e. point in X, for each u, w ∈ X. Then, for each u, w ∈ X we have |Tu − T w| ≤ T(u − w) at a.e. point in X, hence

1.2 Quasi-Normed Spaces

7

Tu − T w Y ≤ T(u − w) Y ≤ T X→Y u − w X thanks to using the lattice property in Y . Consequently, if X, Y are as above then a sub-linear map T : X → Y (1.2.15) is continuous if and only if one has T X→Y < +∞.



Next, assuming that X, · X as well as Y, · Y are quasi-normed vector spaces, we set   Bd(X → Y ) := T : X → Y : T linear mapping with T X→Y < +∞ , (1.2.16) and occasionally write T Bd(X→Y) in place of T X→Y . This is a quasi-normed space which is compatible with (1.1.9), in the sense that if X, Y are quasi-normed spaces then B(X → Y ) and Bd(X → Y ) are identical. Hence, in light of (1.1.12), we have Bd(X → Y ) = B(X → Y ) = L(X → Y ) (1.2.17) whenever X, Y are quasi-normed spaces. We also agree to abbreviate Bd(X) := Bd(X → X).

(1.2.18)

In particular, given a linear and bounded operator T mapping a quasi-normed space (X, · X ) into itself, T Bd(X) =

sup

x ∈X\{0}

T x X = x X

sup

x ∈X, x X =1

T x X .

(1.2.19)

Note that if (X, · X ) is a quasi-normedspace, T ∈ Bd(X), and Y is a linear subspace of X which is invariant under T, then T Y belongs to Bd(Y ) where Y is regarded as a quasi-normed space equipped with the quasi-norm · X , and  T  ≤ T Bd(X) . (1.2.20) Y Bd(Y) As seen from definitions, ∗ if X is a quasi-normed space then Bd(X   → C) = X and Λ X ∗ = sup |Λ(x) : x ∈ X, x X = 1 for each Λ ∈ X ∗ .

(1.2.21)

A companion result, itself a well-known consequence of the Hahn-Banach Theorem, is that if X is a normed space then, for each x ∈ X,  (1.2.22) x X = sup |Λ(x) : Λ ∈ X ∗, Λ X ∗ = 1 . The following elementary result is going to be useful later on. Lemma 1.2.1 Assume that X, Y are quasi-normed spaces, such that Y ⊆ X and the inclusion j : Y → X is continuous with dense range. Then the adjoint operator

8

1 Preliminary Functional Analytic Matters

ι := j ∗ : X ∗ −→ Y ∗ acting according to ι(Λ) = j ∗ (Λ) = Λ ◦ j for every Λ ∈ X ∗ (cf. (1.1.20))

(1.2.23)

is well defined, linear, continuous, and injective. In particular, this induces a continuous embedding (1.2.24) X ∗ → Y ∗ . In addition, the embedding in (1.2.24) is in such a way that the duality pairings X ∗ ·, · X and Y ∗ ·, ·Y are compatible, in the sense that X ∗ Λ,

Λ∈

yX = Y ∗ Λ, yY for every

X∗

(1.2.25)

→ Y ∗ and y ∈ Y → X.

Finally, if actually X, Y are Banach spaces and Y is reflexive, then the embedding (1.2.24) also has dense range.

(1.2.26)

Proof That ι is well defined, linear, and continuous is clear from definitions. To show that ι is injective, assume Λ ∈ X ∗ is such that ι(Λ) = 0 in Y ∗ . Then for each y ∈ Y we have 0 = Y ∗ ι(Λ), yY = Y ∗  j ∗ (Λ), yY =

X ∗ Λ,

j(y)X

(1.2.27)

hence the functional Λ ∈ X ∗ vanishes on the image of j : Y → X. Since the latter is a dense subset of X, we conclude that Λ = 0. This proves that ι is indeed injective. As regards (1.2.25), suppose Λ ∈ X ∗ and y ∈ Y are arbitrary. Regard Λ as a functional in Y ∗ by identifying it with j ∗ (Λ), and regard y as a vector in X by identifying it with j(y) ∈ X. Then X ∗ Λ,

yX =

X ∗ Λ,

j(y)X = Y ∗  j ∗ (Λ), yY = Y ∗ Λ, yY ,

(1.2.28)

proving (1.2.25). To justify the very last claim in the statement, work under the stronger assumptions that X is a Banach space and Y is a reflexive Banach space. As a consequence of the Hahn-Banach Theorem, it suffices to show that if ∈ (Y ∗ )∗ is a functional which vanishes identically on j ∗ (X ∗ ) then necessarily = 0. Since Y is reflexive, this comes down to checking that a given vector y ∈ Y is necessarily zero if ∗ ∗ (1.2.29) Y y, j (Λ)Y ∗ = 0 for each Λ ∈ X . To this end, note that (1.2.29) implies that for each Λ ∈ X ∗ we have 0 = Y y, j ∗ (Λ)Y ∗ = X  j(y), ΛX ∗

(1.2.30)

hence j(y) = 0 by the Hahn-Banach Theorem. Since j is injective, this ultimately forces y = 0, as wanted.  Given a vector space X along with a linear subspace Y , of X, the quotient map

1.2 Quasi-Normed Spaces

π : X → X/Y,

9

π(x) := [x]X/Y = x + Y ∈ X/Y for each x ∈ X,

(1.2.31)

is a well-defined linear operator. Assume next that (X, τX ) is a Hausdorff topological vector space and that Y is a closed linear subspace of X. Then   τX/Y := A ⊆ X/Y : π −1 (A) ∈ τX (1.2.32) is a topology on X/Y , called the quotient topology, which renders (X/Y, τX/Y ) a topological vector space. In addition, τX/Y enjoys the properties listed in the lemma below (for a proof, see [165, Theorem 1.41, p. 31]). Lemma 1.2.2 Let Y be a closed linear subspace of a Hausdorff topological vector space (X, τX ). Then, in relation to the quotient topology on X/Y , defined as in (1.2.32), the following statements are true. (1) The quotient map π : (X, τX ) → (X/Y, τX/Y ) is linear, continuous, surjective, and open. (2) If BX is a local base for τX , then {π(U)}U ∈B X is a local base for τX/Y . (3) Each of the following properties of (X, τX ) is inherited by (X/Y, τX/Y ): local convexity, local boundedness, metrizability, normability. (4) If X is an F-space4, or a Fréchet space5, or a Banach space, then so is X/Y .

Next, consider a quasi-normed space X, · X and, given a linear subspace Y of X, define [x]X/Y = x + Y X/Y := inf x + y X for each x ∈ X. (1.2.33) X/Y y ∈Y

If Y is closed, then (1.2.33) becomes a quasi-norm on the quotient space X/Y , and the associated modulus of concavity (cf. (1.2.1)) satisfies ρ(X/Y ) = ρ(X).

(1.2.34)

Moreover, the canonical projection



π : X, · X −→ X/Y, · X/Y defined by π(x) := [x]X/Y = x + Y ∈ X/Y for each x ∈ X,

(1.2.35)

is a well-defined, linear, and bounded operator. Finally, we remark that if · X is a p-norm for some p ∈ (0, 1] then · X/Y is also a p-norm, and that replacing · X in (1.2.33) by an equivalent quasi-norm on X produces an equivalent quasi-norm on the quotient space X/Y .

4 recall that an F-space is a Hausdorff topological vector space (X, τX ) whose topology τX is induced by a complete invariant metric 5 recall that a Hausdorff topological vector space (X, τX ) is said to be a Fréchet space if X is a locally convex F-space

10

1 Preliminary Functional Analytic Matters

space Lemma 1.2.3 Let X, · X be a p-Banach

for some p ∈ (0, 1] and let Y be a closed linear subspace of X. Then X/Y, · X/Y is also a p-Banach space. Also, if Xo is a closed linear subspace of X which contains Y , then Xo /Y is a closed linear subspace of X/Y .

As a corollary, if X, · X is a quasi-Banach space and Y is a closed linear subspace of X, then X/Y, · X/Y is also a quasi-Banach space, and for each closed linear subspace Xo of X which contains Y it follows that Xo /Y is a closed linear subspace of X/Y . Proof From our earlier discussion, we already know that · X/Y is a p-norm on X/Y . Suppose {ξn }n∈N ⊆ X/Y is Cauchy in the quotient quasi-norm. We wish to prove this sequence converges, and for this it is enough to produce a convergent subsequence. Begin by inductively choosing a subsequence {ξnk }k ∈N such that ∞ 

p

ξnk − ξnk+1 X/Y < ∞.

(1.2.36)

k=1

This may be accomplished as follows. Since {ξn }n∈N ⊆ X/Y is Cauchy, there exists p n1 ∈ N with the property that ξn − ξm X/Y < 2−1 whenever n, m ≥ n1 . Also, there p exists n2 > n1 with the property that ξn − ξm X/Y < 2−2 whenever n, m ≥ n2 . Repeat this process, to produce a strictly increasing sequence {nk }k ∈N such that p ξnk − ξnk+1 X/Y < 2−k for all k ∈ N. From this, (1.2.36) follows. Going further, pick x1 ∈ ξn1 and use the definition in (1.2.33) to conclude that for each k ∈ N there exists a vector xk+1 ∈ ξnk+1 − ξnk with xk+1 X ≤ ξnk+1 − ξnk X/Y + 2−k .

∞

(1.2.37)

p

Together, (1.2.36) that k=1 xk X < ∞. From this, the fact that p ensure and (1.2.37)

N N p (1.2.3) implies k=M xk ≤ k=M xk X for all M, N ∈ N with N > M, and the X



completeness of X we then conclude that the series ∞ k=1 xk converges in X, · X . By continuity of the quotient map (1.2.35), the series ξn1 +

∞ 

(ξnk+1 − ξnk ) =

k=1

converges in X/Y to π



∞ k=0 xk+1

ξn M +1 = ξn1 +

∞ 

π(xk+1 )

(1.2.38)

k=0

 . Given that

M 

(ξnk+1 − ξnk ) for each M ∈ N,

(1.2.39)

k=1

this implies that the subsequence {ξn M +1 } M ∈N is convergent in the quotient p-norm, as desired. Ultimately, this establishes that X/Y, · X/Y is indeed a p-Banach space.

1.2 Quasi-Normed Spaces

11

Finally, consider a closed linear subspace Xo of X which contains Y . We wish to show that Xo /Y is closed in X/Y, · X/Y . With this in mind, pick a sequence {ξn }n∈N ⊆ Xo /Y which converges in X/Y, · X/Y to some ξ ∈ X/Y , with the goal of proving that ξ ∈ Xo /Y . Since {ξn }n∈N is Cauchy in X/Y, · X/Y , we may proceed as in the first part of the proof and produce vectors x 1, x2, . . . , xk , · · · ∈ Xo ∞ satisfying (1.2.37). As before, this ensures that the series xk converges in

k=1 X, · X , and since Xo is closed, it follows that the sum x := ∞ k=0 xk+1 belongs to Xo . Ultimately, ξ = ξn1 +

∞ ∞    (ξnk+1 − ξnk ) = π xk+1 = π(x) ∈ Xo /Y, k=1

(1.2.40)

k=0



as wanted.

The point of our next lemma is that, for a linear continuous operator on quasiBanach spaces, modding out its kernel renders said operator continuous and injective.



Lemma 1.2.4 Let X, · X and Y, · Y be two quasi-Banach spaces and consider a linear continuous operator T : X → Y . Then the operator



 : X/Ker T, · X/Ker T −→ Y, · Y defined by T

 [x]X/Ker T := T x ∈ Y for each x ∈ X, T

(1.2.41)

is well defined, linear, continuous, and injective. Proof We begin by noting that Ker T = Ker(T : X → Y ) is a closed subspace  in (1.2.41) is of X (as the null-space of a continuous linear operator), and that T  well defined, linear, and injective. There remains to show that T in (1.2.41) is also

continuous. Since Lemma 1.2.3 guarantees that X/Ker T, · X/Ker T is a quasiBanach space, thanks to the version of the Closed Graph in [138,  Theorem recorded 

Corollary 6.78, p. 442] it suffices to show that G := [x] , T x : x ∈ X is X/Ker T

closed in the space X/Ker T × Y equipped with the product quasi-norm



X/Ker T × Y [x]X/Ker T , y → [x]X/Ker T X/Ker T + y Y . (1.2.42) To this end, suppose {ξn }n∈N ⊆ X/Ker T is a sequence convergent in X/Ker T, ·

 n )}n∈N converges in X/Ker T to some ξ ∈ X/Ker T, with the property that {T(ξ Y, · Y to some y ∈ Y . The goal is to show that (ξ, y) belongs to G, i.e., that  = y. T(ξ) In view of the Aoki-Rolewicz theorem (cf. (1.2.5)), there is no loss of generality in assuming that · X is actually a p-norm, for some p ∈ (0, 1]. Since {ξn }n∈N ⊆ X/Y is p Cauchy, there exists n1 ∈ N with the property that ξn − ξm X/Ker T < 2−1 whenever p n, m ≥ n1 . Next, there exists n2 > n1 with the property that ξn − ξm X/Ker T < 2−2 whenever n, m ≥ n2 . Inductively, this process produces a strictly increasing sequence p {nk }k ∈N such that ξnk+1 − ξnk X/Ker T < 2−k for all k ∈ N. In particular,

12

1 Preliminary Functional Analytic Matters ∞ 

p

ξnk+1 − ξnk X/Ker T < ∞.

(1.2.43)

k=1

Next, pick an arbitrary vector x1 ∈ ξn1 and, using the definition in (1.2.33), for each k ∈ N select some vector xk+1 ∈ ξnk+1 − ξnk such that xk+1 X < ξnk+1 − ξnk X/Ker T + 2−k . (1.2.44)

p From (1.2.43) and (1.2.44) we deduce that ∞ k=1 xk X < ∞. Also, iterations of

N N p p (1.2.3) imply that k=M xk ≤ k=M xk X for all M, N ∈ N with N > M. X Based and the completeness of X we conclude that the series

∞ on these observations x converges in X, · X to some x ∈ X. In addition, from the definition of k=1 k xk and (1.2.41) we see that  n1 ) = T x1 and T(ξ  nk+1 − ξnk ) = T xk+1 for each k ∈ N. T(ξ

(1.2.45)

In particular, ξn1 + ξn M +1 = T T

M M    ξnk+1 − T ξnk ) = T x1 + T (T xk+1 k=1

=T

 M+1 

k=1

 xk for each M ∈ N.

(1.2.46)

k=1

→ ∞ and the mapping in X,  · X as M

M+1  T : X → Y is continuous, we conclude that T k=1 xk converges to T x in the

space Y, · Y as M → ∞. Passing to limit as M → ∞ in (1.2.46) then yields ξn M +1 = T x, as wanted. y = lim M→∞ T  Since

M+1 k=1

xk converges to x =

∞

k=1 xk

In turn, Lemma 1.2.4 is one of the ingredients in the proof of the following isomorphism result.



Lemma 1.2.5 Let X, · X and Y, · Y be two quasi-Banach spaces and consider a linear continuous operator T : X → Y whose range Im(T : X → Y ) is closed in

Y, · Y . Then the operator



 : X/Ker T, · X/Ker T −→ Im(T : X → Y ), · Y given by T

 [x]X/Ker T := T x ∈ Y for each x ∈ X, T

(1.2.47)

is well defined, linear, continuous, bijective, with a continuous inverse.  in (1.2.47) is well defined, linear, continuous, and bijective Proof That the operator T is a consequence of definitions and Lemma 1.2.4. The continuity of its inverse comes  being open. That this is indeed the case follows from the version of the down to T

1.2 Quasi-Normed Spaces

13

Open Mapping Theorem recorded

in [138, Corollary 6.62, p. 423],

bearing in mind that both X/Ker T, · X/Ker T as well as Im(T : X → Y ), · Y are quasi-Banach spaces, thanks to Lemma 1.2.3 and (1.2.2).  Here is a result which further builds on the previous lemma.



Lemma 1.2.6 Let X, · X and Y, · Y be two quasi-Banach spaces and consider a linear continuous operator T : X → Y whose range Im(T : X → Y ) is closed

Z of X, · X which contains in Y, · Y . Then for each closed linear subspace

Ker(T : X → Y ) it follows that T(Z) is closed in Y, · Y .

Proof Fix a linear subspace Z of X, · X containing Ker T = Ker(T : X → Y ). Lemma 1.2.3 implies that Z/Ker T is a closed subspace of X/Ker T, · X/Ker T .  defined as in (1.2.47) Also, by virtue of Lemma 1.2.5, the corresponding operator T  is As a result, T(Z) = T(Z/Ker T) is closed in the space a topological isomorphism.



Im(T : X → Y ), · Y , hence in Y, · Y (given that T has closed range).  We are now in a position to show that a linear continuous operator on quasiBanach spaces, the quality of having a finite dimensional cokernel entails closed range.



Lemma 1.2.7 Let X, · X and Y, · Y be two quasi-Banach spaces and consider a linear continuous operator T : X → Y with dim(Y /Im T) < ∞. Then T has a closed range, i.e., Im(T : X → Y ) is a closed linear subspace of Y . Proof First we observe that there exists a finite-dimensional linear subspace M of Y , whose dimension is n := dim(Y /Im T), for which M + Im(T : X → Y ) = Y and M ∩ Im(T : X → Y ) = {0}.

(1.2.48)

Indeed, if {ξ1, . . . , ξn } is a basis Y /Im T, then each ξ j is of the form y j + Im T for some y j ∈ Y , and one may check without difficulty that {y1, . . . , yn } are linearly independent and span a finite-dimensional space M satisfying all desired properties. Going further, since dim M < ∞ it follows that M is a closed linear subspace of Y . Consequently, X1 := X × M equipped with the product quasi-norm X × M (x, y) −→ (x, y) X1 := x X + y Y

(1.2.49)

is a closed subspace of X × Y , hence itself a quasi-Banach space. Also, the operator



T1 : X1, · X1 −→ Y, · Y defined by (1.2.50) T1 (x, y) := T x + y ∈ Y for each (x, y) ∈ X1 = X × M, is well defined, linear, continuous, and surjective (cf. (1.2.48)); in particular, T1 has closed range. Moreover, (1.2.48) implies that Ker (T1 : X1 → Y ) is equal to

Ker (T : X → Y ) × {0}. Because Z := X × {0} is a closed subspace of X1, · X1 which contains Ker (T1 : X1 → Y ), we then conclude from Lemma 1.2.6 that  T1 (Z) = T(X) is a closed subspace of Y, · Y .

14

1 Preliminary Functional Analytic Matters

We shall also need the following self-improvement of Lemma 1.2.7.



Lemma 1.2.8 Let X, · X and Y, · Y be two quasi-Banach spaces and consider a linear continuous operator T : X → Y with dim (Y /Im T) < ∞. Also, let M be a finite dimensional linear subspace of Y . Then M + Im (T : X → Y ) is a closed linear subspace of Y . Proof This is a consequence of Lemma 1.2.7 and item (1) in Theorem 1.1.3. Another proof goes as follows. With M having the current significance, consider the space X1 := X × M equipped with the product quasi-norm (1.2.49), and define the operator T1 : X1 → Y as in (1.2.50). As before, X1 is a quasi-Banach space and T1 is a linear and continuous operator. In addition, Im (T : X → Y ) ⊆ Im (T1 : X1 → Y ) which implies dim (Y /Im T1 ) ≤ dim (Y /Im T) < ∞. Lemma 1.2.7 then implies that Im (T1 : X1 → Y ) is a closed linear subspace of Y . Since, as seen from (1.2.50), we have (1.2.51) Im (T1 : X1 → Y ) = M + Im (T : X → Y ), 

the desired conclusion follows.

Recall (1.1.13). If actually X, Y are arbitrary quasi-normed spaces, then from definitions and (1.2.14) it follows that Cp(X → Y ) is a linear subspace of Bd(X → Y ).

(1.2.52)

In addition, (again, see [142]) if X, Y are quasi-Banach spaces then Cp (X → Y ) is a closed subspace of the space Bd (X → Y ).

(1.2.53)

Also, given T ∈ Bd(X → Y ) where X, Y are quasi-normed spaces, the quantity (originally introduced by J. Radon in [156])

ess T X→Y := dist T, Cp(X → Y )   = inf T − K X→Y : K ∈ Cp(X → Y ) (1.2.54) is referred to as the essential norm of the operator T. It is then clear from definitions and (1.2.53) that if X, Y are quasi-Banach spaces then for any T ∈ Bd (X → Y ) ess

we have T X→Y = 0 if and only if T ∈ Cp (X → Y ).

(1.2.55)

It is also apparent from definitions that for any quasi-normed spaces X, Y and any T ∈ Bd(X → Y ) we have ess (1.2.56) T X→Y ≤ T X→Y . ess

Let us also point out that, in the particular case when Y := X, it follows that T X→X is the norm of the equivalence class of T in the Calkin algebra Bd(X)/Cp(X). As is well known,

1.2 Quasi-Normed Spaces

15

a bounded linear operator between Banach spaces is compact if and only if its adjoint is (Schauder’s theorem), and the set of all compact operators from a Banach space to itself form a closed two-sided ideal in the algebra of all bounded linear operators on this space.

(1.2.57)

Here is a lower estimate for the essential norm, complementing (1.2.56). This shows that, given an invertible operator on an infinite dimensional Banach space, its norm cannot be decreased arbitrarily by subtracting compact operators. Lemma 1.2.9 Assume X is an infinite dimensional Banach space and suppose the mapping T : X → X is an isomorphism. Then

−1 ess T X→X ≥ T −1 X→Y .

(1.2.58)

In particular, with I denoting the identity on X, for any compact operator K on X one has I − K X→X ≥ 1. (1.2.59) Proof Pick an arbitrary compact operator K on X. Then (1.2.58) follows as soon as we show that

−1 T − K X→X ≥ T −1 X→Y . (1.2.60) Our strategy is to reason by contradiction. Concretely, define R := T − K and note

−1 that R ∈ Bd (X) satisfies R X→X < T −1 X→Y . Then T −1 R X→X ≤ T −1 X→X R X→X < 1.

(1.2.61)

Consequently, I −T −1 R is invertible (via a Neumann series). Since T is also invertible, it follows that K = T −R = T(I −T −1 R) is invertible. Thus, X has a compact invertible operator which, according to Riesz’ theorem, makes it a finite dimensional space. This contradiction proves (1.2.58). Finally, (1.2.59) follows from (1.2.58) (used with T := I) and (1.2.54).  The essential norm of the composition may be estimated in the manned indicated below. Lemma 1.2.10 Let X j , Yj be quasi-normed spaces for j ∈ {0, 1}, and suppose that S ∈ Bd(X1 → X0 ), T ∈ Bd(X0 → Y0 ), and R ∈ Bd(Y0 → Y1 ). Then ess

ess

R ◦ T ◦ S X1 →Y1 ≤ R Y0 →Y1 T X0 →Y0 S X1 →X0 .

(1.2.62)

Proof Let K ∈ Cp(X0 → Y0 ) be arbitrary. Then R ◦ K ◦ S ∈ Cp(X1 → Y1 ), hence (1.2.54) permits us to estimate ess

ess

ess

R ◦ T ◦ S X1 →Y1 ≤ R ◦ T ◦ S − R ◦ K ◦ S X1 →Y1 = R ◦ (T − K) ◦ S X1 →Y1 ≤ R Y0 →Y1 T − K X0 →Y0 S X1 →X0 .

(1.2.63)

Taking the infimum over K ∈ Cp(X0 → Y0 ) of the most extreme sides in (1.2.63) then yields (1.2.62), on account of (1.2.54). 

16

1 Preliminary Functional Analytic Matters

Let us now take a closer look at the space of bounded operators acting between two given quasi-normed spaces.



Lemma 1.2.11 Whenever X, · X is a quasi-normed space and Y, · Y is a quasi-Banach space, it follows that Bd(X → Y ), · X→Y is a quasi-Banach space. Moreover, if Y is actually a p-Banach space for some p ∈ (0, 1], then Bd(X → Y ) is also a p-Banach space. In fact, with some self-explanatory notation, if · ∗ is an equivalent p-norm on Y for some p ∈ (0, 1], then · (X, · X )→(Y, · ∗ ) is an equivalent p-norm on Bd(X → Y

). As a corollary, if X, · X is a quasi-normed space then its dual (cf. (1.1.18)), i.e., ∗

sup |Λ(x)| for each Λ ∈ X ∗ (1.2.64) X , · X ∗ with Λ X ∗ := x ∈X, x X =1

is a Banach space.

Proof That Bd(X → Y ), · X→Y is a quasi-normed space is clear from definitions. Assume {Tj } j ∈N is a Cauchy sequence in Bd(X → Y ), · X→Y . Since for each x ∈ X and j, k ∈ N we have Tj x − Tk x Y ≤ Tj − Tk X→Y x X it follows that {Tj x} j ∈N is a Cauchy sequence in the quasi-Banach space Y . Hence, T x := lim Tj x j→∞

exists in Y , for each x ∈ X. Clearly, T : X → Y is linear. Also, since for each x ∈ X and j ∈ N we may estimate T x Y ≤ C T x − Tj x Y + C Tj x Y , on the one hand we have T x Y ≤ C lim inf Tj x Y . On the other hand, the fact that {Tj } j ∈N j→∞  

is Cauchy in Bd(X → Y ), · X→Y forces Tj X→Y j ∈N to be bounded. Hence, M := sup Tj X→Y is a finite number, and T x Y ≤ C lim inf Tj x Y ≤ CM x for j→∞

j ∈N

each x ∈ X. This proves that

pick a threshold ε > 0. Given T ∈ Bd(X → Y ). Next, that {Tj } j ∈N is Cauchy in Bd(X → Y ), · X→Y , there exists some N ∈ N with the property that Tk − Tj X→Y ≤ ε whenever j, k ∈ N satisfy j, k ≥ N. Fix now an arbitrary x ∈ X along with an arbitrary j ∈ N satisfying j ≥ N. Since for each k ∈ N we may write T x − Tj x Y ≤ C T x − Tk x Y + C Tk x − Tj x Y it follows that T x − Tj x Y ≤ C lim inf Tk x − Tj x Y k→∞

≤ C lim inf Tk − Tj X→Y x X ≤ Cε x X . k→∞

(1.2.65)

In view of the arbitrariness of x ∈ X, this entails T − Tj X→Y

≤ Cε for each j ≥ N. Hence, {Tj } j ∈N converges to T in Bd(X → Y ), · X→Y , so the latter space is complete. This takes care of the first claim in the statement of the lemma. As regards the second claim in the statement of the lemma, assume · ∗ is an equivalent p-norm on Y for some p ∈ (0, 1]. Then · (X, · X )→(Y, · ∗ ) is an equivalent quasi-norm with · X→Y and for each T, R ∈ Bd(X → Y ) we have

1.2 Quasi-Normed Spaces

17

p

T + R (X, · X )→(Y, · ∗ ) = ≤ ≤

sup

x ∈X, x X =1

p

T x + Rx ∗ 

sup

x ∈X, x X =1

sup

x ∈X, x X =1

p

p

T x ∗ + Rx ∗ p

T x ∗ +

sup



x ∈X, x X =1

p

p

Rx ∗

p

= T (X, · X )→(Y, · ∗ ) + R (X, · X )→(Y, · ∗ ),

(1.2.66)

so · (X, · X )→(Y, · ∗ ) is indeed an equivalent p-norm on Bd(X → Y ). Moreover,

since from the first part of the proof we know that Bd(X → Y ), · X→Y is complete and since · (X, · X )→(Y,   · ∗ ) is equivalent with · X→Y , it follows that Bd(X → Y ), · (X, · X )→(Y, · ∗ ) is complete as well.



Recall (1.1.20). If X, Y are quasi-normed spaces it follows from Lemma 1.2.11 that X ∗, Y ∗ are Banach spaces and for each T ∈ Bd(X → Y ) we have T ∗ ∈ Bd(Y ∗ → X ∗ ) and T ∗ Y ∗ →X ∗ ≤ T X→Y .

(1.2.67)

Moreover, Bd(X → Y ) T −→ T ∗ ∈ Bd(Y ∗ → X ∗ ) is an (antilinear) isometry if Y is actually a normed space.

(1.2.68)

We next discuss Neumann series of operators on quasi-Banach spaces. Lemma 1.2.12 Suppose X is a p-Banach space for some p ∈ (0, 1] and let I denote the identity on X. Also, fix some T ∈ Bd(X) and pick z ∈ C with |z| > T X→X .

(1.2.69)

Then the operator zI −T is invertible on X and its inverse is given by the Neumann series ∞  (zI − T)−1 = z −j−1T j , (1.2.70)



j=0

which converges in Bd(X), · X→X . As a consequence of this and the classical Aoki-Rolewicz Theorem (cf. (1.2.5)), whenever X is a quasi-Banach space and T ∈ Bd(X) it follows that the operator zI − T is invertible on X for any z ∈ C with |z| > T X→X . Moreover, its inverse is given by the Neumann series (1.2.70) which converges in Bd(X), · X→X provided |z| > T X→X .

Proof Since by assumption X, · X is a p-Banach space,

for some p ∈ (0, 1], the last part in Lemma 1.2.11 ensures that Bd(X), · X→X is also a p-Banach space.

18

1 Preliminary Functional Analytic Matters

Granted this, if z is as in (1.2.69), for each N, M ∈ N with N > M we may write N  p z −j−1T j j=M

X→X

≤

N 

|z| −j p−p T j X→X p

j=M

≤ |z| −p

N 

|z| −1 T X→X

jp

(1.2.71)

j=M

which is the tail of a convergent geometric series. Thus, the Neumann series in

(1.2.70) converges in Bd(X), · X→X since the associated sequence of partial sums is Cauchy. If R ∈ Bd(X) denotes the limit operator then, thanks to (1.2.69), (zI − T)R = lim (zI − T) N →∞

N →∞

N →∞

z −j−1T j



j=0

= lim (zI − T) = lim

N 

N 

z −j−1T j



j=0



 I − (T/z) N +1 = I.

(1.2.72)

In a similar fashion, R(zI − T) = I so zI − T is invertible and (zI − T)−1 = R.



As a consequence of the Open Mapping Theorem, surjectivity for a linear operator on quasi-Banach spaces self-improves to a quantitative version of itself, something that may be summarized by saying that plain surjectivity implies “ontoness with control.”



Proposition 1.2.13 Assume X, · X and Y, · Y are two quasi-Banach spaces and suppose T : X → Y is a linear continuous operator with closed range. Then there exists C ∈ (0, ∞) with the property that for each y ∈ Im (T → Y ) one may find x ∈ X such that T x = y and x X ≤ C y Y . Proof Without loss of generality we may assume that T is onto (otherwise replace Y by Im (T → Y ), endowed with the quasi-norm inherited from Y ). In the latter setting, the version of the Open Mapping Theorem for quasi-Banach spaces (cf., e.g., [138,

Corollary 6.62, p. 423]) tells us that T BX (0, 1) is an open neighborhood of 0Y in Y . As such, there exists r ∈ (0, ∞) with the property that

BY (0, r) ⊆ T BX (0, 1) . (1.2.73) Pick now y ∈ Y arbitrary. If y  0, then (r/2)(y/ y Y ) ∈ BY (0, r). As such, (1.2.73) guarantees that there exists z ∈ BX (0, 1) with T z = (r/2)(y/ y Y ). Consequently, x := (2 y Y /r)z ∈ X satisfies T x = y and x X ≤ (2 y Y /r) z X ≤ C y Y if we  define C := 2/r ∈ (0, ∞). Finally, if y = 0Y then x := 0X will do.

1.2 Quasi-Normed Spaces

19

It is useful to note that passing to adjoints does not increase the distance to compact operators. Lemma 1.2.14 Let X, Y be two arbitrary Banach spaces. Then for each operator T ∈ Bd(X → Y ) one has T ∗ ∈ Bd(Y ∗ → X ∗ ) and



dist T ∗, Cp(Y ∗ → X ∗ ) ≤ dist T, Cp(X → Y ) , (1.2.74) with the distance in the left-hand side considered in Bd(Y ∗ → X ∗ ), and the distance in the right-hand side considered in Bd(X → Y ) As a corollary, if X, Y are reflexive Banach spaces, for each T ∈ Bd(X → Y ) one has



(1.2.75) dist T ∗, Cp(Y ∗ → X ∗ ) = dist T, Cp(X → Y ) . With the piece of notation introduced in (1.2.54), we may rephrase the estimate in (1.2.74) as T ∗ Y ∗ →X ∗ ≤ T X→Y for each T ∈ Bd(X → Y ). ess

ess

(1.2.76)

In general, the inequality in (1.2.76) can be strict (see the discussion in [12]). Here is the proof of Lemma 1.2.14. Proof of Lemma 1.2.14 For each compact operator K ∈ Cp(X → Y ), Schauder’s theorem (cf. (1.2.57)) implies that K ∗ ∈ Cp(Y ∗ → X ∗ ), thus for any T ∈ Bd(X → Y ) we may write

dist T ∗, Cp(Y ∗ → X ∗ ) ≤ T ∗ − K ∗ Y ∗ →X ∗ = (T − K)∗ Y ∗ →X ∗ = T − K X→Y .

(1.2.77)

Taking the infimum over all K ∈ Cp(X → Y ) then yields (1.2.74).



As a preamble to the proof of Lemma 1.2.17, the essential norm of the restriction of a given linear continuous operator to the range of projection is looked at in the next lemma. Lemma 1.2.15 Let X be a quasi-normed space and suppose P is a projection6 on X. Set Y := PX (viewed itself as a quasi-normed space, equipped with the quasi-norm inherited from X) and assume T : X → X is a linear continuous operator with the property that T(Y ) ⊆ Y . Finally, denote by T |Y : Y → Y the restriction of T to Y . Then ess



T |Y = dist T |Y , Cp(Y ) ≤ P Y→Y · dist T, Cp(X) , Y→Y ess

= P Y→Y · T X→X

(1.2.78)

where the distances above are considered in Bd(Y ) and Bd(X), respectively. 6 i.e., linear, continuous, idempotent operator

20

1 Preliminary Functional Analytic Matters

Proof Given any K ∈ Cp(X) it follows that K  : Y → Y , given by K  y := P(K y) for each y ∈ Y , is a compact operator on Y . Since

dist T |Y , Cp(Y ) ≤ T |Y − K  Y→Y   = sup T y − K  y Y : y ∈ Y, y Y ≤ 1   = sup P(T y) − P(K y) Y : y ∈ Y, y Y ≤ 1   ≤ P Y→Y · sup T y − K y X : y ∈ Y, y Y ≤ 1   ≤ P Y→Y · sup T x − K x X : x ∈ X, x X ≤ 1 = P Y→Y · T − K X→X ,

(1.2.79)

the estimate claimed in (1.2.78) follows by taking the infimum of the most extreme side in (1.2.79) over K ∈ Cp(X).  The proof of Lemma 1.2.17 also requires the algebraic result described below. Lemma 1.2.16 Let X be a quasi-normed space, and fix Λ1, . . . , Λn ∈ X ∗ which are linearly independent. Then there exist linearly independent vectors x1, . . . , xn ∈ X such that Λ j (xk ) = δ jk for each j, k ∈ {1, . . . , n}. Proof We proceed by induction on n. When n = 1 the claim is trivial. Suppose n + 1 linearly independent functionals Λ1, . . . , Λn, Λn+1 ∈ X ∗ have been given, and assume y1, . . . , yn ∈ X are such that Λ j (yk ) = δ jk for each j, k ∈ {1, . . . , n}. Since Λ1, . . . , Λn, Λn+1 ∈ X ∗ are linearly independent it follows that Λ := Λn+1 − Λn+1 (y1 )Λ1 − Λn+1 (y2 )Λ2 − · · · − Λn+1 (yn )Λn

(1.2.80)

belongs to X ∗ and is not identically zero. Consequently, we may select a vector z ∈ X such that   (1.2.81) Λn+1 − Λn+1 (y1 )Λ1 − Λn+1 (y2 )Λ2 − · · · − Λn+1 (yn )Λn (z) = 1. We then proceed to define yn+1 := z − Λ1 (z)y1 − Λ2 (z)y2 − · · · − Λn (z)yn . 

This entails Λk (yn+1 ) =

1 if k = n + 1, 0 if k ∈ {1, . . . , n}.

(1.2.82)

(1.2.83)

For each fixed i ∈ {1, . . . , n + 1} consider now the (n + 1) × (n + 1) system n+1  j=1

ai j Λk (y j ) = δik ,

k ∈ {1, . . . , n + 1}.

(1.2.84)

1.2 Quasi-Normed Spaces

21

Since this has determinant 1, it follows that for each i ∈ {1, . . . , n + 1} the system (1.2.84) has a (unique) solution (ai j )1≤ j ≤n+1 . Then the vectors xi :=

n+1 

ai j y j for each i ∈ {1, . . . , n + 1}

(1.2.85)

j=1

satisfy Λ j (xk ) = δ jk for each j, k ∈ {1, . . . , n}. Since the latter property enures that  the vectors x1, . . . , xn are linearly independent, the proof is complete. The stage is set for us to prove the following useful result. Lemma 1.2.17 Let X be a quasi-normed space and T : X → X is a linear continuous

operator. Fix a finite dimensional linear subspace Z ⊆ Ker T ∗ : X ∗ → X ∗ and define   Y := x ∈ X : Λ(x) = 0 for each Λ ∈ Z (1.2.86) regarded as a quasi-normed space of its own, equipped with the quasi-norm inherited from X. Then T(Y ) ⊆ Y and there exists a constant CX, Z ∈ (0, ∞) independent of T with the property that, with T |Y : Y → Y denoting the restriction of T to Y , one has ess



T |Y = dist T |Y , Cp(Y ) ≤ CX, Z · dist T, Cp(X) Y→Y ess

= CX, Z · T X→X

(1.2.87)

with the distances above considered in Bd(Y ) and Bd(X), respectively. Proof That T(Y ) ⊆ Y is clear from definitions. To proceed, set n := dim Z ∈ N0 . If n = 0 then Z = {0}, hence Y = X. In this case, (1.2.87) is trivially true as long as CX, Z ≥ 1. Consider the case when n ∈ N. In this scenario, pick a basis Λ1, . . . , Λn for Z. Thus, Λ1, . . . , Λn ∈ X ∗ are linearly independent, so we may invoke Lemma 1.2.16 according to which there exist x1, . . . , xn ∈ X such that Λ j (xk ) = δ jk for each j, k ∈ {1, . . . , n}. We use these to define an operator P : X → X by setting Px := x −

n 

Λ j (x)x j for each x ∈ X.

(1.2.88)

j=1

Then one may check without difficulty that P is linear, continuous, idempotent (hence a projection on X), and that Y = P(X). Granted these properties, Lemma 1.2.15 applies and gives (1.2.87) with CX, Z := P Y→Y .  It is known (see, e.g., the discussion in [142]) that given an arbitrary Hausdorff linear topological space X, it follows that every closed subspace Y of X which has finite codimension in X is topologically complemented in X. Here is a more general version of Lemma 1.2.17.

(1.2.89)

22

1 Preliminary Functional Analytic Matters

Lemma 1.2.18 Let X be a quasi-normed space and consider a closed subspace Y of X of finite codimension in X. Let T : X → X be a linear continuous operator with the property that T(Y ) ⊆ Y , and denote by T |Y : Y → Y the restriction of T to Y (regarded as a quasi-normed space in its own right, equipped with the quasi-norm inherited from X). Then there exists a constant CX,Y ∈ (0, ∞), independent of T with, such that ess



T |Y = dist T |Y , Cp(Y ) ≤ CX,Y · dist T, Cp(X) Y→Y ess

= CX,Y · T X→X

(1.2.90)

with the distances above considered in Bd(Y ) and Bd(X), respectively. The concrete manner in which Lemma 1.2.18 contains Lemma 1.2.17 is as follows. Using notation from the proof of Lemma 1.2.17, the fact that PX = Y implies that if x1, . . . , xn  is the linear span of the vectors x1, . . . , xn then Y + x1, . . . , xn  = X (cf. (1.2.88)).  Thus, Y that has finite codimension in X (in fact, dim(X/Y ) ≤ n). Since Ker Λ, we see that Y is also closed. As such, Y from Lemma 1.2.17 satisfies Y= Λ∈Z

all the hypotheses required of it in Lemma 1.2.18, and (1.2.87) gives (1.2.90). Proof of Lemma 1.2.18 In light of (1.2.89), it follows that there exists a projection P on X such that PX = Y . Granted this, Lemma 1.2.15 applies and yields the desired conclusion.  We continue by establishing the lower bound for the distance to compact operators on Lebesgue spaces of the sort contained in the next lemma. Lemma 1.2.19 Let (X, M, μ) be a measure space, fix some p ∈ (0, ∞), and define  1 if p ≥ 1, Cp := (1.2.91) 1/p−1 2 if p < 1.

Then for each operator T ∈ Bd L p (X, μ) one has  

sup 1 A · T p ≤ Cp · dist T, Cp L p (X, μ) , lim p ε→0+

A∈M μ(A) 0 along with a sequence { A j } j ∈N ⊆ M with lim μ(A j ) = 0 and such that j→∞ 1 A · T p > δ for each j ∈ N. In turn, the last condition ensures that j L (X,μ)→L p (X,μ) p p there exist functions f j ∈ L (X, μ) with f j L (X,μ) = 1 and 1 A · T f j p > δ for each j ∈ N. (1.2.93) j L (X,μ)

1.2 Quasi-Normed Spaces

23

Since we are presently working under the hypothesis that T is compact on L p (X, μ), there is no loss of generality in assuming that {T f j } j ∈N converges in L p (X, μ) to some function g ∈ L p (X, μ).

(1.2.94)

To proceed, define  N := x ∈ X : there exist natural numbers j1 < j2 < · · · < jk < jk+1 < · · · such that x ∈ A jk for each k ∈ N



(1.2.95) and observe that, by design, lim 1 A j (x) = 0 for each x ∈ X \ N.

j→∞

Also, if J is the set of all increasing sequences of natural numbers, then    Aj . J is countable and N = J ∈J

In particular, since μ



j ∈J

(1.2.96)

(1.2.97)

j ∈J

 A j ≤ lim sup μ(A j ) = 0, we conclude from (1.2.97) J j→∞

that μ(N) = 0. In concert with (1.2.96), this proves that lim 1 A j (x) = 0 for μ-a.e. x ∈ X.

j→∞

(1.2.98)

(Parenthetically, we wish to note that (1.2.98) also follows upon observing that |g| p dμ is a measure absolutely continuous with respect to μ.) Having established (1.2.98), Lebesgue’s Dominated Convergence Theorem then ensures that lim 1 A j · g L p (X,μ) = 0. (1.2.99) j→∞

For each j ∈ N we may now write, based on (1.2.93), δ < 1 A j · T f j L p (X,μ) ≤ Cp 1 A j · (T f j − g) L p (X,μ) + Cp 1 A j · g L p (X,μ) ≤ Cp T f j − g L p (X,μ) + Cp 1 A j · g L p (X,μ)

(1.2.100)

which, after passing to limit j → ∞, leads to a contradiction in view of (1.2.94) and (1.2.99). This establishes the fact that  

lim+ sup 1 A · T L p (X,μ)→L p (X,μ) = 0 for all T ∈ Cp L p (X, μ) . (1.2.101) ε→0

A∈M μ(A)0

(1.3.37)

(1.3.38)

and define the intermediate space (for the real method of interpolation) between X0 and X1 as   (1.3.39) (X0, X1 )θ,q := x ∈ X0 + X1 : x (X0,X1 )θ, q < +∞ . From (1.3.39), (1.3.38), and (1.3.35) it is then clear that (X0, X1 )θ,q = (X1, X0 )1−θ,q .

(1.3.40)

It also turns out that the ambient space X plays only a minor role in the definition of (X0, X1 )θ,q . For example, this may be replaced by X0 + X1 . In this vein, let us note that, as is apparent from (1.3.39), (X0, X1 )θ,q → X0 + X1 continuously, for each θ ∈ (0, 1) and q ∈ (0, ∞]. Remark 1.3.3 The following are true (see, e.g., [187, pp. 63–64]).

(1.3.41)

30

1 Preliminary Functional Analytic Matters

(i) If A0 and A1 are two compatible quasi-Banach spaces and θ ∈ (0, 1) and q ∈ (0, ∞] are arbitrary, the real interpolation space (A0, A1 )θ,q is itself a quasi-Banach space. (ii) If X is a Hausdorff topological vector space and A0, A1, B0, B1 are quasiBanach spaces such that A0 → B0 → X and A1 → B1 → X continuously, then for each θ ∈ (0, 1) and q ∈ (0, ∞] it follows that (A0, A1 )θ,q and (B0, B1 )θ,q are quasi-Banach spaces and the embedding (A0, A1 )θ,q → (B0, B1 )θ,q is continuous.

(1.3.42)

Given any x ∈ X0 ∩ X1 along with any t > 0, writing x = x + 0 proves, in the context of (1.3.34), that K(t, x, X0, X1 ) ≤ x X0 , while decomposing x = 0 + x shows, once again the context of (1.3.34), that K(t, x, X0, X1 ) ≤ t x X1 . Together, these observations establish the estimate   K(t, x, X0, X1 ) ≤ min x X0 , t x X1 for each x ∈ X0 ∩ X1 and t > 0. (1.3.43) In turn, (1.3.43) and (1.3.38)-(1.3.39) permit us to conclude that X0 ∩ X1 → (X0, X1 )θ,q continuously, for each θ ∈ (0, 1) and q ∈ (0, ∞].

(1.3.44)

Other basic, general properties satisfied by the intermediate spaces obtained via the real method of interpolation may be found in [15, §3.4]. For example, [15, Theorem 3.4.2(b), p. 47] gives that the set X0 ∩ X1 is dense in the space (X0, X1 )θ,q for each parameters θ ∈ (0, 1) and q ∈ (0, ∞).

(1.3.45)

The next lemma shows that, given two quasi-normed spaces with the property that one embeds continuously into the other, we may define an equivalent quasi-norm on the corresponding intermediate spaces by restricting the parameter t in (1.3.38) to a smaller sub-interval of (0, ∞). Lemma 1.3.4 Assume X0 , X1 are two quasi-normed spaces, and fix θ ∈ (0, 1) along with q ∈ (0, ∞]. Also, pick an arbitrary number t∗ ∈ (0, ∞). Then, if X0 → X1 continuously,

(1.3.46)

an equivalent quasi-norm on the intermediate space (X0, X1 )θ,q is given by

∗ := x (X 0,X1 ) θ, q

∫ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪

∞



t −θ K(t, x, X0, X1 )

t∗

 −θ  ⎪ ⎪ ⎪ ⎪ sup t K(t, x, X0, X1 ) ⎩ t>t∗

for each x ∈ (X0, X1 )θ,q . If, on the other hand,

 q dt  q1 if 0 < q < ∞, t if q = ∞,

(1.3.47)

1.3 Real Interpolation of Quasilinear Operators

X1 → X0 continuously,

31

(1.3.48)

then an equivalent quasi-norm on (X0, X1 )θ,q is given by

∗∗ x (X 0,X1 ) θ, q

 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪

∫

 q dt  q1 if 0 < q < ∞, t 0 :=   ⎪ ⎪ if q = ∞, ⎪ sup t −θ K(t, x, X0, X1 ) ⎪ ⎩ t ∈(0,t∗ ) t∗



t −θ K(t, x, X0, X1 )

(1.3.49)

for each x ∈ (X0, X1 )θ,q . Proof Granted (1.3.46), any x ∈ X0 + X1 = X1 admits the trivial decomposition x = 0 + x with 0 ∈ X0 and x ∈ X1 , it follows from (1.3.34) that K(t, x, X0, X1 ) ≤ t x X1 for each t ∈ (0, ∞) and each x ∈ X0 + X1 = X1 .

(1.3.50)

On the other hand, if x ∈ X0 +X1 = X1 is written as x = x0 +x1 for some x0 ∈ X0 ⊆ X1 and x1 ∈ X1 , then for every t ∈ (t∗, ∞) we may estimate, using the quasi-triangle inequality and the continuity of the embedding X0 → X1 ,



x X1 ≤ C x0 X1 + x1 X1 ≤ C x0 X0 + x1 X1

(1.3.51) ≤ C · max{1, t∗−1 } · x0 X0 + t x1 X1 . Then, in view of (1.3.34), taking the infimum over all such decompositions of x (with t fixed) yields K(t, x, X0, X1 ) ≥ c x X1 for each t ∈ (t∗, ∞) and each x ∈ X0 + X1 = X1,

(1.3.52)

where c ∈ (0, ∞) depends only on X0, X1 and t∗ . Consequently, assuming q ∈ (0, ∞), for each vector x ∈ X0 + X1 = X1 we may estimate  ∫ t∗   ∫ t∗   q dt  q1  q dt  q1  t −θ K(t, x, X0, X1 ) t 1−θ ≤ x X1 · = Cq,θ,t x X1 ∗ t t 0 0  ∫ ∞   dt  q1 q  t −θ = Cq,θ,t∗ x X1 · t t∗ ∫ ∞   q dt  q1 −1  t −θ K(t, x, X0, X1 ) ≤ c · Cq,θ,t∗ t t∗ (1.3.53)   with c as in (1.3.52) and Cq,θ,t , Cq,θ,t > 0 some finite constants depending only ∗ ∗ on q and θ, where the first inequality uses (1.3.50) and the last inequality is based on (1.3.52). Similarly, corresponding to q = ∞, for each x ∈ X0 + X1 = X1 we may

32

1 Preliminary Functional Analytic Matters

estimate sup

t ∈(0,t∗ )



   t −θ K(t, x, X0, X1 ) ≤ x X1 · sup t 1−θ = t∗1−θ x X1 t ∈(0,t∗ )



= t∗ x X1 · sup

t ∈(t∗,∞)

≤ c−1 t∗ · sup

t ∈(t∗,∞)



t −θ



 t −θ K(t, x, X0, X1 ) .

(1.3.54)

A combination of (1.3.38) with (1.3.53)-(1.3.54) then proves that (1.3.47) is indeed an equivalent quasi-norm on the intermediate space (X0, X1 )θ,q if (1.3.46) holds. In the second part of the proof assume (1.3.48) holds. Since any x ∈ X0 + X1 = X0 admits the trivial decomposition x = x + 0 with x ∈ X0 and 0 ∈ X1 , the definition of the K-functional implies K(t, x, X0, X1 ) ≤ x X0 for each t ∈ (0, ∞) and each x ∈ X0 + X1 = X0 .

(1.3.55)

Suppose next that some x ∈ X0 +X1 = X0 has been given. Then for any decomposition x = x0 + x1 with x0 ∈ X0 and x1 ∈ X1 ⊆ X0 and any t ∈ (0, t∗ ) we may estimate



t x X0 ≤ Ct x0 X0 + x1 X0 ≤ Ct x0 X0 + x1 X1

(1.3.56) ≤ C · max{1, t∗ } · x0 X0 + t x1 X1 , based on the quasi-triangle inequality and the continuity of the embedding X1 → X0 . After taking the infimum over all such decompositions of x (with t ∈ (0, t∗ ) fixed) we arrive at the conclusion that K(t, x, X0, X1 ) ≥ c t x X0 for each t ∈ (0, t∗ ) and each x ∈ X0 + X1 = X0,

(1.3.57)

for some c ∈ (0, ∞) depending only on X0, X1 and t∗ . At this stage, if q ∈ (0, ∞) then   , Cq,θ,t ∈ (0, ∞) depending for each x ∈ X0 + X1 = X0 we may write, with Cq,θ,t ∗ ∗ only on q, θ, and t∗ , ∫ ∞   ∫ ∞   dt  q1  q dt  q1 q  t −θ K(t, x, X0, X1 ) t −θ ≤ x X0 · = Cq,θ,t x X0 ∗ t t t∗ t∗ 1  ∫ t∗    1−θ q dt q t = Cq,θ,t x · X0 ∗ t 0 ∫  t∗   q dt  q1  −θ t ≤ c−1 · Cq,θ,t K(t, x, X , X ) 0 1 ∗ t 0 (1.3.58)

1.3 Real Interpolation of Quasilinear Operators

33

with the first inequality supplied by (1.3.55) and the last inequality originating in (1.3.57). In an analogous fashion, if q = ∞ then for each x ∈ X0 + X1 = X0 we have    sup t −θ K(t, x, X0, X1 ) ≤ x X0 · sup t −θ ] = t∗−θ x X0 t ∈(t∗,∞)

t ∈(t∗,∞)

= t∗−1 x X0 · sup



t ∈(0,t∗ )

≤ c−1 t∗−1 · sup

t ∈(0,t∗ )



t 1−θ



 t −θ K(t, x, X0, X1 ) .

(1.3.59)

From (1.3.38) and (1.3.58)-(1.3.59) we may now conclude that if (1.3.48) holds then (1.3.49) is an equivalent quasi-norm on (X0, X1 )θ,q .  We proceed by introducing quasi-normed lattices of functions on a given measure space (Σ, μ). The reader is reminded that L 0 (Σ, μ) denotes the linear space consisting of (equivalence classes of) μ-measurable functions which are finite μ-a.e. on Σ (see the discussion in [133, S 3.2]). Definition 1.3.5 Let (Σ, μ) be a measure space and assume that Y is a quasi-normed space. Call Y a quasi-normed lattice of functions if Y → L 0 (Σ, μ) continuously,

(1.3.60)

and if there exists some constant C0 ∈ (0, ∞) with the property that whenever f ∈ L 0 (Σ, μ) and g ∈ Y satisfy | f (x)| ≤ |g(x)| at μ-a.e. point x ∈ Σ then necessarily f ∈ Y and f Y ≤ C0 g Y . Parenthetically we note that if Y is as above then f ∈ Y is zero in Y if and only if f (x) = 0 at μ-a.e. point x ∈ Σ. Definition 1.3.6 Given a vector space X and assuming that Y is a linear subspace of L 0 (Σ, μ), call the operator T : X → Y quasi-subadditive provided there exists a finite constant C1 > 0 such that for all f , g ∈ X,

(1.3.61) |T( f + g)| ≤ C1 |T f | + |T g| at μ-a.e. point in Σ. We are now in position to state and prove the following real interpolation result for quasi-subadditive operators. Proposition 1.3.7 Assume that X0 , X1 are two compatible quasi-normed spaces and that Y0 , Y1 are two quasi-normed lattices of functions relative to a common measure space (Σ, μ). Suppose (1.3.62) T : X0 + X1 −→ Y0 + Y1 is a quasi-subadditive operator satisfying T Xi ⊆ Yi for i ∈ {0, 1}, and such that there exist M0 , M1 positive finite constants with the property that T x Yi ≤ Mi x Xi for all x ∈ Xi, with i ∈ {0, 1}.

(1.3.63)

34

1 Preliminary Functional Analytic Matters

Then for each θ ∈ (0, 1) and q ∈ (0, ∞], the operator T maps (X0, X1 )θ,q into (Y0, Y1 )θ,q and T x (Y0,Y1 )θ, q ≤ C0 C1 M01−θ M1θ x (X0,X1 )θ, q for every x ∈ (X0, X1 )θ,q

(1.3.64)

where C0 ∈ (0, ∞) is a constant which works both for Y0 and Y1 as in Definition 1.3.5, while the constant C1 ∈ (0, ∞) is associated with T as in Definition 1.3.6. Before presenting the proof of this proposition we wish to make four comments. First, if X is a Hausdorff topological vector space with the property that Xi → X continuously for i ∈ {0, 1} (whose existence is guaranteed by the fact that X0 , X1 are compatible quasi-normed spaces), then in place of (1.3.62) we may initially assume that (1.3.65) T : X0 + X1 −→ X and the same proof will work. Our second comment is that, in a natural sense, having an additive mapping T : X0 + X1 → Y0 + Y1 which satisfies T Xi ⊆ Yi for i ∈ {0, 1} is equivalent to having two additive maps Ti : Xi → Yi for i ∈ {0, 1} that are compatible with one another, in the sense that T0 X0 ∩X1 = T1 X0 ∩X1 ; in the latter scenario, the operators induced by the mappings T0, T1 from (X0, X1 )θ,q into (Y0, Y1 )θ,q for various values of θ ∈ (0, 1) and q ∈ (0, ∞] are additive and compatible with one another (as they all agree with T : (X0, X1 )θ,q → (Y0, Y1 )θ,q ).

(1.3.66)

The third comment is that we may recast (1.3.64) simply as

1−θ

θ · T X1 →Y1 . T (X0,X1 )θ, q →(Y0,Y1 )θ, q ≤ C0 C1 · T X0 →Y0

(1.3.67)

Our fourth (and final) comment is that a version of Proposition 1.3.7 for linear operators taking values in quasi-normed spaces is discussed in Remark 1.3.8. Here is the proof of Proposition 1.3.7. Proof of Proposition 1.3.7 Let x ∈ (X0, X1 )θ,q ⊆ X0 + X1 and consider x0 ∈ X0 , x1 ∈ X1 such that x = x0 + x1 . Thus, by quasi-additivity of T,

|T x| ≤ C1 |T x0 | + |T x1 | at μ-a.e. point on Σ. (1.3.68) Introduce

  E := ξ ∈ Σ : |T x0 |(ξ) + |T x1 |(ξ) = 0 ,

(1.3.69)

so that E is μ-measurable and since (1.3.68) holds, it follows that T x = 0 at μ-a.e. point in E. Next, for i ∈ {0, 1} set

(1.3.70)

1.3 Real Interpolation of Quasilinear Operators

35

 ⎧ ⎪ ⎨ ⎪

 |T xi | T x on Σ \ E, |T x0 | + |T x1 | yi := ⎪ ⎪ 0 on E. ⎩

(1.3.71)

Then each yi is μ-measurable and, thanks to the quasi-subadditivity of T, for each i ∈ {0, 1} we have the pointwise inequality |yi | ≤





|T xi | · C1 |T x0 | + |T x1 | = C1 |T xi | |T x0 | + |T x1 |

(1.3.72)

at μ-a.e. point in Σ. Consequently, since each Yi is a quasi-normed lattice of functions on (Σ, μ), we may conclude (using the boundedness of T) that yi ∈ Yi and yi Yi ≤ C0 C1 T xi Yi ≤ C0 C1 Mi xi Xi ,

i ∈ {0, 1}.

(1.3.73)

Also, analyzing what happens on Σ \ E and E separately and using (1.3.70), we see that y0 + y1 = T x at μ-a.e. point in Σ, (1.3.74) which shows that T x ∈ Y0 + Y1 . As a result, K(t, T x, Y0, Y1 ) ≤ y0 Y0 + t y1 Y1   tM  1 x1 X1 . ≤ C0 C1 M0 x0 X0 + M0

(1.3.75)

Taking the infimum over all representations of x = x0 + x1 yields K(t, T x, Y0, Y1 ) ≤ C0 C1 M0 K

tM

1

M0

 , x, X0, X1 .

(1.3.76)

Now, if 0 < q < ∞ and θ ∈ (0, 1), ∫ T x (Y0,Y1 )θ, q ≤ C0 C1 M0 = C0 C1 M0 =

∞



t −θ K

tM

1

M0

0

 M  −θ ∫ 0

M1

0

∞



, x, X0, X1

  q dt  q1 t



 q dτ τ −θ K τ, x, X0, X1 τ

C0 C1 M01−θ M1θ x (X0,X1 )θ, q

< ∞,

 q1

(1.3.77)

i.e., (1.3.64) holds. In concert with the membership T x ∈ Y0 + Y1 , this implies  T x ∈ (Y0, Y1 )θ,q , which completes the proof. Remark 1.3.8 A similar result to that established in Proposition 1.3.7 holds in the case when T is linear and Y0, Y1 are now two compatible quasi-normed spaces. Indeed, a cursory inspection shows that the previous proof works virtually verbatim with the added benefit that we may now take C0 = C1 = 1, i.e., in place of (1.3.64) we now have

36

1 Preliminary Functional Analytic Matters

T x (Y0,Y1 )θ, q ≤ M01−θ M1θ x (X0,X1 )θ, q for every x ∈ (X0, X1 )θ,q .

(1.3.78)

See also [15, p. 41] in this regard. We close by recording a versatile version of the Reiteration Theorem, for quasinormed Abelian couples. Theorem 1.3.9 Let A0, A1 be two quasi-normed Abelian groups which are subgroups of some larger ambient Abelian group A, and which satisfy the following separability axiom (or compatibility of convergence condition): if the sequence {x j } j ∈N ⊆ A0 ∩ A1 and a0 ∈ A0 and a1 ∈ A1 are such that lim x j = a0 in A0 and lim x j = a1 in A1 , then necessarily j→∞ j→∞ a0 = a1 .

(1.3.79)

Then for each θ 0, θ 1, θ ∈ (0, 1) with θ 0  θ 1 and each q0, q1, q ∈ (0, ∞] one has

(1.3.80) (A0, A1 )θ0,q0 , (A0, A1 )θ1,q1 θ,q = (A0, A1 )(1−θ)θ0 +θθ1,q . In addition, the following two end-point versions of (1.3.80) (corresponding, roughly speaking, to the cases θ 0 = 0 and θ 1 = 1, respectively) also hold:

A0, (A0, A1 )θ1,q1 θ,q = (A0, A1 )θθ1,q and (1.3.81)

(A0, A1 )θ0,q0 , A1

θ,q

= (A0, A1 )(1−θ)θ0 +θ,q .

(1.3.82)

Moreover, formulas (1.3.80)-(1.3.82) are quantitative in the sense that, in each case, the quasi-norm of the space on the left is equivalent to the quasi-norm of the corresponding space on the right, with constants of equivalence depending only on θ 0, θ 1, θ, q0, q1, q as well as the constants in the quasi-triangle inequalities in A0 and A1 . Theorem 1.3.9 is proved for quasi-normed spaces in [98] but essentially the same proof works for quasi-normed Abelian groups. Other variants and generalizations of Theorem 1.3.9 may be found in [20] and [154].

1.4 Complex Interpolation on Quasi-Banach Spaces The presentation in this section follows closely [111]. Throughout, (1.2.10) is tacitly employed. To set the stage for adapting Calderón’s original complex method of interpolation to the setting of quasi-Banach spaces, we first review some basic results from the theory of analytic functions with values in quasi-Banach spaces as developed in [193], [108], [109]. Recall that if X is a topological vector space and U is an open subset of the complex plane then a function f : U → X is called analytic if given z0 ∈ U there exists η > 0 so that there is a power series expansion

1.4 Complex Interpolation on Quasi-Banach Spaces

f (z) =

37

∞  (z − z0 )n xn, xn ∈ X, uniformly convergent for |z − z0 | < η.

(1.4.1)

j=0

As explained in [108], in the context of quasi-Banach spaces, this is the most natural definition. Indeed, there are simple examples which show that complex differentiability leads to an unreasonably weaker concept of analyticity (see also [193] and [8] in this regard). Proposition 1.4.1 Suppose 0 < p ≤ 1 and m ∈ N is such that m > p1 . Then there is a constant C = C(m, p) with the following significance. If X is a p-normed quasiBanach space and f : D¯ → X is a continuous function which is analytic on the unit

f (n) (0) n disk D := {z : |z| < 1} then for z ∈ D we have f (z) = ∞ n=0 n! z , and f (n) (0) X ≤ C(m + n)! sup f (z) X . z ∈D

(1.4.2)

This is [108, Theorem 6.1]. Proposition 1.4.2 Let X be a quasi-Banach space and let U be an open subset of the complex plane. Let fn : U → X, n ∈ N, be a sequence of analytic functions. If lim fn = f uniformly on compacta then f is also analytic. n→∞

This follows from [108, Theorem 6.3]. Proposition 1.4.3 Suppose X is a quasi-Banach space and that U is an open subset of the complex plane. Let f : U → X be a locally bounded function. Suppose there is a weaker Hausdorff vector topology τ0 on X which is locally p-convex for some 0 < p < 1 and such that f : U → (X, τ0 ) is analytic. Then f : U → X is analytic. This is [111, Theorem 3.3]. It shows that, many times in practice, the ambient space (within which the interpolation process is carried out) plays only a minor role in the setup. More specifically, assume that Y is a space of distributions in which a quasi-Banach space X is continuously embedded. Then, having an X-valued function analytic for the quasi-norm topology is basically the same as requiring analyticity for the weak topology (induced on X from Y ). We are now prepared to elaborate on the complex method of interpolation for pairs of quasi-Banach spaces. Consider a compatible couple (or pair) of quasi-Banach spaces X0, X1 , i.e., X j with j ∈ {0, 1} are continuously embedded into a larger topological vector space Y . Also, let U stand for the strip {z ∈ C : 0 < Re z < 1}. Definition 1.4.4 A family F of functions which map U into X0 + X1 is called admissible provided the following axioms are satisfied: (i) F is a (complex) vector space endowed with a quasi-norm · F with respect to which it is complete (i.e., F is a quasi-Banach space); (ii) the point-evaluation mappings evw : F → X0 + X1 , w ∈ U, defined at each f ∈ F by evw ( f ) := f (w), are continuous;

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1 Preliminary Functional Analytic Matters

(iii) for any given compact subset K of U there exists some finite positive constant C with the property that, for any fixed point w ∈ K and any fixed function f ∈ F with f (w) = 0, the mapping U \ {w} z → f (z)/(z − w) ∈ X0 + X1 extends to an element in F and f (z) (1.4.3) z − w ≤ C f F . F Conditions (i)-(iii) in Definition 1.4.4 are the minimal requirements needed in order develop a reasonable interpolation theory at an abstract level. In practice, mimicking the Banach space theory, a common choice for F , the class of admissible functions, is the space of bounded, analytic functions f : U → X0 + X1 , which is extended continuously to the closure of the strip such that the traces t → f ( j + it) are bounded continuous functions into X j , j ∈ {0, 1}. We endow F with the quasi-norm ! f F := max sup f (it) X0 , sup f (1 + it) X1 . (1.4.4) t

t

However, there is an immediate problem that in general the evaluation maps evw are not necessarily bounded on F and so this class is not always admissible. In fact in the special case when X0 = X1 boundedness of the evaluation maps is equivalent to the validity of a form of the Maximum Modulus Principle. A quasi-Banach space X is analytically convex if there is a constant C such that for every polynomial P : C → X we have P(0) X ≤ C max |z |=1 P(z) X . It is shown in [109] that if X is analytically convex it has an equivalent quasi-norm which is plurisubharmonic (i.e., we can insist that the constant C above can be taken to be 1). Let us also point out that being analytically convex is equivalent to the condition that max f (z) X ≤ C

0 0, bearing in mind that ρ# yields the same topology on V as ρ (cf. [133, Theorem 7.1.2]).  Lemma 1.4.12 Let (Σ, M, μ) be a measure space with the property that μ is complete, and let V be a topological vector space whose topology is induced by a quasimetric. Suppose { f j } j ∈N is a sequence of strongly μ-measurable V -valued functions defined on Σ with the property that there exists some μ-nullset A ∈ M such that f (ω) = lim f j (ω) in V for each ω ∈ Σ \ A. j→∞

Then there exists a sequence {g j } j ∈N of countably simple functions which converges to f pointwise on Σ \ A, in a uniform fashion. In particular, f is a strongly μ-measurable function. Proof From assumptions and parts (iii), (ii) in Remark 1.4.8 we see that f is μmeasurable. Also, from Lemma 1.4.10 we know that for each j ∈ N there exists some μ-nullset A j ∈ M such that f j (Σ \ A j ) is a separable subset of V . If for each j is a μ-nullset. Also, j := A ∪ A j ∈ M, it follows that A j ∈ N we now define A since separability in a quasi-metric space is hereditary (cf. [133, Lemma 7.1.1]), it  $ j ) is a separable subset of V . Then A  := j ∈N A j ∈ M is follows that each f j (Σ \ A  is a separable subset of V . With a μ-nullset and Lemma 1.4.9 implies that f (Σ \ A) this in hand, Lemma 1.4.11 applies and yields all desired conclusions. 

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1 Preliminary Functional Analytic Matters

Given a complete measure space (Σ, M, μ) along with a quasi-Banach space (X, · X ), for each p ∈ (0, ∞] we shall denote by L p (Σ, μ) ⊗ X the space of X-valued strongly μ-measurable functions on Σ which are p-th power integrable with respect to μ. More specifically, L p (Σ, μ) ⊗ X is the collection of (equivalence classes of) strongly μ-measurable functions f : Σ −→ X such that the (1.4.12) mapping Σ ω −→ f (ω) X ∈ [0, ∞) belongs to L p (Σ, μ).

a j 1 A j is a countably A few comments are in order here. First, observe that if f = j ∈N

a j X 1 A j becomes an ordinary scalar-valued simple function then f (·) X = j ∈N

μ-measurable function on Σ. In particular, from this observation, Definition 1.4.7, the convention made in (1.2.10), and [133, item (iii) in Remark 3.1.2] we conclude that given any strongly μ-measurable function f : Σ −→ X it follows that the mapping Σ ω −→ f (ω) X ∈ [0, ∞) is a (ordinary, real-valued) μ-measurable function on Σ.

(1.4.13)

Second, in light of (1.4.13) we may rephrase the definition of L p (Σ, μ) ⊗ X given in (1.4.12) as follows L p (Σ, μ) ⊗ X is the collection of all (equivalence classes of) strongly μ-measurable X-valued functions f defined on Σ with the property that  1/p ∫ p f (ω) X dμ(ω) < ∞ (with the usual alteration when p = ∞).

(1.4.14)

Σ

Third, it is clear that L p (Σ, μ) ⊗ X = L p (Σ, μ) ⊗ (X, · X ) is a vector space, and L p (Σ, μ) ⊗ X f −→ f L p (Σ, μ)⊗X := f (·) X p (1.4.15) L (Σ, μ)

is a quasi-norm on the space L p (Σ, μ) ⊗ X. Also, it is clear from definitions that for each r > 0 we have f Lr p (Σ, μ)⊗(X, · X ) = f (·) Xr p/r L (Σ, μ) (1.4.16) p for all f ∈ L (Σ, μ) ⊗ (X, · X ). Fourth, if · X is an r-norm on X for some finite r ∈ (0, p], then · L p (Σ, μ)⊗(X, · X ) is an r-norm on L p (Σ, μ) ⊗ (X, · X ). Indeed, for each f , g ∈ L p (Σ, μ) ⊗ (X, · X ) we may estimate

(1.4.17)

1.4 Complex Interpolation on Quasi-Banach Spaces

f + g Lr p (Σ, μ)⊗(X, · X )

43

r = ( f + g)(·) X

L p/r (Σ, μ)

≤ f (·) Xr + g(·) Xr ≤ f (·) Xr

L p/r (Σ, μ)

L p/r (Σ, μ)

+ g(·) Xr

L p/r (Σ, μ)

= f Lr p (Σ, μ)⊗(X, · X ) + g Lr p (Σ, μ)⊗(X, · X ),

(1.4.18)

from which (1.4.17) follows. The space L p (Σ, μ) ⊗ X also turns out to be complete. Proving this takes some work, and we address this issue separately, in Proposition 1.4.14 stated a little later below. As a preamble, we first prove the following lemma. Lemma 1.4.13 Let (V, · ) be a quasi-normed vector space with the property that · is an r-norm for some r > 0. Also, assume that whenever {v j } j ∈N ⊆ V satisfies that the series

∞

j=1

∞

j=1

v j r < ∞ it follows (1.4.19)

v j converges in the space (V, · ).

Then (V, · ) is complete. Proof Let {w j } j ∈N ⊆ V be a Cauchy sequence in (V, · ). Then there exists a family of integers 1 ≤ n1 < nn < · · · such that for each j ∈ N we have wn − wm < 2−j/r whenever n, m ≥ n j . Define v1 := wn1 and v j := wn j − wn j−1 for each j ≥ 2. Then k ∞ ∞

v j = wnk and v j r ≤ wn1 r + 2−(j−1) < +∞. for each k ∈ N we have j=1

j=1

j=2

Granted these properties, (1.4.19) guarantees that {wnk }k ∈N converges in (V, · ). Hence the Cauchy sequence {w j } j ∈N has a convergent sub-sequence in (V, · ). This readily implies that actually the whole sequence {w j } j ∈N converges in (V, · ), proving that (V, · ) is indeed complete.  Here is the completeness result referred to above. Proposition 1.4.14 Let (Σ, M, μ) be a complete measure  space and consider a quasi- Banach space (X, · X ). Also, fix p ∈ (0, ∞]. Then L p (Σ, μ) ⊗ X, · L p (Σ, μ)⊗X is if X is Banach and p ∈ [1, ∞] then the space   a quasi-Banach space. Moreover, p L (Σ, μ) ⊗ X, · L p (Σ, μ)⊗X is in fact Banach.   Proof From earlier comments we already know that L p (Σ, μ) ⊗ X, · L p (Σ, μ)⊗X is a quasi-normed vector space. To prove that this is also complete, consider first the case when p < ∞. From (1.2.6) and (1.4.17) we know that there exists some

r ∈ 0, min{p, (1 + log2 ρ(X))−1 } with the property

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1 Preliminary Functional Analytic Matters

· X is an r-norm on the space X, and also · L p (Σ, μ)⊗X is an r-norm on L p (Σ, μ) ⊗ X.

(1.4.20)

The idea is to use the criterion for completeness established earlier in Lemma 1.4.13 for the space V := L p (Σ, μ) ⊗ X and the r-norm · L p (Σ, μ)⊗X . With this strategy in mind, consider a sequence { f j } j ∈N ⊆ L p (Σ, μ) ⊗ X satisfying ∞ 

M :=

f j Lr p (Σ, μ)⊗X < +∞.

(1.4.21)

j=1

The goal is to prove that the series

∞ 

  f j converges in L p (Σ, μ) ⊗ X, · L p (Σ, μ)⊗X .

(1.4.22)

j=1

To this end, define G n :=

n 

f j (·) Xr : Σ → [0, ∞) for each n ∈ N,

(1.4.23)

j=1

and G :=

∞ 

f j (·) Xr : Σ → [0, ∞].

(1.4.24)

j=1

From (1.4.13) we know that each G n is μ-measurable, and G is μ-measurable as well. Then for each n ∈ N we may estimate (bearing in mind that p/r > 1) G n L p/r (Σ, μ) ≤

n  f j (·) Xr j=1

L p/r (Σ, μ)

≤

∞ 

f j Lr p (Σ, μ)⊗X = M < +∞,

j=1

(1.4.25) thanks to (1.4.16) and (1.4.21). Since we also have G n  G pointwise on Σ as n → ∞, Lebesgue’s Monotone Convergence Theorem implies that G L p/r (Σ, μ) = lim G n L p/r (Σ, μ) ≤ M < +∞.

(1.4.26)

G ∈ L p/r (Σ, μ).

(1.4.27)

n→∞

Consequently, In particular, G(ω) < +∞ for μ-a.e. ω ∈ Σ which, in light of (1.4.24), goes to show that ∞  f j (ω) Xr < +∞ for μ-a.e. ω ∈ Σ. (1.4.28) j=1

If for each n ∈ N we now define

1.4 Complex Interpolation on Quasi-Banach Spaces

Fn (ω) :=

n 

45

f j (ω) for each ω ∈ Σ,

(1.4.29)

j=1

then whenever m, n ∈ N are such that m ≥ n it follows from the first assertion in (1.4.20) that Fm (ω) − Fn (ω) Xr ≤

m 

f j (ω) Xr for each ω ∈ Σ.

(1.4.30)

j=n

Together, (1.4.28) and (1.4.30) imply that for μ-a.e. ω ∈ Σ we have that {Fn (ω)}n∈N is a Cauchy sequence in (X, · X ). Given that the latter space is complete, it is then meaningful to define F(ω) := lim Fn (ω) = n→∞

∞ 

f j (ω) in (X, · ), for μ-a.e. ω ∈ Σ.

(1.4.31)

j=1

Since each f j is strongly μ-measurable, Lemma 1.4.12 guarantees that F is strongly μ-measurable, while the first assertion in (1.4.20) ensure that F(ω) Xr ≤

∞ 

f j (ω) Xr = G(ω) for μ-a.e. ω ∈ Σ.

(1.4.32)

j=1

Based on these properties and (1.4.27) we deduce that F ∈ L p (Σ, μ) ⊗ X.

(1.4.33)

Relying on the first assertion in (1.4.20), (1.4.24), and (1.4.32) we also see that for each n ∈ N we have n n r   f j (ω) ≤ F(ω) Xr + f j (ω) Xr F(ω) − j=1

X

j=1

≤ F(ω) Xr +

∞ 

f j (ω) Xr

j=1

≤ 2G(ω) at μ-a.e. point ω ∈ Σ.

(1.4.34)

Upon recalling from (1.4.27) that G belongs to L p/r (Σ, μ), Lebesgue’s Dominated Convergence Theorem applies and, in concert with (1.4.16), gives that n n r r   fj p = lim F(·) − f j (·) = 0. (1.4.35) lim F − n→∞ n→∞ L (Σ, μ)⊗X X p/r j=1

j=1

L

(Σ, μ)

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1 Preliminary Functional Analytic Matters

Ultimately, from (1.4.33) and (1.4.35) we conclude that (1.4.22) holds, and this finishes the proof in the case when p < ∞. The case when p = ∞ is similar (and simpler).   Finally, the claim that L p (Σ, μ) ⊗ X, · L p (Σ, μ)⊗X is in fact a Banach space whenever X is Banach and p ∈ [1, ∞], is clear from what we have proved so far.  At long last, we are now in a position to state and prove the following result which provides a multitude of examples of analytically convex quasi-Banach spaces. Proposition 1.4.15 Let (Σ, M, μ) be a complete measure space and assume that (X, · X ) is an analytically convex quasi-Banach space. Then for each p ∈ (0, ∞] it follows that L p (Σ, μ) ⊗ X (see (1.4.12)) is an analytically convex quasi-Banach space.   Proof From Proposition 1.4.14 we already know that L p (Σ, μ) ⊗ X, · L p (Σ, μ)⊗X is a quasi-Banach space, so there remains to show that this is analytically convex. To this end, assume first that 0 < p < ∞. Then, by Theorem 1.4.5, there exists an equivalent quasi-norm · on X so that · p is plurisubharmonic. It follows that for any functions f , g ∈ L p (Σ, μ) ⊗ X and each fixed ω ∈ Σ, the function uω (z) := f (ω) + zg(ω) p is subharmonic. Hence, so is ∫ ∫ uω (z) dμ(ω) = f (ω) + zg(ω) p dμ(ω) U(z) : = Σ

= f +

Σ p zg L p (Σ, μ)⊗(X, · ) .

(1.4.36)

Now, the desired conclusion follows from Theorem 1.4.5. When p = ∞, Theorem 1.4.5 ensures that there exists an equivalent quasi-norm · on X so that · is plurisubharmonic, and we simply write 1 2π

∫ 0

2π

 1 ∫ 2π  f (ω) + eiθ g(ω) dθ ω ∈Σ 2π 0

sup f (ω) + eiθ g(ω) dθ ≥ sup

ω ∈Σ

≥ sup f (ω) . ω ∈Σ

The proof of the lemma is finished.

(1.4.37) 

We now discuss several criteria for analytic convexity in the context of quasiBanach lattices of functions. To set the stage, assume that (Σ, M, μ) is a sigma-finite measure space and denote by L0 the space of all complex-valued, μ-measurable functions on Ω. Then a quasi-Banach function space X on (Σ, M, μ), equipped with a quasi-norm · X so that (X, · X ) is complete, is an order-ideal in the space L0 if it contains a strictly positive function and if f ∈ X and g ∈ L0 with |g| ≤ | f | at μ-a.e. point implies g ∈ X with g X ≤ f X . Going further, a quasi-Banach lattice of functions (X, · X ) is called lattice r-convex if

1.4 Complex Interpolation on Quasi-Banach Spaces

47

m m    1/r  1/r  | f j |r f j Xr ≤ j=1

X

(1.4.38)

j=1

for any finite family { f j }1≤ j ≤m of functions from X (see, e.g., [107]; cf. also [119, Vol. II]). This implies that the space   [X]r := f measurable : | f | 1/r ∈ X , (1.4.39) normed by f [X]r := | f | 1/r Xr , is a Banach function space, called the r-convexification of X (cf. also [119, Vol. II, pp. 53–54], at least if r > 1). Remark 1.4.16 Let (Σ, M, μ) be a sigma-finite measure space and assume that X, Y are two quasi-Banach lattices of functions on (Σ, M, μ), which are r-convex for some r > 0. It is then clear from the above definition that [X]r = [Y ]r if and only if X = Y . The theorem below was first proved in [107], [109], though in this particular form it is stated in [127]. Theorem 1.4.17 Let X be a (complex) quasi-Banach lattice of functions and denote by ρ(X) its modulus of concavity. Then the following assertions are equivalent: (i) X is analytically convex; (ii) X is lattice r-convex for some r > 0;

−1 (iii) X is lattice r-convex for each 0 < r < 1 + log2 ρ(X) . Proof This follows directly from [109, Theorem 4.4] and [107, Theorem 2.2] pro

−1 vided X satisfies an upper p-estimate with p := 1 + log2 ρ(X) . That is, for some equivalent quasi-norm · and some constant C > 0, |x1 | ∨ · · · ∨ |xn | p ≤ C

n 

xj p

(1.4.40)

j=1

for any finite collection x1, . . . , xn ∈ X. However, this is a simple consequence of the fact that in our case |x1 | ∨ · · · ∨ |xn | ≤ |x1 | + · · · + |xn | and the Aoki-Rolewicz theorem (recalled earlier).  Returning to the task of discussing the complex method of interpolation for a compatible couple of quasi-Banach spaces X0, X1 , recall the class of admissible functions F from Definition 1.4.4, and now make the additional assumption that X0 + X1 is analytically convex. This entails   sup f (z) X0 +X1 : 0 < Re z < 1 ≤ C f F, (1.4.41) for each f ∈ F . With this in hand, all the aforementioned deficiencies of the complex method in the context of quasi-Banach spaces (such as the continuity of evaluation functions and the completeness of space F ) are easily corrected. In this setting, we

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1 Preliminary Functional Analytic Matters

define the (outer) complex interpolation spaces Xθ := [X0, X1 ]θ for θ ∈ (0, 1) as Xθ := [X0, X1 ]θ   := x ∈ X0 + X1 : there exists f ∈ F such that x = f (θ)

(1.4.42)

and introduce x [X0,X1 ]θ := inf { f F : f ∈ F so that f (θ) = x},

∀x ∈ Xθ .

(1.4.43)

It then follows that Xθ is a quasi-Banach space for 0 < θ < 1. Let us note at this point that there is an alternative choice for the class of admissible functions. Specifically, define F0 to be a subspace of F consisting of the closure of the space of functions f ∈ F with the property that f (w) ∈ X0 ∩ X1 for all w ∈ U. We will use this class to define the inner complex interpolation spaces, Xθi := [X0, X1 ]iθ for θ ∈ (0, 1), by asking that x ∈ Xθi if and only if there exists f ∈ F0 such that x = f (θ), and set x [X0,X1 ]i := inf { f F : f ∈ F0 so that f (θ) = x}, θ

∀x ∈ Xθi .

(1.4.44)

The inner spaces have the advantage that it is an immediate consequence of the definition that X0 ∩ X1 is dense in each Xθi . Indeed, given any x ∈ Xθi , we have x = f (θ) for some f ∈ F0 . The latter membership implies the existence of a sequence { f j } j ∈N ⊆ F with f j (U) ⊆ X0 ∩ X1 for each j ∈ N with f j − f F → 0 as j → ∞. Since, obviously, for each j ∈ N both f j and f j − f belong to the subspace F0 , it follows that the vector x j := f j (θ) ∈ X0 ∩ X1 satisfies the norm estimate x j − x0 [X0,X1 ]i = ( f j − f )(θ) [X0,X1 ]i ≤ f j − f F for each j ∈ N. The latter, θ θ when combined with lim j→∞ f j − f F = 0, allows us to conclude that X0 ∩ X1 is dense in Xθi . If X0 and X1 are Banach spaces the inner and outer complex methods yield exactly the same spaces (isometrically). But the argument for this depends essentially on the fact that X0 and X1 are Banach spaces (see [110] for more on this topic). Thus, it is far from clear whether the inner and outer complex methods will always yield the same result in our setting. However, in special cases the inner and outer methods do yield the same result, as we will see below. Let us here note that complex interpolation of quasi-Banach spaces contained in an analytically convex ambient space (not necessarily X0 + X1 ) was first studied in [16]. To state the next result, recall that, given two quasi-Banach lattices of functions (X j , · X j ), j ∈ {0, 1}, the Calderón product X01−θ X1θ , with 0 < θ < 1, is   X01−θ X1θ := h ∈ L0 : there exist f ∈ X0, g ∈ X1 such that |h| ≤ | f | 1−θ |g| θ ,   θ : | f | ≤ | f | 1−θ | f | θ , f ∈ X , j ∈ {0, 1} . f X 1−θ X θ := inf f0 X1−θ f 1 0 1 j j X 0 1 0 1 (1.4.45)

1.4 Complex Interpolation on Quasi-Banach Spaces

49

A simple yet important feature for us here is that the Calderón product “commutes" with the process of convexification. More concretely, if X0 , X1 are as above and, in addition, X0 , X1 are also lattice r-convex for some r > 0, it is straightforward to check that (1.4.46) [X01−θ X1θ ]r = ([X0 ]r )1−θ ([X1 ]r )θ , ∀θ ∈ (0, 1), in the sense of equivalence of quasi-norms. It has been pointed out in [111] that the complex method described above gives the result predicted by the Calderón formula for nice pairs of function spaces. Let us record a specific result, building on earlier work in [69] and which has been proved in [111] for what we now call the outer method. To state it, recall that a Polish space is a topological space that is homeomorphic to some complete separable metric space. Theorem 1.4.18 Let Ω be a Polish space and let μ be a sigma-finite Borel measure on Ω. Let X0, X1 be a pair of quasi-Banach function spaces on (Ω, μ). Suppose that both X0 and X1 are analytically convex and separable. Then X0 + X1 is analytically convex and, for each θ ∈ (0, 1), [X0, X1 ]θ = [X0, X1 ]iθ = X01−θ X1θ

(1.4.47)

in the sense of equivalence of quasi-norms. Remark 1.4.19 As pointed out in [111], the hypothesis of separability in this case is equivalent to sigma-order continuity. For a general quasi-normed space X, this property asserts that a non-negative, non-increasing sequence of functions in X which converges a.e. to zero also converges to zero in the quasi-norm topology of X (cf., e.g., [119, Vol. II]). An equivalent reformulation is that if g ∈ X and | fn | ≤ |g| for each n ∈ N and fn → f a.e. as n → ∞ then fn − f X → 0 as n → ∞. For us, it is of interest to also note a result, proved in [39, Theorem 1.29], to the effect that one of the lattices X0, X1 X 1−θ X1θ is sigma-order continuous =⇒ 0 is sigma-order continuous for each index θ belonging to (0, 1).

(1.4.48)

Let us briefly indicate why the inner and outer methods agree here. It fact the argument in [111] shows that for any given function f ∈ [X0, X1 ]θ a nearly optimal choice for F ∈ F with F(θ) = f is of the form F(z) = u| f0 | 1−z | f1 | z where f0 ∈ X0 , f1 ∈ X1 and |u| = 1 a.e. But we can select a sequence of Borel sets En  Ω with the property that 1En f0, 1En f1 ∈ X0 ∩ X1 , and consider Fn := 1En F. Thus Fn ∈ F0 and using order-continuity one sees that Fn − F F → 0 as n → ∞. Thus F ∈ F0 . Remark 1.4.20 For sequence spaces, Theorem 1.4.18 continues to hold in the case when just one of the two quasi-Banach lattices X0 , X1 is separable. Indeed, in [111], the separability hypotheses on X j , j ∈ {0, 1}, was used to ensure that if f0 ∈ X0 and f1 ∈ X1 then the function z → | f0 | 1−z | f1 | z is admissible (i.e. belongs to F ). In fact, the one property which is not immediate is its continuity on the closure of the strip 0 < Re z < 1. Nonetheless, this issue can be handled as follows. If we now set g := | f0 | 1−θ | f1 | θ ∈ X01−θ X1θ for f j ∈ X j and 0 < θ < 1, then

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1 Preliminary Functional Analytic Matters

g1E = | f0 1E | 1−θ | f1 1E | θ for any E ⊂ Ω. In particular, if E is finite then, clearly, z → | f0 1E | 1−z | f1 1E | z is admissible. Thus, as in [111], we have that g1E ∈ [X0, X1 ]θ and g1E [X0,X1 ]θ ≤ C g1E X 1−θ X θ . Consider now En  Ω, a nested family of 0 1 finite sets exhausting Ω (which can be arranged if the X j ’s are sequence spaces). Replacing E by E j \ Ek and using the fact that, by (1.4.48), g1En → g in X01−θ X1θ as n → ∞, ultimately gives that {g1En }n is Cauchy in [X0, X1 ]θ . The same argument further yields that {g1En }n converges to g in X0 + X1 . Hence, g ∈ [X0, X1 ]θ and g [X0,X1 ]θ ≤ C g X 1−θ X θ . From this point on, one proceeds as in the proof of [111, 0 1 Theorem 3.4]. The theorem below originates in [111]; see also [110] where other related results can be found. Theorem 1.4.21 Let X0, X1 and Y0, Y1 be two compatible couples of quasi-Banach spaces and assume that X0 + X1 and Y0 + Y1 are analytically convex. Also, consider a bounded, linear operator T : X j → Yj , j ∈ {0, 1}. Then the following results are true. (a) In the case when Xθ := [X0, X1 ]θ and Yθ := [Y0, Y1 ]θ , or if Xθ := [X0, X1 ]iθ and Yθ := [Y0, Y1 ]iθ , it follows that T induces a bounded linear operator Tθ : Xθ −→ Yθ for each θ ∈ (0, 1),

(1.4.49)

in a natural fashion. Moreover, Tθ Xθ →Yθ ≤ T X1−θ T Xθ 1 →X1 for each θ ∈ (0, 1). 0 →X0

(1.4.50)

(b) Assume that there exists θ o ∈ (0, 1) such that Tθo is an isomorphism. Then there exists ε > 0 such that Tθ continues to be isomorphism whenever |θ − θ o | < ε. (c) If I is any open sub-interval of (0, 1) with the property that the inverse Tθ−1 exists  for each θ ∈ I, then Tθ−1 agrees with Tθ−1  on Yθ ∩ Yθ  for every θ, θ ∈ I. We continue by presenting a general interpolation result with constraints.





Theorem 1.4.22 Let Xi, · Xi , Yi, · Yi , Zi, · Zi , where i ∈ {0, 1}, be quasiBanach spaces such that X0 ∩ X1 is dense in both X0 and X1 , and Z0 ∩ Z1 is dense in both Z0 and Z1 . Suppose that Yi → Zi continuously for i ∈ {0, 1} and that there exists a linear operator D such that D : Xi → Zi boundedly for i ∈ {0, 1}. Define the spaces   i ∈ {0, 1}, (1.4.51) Xi (D) := u ∈ Xi : Du ∈ Yi , equipped with the graph norm, i.e. u Xi (D) := u Xi + Du Yi for each u ∈ Xi (D). Finally, suppose that there exist two continuous linear mappings, G : Zi → Xi and K : Zi → Yi , with the property D ◦ G = I + K on Zi for i ∈ {0, 1}. Then, whenever 0 < θ < 1 and 0 < q ≤ ∞, one has  

(1.4.52) X0 (D), X1 (D) θ,q = u ∈ (X0, X1 )θ,q : Du ∈ (Y0, Y1 )θ,q .

1.4 Complex Interpolation on Quasi-Banach Spaces

51

Furthermore, if X0 + X1 and Y0 + Y1 are analytically convex, then X0 (D) + X1 (D) is also analytically convex and     X0 (D), X1 (D) θ = u ∈ [X0, X1 ]θ : Du ∈ [Y0, Y1 ]θ , θ ∈ (0, 1). (1.4.53) Proof For the real interpolation method, the crux of the matter is establishing the following estimate involving the K-functionals of the pairs (X0 (D), X1 (D)), (X0, X1 ), (Y0, Y1 ): K(t, a; X0 (D), X1 (D)) ≈ K(t, a; X0, X1 ) + K(t, Da; Y0, Y1 ),

(1.4.54)

uniformly in t and a. One direction is, of course, trivial. For the other one, given a ∈ X0 + X1 along with t > 0, let a = x0 + x1, xi ∈ Xi, and Da = y0 + y1, yi ∈ Yi,

i ∈ {0, 1},

(1.4.55)

be nearly optimal splittings so that x0 X0 + t x1 X1 ≈ K(t, a, X0, X1 )

(1.4.56)

y0 Y0 + t y1 Y1 ≈ K(t, Da, Y0, Y1 ).

(1.4.57)

and We then define a new splitting a = x0 + x1 , where xi := xi − GDxi + Gyi, Then Also so that

xi Xi ≤ C( xi Xi + yi Yi ), Dxi = −K Dxi + yi + K yi, Dxi Yi ≤ C( xi Xi + yi Yi ),

i ∈ {0, 1}. i ∈ {0, 1}. i ∈ {0, 1}, i ∈ {0, 1},

(1.4.58) (1.4.59) (1.4.60) (1.4.61)

since K D maps Xi boundedly into Yi for i ∈ {0, 1}, and K maps Yi boundedly to itself, i ∈ {0, 1}. Then (1.4.59)-(1.4.60) justify the equivalence in (1.4.54). Formula (1.4.53), regarding complex interpolation, is due to J.-L. Lions and E. Magenes (cf. [120]) when all spaces involved are Banach. However, their argument goes through with minor modifications for quasi-Banach spaces given the analytic convexity assumptions made in this portion of our theorem. The only thing we need to check is that is that the space X0 (D) + X1 (D) is analytically convex, so that the complex interpolation method outlined in the first part of this section applies to the couple X0 (D), X1 (D). In order to justify this we first note that   X0 (D) + X1 (D) = (X0 + X1 )(D) := u ∈ X0 + X1 : Du ∈ Y0 + Y1 , (1.4.62)

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where the rightmost space is equipped with the natural graph norm. Indeed, (1.4.62) follows readily from the decompositions (1.4.55), (1.4.58). Thus, it suffices to prove that (X0 + X1 )(D) is analytically convex. To this end, let f : U → (X0 + X1 )(D) be an analytic function which extends by continuity to U (where U is the open unit strip {z : 0 < Re z < 1} in the complex plane). Since the inclusion mapping ι : (X0 + X1 )(D) → X0 + X1 is linear and bounded, f may also be regarded as a (X0 + X1 )-valued analytic function in U, extendible by continuity to U. Similarly, the operator D : (X0 + X1 )(D) → Y0 + Y1 is linear and bounded, thus the function D f is a (Y0 + Y1 )-valued analytic function in U, which extends by continuity to U. Consequently, given that X0 + X1 and Y0 + Y1 are analytically convex, we may write max f (z) (X0 +X1 )(D) ≈ max f (z) X0 +X1 + max D f (z) Y0 +Y1

0 0 with the property that Bψ ( f , r) ⊆ O, where Bψ ( f , r) := {g ∈ X : ψ( f − g) < r }.

(1.5.6)

In such a setting, call a sequence { fn }n∈N ⊆ X Cauchy provided for every ε > 0 there exists Nε ∈ N such that ψ( fn − fm ) < ε whenever n, m ∈ N are such that n, m ≥ Nε . Also, call (X, τψ ) complete if any Cauchy sequence in X is convergent in τψ to some element in X. Our second definition introduces a drastically weakened notion of measure. Definition 1.5.2 Given a measurable space (Σ, M), call μ : M → [0, +∞] a feeble measure provided the collection of its null-sets, i.e., Nμ := { A ∈ M : μ(A) = 0} contains , is closed under countable union, and satisfies A ∈ Nμ whenever A ∈ M and there exists B ∈ Nμ such that A ⊆ B. Let (Σ, M) be a measurable space and let μ be a feeble measure on M. As in the case of genuine measures, we shall say that a property is valid μ-a.e. provided the property in question is valid with the possible exception of a set in Nμ . Identifying functions coinciding μ-a.e. on Σ then becomes an equivalence relation, and we shall denote by M (Σ, M, μ) the collection of all equivalence classes of scalar-valued, M-measurable functions on Σ. Lastly, we define   (1.5.7) M+ (Σ, M, μ) := f ∈ M (Σ, M, μ) : f ≥ 0 at μ-a.e. point on Σ . Here is the abstract completeness criterion alluded to earlier. More general results of this nature are established in [140], [138].

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57

Theorem 1.5.3 Assume that (Σ, M) is a measurable space and that μ is a feeble measure on M. Suppose that the function · : M+ (Σ, M, μ) −→ [0, +∞],

(1.5.8)

satisfies the following properties: (1) [Quasi-subadditivity] There exists a constant C0 ∈ [1, +∞) with the property that f + g ≤ C0 max{ f , g },

∀ f , g ∈ M+ (Σ, M, μ);

(1.5.9)

(2) [Pseudo-homogeneity] There exists a function ϕ : (0, +∞) → (0, +∞) satisfying λ f ≤ ϕ(λ) f ,

∀ f ∈ M+ (Σ, M, μ),

∀λ ∈ (0, +∞),

(1.5.10)

and such that   sup ϕ(λ)ϕ(λ−1 ) < +∞ and lim+ ϕ(λ) = 0; λ>0

λ→0

(1.5.11)

(3) [Non-degeneracy] There holds f = 0 ⇐⇒ f = 0,

∀ f ∈ M+ (Σ, M, μ);

(1.5.12)

(4) [Quasi-monotonicity] There exists some constant C1 ∈ [1, +∞) such that for any f , g ∈ M+ (Σ, M, μ) satisfying f ≤ g μ-a.e. on Σ there holds f ≤ C1 g ; (5) [Weak Fatou property] If { fi }i ∈N ⊆ M+ (Σ, M, μ) is a sequence of functions satisfying fi ≤ fi+1 μ-a.e. on Σ for each i ∈ N and such that sup fi < +∞, i ∈N then sup fi < +∞. i ∈N

Finally, define   L := f ∈ M(Σ, M, μ) : f L := | f | < +∞ .

(1.5.13)

Then functions in L are finite μ-a.e. on Σ and, with the topology τ · L considered in the sense of Definition 1.5.1 (relative to the additive group structure on L),

L, τ · L is a Hausdorff, complete, metrizable, topological vector space. (1.5.14) Formula (1.5.13) may be regarded as a general recipe according to which large classes of function spaces are defined in practice. Of course, any genuine quasinorm · on the vector space L 0 (Σ, M, μ), consisting of (classes of) functions from M(Σ, M) which are finite μ-a.e., satisfies the axioms (1)-(3). Any function of the form ϕ(λ) := λ θ , with θ ∈ (0, ∞) fixed, satisfies ∫ (1.5.11). Such an example arises naturally if, e.g., μ is a measure and f := Σ f θ dμ for each f ∈ M+ (Σ, M, μ) (note that · satisfies all hypotheses of Theorem 1.5.3).

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We also wish to note that the weak Fatou property mimics (in abstract, and with a weaker conclusion) the familiar Fatou’s Lemma in the standard setting of Lebesgue spaces. Indeed, in [140], [138] it has been proved that, given a feeble measure μ on a measurable space (Σ, M) and a function · : M+ (Σ, M, μ) → [0, +∞] which is quasi-subadditive and quasi-monotone, the weak Fatou property stated in item (5) of Theorem 1.5.3 is equivalent to the demand that ∀{ fi }i ∈N ⊆ M+ (Σ, M, μ) such that we have lim inf i→∞

lim inf i→∞ fi < +∞ fi < +∞

(1.5.15)

and, in fact, a quantitative version of this implication holds. In addition to completeness, we are also interested in abstract results pertaining to the pointwise behavior of sequences of functions which are convergent in topological vector spaces created according to formula (1.5.13). Specifically, the following theorem appears in [140], [138]. Theorem 1.5.4 Retain the same hypotheses as in Theorem 1.5.3 and recall the vector space L from (1.5.13). Then any sequence { f j } j ∈N in L which is convergent to some f ∈ L in the topology τ · L has a sub-sequence which converges to f pointwise μ-a.e. on Σ. In particular, the positive cone in L equipped with the partial order induced by the pointwise μ-a.e. inequality of functions, i.e., L + := { f ∈ L : f ≥ 0 μ-a.e. on Σ},

is closed in L, τ · L . The work in [140], [138] also addresses the issue of having a continuous embedding of the vector space L from (1.5.13) equipped with the topology τ · L into the space of measurable, a.e. finite functions equipped with the topology of convergence in measure. Theorem 1.5.5 Suppose that (Σ, M, μ) is a measure space and consider a mapping · : M+ (Σ, M, μ) → [0, +∞] satisfying properties (1)-(4) from Theorem 1.5.3. In addition, assume that there exists some sequence {K j } j ∈N ⊆ M satisfying

∞ $ j=1

Kj = Σ

(1.5.16)

as well as K j ⊆ K j+1, μ(K j ) < +∞, 1K j < +∞ for each j ∈ N. In this context, let L 0 (Σ, M, μ) := { f ∈ M(Σ, M, μ) : | f | < +∞ μ-a.e. on Σ} and denote by τμ the topology on this space induced by convergence in measure on sets of finite measure. Then, if L is as in (1.5.13), it follows that



(1.5.17) L, τ · L → L 0 (Σ, M, μ), τμ continuously. Regarding the separability of the topological vector space from (1.5.14), we recall the following result which appears in [138, Theorem 5.5, p. 300] (the reader is referred to [133, Definition 3.6.1] for the notion of separable measure).

1.5 Köthe Function Spaces and Other Categories of Topological Vector Spaces

59

Theorem 1.5.6 Retain the hypotheses of Theorem 1.5.5 and, in addition, assume that · is absolutely continuous, in the sense that for any given f ∈ L there holds & ∀(A j ) j ∈N ⊆ M such that 1 A j → 0 (1.5.18) =⇒ lim | f | · 1 A j = 0. j→∞ pointwise μ-a.e. on Σ as j → ∞

Then L, τ · L is a separable topological space whenever the measure μ is separable.

In the case when L, · L is a Banach function space, it follows from [14, Theorem 5.5, p. 27] that both the absolute continuity property stated in (1.5.18) and the separability of the measure μ are

actually necessary conditions for the separability of the topological space L, τ · L . The remaining portion of this section contains material which is going to be useful in justifying the boundedness result presented later, in Theorem 4.4.7, for linear operators taking values in certain categories of topological vector spaces8. We begin by reviewing a number of definitions and preliminary results which will play a significant role in the formulation and proof of Theorem 4.4.7. Definition 1.5.7 Suppose X is a vector space over C. [1] Call a function · : X → [0, ∞) a θ-pseudo-quasi-norm (or simply pseudoquasi-norm) on X provided the following three conditions hold: (i) (nondegeneracy) for each x ∈ X one has x = 0 if and only if x = 0; (ii) (quasi-subadditivity) there exists a constant C0 ∈ [1, ∞) for which x + y ≤ C0 max{ x , y },

∀x, y ∈ X.

(1.5.19)

(iii) (pseudo-homogeneity) there exist C1 ∈ (0, ∞) and θ ∈ R such that λ x ≤ C1 |λ| θ x ,

∀x ∈ X,

∀λ ∈ C \ {0}.

(1.5.20)

[2] The pair (X, · ) (which shall be referred to as a pseudo-quasi-normed space) is said to be a pseudo-quasi-Banach space provided (X, τ · ) is complete in the sense of Definition 1.5.1, where τ · is the topology induced by · on X. [3] Given s ∈ (0, ∞), call a pseudo-quasi-norm · on X s-norm provided it is genuinely homogeneous and satisfies x + y s ≤ x s + y s for all x, y ∈ X.

(1.5.21)

There are many classes of topological vector spaces which are of a basic importance in analysis that are not Banach but merely quasi-Banach. Indeed, take for 8 Recall that we understand by a topological vector space, a pair (X , τ), where X is a vector space over C and τ is a topology on X such that the vector space operations of addition and scalar multiplication are continuous with respect to τ. We stress that under these assumptions, the topological space (X , τ) may not be Hausdorff.

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example the following familiar scales of spaces: sequence spaces, Lebesgue spaces, weak-Lebesgue spaces, Lorentz spaces, Hardy spaces, weak-Hardy spaces, Besov spaces, Triebel-Lizorkin spaces, as well as their weighted versions (just to name a few). The class of pseudo-quasi-Banach spaces, given as in Definition 1.5.7, further generalizes the notion of a quasi-Banach space (hence, the notion of genuine Banach space) by allowing for the relaxation of the homogeneity condition in the manner described in (1.5.20). The following metrization result in the class of pseudo-quasi-norms, borrowed from [138, Theorem 3.39, p. 130], may be regarded as a generalization of the AokiRolewicz Theorem (see [11], [162], [112]). Theorem 1.5.8 Let X be a vector space over C and assume that · : X → [0, ∞) is a function satisfying the following properties: (1) there exists a constant C0 ∈ [1, ∞) for which x + y ≤ C0 max{ x , y },

∀x, y ∈ X;

(1.5.22)

(2) there exist C1 ∈ (0, ∞) and θ ∈ R such that λ x ≤ C1 |λ| θ x , Set

∀x ∈ X,

∀λ ∈ C \ {0};

(1.5.23)

 −1  α := log2 C0 ∈ (0, ∞],

(1.5.24)

and, for each x ∈ X, define x  :=

sup inf |λ| −θ

λ∈C\{0}

N 

λ xi α

 α1

: N ∈ N, and x1, . . . , x N ∈ X are

i=1

such that

N 

! xi = x ,

i=1

(1.5.25) if α < ∞ and, corresponding to the case when α = ∞, x  :=

sup inf |λ| −θ max λ xi : N ∈ N, and x1, . . . , x N ∈ X are

λ∈C\{0}

1≤i ≤ N

such that

N  i=1

Then ·  : X → [0, ∞) satisfies:

! xi = x , (1.5.26)

1.5 Köthe Function Spaces and Other Categories of Topological Vector Spaces

61

C0−2 x ≤ x  ≤ C1 x for all x ∈ X,

(1.5.27)

λ x  = |λ| θ x  for all x ∈ X and all λ ∈ C \ {0},

(1.5.28)

β

β

β

x + y  ≤ x  + y  for all x, y ∈ X and each finite β ∈ (0, α],

(1.5.29)

x + y  ≤ C0 max{ x , y  } for all x, y ∈ X.

(1.5.30)

Moreover, if in addition to (1.5.22)-(1.5.23), the function · has the property that for every x ∈ X one has x = 0 ⇔ x = 0 (i.e., if · is actually a pseudoquasi-norm on X), then the function β

d : X × X → [0, ∞) given by d(x, y) := x − y  for all x, y ∈ X

(1.5.31)

is a genuine distance on X such that τd = τ ·  = τ · . In particular, the function ·  is continuous on (X, τ · ). Hence, the balls with respect to ·  (see (1.5.6)) are open in τ · . We continue by making a few remarks of general nature. For k ∈ {1, 2}, suppose (Xk , τk ) is a topological vector space. Recall that a linear operator T : X1 → X2 is said to be bounded provided T maps topologically bounded subsets of X1 into topologically bounded subsets of X2 . In particular, if for k ∈ {1, 2}, the function · k : Xk → [0, ∞) is a θ k -pseudo-quasi-norm on Xk (for some θ k ∈ (0, ∞)) such that τ · k = τk , then by the homogeneity conditions for · k as well as the coincidence between notions of topologically and geometrically bounded sets one has the linear operator T : X1 → X2 is bounded if and only if there exists θ /θ C ∈ (0, ∞) such that T f X2 ≤ C f X21 1 for every f ∈ X1 .

(1.5.32)

Another property of general pseudo-quasi-normed spaces which will be of relevance shortly is as follows: if (X, · ) is a pseudo-quasi-normed vector space then there exists C ∈ [1, ∞) such that if lim x j = x∗ in X, in the topology induced on X by · , then j→∞

C −1 x∗ ≤ lim inf x j ≤ lim sup x j ≤ C x∗ . j→∞

j→∞

(1.5.33) The justification of (1.5.33) makes use of the continuity of the function · , given as in Theorem 1.5.8, as well as (1.5.27). We conclude with the following definition which is relevant later, in the context of Theorem 4.4.7. Definition 1.5.9 Two given topological vector spaces, (Xk , τk ) with k ∈ {1, 2} are said to be weakly compatible provided there exists a topological vector space (X , τX ) satisfying the following two conditions:

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(i) every convergent sequence of points in X has a unique limit; (ii) for each k ∈ {1, 2} there exists an injective linear mapping ιk : Xk → X satisfying ⎪ for all {x j } j ∈N ⊆ Xk and x ∈ Xk ⎫ ⎪ ⎬ such that lim x j = x in (Xk , τk ) ⎪ ⎪ j→∞ ⎭

=⇒ lim ιk (x j ) = ιk (x) in (X , τX ). j→∞

(1.5.34)

Since not all topological vector spaces considered in this work are necessarily Hausdorff, the additional demand on (X , τX ) in part (i) of Definition 1.5.9 is not redundant. Also, the mapping ιk : (Xk , τk ) → (X , τX ) in part (ii) may not be continuous given the minimal assumptions on the topological spaces (Xk , τk ). This being said, if ιk is continuous then it necessarily satisfies (1.5.34). Finally, in light of the injectivity of the mapping ιk in part (ii), there is no ambiguity in identifying x ≡ ιk (x) ∈ X whenever x ∈ Xk .

Chapter 2

Abstract Fredholm Theory

With an eye on eventually employing Fredholm theory in the treatment of boundary value problems for elliptic systems in rough domains (later on, in [137]), here we review basic material on the topic of Fredholm theory, from a purely functional analytic point of view. First, in §2.1, we recall useful results of this nature in the classical setting of Banach spaces. This is where Fredholm theory is at its zenith of beauty but, elegant as this may be, Fredholm theory is vastly restricted to such a setting everywhere in the literature. This is because virtually all treatments make essential use of the fact that Banach spaces have a rich duality theory, and rely on the availability of the Hahn-Banach Theorem together with its usual plethora of very useful consequences. Since a variety of function spaces naturally employed in the formulation of the boundary value problems we have in mind, such as certain Hardy, Besov, and Triebel-Lizorkin spaces, fail to be Banach (as their “norms” only satisfy a quasitriangle inequality), we also require a refinement of the “standard” Fredholm theory on Banach spaces which is applicable to more general topological vector spaces (which are not necessarily locally convex). In §2.2 we recall some basic results to this effect, following work in [142] (where a “dual-less” approach to Fredholm theory has been developed) which shows that the core principle of the theory, namely that a Fredholm operator is a linear homeomorphism modulo finite-dimensional spaces is pervasive. Along the way, a number of new useful results are deduced.

2.1 Fredholm Theory in Banach Spaces To get started, recall the following basic definition. Definition 2.1.1 Let X, Y be two Banach spaces. Define the space of Fredholm operators from X to Y as

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1_2

63

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 Φ(X → Y ) := T ∈ Bd(X → Y ) : T has a finite-dimensional cokernel  and a finite-dimensional kernel .

(2.1.1)

Also, consider the index function ind : Φ(X → Y ) −→ Z defined by ind T := dim (Ker T) − codim (Im T) for each T ∈ Φ(X → Y ).

(2.1.2)

Next, set  Φ+ (X → Y ) := T ∈ Bd(X → Y ) : T has closed range  and a finite-dimensional kernel ,

(2.1.3)

and   Φ− (X → Y ) := T ∈ Bd(X → Y ) : T has a finite-dimensional cokernel . (2.1.4) The set of semi-Fredholm operators from X to Y is then defined as the union Φ− (X → Y ) ∪ Φ+ (X → Y ). The index function (2.1.2) may be extended to the set of all semi-Fredholm operators by setting index : Φ− (X → Y ) ∪ Φ+ (X → Y ) −→ Z ∪ {±∞} defined as index T := dim (Ker T) − dim (coker T)

(2.1.5)

for each operator T ∈ Φ− (X → Y ) ∪ Φ+ (X → Y ). Occasionally, if we desire to emphasize the spaces on which the operator T is defined, we shall write index (T : X → Y ) in place of the less informative symbol ind T. The following theorem summarizes various standard properties of Fredholm and semi-Fredholm operators in Banach spaces which we will find useful later on. In this functional analytic setting, the literature is enormous; see, e.g., [17, Chs. 1, 3], [65, Chs. XI, XVII], [66, Vol. I, Sects. IV.6, IV.10], [128, Sect. I.3], [151, Ch. 2], [153, Ch. III], [171, Chs. 5, 7]. Theorem 2.1.2 The following assertions hold for any Banach spaces X, Y, Z: (1) Any semi-Fredholm operator T ∈ Φ− (X → Y ) ∪ Φ+ (X → Y ), hence any Fredholm operator T ∈ Φ(X → Y ), has closed range. (2) If T ∈ Φ± (X → Y ) and S ∈ Φ± (Y → Z) then ST ∈ Φ± (X → Z) and index (ST) = index (S) + index (T).

(2.1.6)

In particular, for any T ∈ Φ(X → Y ) and S ∈ Φ(Y → Z) it follows that ST ∈ Φ(X → Z) and that (2.1.6) holds. (3) If T ∈ Φ± (X → Y ) then T ∗ ∈ Φ∓ (Y ∗ → X ∗ ) and index (T) = −index (T ∗ ). In particular, if T ∈ Φ(X → Y ) then T ∗ ∈ Φ(Y ∗ → X ∗ ) and their indexes satisfy index (T) = −index (T ∗ ).

2.1 Fredholm Theory in Banach Spaces

65

(4) Given T ∈ Bd(X → Y ), one has T ∈ Φ+ (X → Y ) if and only if T is bounded from below modulo compact operators, i.e., there exist a Banach space Z, a compact operator K : X → Z, and a constant C ∈ (0, ∞) such that  xX ≤ CT xY + K x Z for any x ∈ X.

(2.1.7)

In particular, Φ+ (X → Y ) is both open in Bd(X → Y ) and stable under addition of compact operators. Moreover, the index is unaffected by addition of compact operators. Another consequence of (2.1.7) is the fact that if X0 is a closed subspace of X and T ∈ Φ+ (X → X) with T X0 ⊆ X0 , then also T |X0 ∈ Φ+ (X0 → X0 ).

(2.1.8)

(5) The set Φ− (X → Y ) is open in Bd(X → Y ) and Φ− (X → Y ) is stable under addition of compact operators. Furthermore, the index is unaffected by addition of compact operators. (6) Under addition of compact operators, the set Φ(X → Y ) is stable and the index is unaffected. Also, Φ(X → Y ) = Φ− (X → Y ) ∩ Φ+ (X → Y ).

(2.1.9)

(7) Given T ∈ Bd(X → Y ), one has T ∈ Φ(X → Y ) if and only if there exist S1, S2 ∈ Bd(Y → X) and K1 ∈ Cp(Y → Y ) and K2 ∈ Cp(X → X) such that T S1 = IY + K1 and S2T = IX + K2

(2.1.10)

where IX , IY are the identity operators on X, Y . In fact, one may actually take S1 = S2 ∈ Φ(X → Y ), i.e., T is Fredholm if and only if it is invertible modulo compact operators. (8) The index function (2.1.5) is continuous, hence locally constant. Several abstract results will play an important role. The first such result is an index estimate, resembling a monotonicity property of the index on a nested scale of spaces, is recalled below. For a proof see [139] (cf. also [144, Lemma 11.9.21, p. 206]). Lemma 2.1.3 Let X j , Yj , j = 0, 1, be two pairs of Banach spaces such that both inclusions X0 → X1 and Y0 → Y1 are continuous with dense range. If T is a linear operator which is Fredholm both from X0 into Y0 and from X1 into Y1 , then index (T : X0 → Y0 ) ≤ index (T : X1 → Y1 ).

(2.1.11)

Furthermore, (2.1.11) holds with equality if and only if Ker (T : X0 → Y0 ) = Ker (T : X1 → Y1 ) and Ker (T ∗

:

Y0∗

→

X0∗ )

=

Ker (T ∗

(2.1.12) :

Y1∗

→

X1∗ ).

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2 Abstract Fredholm Theory

We shall also need the following abstract regularity result from [139, Lemma 3.10, pp. 136-137]. Lemma 2.1.4 Let X0 , X1 , be two Banach spaces such that X0 → X1 continuously and densely. Assume that T is a linear operator which is Fredholm both from X0 into X0 and from X1 into X1 , with the property that index (T : X1 → X1 ) = index (T : X0 → X0 ).

(2.1.13)

x1 ∈ X1 and T x1 ∈ X0 =⇒ x1 ∈ X0 .

(2.1.14)

X0 ∩ Im (T : X1 → X1 ) = Im (T : X0 → X0 ),

(2.1.15)

Ker (T : X1 → X1 ) = Ker (T : X0 → X0 ).

(2.1.16)

Then As a corollary,

Given a Banach space X, the annihilator V ⊥ of a subspace V of X is defined as   V ⊥ := Λ ∈ X ∗ : Λ(x) = 0 for all x ∈ V , (2.1.17) and the annihilator ⊥W of a subspace W of X ∗ is defined as   ⊥ W := x ∈ X : Λ(x) = 0 for all Λ ∈ W .

(2.1.18)

For example, one of the standard corollaries of the Hahn-Banach Extension Theorem gives V ⊥ = {0} ⇐⇒ V = X, and V ⊥ = X ∗ ⇐⇒ V = {0}. (2.1.19) It is also well known (cf., e.g., [165, Theorem 4.7, p. 96]) that, in the above context, we have ⊥ ⊥ (V ) is the norm-closure of V in X, (2.1.20) and

(⊥W)⊥ is the weak∗ -closure of W in X ∗ . X∗

(2.1.21)

V∗

Also, since  Λ → Λ|V ∈ is a linear mapping which is surjective (thanks to the Hahn-Banach Extension Theorem; cf. [165, Theorem 3.3, p. 58]) and whose kernel is precisely V ⊥ , the First Group Isomorphism Theorem ensures that we may identify X∗ = V ∗. (2.1.22) V⊥ Hence,   (2.1.23) dim X ∗ /V ⊥ = dim V ∗, and, in particular,   dim X ∗ /V ⊥ = dim V if V is finite dimensional.

(2.1.24)

2.1 Fredholm Theory in Banach Spaces

67

A related (albeit weaker) result holds for the brand of annihilator introduced in (2.1.18). Specifically, since X  x → Λx ∈ W ∗ where Λx () := (x) for each  ∈ W is a linear mapping whose kernel is precisely ⊥W, the First Group Isomorphism Theorem gives   (2.1.25) dim X/⊥W ≤ dim W ∗ . In relation to this we claim that, in fact,   dim X/⊥W = dim W if W is finite dimensional.

(2.1.26)

To prove (2.1.26), suppose W is finite dimensional and let Λ1, . . . , Λn ∈ X ∗ be a basis for the space W. Then Lemma 1.2.16 ensures that there exist some linearly independent vectors x1, . . . , xn ∈ X such that Λ j (xk ) = δ jk for each j, k ∈ {1, . . . , n}. If x1, . . . , xn  denotes the linear span of the vectors x1, . . . , xn , then dimx1, . . . , xn  = n and we have ⊥

W ∩ x1, . . . , xn  = {0}.

(2.1.27)

Indeed, if x0 ∈ ⊥W ∩ x1, . . . , xn  then x0 = a1 x1 +· · ·+an xn for some a1, . . . , an ∈ C and for each j ∈ {1, . . . , n} we have 0 = Λ j (x0 ) = a1 Λ j (x1 ) + · · · + an Λ j (xn ) = a j which forces x0 = 0. Having proved (2.1.27), we then conclude that   ultimately dim X/⊥W ≥ n. Since the opposite inequality is implied by (2.1.25), the claim made in (2.1.26) follows. Lemma 2.1.5 Let X be a Banach space, and let V ⊆ X be a closed linear subspace of X. Denote by π the canonical projection π : X −→

X V

given by

(2.1.28)

π(x) := x + V, for all x ∈ X, and denote by ι the canonical inclusion ι : V ⊥ → X ∗ . Then Φ : V ⊥ −→

 ∗ X V

(2.1.29)

given by

Φ(Λ), x + V := Λx, for all Λ ∈ V ⊥ and x ∈ X, is an isomorphism, whose inverse is given by  ∗ Φ−1 : VX −→ V ⊥ Φ−1 () :=  ◦ π, for all  ∈

 ∗ X V

.

(2.1.30)

(2.1.31)

As a consequence, Φ allows the identification  X ∗ V

= V ⊥.

(2.1.32)

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2 Abstract Fredholm Theory

In turn, under the identification made in (2.1.32), it follows that π ∗ : (X/V)∗ → X ∗ , the adjoint of π in (2.1.28), is in fact the canonical inclusion operator ι : V ⊥ → X ∗ .

(2.1.33)

Finally, if X is reflexive then  ∗ −→ V ⊥ given by  ∗  x + V → Ψ(x) := λx ∈ V ⊥ Ψ:

X V

X V

(2.1.34)

where λx (Λ) := Λx for all Λ ∈ V ⊥ is an isomorphism which allows the identification  ⊥∗ X V = . V

(2.1.35)

Proof The claims made in relation to (2.1.30)-(2.1.31) are seen from definitions. In particular, we have (2.1.32). Next, the claim in (2.1.33) comes down to checking that π ∗ ◦ Φ = ι on V ⊥ , which is then seen from definitions. There remains to prove that, under the additional hypothesis that X is reflexive, (2.1.34) is an isomorphism. Having Ψ injective reduces to verifying that if x ∈ X is such that Λx = 0 for all Λ ∈ V ⊥ then necessarily x ∈ V. This, however, is a consequence of one of the corollaries of the Hahn-Banach Extension Theorem (as may be seen reasoning by contradiction, keeping in mind fact that V is  the ∗ closed). To check that Ψ is surjective, fix an arbitrary λ ∈ V ⊥ . Use the Hahn ∈ (X ∗ )∗ with λ  ⊥ = λ. Since X is Banach Extension Theorem to produce some λ V presently assumed to be reflexive, the membership just mentioned implies that there  = Λ(xo ) for each Λ ∈ X ∗ . Then exists some xo ∈ X with the property that λ(Λ) ∗ ⊥ satisfies λxo (Λ) = Λ(xo ) = λ(Λ) for all Λ ∈ V ⊥ , i.e., λxo = λ. λxo = Ψ(xo ) ∈ V  Thus, Ψ(xo ) = λ, proving that Ψ is indeed surjective. A useful generalization of Lemma 2.1.5 is presented below. Lemma 2.1.6 Let X be a Banach space, and let V, W ⊆ X be closed linear subspaces of X with the property that W ⊆ V. Then one has the identification  V ∗ W

=

W⊥ . V⊥

(2.1.36)

The special case W = 0 of (2.1.36) reads V∗ =

X∗ V⊥

(2.1.37)

while the special case V = X of (2.1.36) corresponds precisely to (2.1.32) (after re-denoting W as V). Proof of Lemma 2.1.6 Denote by π the canonical projection

2.1 Fredholm Theory in Banach Spaces

π : V −→

69 V W

given by

(2.1.38)

π(x) := x + W, for all x ∈ V . Given any functional f ∈



∗

, it follows that f ◦ π : V → R is linear and f ◦ π V = f ◦ π (the Hahn-Banach continuous. Pick any

f ◦ π ∈ X ∗ such that

Theorem ensures that such an extension always exists). It follows that

f ◦ π W ≡ 0 which ultimately shows that

f ◦ π ∈ W ⊥ . Consider now the mapping  ∗ ⊥ V Φ: W −→ W V ⊥ given by (2.1.39)  ∗ V Φ( f ) :=

f ◦ π + V ⊥, for each f ∈ W . V W

Note that this is meaningfully and unambiguously defined (since the difference of any two extensions of f ◦ π ∈ V ∗ to functionals in X ∗ always belongs to V ⊥ ). In particular, this makes Φ linear and bounded. In the opposite direction, define  ∗ ⊥ V Ψ:W −→ by setting ⊥ W V (2.1.40) Ψ(Λ + V ⊥ )(x + W) := Λ(x), for each Λ ∈ W ⊥ and x ∈ V . This is a well defined, linear, and bounded mapping. Then for each f ∈ may write



V W

∗

we

  Ψ Φ( f ) (x + W) = (

f ◦ π)(x) = ( f ◦ π)(x) = f (x + W) for each x ∈ V, (2.1.41) hence

 V ∗   Ψ Φ( f ) = f for each f ∈ . W

Finally, pick an arbitrary Λ ∈ W ⊥ ⊆ X ∗ and set f := Ψ(Λ + V ⊥ ) ∈ f ◦ π agrees with Λ on V, since

(2.1.42) 

( f ◦ π)(x) = f (x + W) = Ψ(Λ + V ⊥ )(x + W) = Λ(x) for each x ∈ V .

V W

∗

. Then

(2.1.43)

As such, Λ ∈ X ∗ is an extension of f ◦ π, hence Φ( f ) = Λ + V ⊥ . Ultimately, this goes to show that   Φ Ψ(Λ + V ⊥ ) = Λ + V ⊥ for each Λ ∈ W ⊥ . (2.1.44) Thus, Φ and Ψ are inverse to one another, and this finishes the proof of (2.1.36).  Next, if X, Y are two Banach spaces and T : X → Y is a linear and bounded operator, then (cf., e.g., [165, Theorem 4.12, p. 99])

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2 Abstract Fredholm Theory



⊥ Ker (T ∗ : Y ∗ → X ∗ ) = Im (T : X → Y ) ,  Ker (T : X → Y ) = ⊥ Im (T ∗ : Y ∗ → X ∗ ) ,

(2.1.45) (2.1.46)

and (cf., e.g., [165, Theorem 4.14, p. 101]) Im (T : X → Y ) is closed in Y ⇐⇒ Im (T ∗ : Y ∗ → X ∗ ) is weak∗ -closed in X ∗ ⇐⇒ Im (T ∗ : Y ∗ → X ∗ ) is normed-closed in X ∗ . (2.1.47) As a consequence of (2.1.46) and (2.1.21) we have  ⊥ Ker (T : X → Y ) is the weak∗ -closure of Im (T ∗ : Y ∗ → X ∗ ) in X ∗,

(2.1.48)

and as a consequence of (2.1.45) and (2.1.20) we have  ⊥ Ker (T ∗ : Y ∗ → X ∗ ) is the norm-closure of Im (T : X → Y ) in Y .

(2.1.49)

In relation to (2.1.47)-(2.1.49) we also wish to note a couple of related results. First, it is clear from definitions that if X is a reflexive Banach space then the weak∗ -topology on X ∗ coincides with the weak topology on X ∗ (regarded as a Banach space in its own right).

(2.1.50)

Second, given a locally convex Hausdorff topological vector space (X, τ) along with a linear subspace V of X, the closure of V in (X, τ) coincides with the closure of V in the weak topology on X; in particular, V is τ-closed if and only if V is closed in the weak topology on X.

(2.1.51)

To justify this, denote by V the closure of V in (X, τ). In one direction, if xo ∈ X but xo  V then the Hahn-Banach Theorem ensures the existence of some functional Λ ∈ X ∗ with Λ(xo ) = 1 and such that Λ|V ≡ 0. Define O := {x ∈ X : |Λ(x)| < 1}. Then O + xo is a neighborhood of xo in the weak topology on X which is disjoint from V. Hence, xo does not belong to the closure of V in the weak topology on X. Ultimately, this shows that the closure of V in the weak topology on X is contained in V. In the opposite direction, if x ∈ V and O is any open set in the weak topology on X, then O ∈ τ, hence O ∩ V  . This proves that x belongs to the closure of V in the weak topology on X. Thus, V is contained in the closure of V in the weak topology on X. Third, combining (2.1.50) and (2.1.51) yields the following result: if X is a reflexive Banach space and V is a linear subspace of X ∗ , then the weak∗ -closure of V coincides with the (norm) closure of V in the Banach space X ∗ .

(2.1.52)

2.1 Fredholm Theory in Banach Spaces

71

Hence, as a consequence of (2.1.48) and (2.1.52), we have if X, Y are two Banach spaces, X being reflexive, and T : X → Y is a  ⊥ linear and bounded operator, then Ker (T : X → Y ) is the (norm) closure of Im (T ∗ : Y ∗ → X ∗ ) in the Banach space X ∗ .

(2.1.53)

In particular, from (2.1.53) and (2.1.19) we see that if X, Y are two Banach spaces, X being reflexive, and T : X → Y is a linear and bounded operator, then T is injective if and only if T ∗ has dense range.

(2.1.54)

Finally, we wish to note that the reflexivity assumption is essential in the above context. Indeed, take X :=  1 , Y :=  2 , and take T to be the canonical inclusion ι :  1 →  2 . Then T is injective yet T ∗ , which becomes the canonical inclusion of  2 into  ∞ (cf. Lemma 1.2.1), fails to have dense range. Thus, the conclusion in (2.1.54) may fail when the Banach space X is not reflexive. Lemma 2.1.7 Let X be a Banach space, T : X → X a linear bounded operator and V ⊆ X a closed subspace, invariant under T. Then the annihilator V ⊥ is a closed subspace of X ∗ which is invariant under the adjoint T ∗ : X ∗ → X ∗

(2.1.55)

and the quotient map [T] :

X X −→ , V V

[T](x + V) := (T x) + V,

∀x ∈ X,

(2.1.56)

is well defined, linear and bounded. Moreover, under the identification made in (2.1.32), it follows that [T]∗ : (X/V)∗ → (X/V)∗ , the adjoint of (2.1.56), is the operator T ∗ V ⊥ : V ⊥ → V ⊥ .

(2.1.57)

Finally, if the Banach space X is actually reflexive then, under the identifications made in (2.1.32) and (2.1.35), it follows that the adjoint of T ∗ V ⊥ : V ⊥ → V ⊥ (2.1.58) is the operator [T] : X/V → X/V, and

T ∗ maps V ⊥ isomorphically onto itself if and only if the quotient map [T] is an isomorphism in (2.1.56).

(2.1.59)

Proof That V ⊥ is a closed subspace of X ∗ , invariant under T ∗ , and the quotient map in (2.1.56) is well defined, linear and bounded, are all routine consequences of definitions. Next, observe that π ◦ T = [T] ◦ π where π is as in (2.1.28). Taking

72

2 Abstract Fredholm Theory

adjoints and using (2.1.33) then yields T ∗ ◦ ι = ι◦[T]∗ on (X/V)∗ = V ⊥ (cf. (2.1.32)), where ι is the canonical inclusion (2.1.29). From this, the claim in (2.1.57) readily follows. Moving on, work under the assumption that X is reflexive. Then the identifications made in (2.1.32) and (2.1.35) give  X  ∗∗ V

 ∗ X = V⊥ = , V

(2.1.60)

hence X/V is reflexive. Granted this, the claim made in (2.1.58) follows from (2.1.57). There remains to deal with the equivalence in (2.1.59). In one direction, assume T ∗ : V ⊥ −→ V ⊥ isomorphically.

(2.1.61)

Let x + V ∈ Ker [T]. Then x ∈ X is such that T x ∈ V. We wish to show that x ∈ V. Since V = V, according to one of the standard corollaries of the Hahn-Banach Theorem the latter membership is equivalent to proving that Λ(x) = 0 for each Λ ∈ V ⊥ .

(2.1.62)

To this end, fix Λ ∈ V ⊥ arbitrary. Thanks to (2.1.61), there exists Ψ ∈ V ⊥ such that T ∗ Ψ = Λ. Hence, (2.1.63) Λ(x) = (T ∗ Ψ)(x) = Ψ(T x) = 0, where the last equality is based on the fact that T x ∈ V and Ψ ∈ V ⊥ . This establishes (2.1.62) which, in turn, finishes the proof of the fact that x ∈ V. In summary, x + V ∈ Ker [T] forces x ∈ V, which goes to show that [T] is injective. To prove that [T] is surjective, consider an arbitrary y ∈ X. Define    −1 (2.1.64) Θ : V ⊥ −→ C, Θ(Λ) := T ∗ V ⊥ Λ (y) for each Λ ∈ V ⊥ . Given that Θ is a linear and continuous functional, the Hahn-Banach Theorem ensure  from the subspace V ⊥ to the entire X ∗ . Hence, that it has an extension, call it Θ, ∗ ∗  Θ ∈ (X ) and since we are presently assuming that X is reflexive, there exists x ∈ X  with the property that Θ(Λ) = Λ(x) for every Λ ∈ X ∗ . In particular,    −1 (2.1.65) Λ(x) = T ∗ V ⊥ Λ (y) for each Λ ∈ V ⊥, hence

T ∗ Λ(x) = Λ(y) for each Λ ∈ V ⊥,

(2.1.66)

Λ(T x − y) = 0 for each Λ ∈ V ⊥ .

(2.1.67)

which further implies

From this and the same standard corollary of the Hahn-Banach Theorem invoked earlier we deduce that T x−y ∈ V = V. Thus, ultimately, [T](x+V) = (T x)+V = y+V

2.2 Fredholm Theory in Topological Vector Spaces

73

which goes to show that [T] is indeed surjective. At this stage, we may therefore conclude that if T ∗ is an isomorphism in the context of (2.1.61) then [T] is an isomorphism in (2.1.56). Conversely, assume [T] is an isomorphism in (2.1.56). Consider a functional Λ ∈ V ⊥ such that T ∗ Λ = 0. Pick an arbitrary y ∈ X. Since [T] is surjective, there exists x ∈ X with the property that T x − y ∈ V. As such, we may write 0 = Λ(T x − y) =⇒ Λ(y) = Λ(T x) = T ∗ Λ(x) = 0. (2.1.68) Thus, Λ = 0 which proves that T ∗ V ⊥ is injective. There remains to show that T ∗ : V ⊥ → V ⊥ is surjective. As a preamble, we introduce some notation. For every y ∈ X, define Wy := {x ∈ X : (T x) − y ∈ V }. The fact that [T] is surjective implies Wy  . Moreover, since [T] is injective if x1, x2 ∈ Wy then necessarily x1 − x2 ∈ V. Also, since V is invariant under T, we have V + Wy = Wy . The fact that [T] is an isomorphism also implies Wy1 +y2 = Wy1 + Wy2 for every y1, y2 ∈ X, and Wy = V whenever y ∈ V. Finally, WT x = x + V for every x ∈ X. Returning to the main topic of conversation, pick an arbitrary functional Λ ∈ V ⊥ and define the mapping Φ : X → C by setting Φ(y) := Λ(x), for each y ∈ X and x ∈ Wy .

(2.1.69)

The earlier digression ensures that Φ is a well-defined linear mapping whose restriction to V vanishes identically. In concert, the continuity of [T] and the continuity of Λ imply that Φ is continuous, hence Φ ∈ V ⊥ . In addition, for each x ∈ X and each z ∈ WT x = x + V we have (T ∗ Φ)(x) = Φ(T x) = Λ(z) = Λ(x).

(2.1.70)

Hence, T ∗ Φ = Λ proving that the operator T ∗ : V ⊥ → V ⊥ is indeed surjective (thus, ultimately, an isomorphism). 

2.2 Fredholm Theory in Topological Vector Spaces In this section we refine some of the results in §2.1 through the consideration of topological vector spaces more general than Banach spaces. The following definition (which should be contrasted with Definition 2.1.1) plays a fundamental role. Definition 2.2.1 Given two linear topological spaces X, Y , a linear continuous mapping T : X → Y is said to be Fredholm provided it verifies the following three axioms: (F1) Im T := Im (T : X → Y ) is a closed subspace of Y , of finite codimension in Y . (F2) Ker T := Ker (T : X → Y ) is finite dimensional and topologically complemented in X.

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2 Abstract Fredholm Theory

(F3) T is a relatively open map (meaning that T(O) is a relatively open subset of Im (T : X → Y ) whenever O is open in X). Set

  Φ(X → Y ) := T : X → Y : T Fredholm

(2.2.1)

and consider the index function index : Φ(X → Y ) −→ Z defined by index T := dim (Ker T) − codim (Im T) for each T ∈ Φ(X → Y ).

(2.2.2)

As in the case of Banach spaces, if we wish to stress the spaces on which the operator T is considered, we may write index (T : X → Y ) in place of the less revealing symbol index T. In relation to the very general definition given above, it reassuring to note that in a more restrictive setting we can get away with assuming less, namely, if X and Y are actually Banach spaces, then T ∈ Bd(X → Y ) is Fredholm in the sense of Definition 2.2.1 if and only if T has a finite codimensional range, and a finite-dimensional kernel (hence, T is Fredholm in the sense of Definition 2.1.1).

(2.2.3)

Specifically, we make the following remark: Remark 2.2.2 (i) Regarding condition (F1) in Definition 2.2.1, the reader is alerted to the fact that if X, Y are quasi-Banach spaces and T ∈ Bd(X → Y ) has a range of finite codimension in Y , then necessarily Im (T : X → Y ) is a closed subspace of Y (see Lemma 1.2.7). (ii) Regarding condition (F2) in Definition 2.2.1, recall that a finite-dimensional subspace of a Hausdorff linear topological space is always closed, but it need not necessarily be topologically complemented. On the other hand, if X has enough continuous linear functionals to separate points (e.g., if X is a normed space, or more generally is locally convex) then every finite-dimensional subspace is topologically complemented. (iii) Regarding condition (F3) in Definition 2.2.1, we wish to note that for quasiBanach spaces the Open Mapping Theorem (for quasi-Banach spaces; cf., e.g., [138, Corollary 6.62, p. 423]) provides this relative openness property once we know that the range of T is closed. In stark contrast with the case of Banach spaces just considered above, the topic of Fredholm theory in non-locally convex spaces has remained largely unexplored. Yet recent work in [142] shows that the core principle of the theory, namely that a Fredholm operator is a linear homeomorphism modulo finite-dimensional spaces is pervasive. This is particularly evident in the next result, a refinement of Atkinson’s classical result in Banach spaces (cf. item (7) in Theorem 2.1.2), which should be viewed as the “Fundamental Theorem of Fredholm Theory” in linear topological spaces.

2.2 Fredholm Theory in Topological Vector Spaces

75

Theorem 2.2.3 Given a pair of Hausdorff linear topological spaces X, Y , along with an operator T ∈ L (X → Y ), the following statements are equivalent: (a) T is a Fredholm operator (cf. Definition 2.2.1). (b) There exist S1, S2 ∈ L (Y → X) such that both IX − S1T and IY − T S2 are compact operators. (c) The operator T is invertible modulo compacts, in the sense that there exists some operator S ∈ L (Y → X) such that both IX − ST and IY − T S are compact operators. (d) The operator T is invertible modulo operators with finite dimensional ranges, in the sense that there exists S ∈ L(Y → X) such that both IX − ST and IY − T S have finite dimensional ranges. Moreover, if the operator T ∈ L(X → Y ) is Fredholm then matters may be arranged so that the operators S ∈ L(Y → X) appearing in items (c) and (d) above are also Fredholm. As a consequence of the above theorem we see that if X is a Hausdorff linear topological space and T ∈ L (X), then T is a Fredholm operator on X if and only if its equivalence class is an invertible element of the Calkin algebra L (X)/Cp (X).

(2.2.4)

We wish to augment Theorem 2.2.3 with the following list of additional properties of Fredholm operators in linear topological spaces (again, see [142] for a proof). Theorem 2.2.4 Let X, Y , Z be Hausdorff linear topological spaces. Then the following properties hold. (a) If T ∈ L (X → Y ) is a Fredholm operator (in the sense of Definition 2.2.1) then for any K ∈ Cp (X → Y ) the operator T + K is Fredholm and one has index (T + K) = index (T). Succinctly put, the class of Fredholm operators is stable under compact perturbations and so is the index. In particular, if T is a compact perturbation of the identity on X, then the operator T is Fredholm with index zero, hence the Fredholm Alternative holds: T is one-to-one if and only if T is onto. More broadly, the following Generalized Fredholm Alternative holds: if an operator T ∈ L (X → Y ) is invertible and K ∈ Cp (X → Y ), then T + K is a Fredholm operator and index (T + K) = 0, hence T + K is one-to-one if and only if it maps X onto Y . (b) If T ∈ L (X → Y ) and S ∈ L (Y → Z) are Fredholm operators, then so is ST and index (T S) = index (T) + index (S). (2.2.5) (c) Suppose Y ⊆ X a closed subspace of finite codimension in X. Assume that T ∈ L (X) is a Fredholm operator on X such that T(Y ) ⊆ Y and define   (2.2.6) T Y : Y → Y, T Y (x) := T x, ∀x ∈ Y .

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2 Abstract Fredholm Theory

  Then the operator T Y is a Fredholm operator on Y and index T Y = index T. (d) Assume T ∈ L (X) is a Fredholm operator and V ⊆ X is a finite dimensional subspace of X such that T(V) ⊆ V. Then the quotient map [T] :

X X −→ , V V

[T](x + V) := (T x) + V,

∀x ∈ X,

(2.2.7)

is a Fredholm operator on X/V and index [T] = index T. (e) If Y ⊆ X is a closed subspace of finite codimension in X, it follows that the canonical inclusion ι : Y → X is a Fredholm operator with index ι = −dim (X/Y ). Also, if V ⊆ X is a finite dimensional subspace of X then the canonical quotient projection π : X → X/V is a Fredholm operator with index π = dim V. (f) Let Y be a Hausdorff topological vector spaces such Y is a dense subset of X. Consider a Fredholm operator T : X → X for which Y is an invariant subspace and such that its restriction to Y , i.e., T Y defined as in (2.2.6), is a Fredholm operator on Y . Then     index T Y = index (T) =⇒ ker T = ker T Y . (2.2.8) (g) If T ∈ L (X → Y ) is a Fredholm operator and F is a finite dimensional linear subspace of Y , then T −1 F is a finite dimensional linear subspace of X. (h) If actually X, Y are quasi-Banach spaces, then the set Φ(X → Y ) of all Fredholm operators from X into Y is open in Bd (X → Y ), and the index function index : Φ(X → Y ) −→ Z

(2.2.9)

is continuous, where Z is equipped with the discrete topology. Consequently, for each fixed n ∈ Z, the set of all Fredholm operators of index n from X into Y is both (relatively) open and closed in Φ (X → Y ). For further use let us note that, as may be seen from Theorem 2.2.3 and Theorem 2.2.4, if X, Y are Hausdorff linear topological spaces and if T ∈ L (X → Y ) and S ∈ L (Y → Z) are such that ST is a Fredholm operator, then T is Fredholm if and only if S is Fredholm.

(2.2.10)

The notion of Fredholm radius, recalled below, has been originally introduced by J. Radon in [156]. Definition 2.2.5 Let X be a quasi-Banach space and denote by I the identity operator on X. Given an operator T ∈ Bd (X), define its spectral radius as  ρinv (T; X) := inf r > 0 : zI − T :X → X is a homeomorphism  for each z ∈ C \ B(0, r) . (2.2.11)

2.2 Fredholm Theory in Topological Vector Spaces

77

Also, define the Fredholm radius1 of T as   ρFred (T; X) := inf r > 0 : zI − T ∈ Φ(X → X) for each z ∈ C \ B(0, r) . (2.2.12) Lemma 1.2.12 implies that the infima in (2.2.11)-(2.2.12) are meaningful, hence both ρinv (T; X) and ρFred (T; X) are well-defined numbers in [0, ∞). In fact, 0 ≤ ρFred (T; X) ≤ ρinv (T; X) ≤ T X→X .

(2.2.13)

It is also clear from Definition 2.2.5 that for any T ∈ Bd (X) we have ρFred (T; X) = 0 whenever T ∈ Cp (X).

(2.2.14)

It is also useful to note that item (3) of Theorem 2.1.2 implies that if X is a Banach space, then for each operator T ∈ Bd (X) we have ρFred (T ∗ ; X ∗ ) ≤ ρFred (T; X) (with equality if X is reflexive).

(2.2.15)

Lemma 2.2.6 Let X be a p-Banach space with p ∈ (0, 1] and consider an operator T ∈ Bd (X). Then ess

ρFred (T; X) ≤ T X→X ≤ T X→X

(2.2.16)

zI − T is a Fredholm operator with index zero whenever z ∈ C satisfies |z| > ρFred (T; X).

(2.2.17)

and

Moreover, if X is actually a Banach space and X0 is a closed subspace of X which is invariant under T (i.e., T X0 ⊆ X0 ), then ρFred (T |X0 ; X0 ) ≤ ρFred (T; X). ess

(2.2.18)

Proof Fix an arbitrary real number λ > T X→X . Then (1.2.54) guarantees the existence of some operator K ∈ Cp(X → X) with the property that T −K X→X < λ. Then, for each z ∈ C with |z| ≥ λ, the operator zI − (T − K) is invertible (via a Neumann series; cf. Lemma 1.2.12). Consequently, by item (a) in Theorem 2.2.4, we have zI −T = zI − (T − K) − K ∈ Φ(X → X). In view of (2.2.12), this proves that ρFred (T; X) ≤ λ. With this in hand, the first inequality in (2.2.16) follows by letting ess λ  T X→X . The second inequality in (2.2.16) is a particular case of (1.2.56). Finally, the mapping   z ∈ C : |z| > ρFred (T; X)  z −→ index (zI − T) ∈ Z (2.2.19)

1 sometimes referred to as the essential spectral radius

78

2 Abstract Fredholm Theory

is continuous, hence constant (cf. item (h) in Theorem 2.2.4), and takes the value zero whenever |z| is large enough (cf. Lemma 1.2.12). As such, said mapping is identically zero. This finishes the proof of (2.2.17). As regards (2.2.18), assume X is actually a Banach space and suppose X0 is a closed subspace of X which is invariant under T. From (2.2.17) we see that zI − T ∈ Φ+ (X → X) for each z ∈ C with |z| > ρFred (T; X). Since X0 is a closed subspace of X which is invariant under zI − T, from (2.1.8) we conclude that z ∈ C with |z| > ρFred (T; X) we have zIX0 − T |X0 ∈ Φ+ (X0 → X0 ), where IX0 is the identity operator on X0 . Hence, the mapping   z ∈ C : |z| > ρFred (T; X)  z −→ index (zIX0 − T |X0 ) ∈ Z (2.2.20) is continuous, hence constant (cf. item (8) in Theorem 2.1.2), and takes the value zero whenever |z| is large enough (cf. Lemma 1.2.12). As a consequence, the mapping (2.2.20) is identically zero. This implies that for each z ∈ C with |z| > ρFred (T; X) we have     dim coker (zIX0 − T |X0 ) = dim Ker (zIX0 − T |X0 ) < ∞, (2.2.21) thus zIX0 − T |X0 ∈ Φ(X0 → X0 ) for each z ∈ C with |z| > ρFred (T; X). Ultimately, this proves that ρFred (T |X0 ; X0 ) ≤ ρFred (T; X).

(2.2.22) 

It has been known for a while that the Fredholm radius (aka essential spectral radius; cf. Definition 2.2.5) behaves naturally under real interpolation of Banach spaces. Specifically, [30, Corollary 1.3, p. 37] reads as follows: Proposition 2.2.7 Suppose X0, X1 are two compatible Banach spaces and consider a linear operator T : X0 + X1 → X0 + X1 such that its restrictions T : X j → X j are bounded for j = 0, 1. Then for each θ ∈ (0, 1) and q ∈ [1, ∞] one has    ρFred T; (X0, X1 )θ,q ≤ ρFred (T; X0 )1−θ · ρFred T; X1 )θ . (2.2.23) This extends earlier work in [7] where the case 1 ≤ q < ∞ (a scenario which ensures that X0 ∩ X1 is dense in (X0, X1 )θ,q ) has been treated via a different approach. All these results deal with Banach spaces and the real method of interpolation. In this regard, Proposition 2.2.7 should be compared with Proposition 1.4.24 where a related interpolation result for the essential norm is established, both for the real and complex interpolation method, for linear and bounded operators on quasi-Banach spaces. The notion of spectral radius makes sense in a general unital Banach algebra (A,  · ). Specifically, if 1 denotes the unit in A then for each x ∈ A we define   ρinv (x) := inf r > 0 : z1 − x invertible in A for each z ∈ C \ B(0, r) . (2.2.24) In relation to this, we have the following classical result:

2.2 Fredholm Theory in Topological Vector Spaces

79

Theorem 2.2.8 (Spectral Radius Formula) Let (A,  · ) be a unital Banach algebra. Then for each x ∈ A the limit lim  x n  1/n exists and satisfies n→∞

lim  x n  1/n = ρinv (x) = inf  x n  1/n .

n→∞

n∈N

(2.2.25)

Let (X,  · X ) be a Banach space. We may then define ρinv (T; X) to be the spectral radius of T ∈ Bd (X), in the sense of (2.2.24), regarding Bd (X) a unital Banach algebra.

(2.2.26)

Also, the Calkin algebra A := Bd (X)/Cp (X) equipped with the norm ess

Bd (X)/Cp (X)  [T] → [T] A := T X→X

(2.2.27)

is a unital Banach algebra, and invertible elements in A are precisely equivalence classes [T] of Fredholm operators T on X. In view of this and (2.2.12), we then conclude from Theorem 2.2.8 that for each T ∈ Φ(X → X) we have  n ess  1/n ρFred (T; X) = lim [T]n  1/n A = lim T  X→X

(2.2.28)

 n ess  1/n ρFred (T; X) = inf [T]n  1/n . A = inf T  X→X

(2.2.29)

n→∞

n→∞

and n∈N

n∈N

Chapter 3

Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

In this chapter we study functions whose oscillations vanish at infinitesimal scales. When said oscillations are measured in a mean (i.e., integral) sense, this gives rise to the Sarason space VMO, discussed in §3.1. When the oscillations in question are taken in a pointwise sense (via a local Hölder semi-norm), we come across a new brand of Hölder spaces, defined in §3.2. Originally, Sarason’s space VMO has been defined as the closure of uniformly continuous functions in BMO. Via a PDE approach (relying on these well-posedness of the Dirichlet Problem in the upper half-space with data in BMO) in [124] the authors succeed in producing characterizations of the space VMO as the closure in BMO of classes of smooth functions contained in BMO within which uniform continuity may be suitably quantified, such as the class of smooth functions satisfying a Hölder or Lipschitz condition (thus improving Sarason’s classical result describing VMO as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations). In turn, this stronger density result was used in [124] to show that any Calderón-Zygmund operator T satisfying T(1) = 0 extends as a linear and bounded mapping from VMO (modulo constants) into itself. One of the main goals in §3.1 is to give a new proof of the aforementioned density result of a purely real analysis flavor, which works in the general context of spaces of homogeneous type. Subsequently, in §3.2 we make further use of this technology to study a related scale, namely a new brand of Hölder spaces, obtained by imposing the condition that the local Hölder semi-norm vanishes as the scale (at which this is considered) approaches zero.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1_3

81

82

3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

3.1 Functions of Vanishing Mean Oscillations on Measure Metric Spaces In this section we shall concern ourselves with the space VMO which, heuristically, should be thought of as an integral version1 of uniform continuity. Specifically, given a measure metric space (X, ρ, μ), define the Sarason space VMO(X, μ) of functions of vanishing mean oscillations on X (cf. [170]) as VMO(X, μ) := the closure of UC(X, ρ) ∩ BMO(X, μ) in BMO(X, μ),

(3.1.1)

where UC(X, ρ) stands for the space of uniformly continuous functions on the metric space (X, ρ). That is, VMO(X, μ) is the linear subspace of BMO(X, μ) consisting of f ∈ BMO(X, μ) for which one may find { f j } j ∈N ⊆ UC(X, ρ) ∩ BMO(X, μ) with the property that lim  f − f j BMO(X,μ) = 0. Note that j→∞

if X is bounded then VMO(X, μ) is a closed subspace of BMO(X, μ), a scenario in which VMO(X, μ) is a Banach space when equipped with the norm  · BMO(X,μ) ; hence, VMO(X, μ) → BMO(X, μ) isometrically in this case. In the case when X is unbounded, the quotient space     VMO(X, μ) := VMO(X, μ) ∼ = [ f ] : f ∈ VMO(X, μ)

(3.1.2)

(3.1.3)

 equipped with the norm inherited from BMO(X, μ) (cf. [133, (7.4.95)]) is complete, hence Banach. It is known that the BMO space remains unchanged if the underlying measure is multiplied by an A∞ weight. Here is a version of this result for functions of vanishing mean oscillations. Lemma 3.1.1 Let (X, ρ, μ) be a measure metric space. Then for each w ∈ A∞ (X, μ) one has VMO(X, μ) = VMO(X, w). (3.1.4) Proof This is a direct consequence of [133, Lemma 7.7.5] and (3.1.1).



Even though almost everything in this section is valid in the context of arbitrary spaces of homogeneous type, we find it convenient to restrict ourselves to working on a closed Ahlfors regular subset Σ of Rn , which we equip with the “surface 1 (Σ, σ) measure” σ := H n−1 Σ. To set the stage for what follows, for each f ∈ Lloc and p ∈ [1, ∞) abbreviate

1 as opposed to a pointwise version

3.1 Functions of Vanishing Mean Oscillations on Measure Metric Spaces

Mp ( f ; R) := sup sup

⨏

x ∈Σ r ∈(0,R)

Δ(x,r)

  f −

⨏ Δ(x,r)

83

 p  1/p  f dσ  dσ for all R ∈ (0, ∞),

where Δ(x, r) := Σ ∩ B(x, r) for each x ∈ Σ and r ∈ (0, ∞). (3.1.5) Obviously, Mp ( f ; R) is non-decreasing in R. Also, for each fixed p ∈ [1, ∞), we see from [133, (7.4.112)] that  f BMO(Σ,σ)

⎧ if Σ is unbounded, ⎪ ⎨ supR>0 Mp ( f ; R) ⎪  ≈  ∫  ⎪ ⎪  Σ f dσ  + supR>0 Mp ( f ; R) if Σ is bounded, ⎩

(3.1.6)

1 (Σ, σ). Since for each p ∈ [1, ∞) we have uniformly in f ∈ Lloc

lim Mp ( f ; R) = 0 whenever f is uniformly continuous on Σ,

R→0+

(3.1.7)

it follows from (3.1.1), (3.1.7), and (3.1.6) that lim Mp ( f ; R) = 0 for each f ∈ VMO(Σ, σ) and p ∈ [1, ∞).

R→0+

(3.1.8)

Next, the goal is to prove a quantitative characterization of Sarason’s space VMO, in which uniformly continuous functions are replaced by Hölder functions. Such a version is contained in Theorem 3.1.3, formulated a little further below. As a preamble, we establish the result in the following lemma. Lemma 3.1.2 Assume Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, fix some β ∈ (0, ∞). Then the collection of all functions f ∈ BMO(Σ, σ) with the property that there exists some r0 = r0 ( f ) ∈ (0, ∞) such that | f (x) − f (y)| < +∞ (3.1.9) sup |x − y| β x,y ∈Σ 0< |x−y | 0 and λ > 0, such that Δ(xαk , c t) ⊆ Q αk ⊆ Δ(xαk , λt) for each α ∈ Ik .

(3.1.29)

In turn, from (3.1.29) and the fact that we are currently assuming R ≥ t we conclude that having Q αk ∩ Δ(x, R)   for some α ∈ Ik (3.1.30)   forces Q αk ⊆ Δ x, (2λ + 1)R . In particular,      Q αk ⊆ Δ x, (2λ + 1)R if J(k, x, R) := α ∈ Ik : Q αk ∩ Δ(x, R)   . α∈J(k,x,R)

(3.1.31) Granted these properties, we may now estimate2

2 alternatively, we could have used Vitali’s Covering Lemma (see (3.2.32) in this regard)

88

3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

⨏ Δ(x,R)

⨏   ( f − gt ) −

Δ(x,R)

⨏ ≤2

| f (y) − gt (y)| dσ(y)

Δ(x,R)

2 ≤ σ(Δ(x, R)) ∫ − ≤C

  ( f − gt ) dσ  dσ

α∈J(k,x,R)

⨏



Σ

St (y, z) f (z) −

 α∈J(k,x,R)

Δ(xαk ,(λ+C0 )t)

Δ(xαk ,(λ+C0 )t)

⨏   f −

  ≤ C M1 f ; (λ + C0 )t

Δ(xαk ,(λ+C0 )t)

 α∈J(k,x,R)

 = C M1 f ; (λ + C0 )t 

   f dσ dσ(z) dσ(y)  

  σ Δ(xαk , (λ + C0 )t) × σ(Δ(x, R))

⨏

×

 ⨏    f dσ  f (y) − k  Qα Δ(xαk ,(λ+C0 )t)

∫



σ

 

  f dσ  dσ

σ(Q αk )  σ(Δ(x, R))

k α∈J(k,x,R) Q α

σ(Δ(x, R))

 σ Δ(x, (2λ + 1)R) ≤ C M1 f ; (λ + C0 )t σ(Δ(x, R))   ≤ C M1 f ; (λ + C0 )t ,





(3.1.32)

for some geometric constant C ∈ (0, ∞), independent of f , x, and R. Specifically, the first inequality in (3.1.32) is obvious. The second inequality in (3.1.32) is obtained by unpacking the integral average on Δ(x, R), using the definition of gt from (3.1.18), the last property in (3.1.19), the fact that the “dyadic cubes” {Q αk }α∈Ik cover Σ up to a σ-nullset (cf. (3.1.28)), and the definition of the index set J(k, x, R) given in (3.1.31). The third inequality in (3.1.32) is based on the first property in (3.1.19) and the Ahlfors regularity of Σ, the fact that thanks to the second inclusion in (3.1.29) we have   (3.1.33) Q αk ⊆ Δ(xαk , λt) ⊆ Δ xαk , (λ + C0 )t , plus the observation that if y ∈ Q αk and z ∈ Σ are such that |y − z| < C0 t then |z − xαk | ≤ |xαk − y| + |y − z| < λt + C0 t = (λ + C0 )t (3.1.34)  k  so we necessarily have z ∈ Δ xα, (λ + C0 )t . The fourth inequality in (3.1.32) is seen from the definition made in (3.1.5), the fact that Σ is Ahlfors regular, and (3.1.29). The subsequent equality in (3.1.32) is a result of the fact that the “dyadic cubes”

3.1 Functions of Vanishing Mean Oscillations on Measure Metric Spaces

89

{Q αk }α∈Ik are mutually disjoint. The penultimate inequality in (3.1.32) comes from (3.1.31). The final inequality in (3.1.32) uses the fact that the measure σ is doubling. In summary, the implication in (3.1.26) also holds (with possibly different constants) when R ≥ t. We may therefore allow R ∈ (0, ∞) arbitrary, and taking the supremum in R yields (3.1.22), on account of (3.1.5). With (3.1.22) in hand, we then conclude from (A.0.8) and (3.1.5) that   . ≤ C3 M1 ( f ; C4 t) for each t ∈ 0, 2 diam Σ . (3.1.35)  f − gt BMO(Σ,σ)   As a consequence, for each t ∈ 0, 2 diam Σ we have

. . . gt BMO(Σ,σ) ≤ gt − f BMO(Σ,σ) +  f BMO(Σ,σ) . ≤ C3 M1 ( f ; C4 t) +  f BMO(Σ,σ) < +∞,

(3.1.36)

from which we conclude that   gt ∈ BMO(Σ, σ) for each t ∈ 0, 2 diam Σ .

(3.1.37)

Collectively, (3.1.37), (3.1.20), and Lemma 3.1.2 (with β := 1) then show that

.

gt ∈ C α (Σ) ∩ BMO(Σ, σ) for each   t ∈ 0, 2 diam Σ and α ∈ (0, 1).

(3.1.38)

Passing to limit in (3.1.35) also proves that for each function h ∈ VMO(Σ, σ) we have

. lim sup  f − gt BMO(Σ,σ) ≤ C3 lim+ M1 ( f ; C4 t) t→0+

t→0

≤ C3 lim+ M1 ( f − h; C4 t) + C3 lim+ M1 (h; C4 t) t→0

. , ≤ C3  f − hBMO(Σ,σ)

t→0

(3.1.39)

where the last step takes into account (3.1.8) and (A.0.8) written for f − h in place of f . After taking the infimum over all h ∈ VMO(Σ, σ) in (3.1.39) we ultimately arrive at the conclusion that   . lim sup  f − gt BMO(Σ,σ) ≤ C3 · dist f , VMO(Σ, σ) , (3.1.40) t→0+

. where the distance is measured with respect to  · BMO(Σ,σ) . Having established this, the desired conclusion readily follows in the case when Σ is unbounded, since in this . case  · BMO(Σ,σ) coincides with  · BMO(Σ,σ) . To finish the proof, there remains to treat the case when Σ is compact. This time, . in addition to the semi-norm  · BMO(Σ,σ) , the norm  · BMO(Σ,σ) contains an extra

90

3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

term (see (A.0.9)), but the same type of argument applies as soon as we show that ∫    (3.1.41) lim+  ( f − gt ) dσ  = 0. t→0

Σ

To justify this, by Hölder’s inequality it suffices to prove that for each function ψ ∈ L 2 (Σ, σ) we have G(ψ; t) −→ ψ in L 2 (Σ, σ) as t → 0+,

(3.1.42)

∫

where G(ψ; t)(x) :=

Σ

St (x, y)ψ(y) dσ(y) for each x ∈ Σ.

(3.1.43)

Since the integral kernel of the operator G(·; t) is an approximation to the identity of the sort described in (3.1.17), the following properties hold: G(ψ; t) → ψ pointwise on Σ, as t → 0+ if ψ ∈ Lip(Σ), and   |G(ψ; t)| ≤ CM Σ ψ on Σ, for each ψ ∈ L 2 (Σ, σ) and t ∈ 0, 2 diam Σ ,

(3.1.44)

where M Σ denotes the Hardy-Littlewood maximal operator on Σ. Hence, the claim in (3.1.42) is implied by (3.1.44), Lebesgue’s Dominated Convergence Theorem, the boundedness of M Σ on L 2 (Σ, σ) (cf. [133, Corollary 7.6.2]), and the density result contained in [133, (3.7.22)]. This finishes the proof of Theorem 3.1.3.  Remark 3.1.4 Our proof of the density result stated in.Theorem 3.1.3 is constructive, and the sequence {gt }0 0 such that | f (x) − f (x )| ≤ ε|x − x  |γ whenever x,

x

∈ Σ satisfy |x −

x|

(3.2.1)

< δ.

Specifically, introduce4   V γ (Σ) := f ∈ C 0 (Σ) : f satisfies (3.2.1) .

(3.2.2)

It may be check without difficulty that V γ1 (Σ) ⊆ V γ2 (Σ) whenever γ1 > γ2 > 0, and that . C β (Σ) ⊆ V γ (Σ) for each β > γ. (3.2.3) It is also apparent from definitions that     γ V γ (Σ) = f ∈ Cloc (Σ) : lim+ sup  f C.γ (B(x,r)∩Σ) = 0 . r→0

x ∈Σ

(3.2.4)

We shall use (3.2.2) to define what we call the homogeneous “vanishing” Hölder .γ (Σ), of order γ on the set Σ, as follows: space Cvan

.

.

γ Cvan (Σ) := C γ (Σ) ∩ V γ (Σ)     . = f ∈ C γ (Σ) : lim+ sup  f C.γ (B(x,r)∩Σ) = 0 . r→0

x ∈Σ

(3.2.5)

It is then clear from this definition and (3.2.3) that

.

.

.

γ C γ (Σ) ∩ C β (Σ) ⊆ Cvan (Σ) for each β > γ.

(3.2.6)

3 think of this as a Hölder-styled version of uniform continuity 4 at least formally, V 0 (Σ) corresponds to UC (Σ), the space of uniformly continuous functions on Σ

3.2 A New Brand of Hölder Spaces

99

Lemma 3.2.1 Let Σ be an arbitrary nonempty subset ofRn , and fix some γ ∈ (0, ∞).  . .γ γ Then Cvan (Σ) is a closed subspace of C (Σ),  · C.γ (Σ) .

.

γ (Σ) equipped with the semi-norm inherHenceforth we shall always consider Cvan .γ ited from the space C (Σ), i.e.,  · C.γ (Σ) .

.

Proof of Lemma 3.2.1 Assume f ∈ C γ (Σ) is such that there exists a sequence .γ (Σ) with the property that  f − f j C.γ (Σ) → 0 as j → ∞. Also, { f j } j ∈N ⊆ Cvan fix an arbitrary threshold ε > 0. Then we may find an integer jε ∈ N for which  f − f jε C.γ (Σ) < ε/2. In addition, since f jε ∈ V γ (Σ), there exists δ > 0 such that | f jε (x) − f jε (x )| ≤ (ε/2)|x − x  |γ whenever x, x  ∈ Σ satisfy |x − x  | < δ. For each pair of points x, x  ∈ Σ satisfying |x − x  | < δ we may then estimate   | f (x) − f (x )| ≤ ( f − f jε )(x) − ( f − f jε )(x ) + | f jε (x) − f jε (x )| ≤  f − f jε C.γ (Σ) · |x − x  |γ + (ε/2)|x − x  |γ ≤ ε|x − x  |γ .

(3.2.7)

.

.

γ (Σ), as This shows that f ∈ V γ (Σ). Hence, ultimately, f ∈ C γ (Σ) ∩ V γ (Σ) = Cvan wanted. 

Retaining the assumptions that Σ is an arbitrary nonempty subset of Rn and that γ ∈ (0, ∞) is an arbitrary number, let us also introduce the inhomogeneous “vanishing” Hölder space γ Cvan (Σ) := C γ (Σ) ∩ V γ (Σ)     = f ∈ C γ (Σ) : lim+ sup  f C.γ (B(x,r)∩Σ) = 0 . r→0

x ∈Σ

(3.2.8)

The same argument as in the proof of Lemma 3.2.1 then shows that γ (Σ) is a closed subspace of C γ (Σ), so it becomes a Cvan Banach space when equipped with the norm  · C γ (Σ) .

(3.2.9)

It is also clear from the definition in (3.2.8) and (3.2.3) that γ C γ (Σ) ∩ C β (Σ) ⊆ Cvan (Σ) for each β > γ.

(3.2.10)

Lastly, we wish to note that

.γ   γ Cvan (Σ) = f ∈ Cvan (Σ) : f bounded .

(3.2.11)

Much like the space of essentially bounded functions (in the context of a measurable space), the ordinary Hölder spaces are notoriously difficult when it comes

100

3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

to issues pertaining to density. This being said, for the brands of Hölder spaces introduced above we have the following remarkable density result. Theorem 3.2.2 Let Σ be a closed Ahlfors regular subset of Rn and set σ := H n−1 Σ. Also, fix an exponent γ ∈ (0, 1). Then

.

.

.

γ C γ (Σ) ∩ C β (Σ) ⊆ Cvan (Σ) densely for each β ∈ (γ, 1).

Equivalently,

.

.

(3.2.12)

.

γ (Σ) is the closure of C γ (Σ) ∩ C β (Σ) in Cvan . the space C γ (Σ), for each β ∈ (γ, 1).

(3.2.13)

γ (Σ) is the closure of C γ (Σ) ∩ C β (Σ) in Cvan the space C γ (Σ), for each β ∈ (γ, 1).

(3.2.14)

Furthermore,

.

.

.

As a by-product, we see that the closure of C γ (Σ) ∩ C β (Σ) in the space C γ (Σ) is independent of the choice of the exponent β ∈ (γ, 1), and so is the closure of C γ (Σ) ∩ C β (Σ) in C γ (Σ). Let us also observe that, as is clear from (3.2.14), in the context of the above theorem the following density result holds: if the set Σ is bounded, then for each β ∈ (γ, 1) the γ (Σ) is the closure of C β (Σ) in C γ (Σ). space Cvan

(3.2.15)

γ (Σ) is the answer to the question of providing an intrinsic description of Hence, Cvan the closure of C β (Σ) in the space C γ (Σ), for each given exponent β ∈ (γ, 1), when Σ is a bounded set.

We now turn to the task of proving Theorem 3.2.2. Proof of Theorem 3.2.2 We shall adapt the argument used in the proof of Theo1 (Σ, σ) define rem 3.1.3. To set the stage, for each function f ∈ Lloc ⨏ ⨏   1   f dσ  dσ for all R ∈ (0, ∞), Hγ ( f ; R) := sup sup γ f − r x ∈Σ r ∈(0,R) Δ(x,r) Δ(x,r) where Δ(x, r) := Σ ∩ B(x, r) for each x ∈ Σ and r ∈ (0, ∞). (3.2.16) As is apparent from definitions,

.

sup Hγ ( f ; R) ≤ 2γ  f C.γ (Σ) for each f ∈ C γ (Σ),

(3.2.17)

R>0

and

.

γ lim+ Hγ ( f ; R) = 0 for each f ∈ Cvan (Σ).

R→0

(3.2.18)

To proceed, bring back the approximation to the identity used in the proof of Theorem 3.1.3, i.e., a family of (σ ⊗ σ)-measurable functions St : Σ × Σ → R

3.2 A New Brand of Hölder Spaces

101

  indexed by a parameter t ∈ 0, 2 diam Σ satisfying the properties listed in (3.1.17) . for some finite constant C0 ≥ 2. Next, let us fix an arbitrary function f ∈ C γ (Σ), and for each t ∈ (0, ∞) define ∫ gt (x) := St (x, y) f (y) dσ(y) for each x ∈ Σ. (3.2.19) Σ

Bearing in mind the fact that  Σ is an Ahlfors regular set, much as in (3.1.19) we conclude that for each t ∈ 0, 2 diam Σ and x, x  ∈ Σ with |x − x  | < C0 t we have ⨏ ⨏     (3.2.20) |gt (x) − gt (x )| ≤ C t −1 |x − x  | f dσ  dσ. f − Δ(x,2C0 t)

Δ(x,2C0 t)

In view of the definition made in (3.2.16), this implies that there exist C1, C2 ∈ (1, ∞) such that |gt (x) − gt (x )| ≤ C1 t −1+γ |x − x  | · Hγ ( f ; C2 t) whenever t ∈ (0, ∞) and x, x  ∈ Σ satisfy |x − x  | < C0 t.

(3.2.21)

  As a consequence, for each t ∈ 0, 2 diam Σ we have sup

x,x  ∈Σ 0< |x−x  |0

  To justify this inequality, fix some number t ∈ 0, 2 diam Σ along with an arbitrary point x ∈ Σ and a radius R ∈ (0, ∞). In the regime 0 < R < t, we estimate ⨏ ⨏   1   f − g (3.2.24) ) − ( f − g ) dσ (  dσ ≤ I + II, t t Rγ Δ(x,R) Δ(x,R) where I :=

1 Rγ

II :=

1 Rγ

and

⨏ Δ(x,R)

⨏ Δ(x,R)

⨏   f −

Δ(x,R)

⨏   gt −

Δ(x,R)

  f dσ  dσ,

(3.2.25)

  gt dσ  dσ.

(3.2.26)

From (3.2.16) and the fact that R ∈ (0, t) is arbitrary it follows that I ≤ Hγ ( f ; t).

(3.2.27)

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3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

Since |y − z| < 2R < C0 t for each y, z ∈ Δ(x, R), thanks to the fact that C0 ≥ 2 and that we are currently assuming R < t, we also have ⨏ ⨏ 1 II ≤ γ |gt (y) − gt (z)| dσ(y) dσ(z) R Δ(x,R) Δ(x,R) ⨏

⨏

≤ C1

t −1+γ Hγ ( f ; C2 t) Rγ

≤ C1

t −1+γ Hγ ( f ; C2 t) 2R ≤ C1 Hγ ( f ; C2 t). Rγ

Δ(x,R)

Δ(x,R)

|y − z| dσ(y) dσ(z)

(3.2.28)

The last step above uses the observation that t −1+γ R = (t −1 R)1−γ < 1, Rγ

(3.2.29)

given that we are presently working in the regime 0 < R < t (hence t −1 R < 1), and 1 − γ > 0. Bearing in mind that Hγ ( f ; ·) is nondecreasing, the estimates on I, II above lead to the following conclusion: ⨏ ⨏   1   0 < R < t =⇒ γ ( f − gt ) dσ  dσ ≤ C1 Hγ ( f ; C2 t). ( f − gt ) − R Δ(x,R) Δ(x,R) (3.2.30) Thus, as far as (3.2.23) is concerned, there remains to consider the case when R ≥ t. In such a scenario, start from the realization that Δ(z, t) : z ∈ Δ(x, R) is a family of surface balls of same (finite) radius covering the bounded set Δ(x, R). Vitali’s Covering Lemma (cf. [133, Lemma 7.5.7], or [133, Lemma 7.5.8]) then guarantees that there exist some dilation parameter λ ∈ (1, ∞) and an at most countable family of points {z j } j ∈J ⊆ Δ(x, R) such that  Δ(z j , λt). (3.2.31) {Δ(z j , t)} j ∈J are mutually disjoint, and Δ(x, R) ⊆ j ∈J

Availing ourselves of these properties, as well as (3.1.17), the definition in (3.2.16), and keeping in mind that (t/R)γ ≤ 1 (given that γ > 0 and we are presently assuming R ≥ t), we may now estimate

3.2 A New Brand of Hölder Spaces

1 Rγ

⨏ Δ(x,R)

≤

2 Rγ

103

⨏   ( f − gt ) −

Δ(x,R)

⨏ Δ(x,R)

| f (y) − gt (y)| dσ(y)

 2 ≤ γ R · σ(Δ(x, R)) j ∈J ∫ −



Σ

  ( f − gt ) dσ  dσ

 ⨏    f dσ  f (y) − Δ(z j ,λt)  Δ(z j ,(λ+C0 )t)

∫

⨏

St (y, z) f (z) − 

Δ(z j ,(λ+C0 )t)

   f dσ dσ(z) dσ(y)  

⨏ ⨏   C  σ Δ(z j , (λ + C0 )t)   ≤ γ f dσ  dσ f − R j ∈J σ(Δ(x, R)) Δ(z j ,(λ+C0 )t) Δ(z j ,(λ+C0 )t)      σ Δ(z j , t)    ≤ C(t/R) Hγ f ; (λ + C0 )t σ Δ(x, R) j ∈J   Δ(z , t) σ j j ∈J     ≤ C Hγ f ; (λ + C0 )t σ Δ(x, R)   ≤ C Hγ f ; (λ + C0 )t , γ



(3.2.32)

for some geometric constant C ∈ (0, ∞), independent of f , x, and R. In summary, the implication in (3.2.30) also holds (with possibly different constants) when R ≥ t. We may therefore allow R ∈ (0, ∞) arbitrary, and taking the supremum in R yields (3.2.23), on account of (3.2.16). With (3.2.23) in hand, we then conclude from [133, Proposition 7.4.9] (bearing in mind [133, Lemma 3.6.4] with s := n − 1) that  f − gt C.γ (Σ) ≤ C supR>0 Hγ ( f − gt ; R) ≤ C Hγ ( f ; C4 t)   for each t ∈ 0, 2 diam Σ .

(3.2.33)

  As a consequence, for each t ∈ 0, 2 diam Σ we have gt C.γ (Σ) ≤ gt − f C.γ (Σ) +  f C.γ (Σ) ≤ C3 Hγ ( f ; C4 t) +  f C.γ (Σ) < +∞,

(3.2.34)

from which we conclude that

.   gt ∈ C γ (Σ) for each t ∈ 0, 2 diam Σ .

(3.2.35)

Focusing now directly on (3.2.12), fix β ∈ (γ, 1) and assume the function f .γ actually belongs to the homogeneous vanishing Hölder space Cvan (Σ). Also, fix

104

3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

t ∈ (0, ∞) and pick two arbitrary points x, x  ∈ Σ. On the one hand, if |x − x  | < C0 t then from (3.2.21) we see that |gt (x) − gt (x )| ≤ C1 t −1+γ Hγ ( f ; C2 t) · |x − x  | = C1 t −1+γ Hγ ( f ; C2 t) · |x − x  | 1−β |x − x  | β ≤ C1 t −1+γ Hγ ( f ; C2 t) · (C0 t)1−β |x − x  | β .

(3.2.36)

On the other hand, if |x − x  | ≥ C0 t then (3.2.35) implies |gt (x) − gt (x )| ≤ gt C.γ (Σ) |x − x  |γ ≤ gt C.γ (Σ) (C0 t)γ−β |x − x  | β

(3.2.37)

Together, (3.2.36)-(3.2.37) and (3.2.35) prove that

. .   gt ∈ C γ (Σ) ∩ C β (Σ) for each t ∈ 0, 2 diam Σ .

(3.2.38)

Passing to limit in (3.2.33) also proves that lim sup  f − gt C.γ (Σ) ≤ C3 lim+ Hγ ( f ; C4 t) = 0, t→0+

t→0

(3.2.39)

where the last inequality takes into account (3.2.18). Having established this, (3.2.12) follows from (3.2.6), (3.2.38), and (3.2.39). In fact, a stronger conclusion holds. This hinges on the observation that if the function f ∈ UC(Σ) (i.e., f is uniformly continuous on Σ) then gt defined in (3.2.19) also satisfies (3.2.40) lim+ sup | f (x) − gt (x)| = 0. t→0 x ∈Σ

Indeed, for each t ∈ (0, ∞) and x ∈ Σ we have ∫      | f (x) − gt (x)| =  St (x, y) f (x) − f (y) dσ(y) Σ

⨏ ≤C

Δ(x,C0 t)

   f (x) − f (y) dσ(y),

(3.2.41)

so (3.2.40) follows in view of the fact that f is uniformly continuous. Consequently, since any Hölder function is uniformly continuous, (3.2.39) may be improved to lim  f − gt C γ (Σ) = 0.

t→0+

(3.2.42)

In particular, the conclusion in (3.2.14) readily follows from this. The proof of Theorem 3.2.2 is therefore complete.  Much like the John-Nirenberg space BMO(Σ, σ) may be regarded as the end. point space of the Hölder scale C γ (Σ) with γ ∈ (0, 1), we may view the Sarason .γ (Σ) space VMO(Σ, σ) as the end-point space of our vanishing Hölder scale Cvan

3.2 A New Brand of Hölder Spaces

105

with γ ∈ (0, 1). This analogy is strengthened by the following result, which mirrors Corollary 3.1.6. Corollary 3.2.3 Let Σ be a closed Ahlfors regular subset of Rn and . abbreviate σ := H n−1 Σ. Also, fix some γ ∈ (0, 1). Then for each function f ∈ C γ (Σ) one has  ⨏ ⨏   .γ 1   f dσ  dσ = 0, f ∈ Cvan (Σ) ⇐⇒ lim+ sup sup γ f − R→0 x ∈Σ r ∈(0,R) r Δ(x,r) Δ(x,r) (3.2.43) while for each function f ∈ C γ (Σ) one has  ⨏ ⨏   1   γ f dσ  dσ = 0. f ∈ Cvan (Σ) ⇐⇒ lim+ sup sup γ f − R→0 x ∈Σ r ∈(0,R) r Δ(x,r) Δ(x,r) (3.2.44) .γ In addition, for each given function f ∈ C (Σ) one has (with the distance mea. sured in C γ (Σ))   ⨏ ⨏ .γ   1 dist f , Cvan (Σ) ≈ lim sup sup γ | f (y) − f (z)| dσ(y) dσ(z) x ∈Σ r r→0+ Δ(x,r) Δ(x,r)  ⨏ ⨏    1   ≈ lim sup γ sup f dσ  dσ f − x ∈Σ Δ(x,r) r→0+ r Δ(x,r)   ⨏ ⨏    1    ≈ lim+ f dσ  dσ , (3.2.45) sup f − R→0 x ∈Σ, r ∈(0,R) r γ Δ(x,r) Δ(x,r) where the implicit proportionality constants depend only on the environment. Finally, a similar result to the one described in (3.2.45) is true for functions f ∈ C γ (Σ) (now measuring the distance in C γ (Σ)).

.

Proof Fix an arbitrary function f ∈ C γ (Σ). The left-to-right implication in (3.2.43) is implied by (3.2.16) and (3.2.18), while the right-to-left implication in (3.2.43) is seen from (3.2.39), (3.2.38), (3.2.6), and Lemma 3.2.1. The equivalence in (3.2.44) is justified in a similar manner, now also keeping in mind (3.2.40). . As regards (3.2.45), pick a function f ∈ C γ (Σ). Using notation employed in the .γ (Σ) we may write proof of Theorem 3.2.2, for each h ∈ Cvan

.γ   (Σ) ≤ lim sup  f − gt C.γ (Σ) ≤ C lim+ Hγ ( f ; t) dist f , Cvan t→0+

t→0

≤ C lim+ Hγ ( f − h; t) + C lim+ Hγ (h; t) t→0

≤ C f − hC.γ (Σ) .

t→0

(3.2.46)

The first step above is a consequence of (3.2.38) and (3.2.6). The second step above is directly implied by the inequality (3.2.39). The penultimate step in (3.2.46) is

106

3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli

simply the triangle inequality. The final step takes in (3.2.46) follows on account of (3.2.17) (written for f − h in place of f ), and (3.2.18) (written for h in place of f ). .γ (Σ) in (3.2.46) then leads to the conclusion Taking the infimum over all h ∈ Cvan that .γ  .γ    dist f , Cvan (Σ) ≤ C lim+ Hγ ( f ; t) ≤ C · dist f , Cvan (Σ) . (3.2.47) t→0

With this in hand, routine estimates yield all equivalences in (3.2.45). Finally, the task of establishing equivalences analogous to those recorded in (3.2.45) in the case when f ∈ C γ (Σ) is dealt with similarly. 

Chapter 4

Hardy Spaces on Ahlfors Regular Sets

The pioneering work of R. Coifman and G. Weiss ([35], [36]) has fundamentally reconfigured the classical theory of Hardy spaces by fully recognizing the usefulness and versatility of a brand of Hardy spaces which places minimal regularity and structural demands on the underlying space. Here we are concerned with Hardy spaces on Ahlfors regular subsets of the Euclidean ambient and, by further building on the work in [9], consider topics such as the Fefferman-Stein grand maximal function, Lebesgue-based and Lorentz-based Hardy spaces, real interpolation, atoms and molecules, duality, weak-∗ convergence, and the compatibility of various duality pairings.

4.1 The Fefferman-Stein Grand Maximal Function Given a closed set Σ ⊆ Rn which is Ahlfors regular, abbreviate σ := H n−1 Σ, and of all fix γ ∈ (0, 1). In this context, for each x ∈ Σ define Tγ (x) to be  the collection  functions ψ ∈ Lipc (Σ) with the property that there exists r ∈ 0, 2 diam Σ such that supp ψ ⊆ Σ ∩ B(x, r),

sup |ψ(z)| ≤ z ∈Σ

|ψ(z1 ) − ψ(z2 )| ≤ and ψC.γ (Σ) := sup γ z 1 , z 2 ∈Σ z1 z2

|z1 − z2 |

1  , σ Σ ∩ B(x, r) 1  . r γ · σ Σ ∩ B(x, r)

That is, for each x ∈ Σ, define    Tγ (x) := ψ ∈ Lipc (Σ) : (4.1.1) holds for some r ∈ 0, 2 diam Σ .

(4.1.1)

(4.1.2)

Regarding this set we note the following useful fact:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1_4

107

108

4 Hardy Spaces on Ahlfors Regular Sets

given φ ∈ Lipc (Σ), not identically zero, and some point x ∈ Σ,  then for each R ∈ 0, 2 diam Σ  such that supp φ ⊆ B(x,  R) ∩ Σ it follows that the function φ Cφ,R · σ(B(x, R) ∩ Σ) belongs the normalization constant is defined as to Tγ (x), provided   Cφ,R := max (RφLip(Σ) )γ (2 · supΣ |φ|)1−γ, supΣ |φ| .

(4.1.3)

Indeed, if φ ∈ Lipc (Σ) then for each x, y ∈ Σ and each R > 0 we have   |φ(x) − φ(y)| ≤ R−1 RφLip(Σ) |x − y| and |φ(x) − φ(y)| ≤ 2 · sup |φ|. (4.1.4) Σ

Raising the first inequality in (4.1.4) to the power γ, the second one to the power 1 − γ, then multiplying the resulting inequalities eventually yields

1−γ C  γ φ,R · σ(B(x, R) ∩ Σ)   . φC.γ (Σ) ≤ R−γ RφLip(Σ) 2 · sup |φ| ≤ γ R · σ Σ ∩ B(x, R) Σ

(4.1.5)

Granted this estimate, it is easy to check that if φ, x, R are as in (4.1.3) then the function defined as ψ := φ/[Cφ,R · σ(B(x, R) ∩ Σ)] satisfies the conditions in (4.1.1) with r := R. This proves (4.1.3).   Pressing on, for any distribution f ∈ Lipc (Σ) define the Fefferman-Stein grand maximal function fγ of f as the function pointwise defined on Σ according to   (4.1.6) fγ (x) := sup  f , ψ , ∀x ∈ Σ, ψ ∈ Tγ (x)

  where ·, · currently stands for the duality paring between Lipc (Σ) and Lipc (Σ) (recall that this is compatible with the ordinary integral pairing on Σ, with respect to the measure σ, when such a pairing is meaningful; see [133, Proposition 4.1.4] and the subsequent comment). From (4.1.3) and (4.1.6) we see that with C ∈ (0, ∞) denoting the upper Ahlfors regularity constant of the set Σ,   given any f ∈ Lipc (Σ) and γ ∈ (0, 1), for each φ ∈ Lipc (Σ) we have      f , φ  ≤ C Rn−1 · max (RφLip(Σ) )γ (2 · sup |φ|)1−γ, sup |φ| · fγ (x) (4.1.7) Σ Σ whenever x ∈ Σ and R > 0 are selected such that supp φ ⊆ B(x, R) ∩ Σ. It is useful to observe that other related variants of the inequality in (4.1.7) are valid for radii which are small (relative to the diameter of Σ). Specifically, if  1/(n−1)  , λΣ := 2 CΣ /cΣ

(4.1.8)

where CΣ , cΣ are, respectively, the upper and lower Ahlfors regularity constants of Σ (cf. [133, Definition 5.9.1]), then we have the following companion result for (4.1.7):

4.1 The Fefferman-Stein Grand Maximal Function



109



given any f ∈ Lipc (Σ) and γ ∈ (0, 1), for each φ ∈ Lipc (Σ) we have        f , φ  ≤ C Rn−1+γ · φLip(Σ) γ · sup |φ| 1−γ · fγ (x) Σ

≤ C R · φLip(Σ) · n

(4.1.9)

fγ (x)

  if x ∈ Σ and R ∈ 0, λΣ−1 · 2 diam Σ are such that supp φ ⊆ B(x, R) ∩ Σ, where now C ∈ (0, ∞) is allowed to depend on n, γ, and the Ahlfors regularity constants  in the context of (4.1.9),  the choice  of λΣ in (4.1.8) ensures  of Σ. Indeed, that σ B(x, λΣ R) ∩ Σ is strictly bigger that σ B(x, R) ∩ Σ . In particular, there exists a point     (4.1.10) x∗ ∈ B(x, λΣ R) ∩ Σ \ B(x, R) ∩ Σ . As such, for each y ∈ supp φ ⊆ B(x, R) ∩ Σ we may estimate |φ(y)| = |φ(y) − φ(x∗ )| ≤ |y − x∗ |φLip(Σ) ≤ 2λΣ RφLip(Σ),

(4.1.11)

hence sup |φ| ≤ 2λΣ RφLip(Σ),

(4.1.12)

Σ

and (4.1.9) readily follows by combining (4.1.7) with (4.1.12). In this vein, we also wish to note that whenever ψ ∈ Lipc (Σ) satisfies   supp ψ ⊆ Σ ∩ B(x, r) for some x ∈ Σ and r ∈ 0, λΣ−1 · 2 diam Σ , and ψC.γ (Σ) ≤

 1 , r γ ·σ Σ∩B(x,r)

then

(4.1.13)

γ

z ∈Σ



λΣ

. (4.1.14) σ Σ ∩ B(x, r)     Indeed, much as in (4.1.10), there exists some x∗ ∈ B(x, λΣ r) ∩ Σ \ B(x, r) ∩ Σ , so for each y ∈ supp ψ ⊆ Σ ∩ B(x, r) we may estimate sup |ψ(z)| ≤

|ψ(y)| = |ψ(y) − ψ(x∗ )| ≤ |y − x∗ |γ ψC.γ (Σ) γ

λΣ 1   =  , ≤ (λΣ r) γ r · σ Σ ∩ B(x, r) σ Σ ∩ B(x, r) γ

(4.1.15)

as wanted. Going further, we remark that (4.1.7) entails that, for each γ ∈ (0, 1),   if f ∈ Lipc (Σ) is such that fγ vanishes at one point on Σ then necessarily f is the zero distribution on Σ.

(4.1.16)

Let us also point out that, as visible from (4.1.1)-(4.1.2) and (4.1.6), we have

110

4 Hardy Spaces on Ahlfors Regular Sets

1γ ≤ 1 on Σ.

(4.1.17)

Another relevant example is discussed below. Example 4.1.1 For each fixed point xo ∈ Σ, recall the Dirac distribution with mass  at xo , i.e., the functional δxo ∈ Lipc (Σ) , defined in [133, (4.1.49)]. Then for each γ ∈ (0, 1) we have (δxo )γ (x) ≈ |x − xo | 1−n, uniformly for x ∈ Σ.

(4.1.18)

Indeed, fix a point x ∈ Σ and pick some ψ ∈ Tγ (x). Then there exists some number   r ∈ 0, 2 diam Σ for which the properties of ψ listed in (4.1.1) are true. In particular, if xo  Σ ∩ B(x, r) then δxo ψ = ψ(xo ) = 0, while if xo ∈ Σ ∩ B(x, r) then | δxo ψ| = |ψ(xo )| ≤ sup |ψ(z)| ≤ z ∈Σ

≤

1   σ Σ ∩ B(x, r)

C C ≤ . r n−1 |x − xo | n−1

(4.1.19)

In light of (4.1.6), the estimate in (4.1.19) shows that (δxo )γ (x) ≤ C|x − xo | 1−n for some constant C ∈ (0, ∞) independent of x and xo . In the opposite direction, fix a   point x ∈ Σ \ {xo } and pick a function θ ∈ Cc∞ B(0, 2) such that θ ≡ 1 on B(0, 1). With ρ := |x − xo | ∈ (0, ∞), define the Lipschitz function ψ(z) := ρ1−n θ (z − x)/ρ for each z ∈ Σ. Then there exists some (typically large) geometric constant C ∈ (0, ∞) such that the properties in (4.1.1) hold for ψ/C with r := 2ρ. Consequently, ψ/C belongs to Tγ (x) which, upon noting that |(xo − x)/ρ| = 1, allows us to write    C · (δxo )γ (x) ≥ | δxo ψ| = |ψ(xo )| = ρ1−n θ (xo − x)/ρ  = ρ1−n = |x − xo | 1−n .

(4.1.20)

To finish the proof of (4.1.18), there remains to show that (δxo )γ (xo ) = +∞, which we prove next. Specifically,   with θ as before, define the family of Lipschitz functions ψt (z) := t 1−n θ (z − xo )/t for each z ∈ Σ and t ∈ (0, ∞). Once again, there exists C ∈ (0, ∞) independent of t such that the properties in (4.1.1) are valid for ψt /C with x := xo and r := 2t. Hence, ψt /C ∈ Tγ (xo ) for each t > 0. As such, for each t > 0 we may write C · (δxo )γ (xo ) ≥ | δxo ψt | = |ψt (xo )| = t 1−n |θ(0)| = t 1−n .

(4.1.21)

Sending t → 0+ then yields (δxo )γ (xo ) = +∞, completing the proof of (4.1.18).   Moving on, from [9, Lemma 4.7, p. 131] we know that for each f ∈ Lipc (Σ) the function (4.1.22) fγ : Σ −→ [0, +∞] is lower-semicontinuous,

4.1 The Fefferman-Stein Grand Maximal Function

111

when the set Σ is equipped with the topology  inherited from the Euclidean ambient. 1 (Σ, σ) ⊂ Lip (Σ) the duality pairing in (4.1.6) becomes In the case when f ∈ Lloc c the integral pairing on Σ which, on account of (4.1.1), then readily implies 1 fγ ≤ M Σ f at σ-a.e. point on Σ, for every f ∈ Lloc (Σ, σ),

(4.1.23)

where M Σ is the Hardy-Littlewood maximal operator on (Σ, | · − · |, σ) (recall (A.0.71)). In the opposite direction we have the following result. Lemma 4.1.2 Let Σ ⊆ Rn be a closed set which is Ahlfors regular. Abbreviate σ := H n−1 Σ and fix γ ∈ (0, 1). Then there exists a geometric constant C ∈ (0, ∞) 1 (Σ, σ) one has with the property that for each function f ∈ Lloc | f | ≤ C fγ at σ-a.e. point on Σ.

(4.1.24)

1 (Σ, σ) and consider a bump-function Proof To get started, pick some f ∈ Lloc ∞ n θ ∈ Cc (R ) satisfying θ ≡ 1 on B(0, 1), supp θ ⊆ B(0, 2), and 0 ≤ θ ≤ 1. Also, fix some x ∈ Σ that point for f and select some small r > 0. Then since  is a Lebesgue  the function θ (· − x)/r is non-negative and is identically one on B(x, r) ∩ Σ, we may estimate   ∫   1 y − x

  dσ(y) | f (x)| ≤  f (x) ·  θ r σ B(x, r) ∩ Σ Σ

 ∫ y − x

  1    f (y)θ ≤  dσ(y) r σ B(x, r) ∩ Σ Σ  ∫     y − x

1   +  f (y) − f (x) θ dσ(y) r σ B(x, r) ∩ Σ Σ =: I1 + I2 . (4.1.25)    Since supp θ (· − x)/r ⊆ B(x, 2r) and 0 ≤ θ (· − x)/r ≤ 1, the set Σ is Ahlfors regular, and x is a Lebesgue point for f , it follows that ∫ C  I2 ≤  | f (y) − f (x)| dσ(y) −→ 0 as r → 0+ . (4.1.26) σ B(x, 2r) ∩ Σ B(x,2r)∩Σ 

Also, (4.1.7) used for φ := θ((·− x)/r) ∈ Lipc (Σ), the point x, and the radius R := 2r, guarantees (keeping in mind that RφLip(Σ) ≤ supz ∈Rn |(∇θ)(z)| and supΣ |φ| ≤ 1) that I1 ≤ C · fγ (x),

(4.1.27)

for some purely geometric constant C ∈ (0, ∞), which is independent of x and r. At this stage, combining (4.1.25)-(4.1.27) yields | f (x)| ≤ C fγ (x) from which (4.1.23)

112

4 Hardy Spaces on Ahlfors Regular Sets

follows given that, according to [133, Proposition 7.4.4] and [133, Lemma 3.6.4], σ-a.e. x ∈ Σ is a Lebesgue point for the function f .  Continue to assume that Σ ⊆ Rn is a closed set which is Ahlfors regular. As before, abbreviate σ := H n−1 Σ and fix γ ∈ (0, 1). Another useful fact about the grand maximal function is the existence of a geometric constant C ∈ (0, ∞) with the property that ∫      f , φ  ≤ C fγ |φ| dσ for every f ∈ Lipc (Σ) and φ ∈ Lipc (Σ). (4.1.28) Σ

This follows from [9, Proposition 4.15, p. 148] where a more general geometric setting was considered. Lemma 4.1.3 Let Σ ⊆ Rn be a closed set which is Ahlfors regular. Abbreviate σ := H n−1 Σ then fix a parameter γ ∈ (0, 1) along with two integrability exponents  p ∈ [1, ∞] and q ∈ (0, ∞]. Finally, pick an arbitrary distribution f ∈ Lipc (Σ) . Then fγ ∈ Lloc (Σ, σ) implies f ∈ Lloc (Σ, σ),

(4.1.29)

fγ ∈ L p (Σ, σ) implies f ∈ L p (Σ, σ),

(4.1.30)

fγ ∈ L p,q (Σ, σ) implies f ∈ L p,q (Σ, σ).

(4.1.31)

p

p

Proof Assume first that fγ ∈ Lloc (Σ, σ) with p ∈ (1, ∞]. In this situation, consider the exponent p ∈ [1, ∞) such that 1/p+1/p = 1, with the understanding that p := 1 when p = ∞. Fix xo ∈ Σ and r ∈ (0, ∞) arbitrary, then abbreviate Δr := B(xo, r) ∩ Σ. Next, define L f : Lipc (Δr ) −→ C by setting (4.1.32) L f φ := f , φ for every φ ∈ Lipc (Δr ). p

Then by design, L f is a well-defined linear functional on Lipc (Δr ), which is a subspace of the Banach space L p (Δr , σ). Also, from (4.1.28) we know that there exists a purely geometric constant C ∈ (0, ∞) with the property that for every φ ∈ Lipc (Δr ) we have ∫   fγ |φ| dσ ≤ C fγ  L p (Δr ,σ) φ| L p (Δr ,σ) . (4.1.33) |L f φ| =  f , φ  ≤ C Δr

By the Hahn-Banach Theorem there exists a linear and bounded functional

L f : L p (Δr , σ) −→ C

(4.1.34)

which extends L f in (4.1.32). Hence, L f belongs to the dual of L p (Δr , σ) which, by the Riesz Representation Theorem, can be identified with L p (Δr , σ), given that p ∈ [1, ∞). This proves that

4.2 Lebesgue-Based and Lorentz-Based Hardy Spaces on Ahlfors Regular Sets

there exists a function gr ∈ L p (Δr , σ) such that ∫

Lf h = hgr dσ for each h ∈ L p (Δr , σ).

113

(4.1.35)

Δr

In particular,

f , φ = Lf φ =

∫ Δr

φgr dσ for all φ ∈ Lipc (Δr ).

(4.1.36)

In concert with [133, Corollary 3.7.3], this shows that whenever 0 < r1 < r2 < ∞  we necessarily have gr2 Δr = gr1 at σ-a.e. point in Δr1 . In turn, this pointwise 1 compatibility property guarantees the  existence of a well-defined function g : Σ → C such that for each r > 0 we have g Δr = gr at σ-a.e. point in Δr . Consequently, ∫

g∈

p Lloc (Σ, σ)

and f , φ =

Σ

φg dσ for all φ ∈ Lipc (Σ),

(4.1.37)

p

which goes to show that, indeed, f ∈ Lloc (Σ, σ).

To address the case p = 1 in (4.1.29), define σf := fγ · σ, which is a positive, locally finite, Borel-regular measure on Σ. By reasoning as in the case p ∈ (1, ∞] using σf in place of σ, and keeping in mind that for each r > 0 we have  1 the measure ∗ L (Δr , σf ) = L ∞ (Δr , σf ), we deduce that ∞ (Σ, σ ) such that there exists g ∈ Lloc f ∫ φ g dσf for all φ ∈ Lipc (Σ).

f , φ =

(4.1.38)

Σ

1 (Σ, σ), from (4.1.38) we conclude Since we are currently assuming that fγ ∈ Lloc ∫  1 that the function g := g · fγ belongs to Lloc (Σ, σ) and satisfies f , φ = Σ φg dσ 1 (Σ, σ), finishing the proof of for all φ ∈ Lipc (Σ). Ultimately, this shows that f ∈ Lloc (4.1.29) for all p ∈ [1, ∞]. Next, (4.1.30) is a direct consequence of (4.1.29) and Lemma 4.1.2. In a similar manner, (4.1.31) is implied by [133, (6.2.36)], (4.1.29), and Lemma 4.1.2. 

4.2 Lebesgue-Based and Lorentz-Based Hardy Spaces on Ahlfors Regular Sets We begin by recalling the classical Hardy space H p (Rn, L n )∫(cf., e.g., [176, Theorem 1, p. 91]). Fix a background function Φ ∈ Cc∞ (Rn ) with Rn Φ dL n  0 and set Φt (x) := t −n Φ(x/t) for all t > 0 and x ∈ Rn . Then for each p ∈ (0, ∞), the classical Hardy space H p (Rn, L n ) consists of all tempered distributions f in Rn with the property that their radial maximal function, defined as

114

4 Hardy Spaces on Ahlfors Regular Sets

  ( frad )(x) := sup |(Φt ∗ f )(x)| = sup  Φt (x − ·), f   for each x ∈ Rn, t>0

(4.2.1)

t>0

satisfies frad ∈ L p (Rn, L n ).

(4.2.2)

To be able to define Hardy spaces in more general geometric settings, we should find a suitable substitute for the radial maximal function used above. In this vein, note that all Φt (x − ·)’s behave like normalized “bump functions” (at scale r ≈ t, centered at x), of the sort considered in (4.1.1). This observation suggests that in place of the radial maximal function frad we should use the more geometrically friendly Fefferman-Stein grand maximal function fγ introduced in (4.1.6). With this motivation in mind, we proceed to define the scale of (Lebesgue-based) Hardy spaces as follows. The reader is reminded that, generally speaking, (a)+ := max{a, 0} for each a ∈ R.

(4.2.3)

Definition 4.2.1 Pick a closed set Σ ⊆ Rn which is Ahlfors regular and abbreviate σ := H n−1 Σ. Having selected p, γ such that 1  n−1 (4.2.4) n < p < ∞ and (n − 1) p − 1 + < γ < 1, define the (Lebesgue-based) Hardy space     H p (Σ, σ) := f ∈ Lipc (Σ) : fγ ∈ L p (Σ, σ)

(4.2.5)

and equip it with the quasi-norm    f  H p (Σ,σ) :=  fγ  L p (Σ,σ),

∀ f ∈ H p (Σ, σ).

(4.2.6)

Note that (4.2.6) is a p-norm (see Definition 1.5.7 for a broader point of view), in the sense that p

p

p

 f + g H p (Σ,σ) ≤  f  H p (Σ,σ) + g H p (Σ,σ),

∀ f , g ∈ H p (Σ, σ).

(4.2.7)

We refer the reader to [9] where this brand of Hardy spaces has been studied at length. Among other things, the Hardy space (4.2.5) is complete (hence, quasiBanach) space, and is independent on the parameter γ (chosen as In  in (4.2.4)).  , 1 (see [62, general, the Hardy space H p (Σ, σ) is not locally convex when p ∈ n−1 n Theorem 6.2, p. 71] for a proof in the Euclidean setting). Bearing in mind item (3) of [133, Proposition 4.1.2], from (4.1.7) we see that     H p (Σ, σ) → Lipc (Σ) continuously, for each p ∈ n−1 (4.2.8) n ,∞ . Also, Lemma 4.1.3, (4.1.23), and [133, (7.6.18)] imply that H p (Σ, σ) coincides, in a quantitative sense, with the Lebesgue space L p (Σ, σ) whenever p belongs to (1, ∞)

(4.2.9)

4.2 Lebesgue-Based and Lorentz-Based Hardy Spaces on Ahlfors Regular Sets

115

and, corresponding to the end-point case p = 1, H 1 (Σ, σ) → L 1 (Σ, σ) continuous, proper inclusion.

(4.2.10)

Lastly, from (4.2.5), (4.1.17), and Lemma 4.1.2 we conclude that, in the context of Definition 4.2.1, 1 ∈ H p (Σ, σ) ⇐⇒ Σ is bounded. (4.2.11) In view of this, in the same setting as that considered in Definition 4.2.1, we may find it useful to define the homogeneous Hardy space   f ∈ H p (Σ, σ) : f , 1 = 0 if Σ is bounded, .p H (Σ, σ) := (4.2.12) H p (Σ, σ) if Σ is unbounded. Finally, we wish to note that, as is apparent from (4.1.22)-(4.1.23), [133, (7.6.18)], and (4.2.5)-(4.2.6), L q (Σ, σ) → H p (Σ, σ) continuously if Σ is bounded and

n−1 n

< p ≤ 1 < q ≤ ∞.

(4.2.13)

There is also a useful characterization of Hardy spaces on a given closed Ahlfors regular set Σ ⊆ Rn , with respect to the associated measure σ := H n−1 Σ, in terms of a sufficiently regular approximation to the identity (aka ATTI). Specifically, in such a setting [9, Theorem 3.22, pp. 102-103] guarantees the existence of a family  of (σ ⊗  σ)-measurable functions St : Σ × Σ → R indexed by a parameter t ∈ 0, diam Σ and satisfying,for some constant C ∈ (0, ∞), the following properties  for all x, x , y, y ∈ Σ and t ∈ 0, diam Σ : 0 ≤ St (x, y) ≤ Ct 1−n and St (x, y) = 0 if |x − y| ≥ Ct, |St (x, y) − St (x , y)| ≤ Ct −n |x − x |,

(4.2.14)

  [St (x, y) − St (x , y)] − [St (x, y ) − St (x , y )] ≤ C |x − x ||y − y | , t n+1 ∫ ∫ St (x, y) = St (y, x) and St (x, z) dσ(z) = 1 = St (z, y) dσ(z). Σ

Σ

Then, according to [9, Theorem 4.11, p. 140], the Hardy scale may be characterized1 as follows:     p given f ∈ Lipc (Σ) and p ∈ n−1 n , ∞ , we have f ∈ H (Σ, σ)     if and only if Σ  x −→ sup Lipc (Σ) St (x, ·), f (Lipc (Σ))  (4.2.15) belongs to L p (Σ, σ).

0 4 and pick an arbitrary λ ∈ (2, λ/2). Then, for each j ∈ N introduce the sets

j := B(x j , λ r j ) ∩ Σ, E j := B(x j , λr j ) ∩ Σ, and observe that E j := B(x j , r j ) ∩ Σ, E they satisfy hypotheses (a)-(d) of [138, Theorem 4.18, pp. 178-179] (with {r j } j ∈N as above). Hence, the latter theorem applies and yields a constant C ∈ [1, ∞) together with a family of real-valued functions {φ j } j ∈N defined on Σ with the property that for each j ∈ N one has φ j ∈ Lip(Σ),

φ j Lip(Σ) ≤ Cr j−1

0 ≤ φ j ≤ 1 on Σ,

φ j ≡ 0 on Σ \ B(x j , λ r j ),

and φ j ≥ 1/C on B(x j , r j ) ∩ Σ, and also

 j ∈N

φ j = 1 j ∈N B(x j ,r j )∩Σ = 1 Oα .

(4.3.12) (4.3.13) (4.3.14) (4.3.15)

This partition of unity enjoys a number of additional properties. First, we claim that there exists a geometric constant C ∈ (0, ∞) such that −γ

φ j C.γ (Σ) ≤ Cr j

for every j ∈ N.

(4.3.16)

Indeed, if x, y ∈ Σ, then (4.3.12)-(4.3.13) imply |φ j (x) − φ j (y)| ≤ Cr j−1 |x − y| and |φ j (x) − φ j (y)| ≤ 2.

(4.3.17)

122

4 Hardy Spaces on Ahlfors Regular Sets

Now (4.3.16) follows by raising the first inequality in (4.3.17) to the power γ, raising the second inequality to the power 1 − γ, and then multiplying the resulting inequalities. A second property we wish to single out is as follows. To set the stage, observe that for each fixed j ∈ N we may, thanks to (4.3.9), select a point   (4.3.18) y j ∈ B(x j , Λr j ) ∩ Σ \ Oα . Since λ < λ/2 < Λ/2, there exists some constant c ∈ (0, ∞), independent of j, such that (4.3.19) if R j := cr j , then B(x j , λ r j ) ⊆ B(y j , R j ). Finally, recall the collection of sets Tγ (x) for x ∈ Σ defined in (4.1.2). Then, in relation to the partition of unity {φ j } j ∈N introduced earlier, the second claim we make is that there exists some large constant C0 ∈ (0, ∞) such that    φ j C0 · σ B(x j , r j ) ∩ Σ belongs to Tγ (y j ), for every j ∈ N. (4.3.20) This is seen by applying (4.1.3) for φ := φ j , R := R j , and x := y j . Specifically, since from (4.3.12)-(4.3.13) and (4.3.19) we have that, for each j ∈ N, the Lipschitz function φ j is supported in B(y j , R j ) ∩ Σ, (4.3.21)     and since sup j ∈N supΣ |φ j | ≤ 1 and sup j ∈N r j · φ j Lip(Σ) < +∞, it follows that the quantity Cφ,R (defined in (4.1.3)) may be bounded independently of j. By also taking into account the fact that, thanks to the Ahlfors regularity of Σ, we have σ(B(x j , r j ) ∩ Σ) ≈ σ(B(y j , R j ) ∩ Σ) uniformly in j, we conclude that C0 ∈ (0, ∞) doing the job in (4.3.20) does exist. The third claim we make is that  ∫      f φ j dσ  ≤ C · α · σ B(x j , r j ) ∩ Σ , for every j ∈ N. (4.3.22)   Σ

1 (Σ, σ), together with To prove (4.3.22), for every j ∈ N, we use the fact that f ∈ Lloc (4.3.20), (4.1.6), (4.3.5), and (4.3.18), to write ∫            φj  f φ j dσ  =  f , φ j   = C0 · σ B(x j , r j ) ∩ Σ  f ,       C0 · σ B(x j , r j ) ∩ Σ  Σ

    ≤ C0 · σ B(x j , r j ) ∩ Σ fγ (y j ) ≤ C0 · α · σ B(x j , r j ) ∩ Σ . At this stage we define the sequence of numbers ∫ f φ j dσ λ j := ∫Σ , for every j ∈ N. φ dσ Σ j

(4.3.23)

(4.3.24)

4.3 Real Interpolation of Hardy Spaces

123

1 (Σ, σ) and the properties of each φ , the λ ’s are Based on the fact that f ∈ Lloc j j meaningfully defined. Also, it is immediate from (4.3.24) that ∫ ( f − λ j )φ j dσ = 0, for every j ∈ N. (4.3.25) Σ

Moreover, (4.3.22) implies |λ j | ≤ C · α, for every j ∈ N. This newly introduced sequence allows us to define the function  g := f · 1Σ\Oα + λ j φ j on Σ.

(4.3.26)

(4.3.27)

j ∈N

As is apparent from (4.3.11) and (4.3.13) (bearing in mind that λ < λ),  in  any point Oα has a neighborhood which intersects at most M sets from the family supp φ j j ∈N . This property implies that the series in (4.3.27) converges uniformly on compact subsets of Oα . As such, the function g is both well-defined and σ-measurable on Σ. In addition, a combination of (4.3.27), (4.3.7), (4.3.26), and (4.3.15), gives |g| ≤ C · α at σ-a.e. point on Σ.

(4.3.28)

1 (Σ, σ) and (4.3.6) In particular, g ∈ L ∞ (Σ, σ). If we now set b := f − g, then b ∈ Lloc is satisfied. To finish the proof of Lemma 4.3.2, we are left with showing that the last estimate in (4.3.6) holds. With this goal in mind, we first remark that (4.3.27) and (4.3.15) imply   b = f − g = f · 1 Oα − λj φj = ( f − λ j )φ j at σ-a.e. point on Σ. (4.3.29) j ∈N

j ∈N

Consequently, if we define b j := ( f − λ j )φ j on Σ, for each j ∈ N, then  1 (Σ, σ) and b = b j at σ-a.e. point on Σ. b j ∈ Lloc

(4.3.30)

j ∈N

Our next goal is to prove that there exists a constant C ∈ (0, ∞) such that, for each j ∈ N and at each x ∈ Σ, the following estimate holds: (b j )γ (x)

⎧  ⎪ if x ∈ B(x j , λr j ) ∩ Σ, ⎪ ⎨ C · fγ (x) ⎪

n−1+γ r ≤ j ⎪ if x ∈ Σ \ B(x j , λr j ). ⎪ ⎪ C · α · |x − x j | ⎩

(4.3.31)

Proving this takes some work. To set the stage, fix j ∈ N, x ∈ Σ, and ψ ∈ Tγ (x). Hence, ψ is Lipschitz, compactly supported on Σ, and satisfies (4.1.1) for some r ∈ (0, ∞), i.e.,

124

4 Hardy Spaces on Ahlfors Regular Sets

supp ψ ⊆ Σ ∩ B(x, r),

sup |ψ(z)| ≤ z ∈Σ

and ψC.γ (Σ) ≤ It is then immediate that

1  , σ Σ ∩ B(x, r)

1  . r γ · σ Σ ∩ B(x, r)

(4.3.32)

∫ Σ

|ψ(y)| dσ(y) ≤ 1.

(4.3.33)

We proceed with the proof of (4.3.31) by analyzing three separate cases, as follows. 1 (Σ, σ) we may write Case 1. Assume x ∈ B(x j , λr j ) ∩ Σ and r ≤ r j . Since b j ∈ Lloc ∫ b j (y)ψ(y) dσ(y)

b j , ψ = Σ

∫

∫ =

Σ

f (y)φ j (y)ψ(y) dσ(y) − λ j

Σ

φ j (y)ψ(y) dσ(y) =: I + I I.

Next, using (4.3.26), (4.3.13), and (4.3.33), we estimate ∫ |I I | ≤ |λ j φ j  L ∞ (Σ,σ) |ψ(y)| dσ(y) ≤ Cα < C fγ (x), Σ

(4.3.34)

(4.3.35)

where the last inequality in (4.3.35) is a consequence of the fact that in the current case x ∈ Oα . To obtain a similar upper bound for |I |, the idea is to show that there exists some constant C ∈ (0, ∞), independent of j, x, and ψ, for which φ j ψ/C ∈ Tγ (x). To see why the later membership holds, first note that (4.3.13) and (4.3.32) imply   supp (φ j ψ) ⊆ Σ ∩ B(x, r) and |φ j ψ| ≤ 1/σ Σ ∩ B(x, r) on Σ. (4.3.36) Second, relying on (4.3.13), (4.3.16), (4.3.32), and the fact that we are currently assuming r ≤ r j , we further estimate φ j ψC.γ (Σ) ≤ φ j  L ∞ (Σ,σ) ψC.γ (Σ) + φ j C.γ (Σ) ψ L ∞ (Σ,σ) ≤ ≤

rγ

1 1 C  + γ ·   · σ Σ ∩ B(x, r) r j σ Σ ∩ B(x, r)

rγ

C . · σ Σ ∩ B(x, r)

 

(4.3.37)

This proves that there exists some constant C ∈ (0, ∞) independent of j, x, and ψ, with the property that φ j ψ/C ∈ Tγ (x). Consequently,   |I | =  f , φ j ψ  ≤ C · fγ (x).

(4.3.38)

4.3 Real Interpolation of Hardy Spaces

125

In concert, (4.3.34), (4.3.35), and (4.3.38) prove that | b j , ψ| ≤ C · fγ (x). Since ψ was arbitrarily selected from Tγ (x), this proves (4.3.31) in the current case. Case 2. Assume x ∈ B(x j , λr j ) ∩ Σ and r > r j . The argument in this case is similar to the one in the proof of Case 1. Specifically, start by writing (4.3.34). The estimate in (4.3.35) continues to be valid for the same reasons. To estimate I, again we claim that there exists some constant C ∈ (0, ∞) independent of j, x, and ψ, for which φ j ψ/C ∈ Tγ (x). In the current case, B(x j , r j ) ⊆ B(x, 2λr j ), hence we may view the function φ j ψ as being supported in B(x, 2λr j ) ∩ Σ. Using the Ahlfors regularity of Σ and the fact that r > r j we obtain     |φ j ψ| ≤ 1/σ Σ ∩ B(x, r) ≤ C/σ Σ ∩ B(x, 2λr j ) on Σ. (4.3.39) To estimate φ j ψC.γ (Σ) , we may again proceed as in the first two lines in (4.3.37), and then rely on the the Ahlfors regularity of Σ and the fact that r > r j to write φ j ψC.γ (Σ) ≤ ≤

rγ

C 1 1  + γ ·   · σ Σ ∩ B(x, r) r j σ Σ ∩ B(x, r)

(2λr j



)γ

C . · σ Σ ∩ B(x, 2λr j ) 

(4.3.40)

Hence, φ j ψ/C ∈ Tγ (x) for some constant C ∈ (0, ∞) independent of j, x, ψ. The proof of estimate (4.3.31) in the current case is now finished exactly as in the end-game of Case 1. Case 3. Assume x ∈ Σ \ B(x j , λr j ). The first observation we make is that the identically on Σ if B(x, r) ∩ B(x j , r j ) ∩ Σ = . Since we function b j ψ vanishes ∫ seek to estimate Σ b j ψ dσ, we may therefore assume that there exists some point y∗ ∈ B(x, r) ∩ B(x j , r j ) ∩ Σ. Hence, (λ − 1)r j = λr j − r j < |x − x j | − |y∗ − x j | ≤ |x − y∗ | < r

(4.3.41)

which, given that λ > 1, further permits us to estimate |x − x j | ≤ |x − y∗ | + |y∗ − x j | ≤ r + r j ≤ Cr, (4.3.42) ∫ for some geometric constant C ∈ (0, ∞). In order to estimate Σ b j ψ dσ, introduce the function ψ j := ψ − ψ(x j ) and invoke the cancellation property (4.3.25) to write ∫

b j , ψ = b j (y)ψ j (y) dσ(y) Σ

∫ =

Σ

∫ f (y)φ j (y)ψ j (y) dσ(y) − λ j

=: I I I + IV .

Σ

φ j (y)ψ j (y) dσ(y) (4.3.43)

126

4 Hardy Spaces on Ahlfors Regular Sets

To treat IV, rely on (4.3.26) and (4.3.13) to first obtain ∫ |IV | ≤ |λ j |φ j  L ∞ (Σ,σ) |ψ j (y)| dσ(y) B(x j ,r j )∩Σ

∫ ≤ Cα

B(x j ,r j )∩Σ

|ψ j (y)| dσ(y).

(4.3.44)

Making use of (4.3.32) we write |ψ j (y)| = |ψ(y) − ψ(x j )| ≤ |y − x j |γ ψC.γ (Σ) γ

≤

rj

  r γ · σ B(x, r) ∩ Σ

for each y ∈ B(x j , r j ) ∩ Σ.

(4.3.45)

Returning with this in (4.3.44) and also employing the Ahlfors regularity of Σ and (4.3.42) we may further estimate γ

r n−1+γ   j  , r ) ∩ Σ ≤ Cα · σ B(x j j r r γ · σ B(x, r) ∩ Σ

n−1+γ r j ≤ Cα · . (4.3.46) |x − x j | 

|IV | ≤ C · α ·

rj

We desire a similar upper bound for |I I I |. To obtain this desired bound recall y j and R j from (4.3.18) and (4.3.19) and observe that (4.3.21) ensures that φ j ψ j ∈ Lipc (Σ) and supp (φ j ψ j ) ⊂ B(y j , R j ) ∩ Σ.

(4.3.47)

Also, the properties of φ j and ψ (from (4.3.12)-(4.3.13) and (4.3.32)) combined with (4.3.42), (4.3.19), and the Ahlfors regularity of Σ give φ j ψ j  L ∞ (Σ,σ) ≤

sup

x ∈B(x j ,r j )∩Σ

|ψ(x) − ψ(x j )| ≤

sup

x ∈B(x j ,r j )∩Σ

|x − x j |γ ψC.γ (Σ)

1   r γ · σ B(x, r) ∩ Σ

n−1+γ r 1 j . ·  ≤C |x − x j | σ B(y j , R j ) ∩ Σ γ

≤ rj ·

(4.3.48)

Moreover, we claim that φ j ψ j C.γ (Σ) ≤ C



r j n−1+γ 1  . · γ |x − x j | R j · σ B(y j , R j ) ∩ Σ

(4.3.49)

Assume (4.3.49) for now. Then (4.3.47), (4.3.48), and (4.3.49) imply that there exists C ∈ (0, ∞) independent of j, x, ψ, such that

4.3 Real Interpolation of Hardy Spaces

C

φ j ψj

n−1+γ rj |x−x j |

127

∈ Tγ (y j ).

(4.3.50)

In turn, we may employ this membership to estimate |I I I | (from (4.3.43)) as   

n−1+γ  r   φ ψ j j j   |I I I | =  f , φ j ψ j   = C   f,

n−1+γ   |x − x j | rj C |x−x j | ≤C



n−1+γ r r j n−1+γ  j fγ (x) ≤ Cα · . |x − x j | |x − x j |

(4.3.51)

This is the same upper bound as the one obtained for IV. Collectively, (4.3.43), (4.3.46), and (4.3.51) give

n−1+γ   rj  b j , ψ  ≤ Cα · |x−x j | for every x ∈ Σ \ B(x j , λr j ) and ψ ∈ Tγ (x).

(4.3.52)

The estimate in (4.3.31) corresponding to x ∈ Σ \ B(x j , λr j ) now follows from (4.3.52). To finish the proof in Case 3 we are left with establishing (4.3.49). As an intermediate step, we rely on the fact that r j ≈ R j , the Ahlfors regularity of Σ, and (4.3.42) to write rγ

1 C C  ≤ n−1+γ ≤ n−1+γ r |x − x · σ B(x, r) ∩ Σ j|

n−1+γ r 1 j  . ≤C · γ |x − x j | R j · σ B(y j , R j ) ∩ Σ 

(4.3.53)

Note that, in light of estimate (4.3.53), in order to conclude (4.3.49) it suffices to prove that, for every y, z ∈ Σ,   (φ j ψ j )(y) − (φ j ψ j )(z) C  . ≤ γ (4.3.54) |y − z|γ r · σ B(x, r) ∩ Σ The proof of (4.3.54) is handled by analyzing separately the following three cases: Case (a): y ∈ Σ \ B(x j , r j ) and z ∈ Σ \ B(x j , r j ); Case (b): z ∈ B(x j , r j ) ∩ Σ; Case (c): y ∈ B(x j , r j ) ∩ Σ. Since supp (φ j ψ j ) ⊂ B(x j , r j ) ∩ Σ, the estimate in (4.3.54) is immediate in Case (a). Also, we observe that the estimate in (4.3.54) is symmetric in y and z, thus its proof in Case (b) is the same as its proof in Case (c). As such, we are only left with treating Case (c). Having fixed y ∈ B(x j , r j ) ∩ Σ and an arbitrary z ∈ Σ, we then write

128

4 Hardy Spaces on Ahlfors Regular Sets

  (φ j ψ j )(y) − (φ j ψ j )(z)       ≤ φ j (z) ψ j (y) − ψ j (z) + ψ j (y) φ j (y) − φ j (z) ≤ |y − z|γ ψC.γ (Σ) + |y − x j |γ ψC.γ (Σ) |y − z|γ φ j C.γ (Σ) ≤ |y − z|γ · ≤

rγ

C|y − z|γ 1 1  + r γj · γ   · γ · σ B(x, r) ∩ Σ r · σ B(x, r) ∩ Σ rj 

C|y − z|γ  , r γ · σ B(x, r) ∩ Σ

(4.3.55)

where for the third inequality we used (4.3.32) and (4.3.16). Hence, (4.3.54) is proved to be true, and this completes the proof of the estimate in (4.3.31) corresponding to Case 3. Having finished the proof of (4.3.31), the next order of business is showing that b (recall (4.3.29)) satisfies the estimate in the last line of (4.3.6). In this regard, we first claim that ∫ Σ

bψ dσ = lim

N →∞

N ∫  j=1

Σ

b j ψ dσ,

∀ψ ∈ Lipc (Σ).

(4.3.56)

To justify this claim, fix ψ ∈ Lipc (Σ) and note that (4.3.30) guarantees that ! N j=1 b j ψ → bψ at σ-a.e. point in Σ, while (4.3.26), (4.3.13), and (4.3.8) collectively imply N N        b j ψ ≤ | f − λ j ||ψ|φ j ≤ M | f | + C · α |ψ|1 Oα ∈ L 1 (Σ, σ).  j=1

(4.3.57)

j=1

Based on these two observations and Lebesgue’s Dominated Convergence Theorem we conclude that the claim made in (4.3.56) is true. Going further, from (4.3.56) and (4.1.6) we see that  bγ ≤ (b j )γ at each point on Σ. (4.3.58) j ∈N

Given that 0 < p < 1, this further implies  p   p bγ ≤ (b j )γ

on Σ.

(4.3.59)

j ∈N

To estimate the L p -(quasi)norm of each (b j )γ , fix j ∈ N and use (4.3.31) to write

4.3 Real Interpolation of Hardy Spaces

∫ Σ



(b j )γ

p

∫ dσ =

129



B(x j ,λr j )∩Σ

(b j )γ

∫

≤C



B(x j ,λr j )∩Σ

fγ

p

p

∫ + Cα Since γ > (n − 1)

1 p



p Σ\B(x j ,λr j )

∫ dσ +



Σ\B(x j ,λr j )

(b j )γ

p

dσ

dσ r j (n−1+γ)p dσ(x). |x − x j |

(4.3.60)

 − 1 , [133, Lemma 7.2.1] applies and [133, (7.2.5)] yields

∫



Σ\B(x j ,λr j )

r j (n−1+γ)p n−1−(n−1+γ)p dσ ≤ Cr j . |x − x j |

(4.3.61)

In turn, (4.3.61), the Ahlfors regularity of Σ, the fact that B(x j , λr j ) ∩ Σ ⊂ Oα , and the definition of Oα may be employed to estimate ∫ r

(n−1+γ)p   j p α dσ(x) ≤ Cα p · r jn−1 ≤ Cα p · σ B(x j , r j ) ∩ Σ Σ\B(x j ,λr j ) |x − x j | ∫ ≤ C α p · 1B(x j ,λr j )∩Σ dσ Σ

∫ ≤C



B(x j ,λr j )∩Σ

fγ

p

dσ.

In concert, (4.3.60) and (4.3.62) imply ∫ ∫  p   p fγ dσ for each j ∈ N. (b j )γ dσ ≤ C Σ

B(x j ,λr j )∩Σ

(4.3.62)

(4.3.63)

Consequently, (4.3.59), (4.3.63), and (4.3.8) may be combined to compute ∫ ∫  ∫  p  p   p fγ 1B(x j ,λr j )∩Σ dσ ≤ C fγ dσ, bγ dσ ≤ (4.3.64) Σ

j ∈N

Σ

Oα

hence the estimate in the last line of (4.3.6) is satisfied. The proof of Lemma 4.3.2 is therefore complete.  Along the way, we find it useful to have a density result of the sort described in the next lemma. Lemma 4.3.3 Let Σ ⊆ Rn be a closed set which is Ahlfors regular and define σ := H n−1 Σ. Then for each p ∈ n−1 n , 1 and q ∈ (0, ∞), H p,q (Σ, σ) ∩

"

p α}   ≤ Cd 1 Oα · fγ  L d,∞ (Σ,σ) . As a byproduct of (4.3.73) and [133, (6.2.16), (6.2.26)] we see that   h ∈ L d (Σ, σ) for each d ∈ n−1 n ,∞ .

(4.3.73)

(4.3.74)

Also, whenever d ∈ (p, ∞), the log-convex estimate accompanying [133, (6.2.47)] guarantees (also keeping [133, (6.2.25)] in mind) the existence of some constant C = C(d, p) ∈ (0, ∞) with the property that   1−θ  θ  1Σ\O · fγ  d ≤ C 1Σ\Oα · fγ  L p, q (Σ,σ) 1Σ\Oα · fγ  L ∞ (Σ,σ) α L (Σ,σ)  1−θ ≤ Cα θ ·  fγ  L p, q (Σ,σ) < +∞,

(4.3.75)

where θ ∈ (0, 1) is such that 1/d = (1 − θ)/p. As a result, 1Σ\Oα · fγ ∈ L d (Σ, σ) for each d ∈ (p, ∞).

(4.3.76)

Combining (4.3.71), (4.3.74), and (4.3.76) yields gγ ∈ L d (Σ, σ) for each d ∈ (p, ∞) which, in light of (4.2.5), further entails " g∈ H d (Σ, σ). (4.3.77) p λ} ≤ t , ∀t > 0.

(4.3.85)

From [133, (6.2.41), (6.2.26)] and (4.2.24) it follows that, for some C ∈ (0, ∞) independent of f , F(t) ≤ Ct −1/pθ  f  H p θ , q (Σ,σ) for each t ∈ (0, ∞).

(4.3.86)

For each fixed t ∈ (0, ∞), the idea is now to apply Lemma 4.3.2 for the threshold α := F(t p ). In this regard, note that (4.3.86) implies α ∈ [0, ∞). Whenever applicable, Lemma 4.3.2 allows us to decompose f as 1 f = gt + bt with gt ∈ L ∞ (Σ, σ) and bt ∈ Lloc (Σ, σ)

(4.3.87)

134

4 Hardy Spaces on Ahlfors Regular Sets

satisfying, for some constant C ∈ (0, ∞) independent of t, |gt | ≤ C · F(t p ) for σ-a.e. point on Σ, and ∫ ∫  p   p (bt )γ dσ ≤ C fγ dσ,

(4.3.88)

  Ut := Oα = x ∈ Σ : fγ (x) > F(t p ) .

(4.3.89)

Σ

where

Ut

Note that Ut is a relatively open subset of Σ, hence σ-measurable. Also, whenever α > 0, Chebytcheff’s inequality gives that ∫   σ(Ut ) = σ Oα ≤ α−p ( fγ ) p dσ < +∞, (4.3.90) Σ

where the last inequality uses the fact that f ∈ H p (Σ, σ). In particular, having σ(Ut ) < +∞ forces Ut to be a proper subspace of the unbounded Ahlfors regular set Σ. This property guarantees the applicability of Lemma 4.3.2. In relation to (4.3.87), it is also useful to observe that thanks to (4.3.88), [133, (6.2.5)] (whose validity is ensured by the fact that F(t p ) < +∞, as seen from (4.3.86)), and (4.3.86) we have ∫ ∫ tp ∫    p p F(s) p ds (bt )γ dσ ≤ C fγ dσ ≤ C Σ

Ut

0

∫ p

≤ C f  H p θ , q (Σ,σ) ∫

s−p/pθ ds

0 tp

p

= C f  H p θ , q (Σ,σ)

tp

s θ−1 ds

0

= C f  H p θ , q (Σ,σ) t θ ·p < +∞. p

(4.3.91)

In view of (4.2.5), this implies that bt ∈ H (Σ, σ) and bt  p

H p (Σ,σ)

≤C

∫

tp

F(s) p ds

1/p

.

(4.3.92)

0

Moving on, recall from (1.3.34) that the K-functional is presently given by   (4.3.93) K(t, f ) := K t, f , H p (Σ, σ), L ∞ (Σ, σ)   = inf b H p (Σ,σ) + t g L ∞ (Σ,σ) : f = b + g, b ∈ H p (Σ, σ), g ∈ L ∞ (Σ, σ) . Then, on account of (4.3.87)-(4.3.88) and (4.3.92), we may estimate

4.3 Real Interpolation of Hardy Spaces

K(t, f ) ≤ C

135

∫

tp

F(s) p ds

1/p

+ Ct · F(t p ).

(4.3.94)

0

Next, we use (1.3.38) to write  f (H p (Σ,σ), L ∞ (Σ,σ))θ, q =

∫

∞



t −θ K(t, f )

0

%∫ ≤C

∞

t

−θq

∫

0

+C

0

∫ 0

∞

 q dt 1/q t

tp

q/p dt F(s) ds t

& 1/q

p

  q dt 1/q t (1−θ)q F(t p ) =: I + I I. t

(4.3.95)

To treat I, we make the change of variables ρ := t p and then apply Hardy’s inequality, as stated in Lemma 4.3.5, to write ∫ ∞ dt 1/q I≤C t (1−θ)q/p F(t)q t 0     = C t 1/pθ F(t) L q (0,∞), dt  = C  fγ  L p θ , q (Σ,σ) . (4.3.96) t

Note that after making the change of variables ρ := t p in I I we also obtain, much as above, ∫ ∞ dρ 1/q II ≤ C ρ(1−θ)q/p F(ρ)q ρ 0     = C  ρ1/pθ F(ρ) L q (0,∞), dt  = C  fγ  L p θ , q (Σ,σ) . (4.3.97) t

Let us summarize our progress. Combining (4.3.95)-(4.3.97) we arrive at the conclusion that there exists some constant C ∈ (0, ∞) with the property that    f (H p (Σ,σ), L ∞ (Σ,σ))θ, q ≤ C  fγ  L p θ , q (Σ,σ) = C f  H p θ , q (Σ,σ), 1 (Σ, σ). for each f ∈ H pθ ,q (Σ, σ) ∩ Lloc

(4.3.98)

Consider now f ∈ H pθ ,q (Σ, σ) arbitrary. The density result from Lemma 4.3.4 1 (Σ, σ) such that ensures the existence of a sequence { f j } j ∈N ⊆ H pθ ,q (Σ, σ) ∩ Lloc f j −→ f in H pθ ,q (Σ, σ) as j → ∞.

(4.3.99)

Thus, { f j } j ∈N is Cauchy in H pθ ,q (Σ, σ) which, in light of (4.3.98), forces it to be a Cauchy sequence in the intermediate space H p (Σ, σ), L ∞ (Σ, σ) θ,q . Given that the latter  Theorem 3.4.2(b), p. 47]), it follows that there exists   is complete (cf. [15, h ∈ H p (Σ, σ), L ∞ (Σ, σ) θ,q such that lim f j = h in H p (Σ, σ), L ∞ (Σ, σ) θ,q . j→∞

Recall that the latter space embeds continuously in the ambient topological space

136

4 Hardy Spaces on Ahlfors Regular Sets

within which the interpolation process takes place (cf. (1.3.1)), which we presently take to be the space of distributions on Σ, i.e.,  p    H (Σ, σ), L ∞ (Σ, σ) θ,q → Lipc (Σ) continuously. (4.3.100)   This further implies that f j → h in Lip  c (Σ)  as j → ∞. Since, thanks to (4.3.99) and (4.2.29), we also have f j → f in Lipc (Σ) as j → ∞, we finally conclude that h = f as distributions on Σ. From this, we ultimately conclude that if Σ is unbounded we have   H pθ ,q (Σ, σ) → H p (Σ, σ), L ∞ (Σ, σ) θ,q continuously. (4.3.101) We claim that the inclusion in (4.3.101) continues to be valid in the case when Σ is bounded. Observe that in such a scenario we have σ(Σ) < +∞ which further implies (4.3.102) L ∞ (Σ, σ) → H p (Σ, σ) continuously, thanks to (4.2.5) and (4.1.23). Granted this embedding, Lemma 1.3.4 applied with   1/p t∗ := σ(Σ) ∈ (0, ∞)

(4.3.103)

gives that  f (H p (Σ,σ), L ∞ (Σ,σ))θ, q ≈

∫ 0

t∗



t −θ K(t, f )

 q dt q1 , t

(4.3.104)

uniformly for f ∈ H p (Σ, σ). Since (4.3.89) and [133, (6.2.3)] (used here with f replaced by fγ and t replaced by t p ) presently imply   σ(Ut ) = σ x ∈ Σ : fγ (x) > F(t p ) ≤ t p for each t ∈ (0, ∞),

(4.3.105)

it follows from (4.3.103) and (4.3.105) that σ(Ut ) < σ(Σ) for every t ∈ (0, t∗ ). Consequently, Ut is a proper subset of Σ, for every t ∈ (0, t∗ ).

(4.3.106)

Having established (4.3.106), we may invoke Lemma 4.3.2 with α := F(t p ) ∈ [0, ∞) for each t in the range (0, t∗ ). As a result, we are still able to decompose f as in (4.3.87)-(4.3.88), whenever t ∈ (0, t∗ ). Then, if q ∈ (0, ∞), on account of (4.3.93) and (4.3.87)-(4.3.88), we may continue to estimate K(t, f ) as in (4.3.94) for each t ∈ (0, t∗ ). With this estimate in hand, we may then run the same argument which has produced (4.3.96)-(4.3.97), by once gain relying on Hardy’s inequality (as in Lemma 4.3.5, now used with M := t∗ ), to conclude that (4.3.98) continues to be valid in the case when Σ is bounded. Ultimately, the inclusion in (4.3.101) remains true when Σ is bounded. Let us record our progress. The argument so far proves that the inclusion in (4.3.101) is valid for Σ as in the statement of the theorem. To establish the opposite

4.3 Real Interpolation of Hardy Spaces

137

inclusion in (4.3.101), observe that the grand maximal function induces well-defined, sub-linear, bounded operators H p (Σ, σ)  f −→ fγ ∈ L p (Σ, σ),

(4.3.107)

thanks to (4.2.1)-(4.2.6), as well as L ∞ (Σ, σ)  f −→ fγ ∈ L ∞ (Σ, σ),

(4.3.108)

thanks to (4.1.23). On account of (4.3.107)-(4.3.108) and the real interpolation result from Proposition 1.3.7, we then conclude that 

H p (Σ, σ), L ∞ (Σ, σ)

 θ,q

 f −→ fγ ∈ L pθ (Σ, σ)

(4.3.109)

is a well-defined bounded sub-linear mapping. In light of the definition of Lorentzbased Hardy spaces, this implies   p (4.3.110) H (Σ, σ), L ∞ (Σ, σ) θ,q → H pθ ,q (Σ, σ) continuously. At this stage, (4.3.1) follows from (4.3.98) and (4.3.110). Step II: The proof of (4.3.2) in the case when q0, q1, q ∈ (0, ∞). In such a scenario, (4.3.2) is a consequence of (4.3.1) and the reiteration theorem for the real method of interpolation recalled in Theorem 1.3.9. Step III: The proof of (4.3.3) in the case when q = ∞. Fix θ ∈ (0, 1) and suppose n−1 n

< p0 < p < p1 < ∞ satisfy

1 p

=

+ θp .

(4.3.111)

∈ (0, 1).

(4.3.112)

1−θ p0

    Also, choose some γ ∈ (n − 1) p10 − 1 , 1 , and define η := θ



1 p0

−

1 p1



∈ (0, ∞),

β := 1 −

p p1

The goal is to prove that, with equivalence of quasi-norms,  p0  H (Σ, σ), H p1 (Σ, σ) θ,∞ = H p,∞ (Σ, σ).

(4.3.113)

We shall first deal with the case when Σ is unbounded. In such a scenario, pick an arbitrary f ∈ H p,∞ (Σ, σ) and denote by F the non-increasing re-arrangement of fγ (cf. (4.3.85)). As was the case with (4.3.86), we presently have F(t) ≤ Ct −1/p  f  H p,∞ (Σ,σ) for each t ∈ (0, ∞).

(4.3.114)

The strategy is to invoke, for each fixed t ∈ (0, ∞), the Calderón-Zygmund type decomposition of the distribution f as in [9, Theorem 5.16, p. 207-209] at level α := F(t η ) ∈ [0, ∞). Specifically, for each t ∈ (0, ∞) this allows us to split   f = gt + bt in Lipc (Σ) , (4.3.115)

138

4 Hardy Spaces on Ahlfors Regular Sets

  for some gt , bt ∈ Lipc (Σ) satisfying (cf. [9, (5.2.18) and (5.2.20), p. 208]) (bt )γ ≤ Cα · ht + C · fγ · 1 Oα on Σ,

(4.3.116)

(gt )γ ≤ Cα · ht + C · fγ · 1Σ\Oα on Σ,

(4.3.117)

where C ∈ (0, ∞) is a geometric constant independent of f and t, the function ht is as in (4.3.69), and the level set Oα is given by   Oα := x ∈ Σ : fγ (x) > α = F(t η ) .

(4.3.118)

The applicability of [9, Theorem 5.16, p. 207-209] (which ensures the existence of such a decomposition) requires that in the case when α > 0 the set Oα does not coincide with Σ. Given that σ(Σ) = +∞ (since we are presently assuming Σ to be unbounded), the latter condition is satisfied since, as seen from Chebytcheff’s inequality, [133, (6.2.5)], and (4.3.114), we have 



σ Oα ≤ α

−p0

∫ Oα

( fγ ) p0

dσ ≤ Cα

≤ Cα−p0  f  H0p,∞ (Σ,σ)

−p0

∫

tη

F(s) p0 ds

0

∫

p

tη

s−p0 /p ds

0

= Cα−p0  f  H0p,∞ (Σ,σ) t θ ·p0 < +∞, p

(4.3.119)

where the last inequality uses the fact that f ∈ H p,∞ (Σ, σ). Having justified the decomposition of the f as in (4.3.115)-(4.3.118),   distribution , ∞ there exists Cd ∈ (0, ∞) indepenwe recall from (4.3.72) that for each d ∈ n−1 n dent of t such that     1/d ht  L d (Σ,σ) ≤ Cd · σ Oα . (4.3.120) Two consequences of (4.3.120) are of a particular interest to us. First, based on (4.3.120) with d := p0 , (4.3.118), and [133, (6.2.21), (6.2.27)], we may estimate    1/p0 α · ht  L p0 (Σ,σ) ≤ Cα · σ Oα    1/p0 = Cα · σ {x ∈ Σ : ( fγ · 1 Oα )(x) > α}     ≤ C  fγ · 1 Oα  L p0,∞ (Σ,σ) ≤ C  fγ · 1 Oα  L p0 (Σ,σ),

(4.3.121)

for some constant C ∈ (0, ∞) independent of t. Second, based on (4.3.120) with d := p1 , (4.3.118), [133, (6.2.21)], and (4.2.24), we conclude that, once again for some C ∈ (0, ∞) independent of t,

4.3 Real Interpolation of Hardy Spaces

139

   1/p1     1/p1 α · ht  L p1 (Σ,σ) ≤ Cα · σ Oα = Cα β · α1−β · σ Oα '    ( 1−β  1−β 1/p = Cα β · α · σ Oα ≤ Cα β  fγ  L p,∞ (Σ,σ) = Cα β  f  H p,∞ (Σ,σ) . 1−β

(4.3.122)

We augment (4.3.121)-(4.3.122) with another useful estimate. Specifically, keeping this in mind that 1/p1 = (1 − β)/p and invoking [133, (6.2.49)] implies (by also taking into account (4.3.118))     fγ · 1Σ\O  α

L p1 (Σ,σ)

1−β β   ≤ C  fγ · 1Σ\Oα  L p,∞ (Σ,σ) ·  fγ · 1Σ\Oα  L ∞ (Σ,σ)  1−β 1−β ≤ C  fγ  L p,∞ (Σ,σ) · α β = Cα β  f  H p,∞ (Σ,σ)

(4.3.123)

for some constant C ∈ (0, ∞) independent of t. The nest step is to make use of (4.3.121)-(4.3.123) to estimate bt  H p0 (Σ,σ) and gt  H p1 (Σ,σ) . First, from (4.3.116), (4.3.121), and [133, (6.2.5)], we obtain   (bt )γ 

L p0 (Σ,σ)

  ≤ Cαht  L p0 (Σ,σ) + C  fγ · 1 Oα  L p0 (Σ,σ)   ≤ C  fγ · 1 Oα  L p0 (Σ,σ) = C ≤C

∫

tη

F(s) p0 ds

1/p0

∫

 Oα

fγ

 p0

dσ

1/p0

.

(4.3.124)

0

In view of (4.3.119) and (4.2.5), this implies that for some constant C ∈ (0, ∞) independent of t we have bt ∈ H p0 (Σ, σ) and bt  H p0 (Σ,σ) ≤ C

∫

tη

F(s) p0 ds

1/p0

.

(4.3.125)

0

Second, from (4.3.117), (4.3.122), and (4.3.123) we see that   1−β (gt )γ  p ≤ Cα β  f  H p,∞ (Σ,σ) . L 1 (Σ,σ)

(4.3.126)

Collectively, (4.2.5), (4.3.126), and the earlier definitions of α and β imply that for some constant C ∈ (0, ∞) independent of t we have   1−(p/p1 ) p/p gt ∈ H p1 (Σ, σ) and gt  H p1 (Σ,σ) ≤ C F(t η ) ·  f  H p,∞1 (Σ,σ) .

(4.3.127)

Pressing on, recall from (1.3.34) that for each t ∈ (0, ∞) the K-functional is presently given by

140

4 Hardy Spaces on Ahlfors Regular Sets

  K(t, f ) := K t, f , H p0 (Σ, σ), H p1 ∞(Σ, σ) (4.3.128)   = inf b H p0 (Σ,σ) + t g H p1 (Σ,σ) : f = b + g, b ∈ H p0 (Σ, σ), g ∈ H p1 (Σ, σ) . On account of (4.3.128), (4.3.115), (4.3.125), and (4.3.127), for each t ∈ (0, ∞) we may therefore estimate (with η as in (4.3.112)) K(t, f ) ≤ C

∫

tη

F(s) p0 ds

1/p0

0

  1−(p/p1 ) p/p + Ct · F(t η ) ·  f  H p,∞1 (Σ,σ) (4.3.129)

for some constant C ∈ (0, ∞) independent of t. Based on (1.3.38) and (4.3.129) we may now estimate    f (H p0 (Σ,σ), H p1 (Σ,σ))θ,∞ = sup t −θ K(t, f ) t>0

)

≤ C · sup t

−θ

∫

tη

p0

F(s) ds

t>0

1/p0

*

0

'   1−(p/p1 ) ( p/p + C f  H p,∞1 (Σ,σ) · sup t (1−θ) · F(t η ) t>0

=: I I I + IV .

(4.3.130)

Note that, thanks to (A.0.60) and (4.2.5), we have ) * ∫ tη 

1/p0  p s1/p F(s) 0 s−p0 /p ds I I I = C · sup t −θ t>0

0



)



≤ C · sup s1/p F(s) · sup t −θ s>0

t>0

∫

tη

s−p0 /p ds

1/p0

*

0

    = C · sup s1/p F(s) = C  fγ  L p,∞ (Σ,σ) s>0

= C f  H p,∞ (Σ,σ),

(4.3.131)

where the first equality uses the fact that, as seen from (4.3.111)-(4.3.112), we have  p0 /p ∈ (0, 1) and −θ + 1 − (p0 /p) (η/p0 ) = 0. In addition, after changing s := t η    −1 and observing that (4.3.111)-(4.3.112) imply (1 − θ)/η 1 − (p/p1 ) = 1/p, we conclude that

4.3 Real Interpolation of Hardy Spaces

141

'

  1−(p/p1 ) p/p IV = C f  H p,∞1 (Σ,σ) · sup s(1−θ)/η · F(s)

(

s>0

  1−(p/p1 ) p/p = C f  H p,∞1 (Σ,σ) · sup s1/p F(s) s>0

p/p

= C f  H p,∞1 (Σ,σ)

 1−(p/p ) ·  fγ  p,∞ 1 L

(Σ,σ)

1−(p/p )

p/p

1 = C f  H p,∞1 (Σ,σ) ·  f  H p,∞ (Σ,σ)

= C f  H p,∞ (Σ,σ) .

(4.3.132)

From (4.3.130)-(4.3.132) we arrive at the conclusion that there exists some constant C ∈ (0, ∞) with the property that  f (H p0 (Σ,σ), H p1 (Σ,σ))θ,∞ ≤ C f  H p,∞ (Σ,σ) for all f ∈ H p,∞ (Σ, σ).

(4.3.133)

Since, by definition,  p0  H (Σ, σ),H p1 (Σ, σ) θ,∞    = f ∈ H p0 (Σ, σ) + H p1 (Σ, σ) ⊆ Lipc (Σ) :

  f (H p0 (Σ,σ), H p1 (Σ,σ))θ,∞ < +∞ ,

this ultimately proves that if Σ is unbounded then   H p,∞ (Σ, σ) → H p0 (Σ, σ), H p1 (Σ, σ) θ,∞ continuously.

(4.3.134)

(4.3.135)

We claim that the inclusion in (4.3.135) remains well-defined and continuous when Σ is bounded. Since in such a scenario (4.2.21) implies H p1 (Σ, σ) → H p0 (Σ, σ) continuously,

(4.3.136)

Lemma 1.3.4 applied with   1/η t∗ := σ(Σ) ∈ (0, ∞)

(4.3.137)

gives that  f (H p0 (Σ,σ), H p1 (Σ,σ))θ,∞ ≈ sup

t ∈(0,t∗ )



 t −θ K(t, f ) ,

(4.3.138)

with proportionality constants independent of f . Since (4.3.89) and [133, (6.2.3)] (presently used with f replaced by fγ and t replaced by t p ) presently imply   σ(Oα ) = σ x ∈ Σ : fγ (x) > F(t η ) ≤ t η for each t ∈ (0, ∞),

(4.3.139)

142

4 Hardy Spaces on Ahlfors Regular Sets

it follows from (4.3.137) and (4.3.139) that σ(Oα ) < σ(Σ) for every t ∈ (0, t∗ ). Hence, (4.3.140) Oα is a proper subset of Σ, for every t ∈ (0, t∗ ). In turn, (4.3.140) guarantees that we are still able to decompose the distribution f as in (4.3.115)-(4.3.118) for each t ∈ (0, t∗ ). Consequently, the estimate for the K-functional in (4.3.129) continues to be valid for t ∈ (0, t∗ ). Granted this estimate, we may then run the same argument which has led to (4.3.131)-(4.3.132). Ultimately, this proves that the inclusion in (4.3.135) is also valid when Σ is bounded. As regards the opposite inclusion in (4.3.135), observe that thanks to (4.2.5) the grand maximal function induces well-defined, sub-linear, bounded operators H p0 (Σ, σ)  f −→ fγ ∈ L p0 (Σ, σ),

(4.3.141)

H p1 (Σ, σ)  f −→ fγ ∈ L p1 (Σ, σ).

(4.3.142)

Based on (4.3.141)-(4.3.142) and the real interpolation result from Proposition 1.3.7 and [133, (6.2.48)] (both used with q = ∞) we then conclude that 

H p0 (Σ, σ), H p1 (Σ, σ)

 θ,∞

 f −→ fγ ∈ L p,∞ (Σ, σ)

(4.3.143)

is a well-defined bounded sub-linear mapping. Upon recalling the definition of Lorentz-based Hardy spaces, this implies  p0  H (Σ, σ), H p1 (Σ, σ) θ,∞ → H p,∞ (Σ, σ) continuously. (4.3.144) Finally, (4.3.135) and (4.3.144) justify (4.3.113). Step IV: The proof of (4.3.3) in full generality. When q ∈ (0, ∞) then the interpolation formula in (4.3.3) follows from (4.3.2) with q0 := p0 and q1 := p1 (a scenario which has been already dealt with in Step II), bearing in mind (4.2.25). Finally, formula (4.3.3) with q = ∞ has been proved in Step III. Step V: The proof of (4.3.2) in full generality. The fact that formula (4.3.2) holds as stated follows from (4.3.3) and the reiteration theorem for the real method of interpolation recorded in Theorem 1.3.9. The proof of Theorem 4.3.1 is therefore complete.  Once Theorem 4.3.1 has been established, from (4.3.3), (1.3.44), and (1.3.45) we conclude that if Σ ⊆ Rn is a closed set which is Ahlfors regular and σ := H n−1 Σ, then H p0 (Σ, σ) ∩ H p1 (Σ, σ) embeds continuously (cf. (1.3.3)) and (4.3.145) densely into the space H p,q (Σ, σ) granted n−1 n < p0 < p < p1 < ∞ and q ∈ (0, ∞). Also, from Theorem 4.3.1 and (1.3.41) we see that

4.4 Atomic Decompositions for Hardy Spaces

143

if Σ ⊆ Rn is a closed set which is Ahlfors regular and σ := H n−1 Σ, then H p,q (Σ, σ) embeds continuously into H p0 (Σ, σ) + H p1 (Σ, σ) (4.3.146) whenever n−1 n < p0 < p < p1 < ∞ and q ∈ (0, ∞].

4.4 Atomic Decompositions for Hardy Spaces We begin by specializing results from [9] pertaining to atomic decompositions of Hardy spaces to the setting of Ahlfors regular subsets of the Euclidean ambient. To set the stage, given any closed subset Σ of Rn along with some β ∈ (0, ∞) define  β C (Σ) if Σ is bounded, L β (Σ) := (4.4.1) .β C (Σ)/∼ if Σ is unbounded, where, as before, the equivalence identifies functions which differ by a constant on Σ. Next, assume that Σ ⊆ Rn is a closed set which is Ahlfors regular and abbreviate σ := H n−1 Σ. In this context, given two exponents p ∈ (0, 1] and q ∈ [1, ∞] with q > p, call a σ-measurable function a : Σ → C a (p, q)-atom provided there exist a point x ∈ Σ and a number r ∈ (0, 2 diam Σ) with the property that ∫   1/q−1/p , a dσ = 0. (4.4.2) supp a ⊆ B(x, r) ∩ Σ, a L q (Σ,σ) ≤ σ B(x, r) ∩ Σ Σ

It is also agreed that in the case when Σ is bounded the constant function a(x) := [σ(Σ)]−1/p for every x ∈ Σ is also considered to be a (p, q)-atom.

(4.4.3)

p,q

We then define the atomic Hardy space Hat (Σ, σ) as   ∗ p,q Hat (Σ, σ) := f ∈ L (n−1)(1/p−1) (Σ) : there exist {λ j } j ∈N ∈  p (N) along with a sequence {a j } j ∈N of (p, q)-atoms such that f =

∞ 

 ∗ λ j a j in L (n−1)(1/p−1) (Σ)

(4.4.4)

j=1

when p ∈ (0, 1) and, corresponding to the case when p = 1,  1,q Hat (Σ, σ) := f ∈ L 1 (Σ, σ) : there exist {λ j } j ∈N ∈  1 (N)and (1, q)-atoms {a j } j ∈N such that f =

∞  j=1

 λ j a j in L 1 (Σ, σ) .

(4.4.5)

144

4 Hardy Spaces on Ahlfors Regular Sets p,q

In all cases, we equip Hat (Σ, σ) with the quasi-norm  f  Hatp, q (Σ,σ) := inf

∞ 

|λ j | p

1/p (4.4.6)

j=1

where the infimum is taken over all writings f = (4.4.4)-(4.4.5). The above definitions imply that p,q

if f ∈ Hat (Σ, σ) is expanded as f =

∞ ! j=1

!∞

j=1

p,q

λ j a j of f ∈ Hat (Σ, σ) as in

  λ j a j in Lipc (Σ) for some

sequence {λ j } j ∈N ∈  1 (N) and with each a j a (p, q)-atom on Σ, then ∞ ! p,q λ j a j also converges to f in Hat (Σ, σ). the series

(4.4.7)

j=1

Theorem 4.4.1 Given a closed set Σ ⊆ Rn which is Ahlfors regular, abbreviate σ := H n−1 Σ, and select p, q, γ such that   n−1 1 ≤ q ≤ ∞, p < q, and (n − 1) p1 − 1 < γ < 1. (4.4.8) n < p ≤ 1, Associated with these parameters, consider the Hardy space H p (Σ, σ) defined using the grand-maximal function as in Definition 4.2.1, as well as the atomic Hardy space p,q Hat (Σ, σ) defined as in (4.4.4)-(4.4.5). p,q Then H p (Σ, σ) may be naturally identified with Hat (Σ, σ) both algebraically and quantitatively (in the sense of equivalence of quasi-norms). In particular, these spaces do not depend on the choice of the exponent q and the index γ as in (4.4.8). Proof This follows by combining [9, Theorem 5.27, p. 263] with [9, Theorem 7.5, p. 302].  As a consequence of Theorem 4.4.1 and (4.4.5) we see that if Σ ⊆ Rn is a closed Ahlfors regular set and σ := H n−1 Σ, then 1 if in addition H 1 (Σ, σ) embeds continuously into ∫  L (Σ,1σ); moreover,  1 Σ is unbounded, then H (Σ, σ) ⊂ f ∈ L (Σ, σ) : Σ f dσ = 0 . In particular, from (4.4.9) and (4.2.12) we deduce that ∫   . H 1 (Σ, σ) = f ∈ H 1 (Σ, σ) : f dσ = 0 . Σ

(4.4.9)

(4.4.10)

Theorem 4.4.1 also implies, in concert with (4.2.21), that if Σ ⊆ Rn is a closed set which is Ahlfors regular and σ := H n−1 Σ then H p1 (Σ, σ) → H p0 (Σ, σ) continuously and densely provided Σ is bounded and

n−1 n

< p0 ≤ p1 < ∞.

(4.4.11)

4.4 Atomic Decompositions for Hardy Spaces

145

For later purposes it is useful to know that Hardy spaces are separable quasiBanach spaces, an issue addressed in our next proposition. Proposition 4.4.2 Let Σ ⊆ Rn be a closed set define  which is Ahlfors regular and p (Σ, σ) is , ∞ , the quasi-Banach space H σ := H n−1 Σ. Then, for each p ∈ n−1 n separable. Proof When p ∈ (1, ∞), the fact that H p (Σ, σ) is separable is a consequence of  , 1 . For now, (4.2.9) and [133, (3.6.27)]. Consider next the case when p ∈ n−1 n suppose Σ is unbounded. Fix a point x0 ∈ Σ along with an exponent r ∈ (1, ∞) and, for each j ∈ N, abbreviate Δ j := B(x0, j) ∩ Σ. From [133, Lemma 3.6.4] we know that L r (Δ j , σ) is separable for each j ∈ N. If we now introduce ∫   r r L0 (Δ j , σ) := f ∈ L (Δ j , σ) : f dσ = 0 , ∀ j ∈ N, (4.4.12) Δj

the fact that any subset of a separable metric space is separable implies that for each j ∈ N the space L0r (Δ j , σ) is separable.

(4.4.13)

Granted this, for each j ∈ N we may pick a set Fj which is a countable and dense subset of L0r (Δ j , σ).

(4.4.14)

Next, extend each function in Fj by zero to the entire Σ and denote the collection of

j is a multiple of a (p, r)-atom on Σ,

j . Then every f ∈ F all these extensions by F p

hence Fj ⊂ H (Σ, σ) for each j ∈ N. Let us record our progress: 

j , if F denotes the collection of all finite sums of functions in j ∈N F (4.4.15) p then F is a countable subset of H (Σ, σ) which is stable under addition. The claim we make at this stage is that F is dense in H p (Σ, σ).

(4.4.16)

Keeping in mind the atomic characterization of H p (Σ, σ) from Theorem 4.4.1, this claim will follow as soon as we prove that for every (p, r)-atom a on Σ, every λ ∈ R, and every ε > 0, there exists some f ∈ F such that λ a − f  H p (Σ,σ) < ε.

(4.4.17)

To this end, fix a, λ, and ε as in (4.4.17). Then there exists some natural number j0 ∈ N such that supp a ⊆ Δ j0 and (λa)Δ j ∈ L0r Δ j0 , σ . Relying on (4.4.14) 0 (presently used with j = j0 ) consider f ∈ Fj0 such that λ a − f  L r (Δ j0 ,σ) < ε · σ(Δ j0 )1/r−1/p .

(4.4.18)

146

4 Hardy Spaces on Ahlfors Regular Sets

j ⊆ F . Moreover, If f denotes the extension of f by zero to Σ, it follows that f ∈F 0

the function g := λ a − f satisfies ∫ g dσ = 0, and g L r (Σ,σ) < ε · σ(Δ j0 )1/r−1/p . (4.4.19) supp g ⊆ Δ j0 , Δ j0

The properties in (4.4.19) imply that ε −1 g is a (p, r)-atom on Σ. Consequently, ε −1 g H p (Σ,σ) ≤ C for some constant C ∈ (0, ∞) which depends only on Σ, p, and r. In turn, this inequality translates into λ a − f  H p (Σ,σ) < Cε. After re-adjusting ε, this completes the proof of (4.4.17) which, in turn, finishes the proof of the fact that F is dense in H p (Σ, σ). The desired conclusion now follows in the case when Σ is unbounded. Finally, when Σ is bounded, the argument proceeds as before with one minor adjustment. Specifically, since we now are also allowing constant functions to be 

j by also including the constant multiples of (p, r)-atoms, we enrich the union j ∈N F functions with rational values. Reasoning as before then shows that taking F to the collection of all finite sums of functions in this family continues yields a countable  and dense subset of H p (Σ, σ). This finishes the proof of Proposition 4.4.2. It turns out that if a given distribution f in some Hardy space additionally belongs to a Lebesgue space, or another Hardy space, then one may perform an atomic decomposition which converges to f simultaneously in all said spaces. This is made precise in the theorem below. Theorem 4.4.3 Let Σ ⊆ Rn be a closed set which is Ahlfors regular and define n−1 σ := H n−1 Σ. Suppose n−1 n < p ≤ 1, n < q < ∞, 1 < r < ∞, and pick p q some arbitrary f ∈ H (Σ, σ) ∩ H (Σ, σ). Then there exist a numerical sequence {λ j } j ∈N ∈  p (N) along with a sequence of (p, r)-atoms {a j } j ∈N on Σ such that ∞ 

|λ j | p

1/p

≤ C f  H p (Σ,σ)

(4.4.20)

j=1

for some finite constant C > 0 which depends only on the ambient, and f =

∞ 

λ j a j with convergence both in H p (Σ, σ) and in H q (Σ, σ).

(4.4.21)

j=1

As a consequence, f =

∞ ! j=1

λ j a j with convergence in H p (Σ, σ) ∩ H q (Σ, σ) e-

(4.4.22)

quipped with the intersection quasi-norm (cf. (1.3.3)). It is also possible to allow r = ∞ in the above theorem. Indeed, as observed in [159], the argument from [176, pp. 107-109] yields just that. Originally carried

4.4 Atomic Decompositions for Hardy Spaces

147

out in the whole Euclidean space as ambient, said argument is robust enough to be adapted to the setting considered here. Proof of Theorem 4.4.3 First we consider the case when Σ is unbounded. We begin by reviewing the technology of decomposing a distributions belonging to a Hardy space as an infinite linear span of atoms. The approach we take hinges on the existence of a family {St }t>0 of integral operators on Σ which constitute an approximation to the identity of order ε1 ∈ (0, 1]. This means that, for each t ∈ (0, ∞), St is an integral operator of the form ∫ St f (x) := St (x, y) f (y) dσ(y), x ∈ Σ, (4.4.23) Σ

with an integral kernel St : Σ × Σ → R satisfying, for some C ∈ (0, ∞) independent of t, the following properties: (i) 0 ≤ St (x, y) ≤ Ct 1−n for all x, y ∈ Σ, and St (x, y) = 0 if |x − y| ≥ Ct; (ii) |St (x, y) − St (x , y)| ≤ Ct −(n−1+ε1 ) |x − x |∫ε1 for every x, x , y ∈ Σ; (iii) St (x, y) = St (y, x) for every x, y ∈ Σ, and Σ St (x, y) dσ(y) = 1 for every x ∈ Σ. The construction of such an approximation to the identity for any given exponent ε1 ∈ (0, 1] is carried out in detail in [9, Theorem 3.22, pp. 102-103], following R. Coifman’s template, outlined in [43]. Specifically, fix a function h ∈ C 1 (R) with the property that 0 ≤ h ≤ 1 pointwise on R, h ≡ 1 on [−1/2, 1/2], and h ≡ 0 on R \ (−2, 2). Next, for each t ∈ (0, ∞), let Tt be the integral operator acting on functions f defined on Σ according to ∫ |x − y|

f (y) dσ(y), x ∈ Σ. (4.4.24) (Tt f )(x) := t −(n−1) h t Σ Based on properties of the function h and the Ahlfors regularity of Σ, it is straightforward to check that there exists a finite constant Co ≥ 1 such that Co−1 ≤ (Tt 1)(x) ≤ Co for each x ∈ X and each t ∈ (0, ∞).

(4.4.25)

Keeping this in mind, for each t ∈ (0, ∞) it is then meaningful to define for each x, y ∈ Σ    ∫  h |x − z|/t) h |z − y|/t t −2(n−1)

dσ(z). (4.4.26) St (x, y) := (Tt 1)(x)(Tt 1)(y) T 1 (z) Σ

t Tt 1

Based on this formula, one may check that properties (i)-(iii) above hold. Moreover, if for each t ∈ (0, ∞) we now define the integral kernel Dt (x, y) := t

 ∂  St (x, y) , ∂t

x, y ∈ Σ,

(4.4.27)

a direct computation shows that Dt (x, y) behaves much like St (x, y), both in terms of size and regularity. Ultimately, this permits us to conclude that the family

148



4 Hardy Spaces on Ahlfors Regular Sets



above, the only Dt (x, y) t>0 satisfies properties similar to those recorded in (i)-(iii) ∫ exception being that we now have the cancellation condition Σ Dt (x, y) dσ(y) = 0 for every x ∈ Σ. Hence, if for each t > 0 we define the integral operator ∫ Dt f (x) := Dt (x, y) f (y) dσ(y), x ∈ Σ, (4.4.28) Σ

it follows that for every ε2 > 0 the family {Dt }t>0 is, in the language of [88, Main Assumption, p. 1510], a Calderón Reproducing Formula family of order (ε1, ε2 ) on Σ (or a (ε1, ε2 )-CRF, for short). This terminology is inspired by the fact that each Dt is a bounded operator on L po (Σ, σ) for every po ∈ (1, ∞) and the following identity, in the spirit of the classical Calderón reproducing formula, holds (see [83]-[84], and also [88, Proposition 2.13, p. 1517]) ∫ ∞ dt Dt2 f , ∀ f ∈ L po (Σ, σ) with po ∈ (1, ∞). (4.4.29) f = t 0 Henceforth fix ε1 ∈ (0, 1]. Having also fixed an arbitrary background parameter N ∈ (0, ∞), to be specified later, it is convenient to further decompose each kernel Dt (x, y) as a weighted superposition of bump functions of the following sort: Dt (x, y) =

∞ 

=0

2−N ϕ2 t (x, y),

x, y ∈ Σ,

t > 0,

(4.4.30)

where each ϕ2 t (x, y) is an adjusted bump function in the variable x associated with the ball B(y, 2 t), which means that there exists some constant C ∈ (0, ∞) with the property that for every  ∈ N0 and t > 0 we have: supp ϕ2 t (·, y) ⊆ B(y, 2 t) for each y ∈ Σ;

1−n each x, y ∈ Σ; |ϕ2 t (x, y)| ≤ C(2 t) for ϕ2 t (·, y) .η ≤ C(2 t)1−n−η for each η ∈ (0, ε1 ] and each y ∈ Σ; C (Σ) ∫ (4) Σ ϕ2 t (x, y) dσ(x) = 0 for each y ∈ Σ.

(1) (2) (3)

Together, (4.4.29) and (4.4.30) permit us to write, for each f ∈ L po (Σ, σ) with po ∈ (1, ∞), ∫ ∞ ∫ ∫ ∞

dt dt = Dt (Dt f )(x) Dt (x, y)(Dt f )(y) dσ(y) f (x) = t t 0 0 Σ ∫ ∞  dt = (4.4.31) 2−N ϕ2 t (x, y)(Dt f )(y) dσ(y) , ∀x ∈ Σ. t Σ×(0,∞)

=0 In turn, formula (4.4.31) is the staring point in the quest of decomposing a given distribution f belonging to the Hardy space H p (Σ, σ) into atoms. To describe this in more details, we shall freely borrow notation and results from [88, pp. 1524-1525] and [185, pp. 347-350].

4.4 Atomic Decompositions for Hardy Spaces

149

As a preamble, we briefly review the characterization of Hardy spaces in terms of the Lusin area-function. Concretely, having fixed some background parameter κ > 0, define the Lusin area-function A κ F of a given (σ ⊗ L 1 )-measurable function F : Σ × (0, ∞) → R as ∫ dσ(y) dt 12 |F(y, t)| 2 , ∀x ∈ Σ, (4.4.32) (A κ F)(x) := tn Υκ (x) where   Υκ (x) := (y, t) ∈ Σ × (0, ∞) : |x − y| < κ t ,

∀x ∈ Σ.

(4.4.33)

In relation to this, first observe that A κ F : Σ −→ [0, ∞] is lower-semicontinuous.

(4.4.34)

To justify this, fix xo ∈ Σ arbitrary and consider a sequence {x j } j ∈N ⊆ Σ convergent as is easily seen by analyzing each of the cases (y, t) ∈ Υκ (xo ) and to xo . Then,  (y, t) ∈ Σ × (0, ∞) \ Υκ (xo ), we have lim inf 1Υκ (x j ) (y, t) ≥ 1Υκ (xo ) (y, t), j→∞

∀(y, t) ∈ Σ × (0, ∞).

(4.4.35)

Granted this, Fatou’s lemma permits us to estimate ∫ dσ(y) dt 12 |F(y, t)| 2 (A κ F)(xo ) = tn Υκ (x o ) ∫ dσ(y) dt 12 = 1Υκ (xo ) (y, t)|F(y, t)| 2 tn Σ×(0,∞) ∫ dσ(y) dt 12 ≤ lim inf 1Υκ (x j ) (y, t)|F(y, t)| 2 tn Σ×(0,∞) j→∞ ∫ dσ(y) dt 12 ≤ lim inf ≤ 1Υκ (x j ) (y, t)|F(y, t)| 2 j→∞ tn Σ×(0,∞) ∫ dσ(y) dt 12 = lim inf |F(y, t)| 2 = lim inf (A κ F)(x j ), (4.4.36) j→∞ j→∞ tn Υκ (x j ) finishing the proof of (4.4.34). Now, with the family {Dt }t>0 as before, here is the promised characterization of the scale of Hardy spaces on Σ (cf., e.g., [88]3):

3 Under the current assumptions, the Hardy spaces used in [88] coincide with those from Definition 4.2.1; see [88, Remarks 2.27,2.30] and Theorem 4.4.1

150

4 Hardy Spaces on Ahlfors Regular Sets

s if n−1 n < s < ∞ then a distribution g on Σ belongs to H (Σ, σ) if and r only if A κ G ∈ L (Σ, σ) where the function G : Σ × (0, ∞) → R is defined as G(x, t) := (Dt g)(x) for each x ∈ Σ and t ∈ (0, ∞); moreover, g H s (Σ,σ) ≈ A κ G L s (Σ,σ) .

(4.4.37)

Continuing the discussion about the Lusin area-function, the goal to show that there exists some constant C ∈ (0, ∞), depending only on Σ, κ, and n, such that for any two (σ ⊗ L 1 )-measurable functions H1, H2 : Σ × (0, ∞) → R we have ∫ ∫ dt |H1 (y, t) H2 (y, t)| dσ(y) ≤ C (A κ H1 )(x)(A κ H2 )(x) dσ(x). (4.4.38) t Σ×(0,∞) Σ First note that, thanks to (4.4.34), the integral in the right-hand side of (4.4.38) is meaningful. Next, the idea is to estimate the expression ∫ ∫ dt

(4.4.39) I := |H1 (y, t) H2 (y, t)| dσ(y) n dσ(x) t Σ Υκ (x) in two ways. On the one hand, using Fubini-Tonelli’s Theorem we may write ∫ ∫

dt I= |H1 (y, t)||H2 (y, t)| 1Υκ (x) (y, t) dσ(x) dσ(y) n t Σ×(0,∞) Σ ∫   dt = |H1 (y, t)||H2 (y, t)|σ B(y, κ t) ∩ Σ dσ(y) n t Σ×(0,∞) ∫ dt (4.4.40) ≈ |H1 (y, t)||H2 (y, t)| dσ(y) , t Σ×(0,∞)   since σ B(y, κ t) ∩ Σ ≈ t n−1 uniformly in y ∈ Σ and t > 0. On the other hand, based on Cauchy-Schwarz’ inequality we may estimate ∫ ∫ ∫ dt 1/2 dt 1/2 I≤ |H1 (y, t)| 2 dσ(y) n |H2 (y, t)| 2 dσ(y) n dσ(x) t t Σ Υκ (x) Υκ (x) ∫ (4.4.41) = (A κ H1 )(x)(A κ H2 )(x) dσ(x). Σ

Now, (4.4.38) follows from (4.4.40) and (4.4.41). We need to make yet another quick detour in order to address a relevant technical issue. Specifically, let M Σ be the Hardy-Littlewood maximal operator associated with the space of homogeneous type (Σ, | · − · |, σ) as in [133, Corollary 7.6.3], and 1 (Σ, σ). While we know from [133, (7.6.17)] that fix some arbitrary function h ∈ Lloc M Σ h : Σ → [0, +∞] is a σ-measurable function, as opposed to the Euclidean case, In particular, given some λ > 0, the level M Σh is not necessarily semicontinuous.  set x ∈ Σ : (M Σ h)(x) > λ , while σ-measurable, is not necessarily open. To rectify this, we propose to work with the following version of M Σ , defined for each σ-measurable function h : Σ → C as

4.4 Atomic Decompositions for Hardy Spaces

1 n−1 r r ∈(0,∞)

+Σ h(x) := sup M

151

∫ |h| dσ,

∀x ∈ Σ.

(4.4.42)

B(x,r)∩Σ

In relation to this, we claim that +Σ h : Σ −→ [0, +∞] is a lower-semicontinuous function M 1 (Σ, σ). for every h ∈ Lloc

(4.4.43)

1 (Σ, σ) along with some λ > 0, the goal being to show To prove (4.4.43), fix h ∈ Lloc that   +Σ h)(x) > λ is relatively open in Σ. (4.4.44) the set x ∈ Σ : (M

With this finality in mind, pick a point xo belonging to the above set and note that, in light of (4.4.42), this entails the existence of some r > 0 with the property that ∫ A := |h| dσ > λ r n−1 . (4.4.45) B(x o ,r)∩Σ

  Granted this, it becomes possible to choose a number s ∈ r, (A/λ)1/(n−1) . Such a choice then forces both s > r and A > λ s n−1 . Consider now an arbitrary point y ∈ B(xo, s − r) ∩ Σ. Then simple geometry gives B(xo, r) ∩ Σ ⊆ B(y, s) ∩ Σ, which then permits us to estimate ∫ ∫ |h| dσ ≥ |h| dσ = A > λ s n−1 . (4.4.46) B(y,s)∩Σ

B(x o ,r)∩Σ

+Σ h(y) > λ for every Having established this, we conclude from (4.4.42) that M   y ∈ B(xo, s − r) ∩ Σ. Thus, B(xo, s − r) ∩ Σ ⊆ x ∈ Σ : (M Σ h)(x) > λ which ultimately proves (4.4.44). Thus, the claim made in (4.4.43) is indeed true. In addition, thanks to the fact that we are currently assuming that Σ is Ahlfors regular, it follows that for each σ-measurable function h : Σ → C the maximal +Σ h from (4.4.42) is pointwise comparable with the standard Hardyfunction M Littlewood maximal function M Σ h defined as in (A.0.71). More specifically, if 0 < co ≤ Co < +∞ are the lower and upper Ahlfors regularity constants for Σ, then for each σ-measurable function h : Σ → C we have +Σ h ≤ Co · M Σ h everywhere on Σ. co · M Σ h ≤ M

(4.4.47)

As a consequence of this and [133, (7.6.18)-(7.6.19)], we have that +Σ : L p (Σ, σ) −→ L p (Σ, σ) is well defined, M sub-linear and bounded for each p ∈ (1, ∞], and, corresponding to the case p = 1, the modified maximal operator

(4.4.48)

152

4 Hardy Spaces on Ahlfors Regular Sets

+Σ : L 1 (Σ, σ) −→ L 1,∞ (Σ, σ) is well defined, sub-linear and bounded. (4.4.49) M Let us also note that, thanks to (4.4.47) and [133, Corollary 7.6.5] (whose applicability is ensured by [133, (3.6.26)] in [133, Lemma 3.6.4]), +Σ h at for each σ-measurable function h : Σ → C we have co · h ≤ M σ-a.e. point on Σ, and this inequality actually holds everywhere if h is the characteristic function of a relatively open subset of Σ.

(4.4.50)

Returning to the main topic of discussion, assume next that f ∈ H p (Σ, σ) with < p ≤ 1 and consider

n−1 n

the function F : Σ×(0, ∞) −→ R defined as F(x, t) := (Dt f )(x) for each x ∈ Σ and t ∈ (0, ∞).

(4.4.51)

Then (4.4.37) implies that for each κ > 0 we have A κ F ∈ L p (Σ, σ). To proceed, for each k ∈ Z define   Ek := x ∈ Σ : (A1 F)(x) > 2k ,

(4.4.52)

(4.4.53)

then, with co ∈ (0, ∞) denoting the lower Ahlfors regularity constant of Σ, introduce   +Σ 1E )(x) > co /2n , (4.4.54) Ok := x ∈ Σ : (M k +Σ stands for the modified Hardy-Littlewood maximal operator defined in where M (4.4.42). Note that (4.4.34) ensures that for each k ∈ Z the set Ek is relatively open (in Σ), while Chebytcheff’s inequality and (4.4.52) allow us to estimate σ(Ek ) ≤ 2−k p A1 F  L p (Σ,σ) < +∞. p

(4.4.55)

In particular, 1Ek ∈ L 1 (Σ, σ) hence, thanks to (4.4.44) and (4.4.49), Ok is relatively open in Σ, and σ(Ok ) ≤ C · σ(Ek ) < +∞,

(4.4.56)

for some C ∈ (0, ∞) independent of k. In particular, σ(Ok ) < +∞ implies, since we are presently assuming that Σ is unbounded (hence σ(Σ) = +∞), Ok is a proper subset of Σ, for each k ∈ Z.

(4.4.57)

Parenthetically, we note that since (4.4.50) implies that for each k ∈ Z we have +Σ 1E )(x) ≥ co for each point x ∈ Ek , we also have Ek ⊆ Ok for each k ∈ Z. (M k Going forward, let us now define the “stopping time” function φ : Σ × (0, ∞) → Z by setting for each (y, t) ∈ Σ × (0, ∞)

4.4 Atomic Decompositions for Hardy Spaces

φ(y, t) to be the largest  integer  k ∈ Z with  the property that σ B(y, t) ∩ Ek > σ B(y, t) ∩ Σ /2.

153

(4.4.58)

Since Ek decreases from Σ to a σ-nullset as k increases from −∞ to +∞, the function φ is well defined. Moreover, φ enjoys the following properties:    (a) the sets φ−1 ({k}) k ∈Z are pair-wise disjoint and k ∈Z φ−1 ({k}) = Σ × (0, ∞); (b) if (y, t) ∈ φ−1 ({k}) then B(y,  t) ∩ Σ ⊆ Ok ;    (c) if (y, t) ∈ φ−1 ({k}) then σ B(y, t) ∩ (Σ \ Ek+1 ) ≥ σ B(y, t) ∩ Σ /2. Indeed, (a) is obvious. To prove (b), pick some (y, t) ∈ φ−1 ({k}) and for each z ∈ B(y, t) ∩ Σ write   ∫ σ B(z, 2t) ∩ E 1 k +Σ 1E )(z) ≥ 1E dσ = (M k (2t)n−1 B(z,2t)∩Σ k (2t)n−1   co co σ B(y, t) ∩ Ek  > n. ≥ n−1 ·  (4.4.59) 2 2 σ B(y, t) ∩ Σ The first inequality in (4.4.59) is clear from (4.4.42) while the subsequent equality is self-evident. The second inequality in (4.4.59) follows by combining the inclusion B(y, t) ⊆ B(z, 2t) for each z ∈ B(y, t) with the fact that co is the lower Ahlfors regularity constant of Σ. The last inequality in (4.4.59) is implied by (4.4.58). In view of (4.4.54), the estimate in (4.4.59) ultimately proves that B(y, t) ∩ Σ ⊆ Ok . Hence, the claim made in (b) is justified. Finally, as far as the in (c) is  estimate   concerned, observe that this is equivalent to σ B(y, t) ∩ Ek+1 ) ≤ σ B(y, t) ∩ Σ /2 which is valid thanks to the manner in which the number k = φ(y, t) has been defined in (4.4.58). Moving on, fix a sufficiently large geometric constant λ ∈ (0, ∞). Given the properties of Ok noted in (4.4.56)-(4.4.57) (and keeping in mind that Σ is Ahlfors regular), for each fixed k ∈ Z for which Ok is not empty we may invoke [133, Proposition 7.5.6] to guarantee the existence of some number Λ ∈ (0, ∞) independent of k along with a sequence of dyadic cubes {Q j,k } j ∈N (relative to a dyadic grit D(Σ) associated with the space of homogeneous type (Σ, | · − · |, σ) as in [133, Proposition 7.5.4]) with the property that the cubes {Q j,k } j ∈N are mutually disjoint, contained  Q j,k = 0, in Ok , and satisfy σ Ok \

(4.4.60)

    B x j,k , λ (Q j,k ) ∩ Σ ⊆ Ok and dist Q j,k , Σ \ Ok ≤ Λ · (Q j,k ) for each j ∈ N, where x j,k and (Q j,k ) denote, respectively, the center and side-length of the dyadic cube Q j,k .

(4.4.61)

j ∈N

and

For each k ∈ Z and j ∈ N we then define

154

4 Hardy Spaces on Ahlfors Regular Sets

 Tj,k := (y, t) ∈ φ−1 ({k}) : y ∈ Q j,k ⊆ Σ × (0, ∞), 

and observe that   Tj,k j ∈N, k ∈Z is a family of pairwise disjoint sets, satisfying , , Tj,k = φ−1 ({k}) for each k ∈ Z, and Tj,k = Σ × (0, ∞), j ∈N

(4.4.62)

(4.4.63)

j ∈N, k ∈Z

up to a null-set for the product measure σ ⊗ L 1 . Indeed, that the sets Tj,k ’s are mutually disjoint is clear from (4.4.62) upon recalling that the cubes {Q j,k } j ∈N are mutually disjoint. The first equality in (4.4.63) is a consequence of (4.4.62), property (b) of the stopping time function φ, and (4.4.60). Finally, the last equality in (4.4.63) is implied by the first equality in (4.4.63) together with property (a) of the stopping time function φ. In the context of (4.4.31), the fact that % & ,, Tj,k = 0 (4.4.64) (σ ⊗ L 1 ) Σ × (0, ∞) \ j ∈N k ∈Z

suggests that we attempt to express the given distribution f as the series ∞  

2−N

∫ Tj, k

=0 j ∈N k ∈Z

ϕ2 t (·, y)(Dt f )(y) dσ(y)

dt . t

In turn, this bring into focus the family of building blocks ∫ dt

a j,k (x) := ϕ2 t (x, y)(Dt f )(y) dσ(y) , ∀x ∈ Σ, t Tj, k

(4.4.65)

(4.4.66)

indexed by  ∈ N0 , k ∈ Z, and j ∈ N. Note that each function a j,k satisfies ∫ Σ

a j,k (x) dσ(x)

∫ = Tj, k

∫

dt ϕ2 t (x, y) dσ(x) (Dt f )(y) dσ(y) = 0, t Σ

(4.4.67)

thanks to property (4) for the adjusted bump function ϕ2 t . To study the supports of a j,k . these building blocks, fix  ∈ N0 , k ∈ Z, and j ∈ N and pick some x ∈ supp Property (1) of ϕ2 t then forces that on the domain of integration in (4.4.66) we have |x − y| ≤ 2 t.

(4.4.68)

In addition, from (4.4.62) we see that if (y, t) ∈ Tj,k then y ∈ Q j,k and (y, t) ∈ φ−1 ({k}).

(4.4.69)

In view of property (b) of the stopping time function φ, the latter membership further guarantees that B(y, t) ∩ Σ ⊆ Ok hence, on the one hand,

4.4 Atomic Decompositions for Hardy Spaces

155

  dist y, Σ \ Ok ≥ t.

(4.4.70)

On the other hand, based on triangle inequality and (4.4.61) we may bound, bearing in mind that y ∈ Q j,k ,     dist y, Σ \ Ok ≤ dist Q j,k , Σ \ Ok + diam(Q j,k ) ≤ C · (Q j,k ), (4.4.71) for some purely geometric constant C ∈ (0, ∞), independent of j, k. Collectively, (4.4.70) and (4.4.71) then give t ≤ C · (Q j,k )

(4.4.72)

which then allows us to estimate (with x, y, t as above; recall that x j,k stands for the center of Q j,k and that  ∈ N0 ) |x − x j,k | ≤ |x − y| + |y − x j,k | ≤ 2 t + diam(Q j,k ) ≤ C · 2 · diam(Q j,k ). (4.4.73) All in all, this establishes that, for some purely geometric constant C > 0, independent of  and j, k,   (4.4.74) supp a j,k ⊆ Δ j,k := B x j,k , C · 2 · diam Q j,k ∩ Σ. Thus, when appropriately normalized, the function a j,k becomes a genuine atom on Σ. Seeking a normalization in L r (Σ, σ), we proceed by duality, following the approach in [185, p. 349]. Specifically, let r ∈ (1, ∞) be the Hölder conjugate exponent of r and pick some g ∈ L r (Σ, σ). Define ∫ G (y, t) := ϕ2 t (x, y)g(x) dσ(x) for y ∈ Σ and t > 0, (4.4.75) Σ

and pick some κ > 0. We claim that for each fixed parameter υ > 0 there exists a finite constant C = Cυ > 0, independent of g, such that A κ G  L r (Σ,σ) ≤ C · 2 υ g L r (Σ,σ) for every  ∈ N0 .

(4.4.76)

The main ingredient in the proof of (4.4.76) is the L p -square function estimate from [95]. To establish a dictionary between the current setting and that used in [95, Theorem 1.1, pp. 6-7], we first observe that if for each  ∈ N0 we introduce   θ (y, t), x := t −υ ϕ2 t (x, y), x, y ∈ Σ, t > 0, (4.4.77) and

∫ (Θ g)(y, t) :=

Σ

  θ (y, t), x g(x) dσ(x),

y ∈ Σ, t > 0,

(4.4.78)

156

4 Hardy Spaces on Ahlfors Regular Sets

then for each  ∈ N0 and x ∈ Σ we have (on account of (4.4.77)-(4.4.78), (4.4.102), and (4.4.32)-(4.4.33)) %

∫

(A κ G )(x) =

  (Θ g)(y, t)2

(y,t)∈Σ×(0,∞) |(x,0)−(y,t) | 0,

(4.4.80)

and note that (4.4.77) and properties (1)-(2) of the adjusted bump function ϕ2 t (x, y) readily imply the estimate      θ (y, t), x  = t −υ ϕ2 t (x, y) ≤

C · 2 υ , |(x, 0) − (y, t)| n−1+υ

∀x, y ∈ Σ, ∀t > 0,

(4.4.81)

for some constant C ∈ (0, ∞) independent of . Finally, from (4.4.77) and property (3) of the adjusted bump function ϕ2 t (x, y) we may easily conclude that once some α ∈ (0, ε] has been fixed then there exists a constant C ∈ (0, ∞) independent of  such that      θ (y, t), x − θ (y, t), x ≤

C · 2 υ |x − x |α |(x, 0) − (y, t)| n−1+α+υ

(4.4.82)

for all x, x, y ∈ Σ and t > 0 satisfying |x − x | ≤ 12 |(x, 0) − (y, t)|. Granted the identification in (4.4.79) and the cancellation, size, and regularity properties established in (4.4.80)-(4.4.82), we may invoke the equivalence (2) ⇔ (12) from [95, Theorem 1.1, pp. 6-7] (with X := Σ × [0, ∞), E := Σ × {0}, d := n − 1, μ := σ ⊗ L 1 , m := n, and r in place of p) in order to conclude that (4.4.76) holds. Pushing forth in our quest of establishing suitable bounds for the L r -norms of the building blocks introduced in (4.4.66), with the function F as in (4.4.51) and the function G as in (4.4.75), for each , j, k fixed we may estimate (with constants

4.4 Atomic Decompositions for Hardy Spaces

157

independent of , j, k and the function g) ∫    a j,k (x)g(x) dσ(x)  Σ

∫  = ∫



Σ×(0,∞)

(4.4.83)

 dt  1Tj, k · F (y, t)G (y, t) dσ(y)  t



 dt 1Tj, k (y, t)|F(y, t)||G (y, t)| dσ(y) t Σ×(0,∞)   ∫   σ B(y, t) ∩ (Σ \ Ek+1 )   × 1Tj, k (y, t) ≤2 σ B(y, t) ∩ Σ Σ×(0,∞)

≤

× |F(y, t)||G (y, t)| dσ(y) ∫ ≤C

Σ×(0,∞)



%∫



1Tj, k (y, t)

Σ\Ek+1





&

1B(y,t)∩Σ (x) dσ(x) × × |F(y, t)||G (y, t)| dσ(y)

∫ ≤C

Σ\Ek+1

%∫ Σ×(0,∞)



dt t

dt tn

   1Tj, k (y, t) 1B(y,t)∩Σ (x)|F(y, t)|× & dt × |G (y, t)| dσ(y) n dσ(x). t

The first equality in (4.4.83) is based on Fubini’s Theorem, and the subsequent inequality is clear. The second inequality relies on property (b) for the stopping time (y, t) ∈ Tj,k (cf. function φ and the fact that we have (y, t) ∈ φ−1 ({k}) whenever  (4.4.62)). The third equality is justified by observing that σ B(y, t) ∩ Σ ≈ t n−1 and   that σ B(y, t) ∩ (Σ \ Ek+1 ) may be recast as the inner integral appearing in the penultimate line of (4.4.83). Finally, the last inequality above is Fubini’s Theorem. Pressing on, observe that there exists a purely geometric constant C ∈ (0, ∞) (in particular, independent of j, k, ) such that for each x, y ∈ Σ and t > 0 we have         (4.4.84) 1Tj, k (y, t) 1B(y,t)∩Σ (x) ≤ 1C ·Q j, k (x) 1B(x,t)∩Σ (y). To justify this inequality it suffices to prove that whenever the left-hand side is nonzero then the right-hand side is nonzero as well. To this end, arguing as in (4.4.68)-(4.4.73) with  = 0 we obtain that whenever x, y ∈ Σ and t ∈ (0, ∞) are such that the left-hand side of (4.4.84) is nonzero then necessarily |x − y| < t and |x − x j,k | ≤ C · diam(Q j,k ). These place x in C · Q j,k and y in B(x, t) ∩ Σ which, in turn, make the right-hand side of (4.4.84) equal to one. Thus, (4.4.84) holds and,

158

4 Hardy Spaces on Ahlfors Regular Sets

when used back in (4.4.83), it permits us to further estimate ∫    a j,k (x)g(x) dσ(x)  Σ

%∫

∫ ≤C

Σ\Ek+1



Σ×(0,∞)

(4.4.85)

   1C ·Q j, k (x) 1B(x,t)∩Σ (y)× & dt × |F(y, t)||G (y, t)| dσ(y) n dσ(x) t

%∫

∫ ≤C

& dt |F(y, t)||G (y, t)| dσ(y) n dσ(x) t Υ1 (x)

C ·Q j, k ∩(Σ\Ek+1 )

%∫

∫ ≤C

dt |F(y, t)| 2 dσ(y) n t Υ1 (x)

C ·Q j, k ∩(Σ\Ek+1 )

& 1/2 ×

%∫ × ∫ =C ≤C



C ·Q j, k ∩(Σ\Ek+1 )

max

x ∈Σ\Ek+1



dt |G (y, t)| dσ(y) n t Υ1 (x) 2

& 1/2 dσ(x)

  A1 F (x) A1 G )(x) dσ(x)



A1 F (x)

∫ C ·Q j, k

A1 G dσ

   1/r  ≤ C 2k+1 A1 G  L r (Σ,σ) σ C · Q j,k  1/r  ≤ C · 2k 2 υ σ 2− · Δ j,k g L r (Σ,σ)  1/r  ≤ C · 2k 2 υ 2− (n−1)/r σ Δ j,k g L r (Σ,σ) . Above, the first inequality is obtained by combining  with (4.4.84). The  (4.4.83) second inequality is based on the observation that 1B(x,t)∩Σ (y)  0 if and only if (y, t) ∈ Υ1 (x) (cf. (4.4.33)). The third inequality use the Cauchy-Schwarz inequality, while the subsequent equality is seen from (4.4.32). The fourth inequality is selfevident, while the next one uses (4.4.53) and Hölder’s inequality. Finally, the first inequality in the last line of (4.4.85) is implied by (4.4.76) and the definition of Δ j,k in (4.4.74), while the very last inequality is a consequence of the Ahlfors regularity of Σ. Having proved (4.4.85) for arbitrary functions g ∈ L r (Σ, σ), from Riesz’ Duality Theorem we may now conclude that

4.4 Atomic Decompositions for Hardy Spaces

   a 

j,k L r (Σ,σ)

159

 1/r  ≤ C 2k 2 υ 2− (n−1)/r σ Δ j,k .

(4.4.86)

Consequently, for each  ∈ N0 , j ∈ N, and k ∈ Z, the function a j,k :=

a j,k λ j,k

is a (p, r)-atom on Σ

(4.4.87)

if we take   1/p λ j,k := C · 2k 2 υ 2− (n−1)/r σ Δ j,k .

(4.4.88)

At this stage, assuming that the parameter N has been selected so that N > υ + (n − 1)(1 − p/r)

(4.4.89)

to begin with, we may estimate 

∞  

=0 j ∈N k ∈Z

 =C



- 1/p



p 2−N p λ j,k 

∞  

−N p k p

2

2

υp

2

− (n−1)p/r

2

σ



=0 j ∈N k ∈Z

 ≤C

∞  

−N p k p

2

2

υp

2

(n−1)(1−p/r)

2

 Δ j,k

- 1/p



σ Q j,k



- 1/p

=0 j ∈N k ∈Z

 ≤C



2

kp



σ Q j,k

j ∈N k ∈Z

 ≤C







- 1/p ≤C



2

kp



σ Ok



- 1/p

k ∈Z

- 1/p 2

kp

σ(Ek )

≤ CA1 F  L p (Σ,σ)

k ∈Z

≤ C f  H p (Σ,σ),

(4.4.90)

where, in the first inequality we have used the definition of Δ j,k from (4.4.74) and the Ahlfors regularity of Σ, in the second inequality we have employed (4.4.89), in the third inequality we have invoked (4.4.60), in the fourth inequality we have recalled the estimate in (4.4.56), in the fifth inequality we have recalled a well-known formula in real variable analysis estimating weighted sums of level sets of a given function in terms of its Lebesgue quasi-norm, and in the last inequality we have appealed to (4.4.37).   belongs to the In particular (4.4.90) shows that the sequence 2−N λ j,k j,k,

space  p . If we now re-index said sequence simply as {λ j } j ∈N , and also re-index the

160

4 Hardy Spaces on Ahlfors Regular Sets

family of (p, r)-atoms from (4.4.87) simply as {a j } j ∈N , then from [88, Lemma 2.23, p. 1523], (4.4.64), (4.4.66), (4.4.87), and (4.4.90), we conclude that f =

∞ 

λ j a j in H p (Σ, σ) and

j=1

∞ 

|λ j | p

1/p

≤ C f  H p (Σ,σ) .

(4.4.91)

j=1

In relation to (4.4.91), let us remark that, thanks to our earlier identifications and (4.4.87), ! the partial sums of the series ∞ λ j a j are the same as !∞j=1! ! (4.4.92) −N the partial sums of the series =0 a j,k j ∈N k ∈Z 2 and that, as seen from (4.4.66), all associated with the series !∞ the ! partial ! sums −N

are independent of the ex2 a j ∈N k ∈Z

=0 j,k ponent p (labeling the Hardy space in which the distribution f originates).

(4.4.93)

Keeping (4.4.92)-(4.4.93) in mind, it follows that if f ∈ H p (Σ, σ) ∩ H q (Σ, σ) with n−1 n < q ≤ 1, then running the same argument that has led to (4.4.91) we also obtain that, with the same {λ j } j ∈N and {a j } j ∈N constructed as before (in relation to f ∈ H p (Σ, σ)), ∞  f = λ j a j in H q (Σ, σ). (4.4.94) j=1

  The claim in (4.4.21) is therefore established in the case when q ∈ n−1 n ,1 . Henceforth, consider the case 1 < q < ∞, i.e., assume f ∈ H p (Σ, σ) ∩ L q (Σ, σ) (see [133, (3.6.27)] in this regard). The goal is to show that, with {λ j } j ∈N and {a j } j ∈N constructed as before we also have the the series !∞ sequence of partial sums of q (Σ, σ). λ a converges to f in L j j j=1

(4.4.95)

In light of (4.4.92), this is further equivalent to proving that the sequence associated with the !∞ !of partial ! sums −N series =0 2 a j,k is convergent j ∈N k ∈Z to the function f in L q (Σ, σ).

(4.4.96)

Justifying the claim in (4.4.96) requires some preparation. For starters, let { fm }m∈N be the sequence of partial sums referred to in (4.4.96). Concretely, in view of (4.4.66) for each m ∈ N we may express fm (x) =

m 

=0

2−N

∫ Om

ϕ2 t (x, y)(Dt f )(y) dσ(y)

dt , t

∀x ∈ Σ,

(4.4.97)

4.4 Atomic Decompositions for Hardy Spaces

where Om :=

161

m , m ,

Tj,k .

(4.4.98)

j=1 k=−m

Observe that {Om }m∈N is a nested increasing sequence of sets which, up to a (σ⊗L 1 )nullset, exhaust Σ × (0, ∞) (cf. (4.4.64)). In this notation, the claim in (4.4.96) may be simply restated as (4.4.99) lim fm = f in L q (Σ, σ). m→∞

To this end, fix some κ > 0 and recall the function F from (4.4.51). Granted the assumptions on f , Proposition 4.2.2 gives (bearing in mind that presently we are assuming p ≤ 1 < q) that f also belongs to the Hardy space H 1 (Σ, σ). With this membership in hand, from (4.4.92)-(4.4.93) and (4.4.91) (used now with p = 1) we conclude that lim fm = f in H 1 (Σ, σ). This further implies lim fm = f in m→∞

m→∞

L 1 (Σ, σ), hence also (by eventually restricting the index m to a sub-sequence of N) lim fm (x) = f (x)

m→∞

for σ-a.e. x ∈ Σ.

(4.4.100)

Having noted (4.4.100), the claim in (4.4.99) follows as soon as we show that { fm }m∈N is a Cauchy sequence in L q (Σ, σ). With this goal in mind, for each m, k ∈ N and each x ∈ Σ write ∫ m  dt 2−N ϕ2 t (x, y)(Dt f )(y) dσ(y) fm+k (x) − fm (x) = t Om+k \Om

=0 +

m+k 

−N

2

∫ Om+k

=m+1

ϕ2 t (x, y)(Dt f )(y) dσ(y)

dt . t

(4.4.101)



Also, fix an arbitrary g ∈ L q (Σ, σ) where q is such that 1/q + 1/q = 1, and for each  ∈ N0 and m ∈ N define ∫ ϕ2 t (x, y)g(x) dσ(x) for every y ∈ Σ and t > 0, G (y, t) := (4.4.102) Σ and Fm := 1 Om · F (with the function F as in (4.4.51)). In this notation, for each m, k ∈ N we may write ∫ ( fm+k − fm )(x)g(x) dσ(x) Σ

=

m 

2−N

∫

  dt G (y, t) Fm+k (y, t) − Fm (y, t) dσ(y) t Σ×(0,∞)

=0

+

m+k 

=m+1

(4.4.103)

−N

2

∫ Σ×(0,∞)

G (y, t)Fm+k (y, t) dσ(y)

dt . t

162

4 Hardy Spaces on Ahlfors Regular Sets

Returning to the mainstream discussion, starting with (4.4.103) and then employing (4.4.38) and Hölder’s inequality, for each m, k ∈ N we may estimate ∫    (4.4.104)  ( fm+k − fm )(x)g(x) dσ(x) Σ

≤C

m 

=0

+C

2−N A κ (Fm+k − Fm ) L q (Σ,σ) A κ G  L q (Σ,σ)

m+k 

=m+1

2−N A κ Fm+k  L q (Σ,σ) A κ G  L q (Σ,σ) .

To proceed, we recall from (4.4.76) (presently used with p in place of r ) that for each fixed parameter υ > 0 there exists a finite constant C = Cυ > 0, independent of g, such that A κ G  L p (Σ,σ) ≤ C · 2 υ g L p (Σ,σ) for every  ∈ N0 .

(4.4.105)

In addition to this basic estimate, we shall also need that 0 ≤ A κ Fm ≤ A κ F on Σ for each m ∈ N,

(4.4.106)

A κ (F − Fm ) → 0 in L q (Σ, σ) as m → ∞.

(4.4.107)

Accepting the properties recorded in (4.4.106)-(4.4.107) for the time being, we may readily conclude the proof of (4.4.99). Indeed, from (4.4.104) and (4.4.105) we see that for each m, k ∈ N we have

4.4 Atomic Decompositions for Hardy Spaces

 fm+k − fm  L q (Σ,σ) =

sup



g ∈L q (Σ,σ), g  L q (Σ, σ) ≤1

≤C

m 

163

∫     ( fm+k − fm )(x)g(x) dσ(x) Σ

2−(N −υ) A κ (Fm+k − Fm ) L q (Σ,σ)

=0

+C

m+k 

2−(N −υ) A κ Fm+k  L q (Σ,σ)

=m+1

≤C

m 

2−(N −υ) A κ (Fm+k − F) L q (Σ,σ)

=0

+C

m 

2−(N −υ) A κ (Fm − F) L q (Σ,σ)

=0

+C

m+k 

2−(N −υ) A κ Fm+k  L q (Σ,σ) .

(4.4.108)

=m+1

Assume that N > υ to begin with. Then from (4.4.108), (4.4.106)-(4.4.107), and (4.4.52) we may conclude that  fm+k − fm  L q (Σ,σ) can be made as small as desired, uniformly in k ∈ N, by taking m sufficiently large. Thus, the sequence { fm }m∈N is indeed Cauchy in L q (Σ, σ). As remarked earlier, this finishes the proof of (4.4.99), modulo the justification of (4.4.76)-(4.4.107). Moving on, (4.4.106) is clear from (4.4.32) and (4.4.102). Finally, to prove (4.4.107), make use of (4.4.52) to conclude that there exists a σ-measurable set E ⊆ Σ satisfying σ(E) = 0 and (A κ F)(x) < +∞ for each x ∈ Σ \ E.

(4.4.109)

For each fixed x ∈ Σ \ E it follows that for every m ∈ N we have ∫   dt 1/2 A κ (F − Fm ) (x) = 1(Σ×(0,∞))\Om (y, t)|F(y, t)| 2 dσ(y) n , t Υκ (x)

(4.4.110)

and the fact that (A κ F)(x) < +∞ implies that 

  dt

1(Σ×(0,∞))\Om |F | Υκ (x) ≤ |F | Υκ (x) ∈ L 2 Υκ (x), σ ⊗ n , t

∀m ∈ N.

(4.4.111)

Since Om  Σ × (0, ∞) as m → ∞ (up to a (σ ⊗ L 1 )-nullset; cf. (4.4.98) and (4.4.64)), it follows that 1(Σ×(0,∞))\Om |F | converges pointwise to zero as m → ∞. As such, Lebesgue’s Dominated Convergence Theorem applies (in the ambient Υκ (x) equipped with the measure σ ⊗ tdtn ) and, in light of (4.4.110), gives that

164

4 Hardy Spaces on Ahlfors Regular Sets



 A κ (F − Fm ) (x) → 0 as m → ∞ for each x ∈ Σ \ E.

(4.4.112)

With this in hand, one more application of Lebesgue’s Dominated Convergence Theorem (bearing in mind (4.4.52), (4.4.106), and the fact that σ(E) = 0) proves (4.4.107). This finishes the proof of the theorem in the case when Σ is unbounded. Finally, we note that the main ingredients in used in the proof so far may be adapted to the case when Σ is bounded. For example, a discrete Calderón-type reproducing formula valid in arbitrary spaces of homogeneous type (bounded and unbounded) may be found in [95, Proposition 2.14, p. 25]. As such, the same type of argument applies with minor natural adjustments to yield similar conclusions in such a scenario. This completes the proof of Theorem 4.4.3.  As illustrated by the next proposition, the space of finite linear combinations of atoms has excellent approximation qualities vis-a-vis to the entire Hardy space. Proposition 4.4.4 Fix a closed set Σ ⊆ Rn which is Ahlfors regular and abbreviate  n−1 p,q n−1 σ := H Σ. Consider p ∈ n , 1 and q ∈ [1, ∞] with q > p, and let Hfin (Σ, σ) stand for the vector space consisting of all finite linear combinations of (p, q)-atoms p,q p,q on Σ. Also, define a quasi-norm on Hfin (Σ, σ) by setting, for each f ∈ Hfin (Σ, σ),   f  H p, q (Σ,σ) := inf fin

N 

|λ j | p

1/p

: N ∈ N and f =

j=1

N 

λ j a j for some

j=1

-

numbers {λ j }1≤ j ≤ N ⊆ C and some (p, q)-atoms {a j }1≤ j ≤ N . (4.4.113) Then   ∫ p,q q Hfin (Σ, σ) = f ∈ Lcomp (Σ, σ) : Σ f dσ = 0 if Σ is unbounded, p,q

Hfin (Σ, σ) = L q (Σ, σ) if Σ is bounded and, in all cases,

(4.4.114)

p,q

Hfin (Σ, σ) is a dense linear subspace of H p (Σ, σ), and under the additional assumption that q < ∞, it follows that  ·  H p, q (Σ,σ) and  ·  H p (Σ,σ) are equivalent quasi-norms on fin p,q the space Hfin (Σ, σ).

(4.4.115)

Proof The two equalities in (4.4.114) are clear from definitions, while the claim in the last line is a consequence of Theorem 4.4.1. Finally, (4.4.115) is implied by [71, Theorem 5.6, p. 2276].  The issue of the separability of the Lorentz-based Hardy spaces is discussed below.

4.4 Atomic Decompositions for Hardy Spaces

165

Proposition 4.4.5 Let Σ ⊆ Rn be a closed  set which is Ahlfors regular and define σ := H n−1 Σ. Then, for each p ∈ n−1 , ∞ and q ∈ (0, ∞), the Lorentz-based Hardy n space H p,q (Σ, σ) is a separable quasi-Banach space.   Proof Consider first the case when we have p ∈ n−1 n , 1 . In such a scenario, pick  n−1  r ∈ (1, ∞) then select p0 ∈ n , p and p1 ∈ (1, r). In relation to the exponent r, bring back the set F from the proof of Proposition 4.4.2 (cf. (4.4.15)) and observe that, according to what has been established there,   F ⊆ H s (Σ, σ) for each s ∈ n−1 (4.4.116) n ,r . In particular, F ⊆ H p0 (Σ, σ) ∩ H p1 (Σ, σ) hence also F ⊆ H p,q (Σ, σ) (cf. (4.3.145)). We now claim that if the quasi-norm  ·  H p0 (Σ,σ)∩H p1 (Σ,σ) is defined as in (1.3.3), then for every (p0, r)-atom a on Σ, every λ ∈ R, and every ε > 0, there exists some f ∈ F such that λ a − f  H p0 (Σ,σ)∩H p1 (Σ,σ) < ε.

(4.4.117)

Recycling notation originally introduced in the proof of Proposition 4.4.2, this may be justified along the lines of (4.4.17), by choosing f ∈ Fj0 such that in place of (4.4.18) we now have   λ a − f  L r (Δ j0 ,σ) < ε · min σ(Δ j0 )1/r−1/p0 , σ(Δ j0 )1/r−1/p1 . (4.4.118)

j ⊆ F . Also, Then, with f denoting the extension of f by zero to Σ, we have f ∈F 0 the function g := λ a − f satisfies ∫ g dσ = 0, g L r (Σ,σ) < ε · σ(Δ j0 )1/r−1/p0 . (4.4.119) supp g ⊆ Δ j0 , Δ j0

Since these properties imply that ε −1 g is a (p0, r)-atom on Σ, it follows that ε −1 g H p0 (Σ,σ) ≤ C for some constant C ∈ (0, ∞) which depends only on Σ, p0 , and r. Thus, λ a − f  H p0 (Σ,σ) < Cε. Also, based on [133, (3.6.27)], (4.4.118), and Hölder’s inequality we may write λ a − f  H p1 (Σ,σ) ≈ λ a − f  L p1 (Σ,σ) = λ a − f  L p1 (Δ j0 ,σ) ≤ λ a − f  L r (Δ j0 ,σ) · σ(Δ j0 )1/p1 −1/r < ε · σ(Δ j0 )1/r−1/p1 · σ(Δ j0 )1/p1 −1/r = ε.

(4.4.120)

Hence, ultimately, λ a − f  H p0 (Σ,σ)∩H p1 (Σ,σ) < Cε which, after re-adjusting ε, completes the proof of the claim made in (4.4.117). In turn, from (4.4.117), (4.4.22), and the fact that F is stable under addition (cf. (4.4.15)) we deduce that

166

4 Hardy Spaces on Ahlfors Regular Sets

F is dense in H p0 (Σ, σ) ∩ H p1 (Σ, σ),  ·  H p0 (Σ,σ)∩H p1 (Σ,σ) .

(4.4.121)

Since, in turn, H p0 (Σ, σ) ∩ H p1 (Σ, σ) embeds continuously and densely into the space H p,q (Σ, σ) (as noted in (4.3.145)), we ultimately conclude that F is dense in H p,q (Σ, σ).

(4.4.122)

Given that F is also known to be a countable set (cf. (4.4.15)), that the   we conclude quasi-Banach space H p,q (Σ, σ) is indeed separable if p ∈ n−1 , 1 . Finally, when n p ∈ (1, ∞) the same conclusion follows from (4.2.30) and [133, Proposition 6.2.7]. This completes the proof of Proposition 4.4.5.  Moving on, assume Σ ⊆ Rn is a closed Ahlfors regular set, and abbreviate σ := H n−1 Σ. The goal now is to characterize the Hardy space H p (Σ, σ) for p ∈ n−1 n , 1 in terms of ions. Following [143, § A], these are defined as follows. Pick α ∈ (0, 1). We say that a function f ∈ L∞ (Σ, σ) is an ion, or an (α, p)-ion,  provided there exist x0 ∈ Σ and r ∈ 0, 2 diam Σ such that ∫    −(n−1)/p supp f ⊆ B(x0, r) ∩ Σ,  f  L ∞ (Σ,σ) ≤ r ,  f dσ  ≤ r α . (4.4.123) Σ

(or an α-charge) provided there exist a Also, call a function h ∈ L ∞ (Σ, σ) a charge  point x0 ∈ Σ along with a radius r ∈ 0, 2 diam Σ such that supp h ⊆ B(x0, r) ∩ Σ and h L ∞ (Σ,σ) ≤ r α−(n−1) .

(4.4.124)

Of course, any (p, ∞)-atom f on Σ is an (α, p)-ion for any α ∈ (0, 1). It is also clear from definitions that if the set Σ is compact then any α-charge is a fixed multiple of an (α, p)-ion.

(4.4.125)

In addition, any α-charge h satisfies, with q := (n − 1)/(n − 1 − α) > 1,   1/q h L q (Σ,σ) ≤ r α−(n−1) σ B(x0, r) ∩ Σ ≤ C 1/q r α−(n−1)+(n−1)/q = C 1/q,

(4.4.126)

where C ∈ (0, ∞) is the upper Ahlfors regularity constant of Σ. We have the following ionic characterization of Hardy spaces, in the spirit of Theorem 4.4.1. Proposition 4.4.6 Let Σ ⊆ Rn be a compact Ahlfors regular set, and abbreviate  σ := H n−1 Σ. Then for each p ∈ n−1 n , 1 one has

4.4 Atomic Decompositions for Hardy Spaces

167

H p (Σ, σ) (4.4.127)      λ j f j in Lip (Σ) : each f j is an (α, p)-ion, and |λ j | p < ∞ = j

j

as sets, and  f  H p (Σ,σ) ≈ inf



|λ j | p

1/p

uniformly for f ∈ H p (Σ, σ),

(4.4.128)

j

where the infimum in the right side is taken over all ionic decompositions of f . p

Proof Temporarily denote the right side of (4.4.127) by Hion (Σ, σ). Since, as observed earlier, any (p, ∞)-atom is an ion, from (4.4.127) and Theorem 4.4.1 we p conclude that H p (Σ, σ) ⊆ Hion (Σ, σ) and there exists C ∈ (0, ∞) such that for each p H (Σ, σ) we have

1/p  inf |λ j | p ≤ C f  H p (Σ,σ), (4.4.129) j

where the infimum in the left side is taken over all ionic decompositions of f . To establish the reverse inclusion together with the opposite inequality in (4.4.129), fix some α ∈ (0, 1). We claim that there exists a constant C ∈ (0, ∞) such that f (α, p)-ion =⇒ f ∈ H p (Σ, σ) and  f  H p (Σ,σ) ≤ C.

(4.4.130)

To justify this claim, start with a function f as in (4.4.123) and introduce ∫ 1  h := λ · 1B(x0,r)∩Σ where λ :=  f dσ. (4.4.131) σ B(x0, r) ∩ Σ Σ Note that |λ| ≤ r −(n−1)/p by the first two conditions in (4.4.123). As such, if we define g := f − h on Σ, (4.4.132) then supp g ⊆ B(x0, r) ∩ Σ,

g L ∞ (Σ,σ) ≤ 2r

−(n−1)/p

∫ ,

Σ

g dσ = 0.

(4.4.133)

Hence, g/2 is a (p, ∞)-atom. In particular, g H p (Σ,σ) ≤ 2. As seen from its definition in (4.4.131), the remainder h satisfies supp h ⊆ B(x0, r) ∩ Σ and h L ∞ (Σ,σ) ≤ r −(n−1) .

(4.4.134)

Since Σ is compact, it follows that h is a β-charge for any β ∈ (0, 1). Consequently, if we define q := (n − 1)/(n − 1 − β) > 1 then (4.4.126) gives

168

4 Hardy Spaces on Ahlfors Regular Sets

h L q (Σ,σ) ≤ C,

(4.4.135)

where C ∈ (0, ∞) depends only on β and the upper Ahlfors regularity constant of Σ. By design, we have f = g + h, so the claim in (4.4.130) follows on account of the fact that we presently have L q (Σ, σ) ⊆ H p (Σ, σ) continuously for each q > 1 (cf. (4.2.13)). p In turn, (4.4.130) is the main ingredient in the proof of Hion (Σ, σ) ⊆ H p (Σ, σ). p Specifically, start with some f ∈ Hion (Σ, σ) and consider an arbitrary decomposition   ! ! f = j λ j f j in Lip (Σ) , in which each f j is an (α, p)-ion and j |λ j | p < ∞. For any two integers M, N ∈ B with M > N we may then estimate M  p   λ j fj  p  j=N

H (Σ,σ)

≤

M 

p

λ j f j  H p (Σ,σ)

j=N

=

M 

p

|λ j | p  f j  H p (Σ,σ) ≤ C

j=N

M 

|λ j | p,

(4.4.136)

j=N

thanks to (4.2.7) and (4.4.130). This proves that the sequence of partial sums in ! the series j λ j f j is Cauchy in H p (Σ, σ). Since the latter space is quasi-Banach   and, according to (4.2.8), embeds continuously into Lip (Σ) (whose topology is Hausdorff; cf. [133, (4.1.39)]), we ultimately conclude that f ∈ H p (Σ, σ) and  f  H p (Σ,σ) ≤ C



|λ j | p

1/p

.

(4.4.137)

j p

This goes to show that Hion (Σ, σ) ⊆ H p (Σ, σ), finishing the proof of (4.4.127). Taking the infimum in (4.4.137) over all ionic decompositions of f also yields  f  H p (Σ,σ) ≤ C inf



|λ j | p

1/p

.

(4.4.138)

j

Together with (4.4.129), this concludes the proof of (4.4.128).



Our next theorem contains a useful boundedness criterion for linear operators defined on Hardy spaces and taking values in certain categories of topological vector spaces. set which is Ahlfors Theorem 4.4.7 Suppose Σ ⊆ Rn is a closed   regular and define n−1 σ := H n−1 Σ. Pick three exponents p ∈ n−1 n ,1 ,q ∈ n , ∞ , and r ∈ (1, ∞). Next, fix a topological vector space (X, τ) along with a pseudo-quasi-Banach space (Y, ·) such that (X, τ) and (Y, τ ·  ) are weakly compatible (in the sense of Definition 1.5.9). Denote by θ ∈ R the parameter quantifying the homogeneity of  ·  (cf. (1.5.20)) and assume % &  f + g sup θ ≥ p · log2 . (4.4.139) max{ f , g} f , g ∈Y not both zero

4.4 Atomic Decompositions for Hardy Spaces

169

Finally, consider a continuous linear operator T : H q (Σ, σ) −→ (X, τ)

(4.4.140)

with the property that there exists some C ∈ (0, ∞) such that T a ∈ Y and T a ≤ C, for every (p, r)-atom a on Σ.

(4.4.141)

Then there exists a unique linear and bounded (hence continuous) operator

: H p (Σ, σ) −→ (Y,  · ), T

(4.4.142)

whose operator norm is bounded by a multiple of the constant C appearing in (4.4.141), and which extends T, in the sense that if (X , τX ) is the ambient topological vector space in which (X, τ) and (Y, τ ·  ) embed as in Definition 1.5.9 then

f = T f in X , for each f ∈ H q (Σ, σ) ∩ H p (Σ, σ). T

(4.4.143)

: H p (Σ, σ) −→ Y acts on any given f ∈ H p (Σ, σ) the operator T

f := lim j→∞ T f j in (Y, τ ·  ) whenever the sequence according to T p,r { f j } j ∈N ⊆ Hfin (Σ, σ) is such that f = lim j→∞ f j in H p (Σ, σ).

(4.4.144)

In fact,

Before presenting the proof of Theorem 4.4.7 a couple of comments are in order. First, in [18] one may find an example of a linear functional which is uniformly bounded on all (1, ∞)-atoms (in the Euclidean setting), yet cannot be extended to a bounded linear functional defined on all of H 1 . Second, if  ·  is actually an s-norm (in the sense of item [3] of Definition 1.5.7) for some s ∈ (0, ∞) then ' (s  f + g s ≤  f  s + g s ≤ 21/s max{ f , g} , ∀ f , g ∈ Y . (4.4.145) This implies that the supremum in (4.4.139) is ≤ 21/s , which means that condition (4.4.139) is automatically satisfied if  ·  is actually an s-norm for some s ∈ (0, ∞) and θ ≥ p/s.

(4.4.146)

Proof of Theorem 4.4.7 The property that T map (p, r)-atoms on Σ into the space Y p,r implies that T f ∈ Y for each f ∈ Hfin (Σ, σ). In relation to this, we will first establish that there exists some C ∈ (0, ∞) such that θ T f  ≤ C f  H p (Σ,σ) for each function f ∈ Hfin (Σ, σ). p,r

(4.4.147)

! To this end, given f as above, consider an arbitrary way of expressing f = N j=1 λ j a j N N on Σ where N ∈ N, {λ j } j=1 ⊆ C, and {a j } j=1 is a sequence of (p, r)-atoms on Σ.

170

4 Hardy Spaces on Ahlfors Regular Sets

We claim that there exists a finite constant C > 0 (independent of f and its atomic decomposition) with the property that T f  ≤ C

# N

|λ j | p

$ θ/p

.

(4.4.148)

j=1

In order to justify (4.4.148) we proceed by considering two cases. Suppose first that the supremum displayed in (4.4.139) is one, i.e., suppose  f + g ≤ max{ f , g} for every f , g ∈ Y . In this scenario write    N     T f  =  λ j T a j  ≤ max λ j T a j  ≤ C max |λ j | θ T a j  1≤ j ≤ N 1≤ j ≤ N j=1

≤ C max |λ j | θ = C 1≤ j ≤ N

≤C

# N

|λ j |

$θ

≤C

j=1



max |λ j |

θ

1≤ j ≤ N

# N

|λ j | p

$ θ/p

,

(4.4.149)

j=1

where the second inequality is a consequence of the pseudo-homogeneity of  · , the third inequality follows from the uniform bound in (4.4.141), and the last inequality makes use of the fact p ≤ 1. Next, assume that the supremum displayed in (4.4.139) is strictly greater than one and let % ) &* −1  f + g ∈ (0, ∞). (4.4.150) sup β := log2 max{ f , g} f , g ∈Y not both zero

Then for some C ∈ (0, ∞) which depends only on β and the proportionality constants in (1.5.27) we may estimate  β β  N   N    N β T f  =  λ j T a j  ≤ C  λ j T a j  ≤ C |λ j | θβ · T a j  j=1

≤C

N  j=1



j=1

|λ j | θβ · T a j  β ≤ C

N 

|λ j | θβ,

j=1

(4.4.151)

j=1

where the first and third inequalities follow from (1.5.27) in Theorem 1.5.8, the second inequality comes from (1.5.28)-(1.5.29) in Theorem 1.5.8, and the last inequality is a consequence of (4.4.141). Note that the usage of (1.5.29) is valid given the definition of β ∈ (0, ∞). Combining the estimate in (4.4.151) with the fact that θ β ≥ p (as a result of (4.4.139) and the definition of β) ultimately permits us to write

4.4 Atomic Decompositions for Hardy Spaces

171

  # # $ 1/β $ θ/p N N   N T f  =  λ j T a j  ≤ C |λ j | θβ ≤C |λ j | p , j=1

j=1

(4.4.152)

j=1

finishing the proof of (4.4.148). Taking the infimum in (4.4.148) over all finite atomic θ decompositions of f then yields T f  ≤ C f  H , from which (4.4.147) p, r (Σ,σ) fin

follows on account of (4.4.115). Given (1.5.32), the pseudo-homogeneity of  · , and the homogeneity of  ·  H p (Σ,σ) , the estimate in (4.4.147) implies that    p,r T  H p, r (Σ,σ) : Hfin (Σ, σ),  ·  H p (Σ,σ) −→ (Y,  · ) fin (4.4.153) is a well-defined, linear, and bounded mapping. p,r

In particular, T maps Cauchy sequences in the space Hfin (Σ, σ) (equipped with the quasi-norm  ·  H p (Σ,σ) ) into Cauchy sequences in (Y, τ ·  ) (in the sense of Definition 1.5.1). Based on this, the density result in (4.4.114), and the completeness of (Y, τ ·  ) (again, in the sense of Definition 1.5.1), it follows that (4.4.153) may

: H p (Σ, σ) → Y as in (4.4.144). That T

is be further extended to an operator T unambiguously defined may be seen by interlacing sequences. In turn, this further

is linear and that T

agrees with T on H p,r (Σ, σ). In addition, from implies that T fin

and the property displayed in (1.5.33) it follows that there exists the definition of T C ∈ (0, ∞) such that p

f  ≤ C f  θ p T H (Σ,σ) for each f ∈ H (Σ, σ).

(4.4.154)

is indeed bounded in the context of In concert with (1.5.32) this shows that T (4.4.142). There remains to justify (4.4.143). Fix a function f ∈ H q (Σ, σ) ∩ H p (Σ, σ). p,r Theorem 4.4.3 implies that there exists a sequence { f j } j ∈N ⊆ Hfin (Σ, σ) such that q p lim f j = f both in H (Σ, σ) and in H (Σ, σ). Relying on the convergence in j→∞

in (4.4.142) and the fact that T

agrees with T on H p (Σ, σ), from the continuity of T p,r Hfin (Σ, σ) we may conclude that

f = lim T

f j = lim T f j in (Y, τ ·  ). T j→∞

j→∞

(4.4.155)

On the other hand, from the H q -convergence and the boundedness of T in (4.4.140) we have (4.4.156) T f = lim T f j in (X, τ). j→∞

Recall that (X, τ) and (Y, τ ·  ) are weakly compatible. Then, if (X , τX ) is as in Definition 1.5.9, from (4.4.155), (4.4.156), and the fact that convergent sequences in (X , τX ) have unique limits, we conclude that

f = lim T

f j = lim T f j = T f in (X , τX ). T j→∞

j→∞

(4.4.157)

172

4 Hardy Spaces on Ahlfors Regular Sets



This justifies (4.4.143), hence the proof of Theorem 4.4.7 is complete.

It is of interest to show that Hardy spaces are local whenever the underlying set is compact, in the following precise sense. Proposition 4.4.8 Fix n ∈ N with n ≥ 2 and suppose Σ ⊆ Rn be a compact Ahlfors regular set. Abbreviate σ := H n−1 Σ and pick some p ∈ n−1 n , 1 . Also, suppose (n − 1)(p−1 − 1) < γ < 1. Then the Hardy space H p (Σ, σ) is a module over the Hölder space C γ (Σ), in the sense that the operator Mϕ of multiplication by any given function ϕ ∈ C γ (Σ) induces a well defined linear and bounded mapping Mϕ : H p (Σ, σ) −→ H p (Σ, σ).

(4.4.158)

Moreover, there exists a constant C = C(Σ, n, p, γ) ∈ (0, ∞) with the property that Mϕ  H p (Σ,σ)→H p (Σ,σ) ≤ CϕC γ (Σ) for each ϕ ∈ C γ (Σ). (4.4.159)   Proof Since γ − (n − 1) p1 − 1 ∈ (0, 1), there exists q ∈ (1, ∞) large enough so that α := γ − (n − 1)

1 p

−1−

1 q



∈ (0, 1).

(4.4.160)

This choice of α also ensures that α

q∗ := q satisfies 1 < q∗ < q, n−1

(4.4.161)

which is going to be relevant shortly. For now, pick a function ϕ ∈ C γ (Σ), then select A, B ∈ (0, ∞) such that sup |ϕ| ≤ A and |ϕ(x) − ϕ(y)| ≤ B|x − y|γ for all x, y ∈ Σ.

(4.4.162)

Σ

With q ∈ (1, ∞) as above, let f be a (p, q)-atom on Σ, Hence, there exist a point x0 ∈ Σ and a number r ∈ 0, diam Σ with the property that ∫   1/q−1/p , f dσ = 0. supp f ⊆ B(x0, r) ∩ Σ,  f  L q (Σ,σ) ≤ σ B(x0, r) ∩ Σ Σ

(4.4.163)

If we set λ := ϕ(x0 ) then |λ| ≤ A and (ϕ − λ) f  L q (Σ,σ) ≤ CB r α−(n−1),

(4.4.164)

where C ∈ (0, ∞) depends only on Σ, p, q. Based on the last property above, the support condition in (4.4.163), (4.4.161), Hölder’s inequality, and the fact that Σ is upper Ahlfors regular, we may estimate   1−(q∗ /q) (ϕ − λ) f  L q∗ (Σ,σ) ≤ (ϕ − λ) f  L q∗ (Σ,σ) · σ B(x0, r) ∩ Σ ≤ C r α−(n−1)+(n−1)(1−(q∗ /q)) = C,

(4.4.165)

4.4 Atomic Decompositions for Hardy Spaces

173

where C ∈ (0, ∞) depends only on the upper Ahlfors regularity constant of Σ as well as B, p, q, and γ. Consequently, the decomposition ϕ f = λ f + g, where g := (ϕ − λ) f ,

(4.4.166)

expresses f as a sum of a multiple of an (p, q)-atom, for some coefficient bounded independently of f , x0, r, and a function whose norm in L qα (Σ, σ) is also bounded independently of f , x0, r. In view of this, (4.4.160), (4.4.126), and the fact that L q∗ (Σ, σ) embeds continuously into H p (Σ, σ) (cf. (4.2.13)), we then conclude that ϕ f belongs to the space H p (Σ, σ) and ϕ f  H p (Σ,σ) ≤ C, for some finite constant C > 0 depending only on A, B, Σ, n, p, and γ. Granted this, the fact that Mϕ induces a well-defined, linear, and bounded operator in the context of (4.4.158) is then a consequence of Theorem 4.4.7, bearing in mind  that Mϕ is a linear and bounded operator on any L s (Σ, σ) with s > 1. We conclude this section with a remark concerning Hardy spaces of vector distributions. To set the stage, suppose Σ ⊆ Rn is a closed  set which is Ahlfors regular and , 1 along with q ∈ [1, ∞] satisfying define σ := H n−1 Σ. Also, pick some p ∈ n−1 n q > p, and fix some integer M ∈ N. The goal is to discuss atomic decompositions  M for distributions belonging to the vector Hardy space H p (Σ, σ) . In this regard, we find it convenient to work with C M -valued (p, q)-atoms on Σ, i.e., functions  M a ∈ L q (Σ, σ) such that for some x ∈ Σ and r ∈ (0, 2 diam Σ) one has   1/q−1/p supp a ⊆ Σ ∩ B(x, r), a[L q (Σ,σ)] M ≤ σ Σ ∩ B(x, r) , ∫ a dσ = 0 ∈ C M . as well as Σ

(4.4.167)

Furthermore, in the case when Σ is bounded we also agree that constant vectors (4.4.168) a ∈ C M with |a| ≤ [σ(Σ)]−1/p are C M -valued (p, q)-atoms on Σ.  M Suppose now that some f = ( fβ )1≤β ≤M ∈ H p (Σ, σ) has been given. Then ! according to Theorem 4.4.1 each fβ has an atomic decomposition fβ = ∞ j=1 λβ j aβ j (no summation on β here) where each aβ j is a (p, q)-atom on Σ (cf. (4.4.2)) and {λβ j } j ∈N ∈  p . Moreover, it may be assumed that said atomic decomposition is quasi-optimal, in the sense that (for some absolute constants)  fβ  H p (Σ,σ) ≈

∞ 

|λβ j | p

 1/p

for each β ∈ {1, . . . , M }.

(4.4.169)

j=1

For each index β ∈ {1, . . . , M } introduce eβ := (δγβ )1≤γ ≤M ∈ C M (using the Kronecker symbol formalism) then write

174

4 Hardy Spaces on Ahlfors Regular Sets

f =

M 

fβ eβ =

β=1

M  ∞ 

λβ j aβ j eβ =

β=1 j=1

M  ∞ 

λβ j Aβ j

(4.4.170)

β=1 j=1

M  with convergence in H p (Σ, σ) , where Aβ j := aβ j eβ for each β ∈ {1, . . . , M } and j ∈ N.

(4.4.171)

M These are C M -valued functions as in (4.4.167)-(4.4.168), hence   C  -valued (p, q) atoms on Σ. If we then relabel the sequences Aβ j 1≤β ≤M and λβ j 1≤β ≤M simply j ∈N j ∈N     as a j j ∈N and λ j j ∈N , respectively, we may re-cast (4.4.169)-(4.4.171) in the form of the following conclusion:

M  may be decomposed as each distribution f ∈ H p (Σ, σ) ∞  M ! f = λ j a j with convergence in H p (Σ, σ) , where each j=1

a j is a C M -valued (p, q)-atom on Σ (cf. (4.4.167)-(4.4.168)), and  f [H p (Σ,σ)] M ≈

∞ ! j=1

|λ j | p

 p1

(4.4.172)

(with absolute constants).

Moreover, as is apparent from the above discussion, the analogues of Theorem 4.4.3 and Proposition 4.4.4 remain valid for vector Hardy space.

(4.4.173)

4.5 Molecules in Hardy Spaces A molecule may be regarded as weakened version of an atom, in which the compact support condition has been replaced by some power-type decay condition over dyadic annuli (while the vanishing moment condition is retained). This is made precise in the definition below (see [9, Definition 6.1, p. 266] for a more general setting). Definition 4.5.1 Suppose Σ ⊆ Rn is a closed set  which is Ahlfors regular and define  σ := H n−1 Σ. Pick two exponents p ∈ n−1 n , 1 and q ∈ [1, ∞] such that q > p, and fix some ε ∈ (0, ∞). Having chosen a point xo ∈ Σ along with some finite number r ∈ (0, diam Σ], call a σ-measurable function m defined on Σ a (p, q, ε)-molecule centered near the ball B(xo, r) on Σ (or, simply, a H p -molecule on Σ) provided for each k ∈ N0 one has ∫

1/q   1/q−1/p |m| q dσ ≤ 2k(n−1)[1/q−1−ε] σ B(xo, r) ∩ Σ (4.5.1) A k (x o ,r)

where

4.5 Molecules in Hardy Spaces

175

⎧ if k = 0, ⎪ ⎨ B(xo, r) ∩ Σ ⎪ Ak (xo, r) :=   ⎪ k k−1 ⎪ ⎩ B(xo, 2 r) \ B(xo, 2 r) ∩ Σ if k ≥ 1, and

(4.5.2)

∫ Σ

m dσ = 0.

(4.5.3)

In addition, in the case when Σ is bounded it is also agreed that the constant function given by m(x) := [σ(Σ)]−1/p for every x ∈ Σ is a (p, q, ε)-molecule on Σ. In the context of Definition 4.5.1, from (4.5.1)-(4.5.2) we conclude that if m is a (p, q, ε)-molecule on Σ then 1 m ∈ L qo (Σ, σ) for each qo ∈ 1+ε ,q .

(4.5.4)

In particular, the case qo := 1 ensures that the cancellation condition in (4.5.3) is meaningfully formulated. Of course, every (p, q)-atom is a (p, q, ε)-molecule for each ε > 0.

(4.5.5)

It turns out (see, e.g., [9, Theorem 6.4, pp. 273-274]) that any H p -molecule m on Σ (cf. Definition 4.5.1) belongs to the Hardy space H p (Σ, σ) and satisfies m H p (Σ,σ) ≤ C for some constant C ∈ (0, ∞) independent of the molecule m.

(4.5.6)

In fact, H p -molecules on Σ are building blocks for the Hardy space H p (Σ, σ), much as in Theorem 4.4.1. See [9, Chapter 6] for more information (including pertinent references) in this regard. In the lemma below we provide a more intrinsic characterization of molecules. Lemma 4.5.2 Let Σ ⊆ Rn be a closed set which is Ahlfors regular. Abbreviate σ := H n−1 Σ and fix some point xo ∈ Σ. Then for each exponents p, q, θ, d such that   p ∈ n−1 q ∈ [1, ∞) with q > p, n ,1 , (4.5.7)  θ ∈ (0, 1), and d := θ −1 (n − 1) q/p − 1), there exists a constant C = C(Σ, n, p, q, θ) ∈ (0, ∞) with the property that every function     m ∈ L q (Σ, σ) ∩ L q Σ, | · −xo | d σ = L q Σ, (1 + | · −xo | d )σ (4.5.8) satisfying

∫ Σ

m dσ = 0

(4.5.9)

176

4 Hardy Spaces on Ahlfors Regular Sets

necessarily is a multiple of a H p -molecule on Σ (hence, in particular m belongs to the Hardy space H p (Σ, σ)) and θ m H p (Σ,σ) ≤ Cm L1−θ q (Σ,σ) · m q L (Σ, | ·−x

o|

d σ)

≤ Cm L q (Σ,(1+ | ·−xo | d )σ) . Conversely, for each exponents p, q, ε, θ, d such that   p ∈ n−1 q ∈ [1, ∞) with q > p, ε > 1/p − 1, n ,1 ,  1/p−1/q −1 1+ε−1/q < θ < 1, and d := θ (n − 1) q/p − 1),

(4.5.10)

(4.5.11)

there exists a constant C = C(Σ, n, p, q, ε, θ) ∈ (0, ∞) with the property that every (p, q, ε)-molecule m centered near the ball B(xo, r) on Σ, for some finite number r ∈ (0, diam Σ], satisfies (4.5.8) and θ m L1−θ q (Σ,σ) · m q L (Σ, | ·−x

o|

d σ)

≤ C.

(4.5.12)

Proof As a preamble, we note that, as may be seen using Hölder’s inequality and [133, Lemma 7.2.1],  if qo ∈ (0, q] and a > (n − 1) q/qo − 1) then (4.5.13)   L q Σ, (1 + | · −xo | a )σ → L qo (Σ, σ).   In turn, since d > (n − 1) q/qo − 1 whenever qo ≥ p, this implies "   L qo (Σ, σ) ⊆ L 1 (Σ, σ). L q Σ, (1 + | · −xo | d )σ ⊆

(4.5.14)

p ≤qo ≤q

Hence, any function m as in (4.5.8) is absolutely integrable on Σ, which goes to show that the property formulated in (4.5.9) is indeed meaningful. Fix now a function m as in (4.5.8)-(4.5.9) and, without loss of generality, assume m does not vanish σ-a.e. on Σ. To show that m is a multiple of a H p -molecule on Σ, in the precise quantitative fashion described in (4.5.10), we distinguish two cases.   Case I: Assume p ∈ n−1 n , 1 . In this scenario, choose ε > 0 such that 1/p = 1 + ε. For some r > 0 to be specified later, recall (4.5.2) and, for each k ∈ N, estimate

4.5 Molecules in Hardy Spaces

∫ A k (x o ,r)

=

|m| q dσ

∫ A k (x o ,r)

177

1/q (4.5.15) |m| q | · −xo | d | · −xo | −d dσ

1/q

≤ C(2k r)−d/q m L q (Σ, | ·−xo | d σ) ≤ C2θ

−1 k(n−1)[1/q−1−ε]

  θ −1 (1/q−1/p) σ B(xo, r) ∩ Σ m L q (Σ, | ·−xo | d σ),

based on the definition of d, ε, and the Ahlfors regularity of Σ. Consequently, ∫

1/q ∫

θ/q ∫

(1−θ)/q |m| q dσ = |m| q dσ |m| q dσ A k (x o ,r)

A k (x o ,r)

A k (x o ,r)



≤ C2k(n−1)[1/q−1−ε] σ B(xo, r) ∩ Σ θ × m L1−θ q (Σ,σ) m q L (Σ, | ·−x

o|

 1/q−1/p

× (4.5.16)

d σ)

for each k ∈ N. To also be able to estimate the left-most side of (4.5.16) when k = 0, choose # $ m L q (Σ, | ·−xo | d σ) q/d ∈ (0, ∞). (4.5.17) r := m L q (Σ,σ) Together with the Ahlfors regularity of Σ, this choice of r permits us to write ∫

1/q |m| q dσ ≤ m L q (Σ,σ) (4.5.18) A0 (x o ,r)

θ = r (n−1)(1/q−1/p) m L1−θ q (Σ,σ) m q L (Σ, | ·−x

o|

d σ)

  1/q−1/p θ ≤ Cσ B(xo, r) ∩ Σ m L1−θ q (Σ,σ) m q L (Σ, | ·−x

o|

d σ)

.

Assume m is does not vanish σ-a.e. on Σ. Then from (4.5.15), (4.5.18), and (4.5.9), we deduce that the function m

:=

m θ Cm L1−θ q (Σ,σ) m q L (Σ, | ·−x o | d σ)

(4.5.19)

is a (p, q, ε)-molecule centered near the ball B(xo, r) on Σ, in the sense of Definition 4.5.1. Hence, m is a multiple of a H p -molecule on Σ. In light of (4.5.6), we conclude that m belongs to the Hardy space H p (Σ, σ) and the first inequality in (4.5.10) follows with the help of (4.5.19). Since the second inequality in (4.5.10) is clear, and since the case when m = 0 at σ-a.e. point in Σ is obvious, the treatment of Case I is complete.

178

4 Hardy Spaces on Ahlfors Regular Sets

Case II: Assume p = 1. Recall that this entails q ∈ (1, ∞) and d = (n−1)θ −1 (q−1). In particular, θ −1 (1 − 1/q) + 1/q > 1 which means that it is possible to choose po ∈



θ −1 (1 − 1/q) + 1/q

 −1 " n − 1

,1 . ,1 n

(4.5.20)

In turn, such a choice implies that θ o :=

θ(1/po − 1/q) belongs to (0, 1) and d = (n − 1)θ o−1 (q/po − 1). 1 − 1/q

(4.5.21)

Starting with the given function m as in (4.5.8)-(4.5.9) we may, granted (4.5.20)(4.5.21), run the same argument as in Case I with p replaced by po and θ replaced  by θ o (while retaining the same q and d). Since po ∈ n−1 n , 1), this yields θo o m ∈ H po (Σ, σ) and m H p o (Σ,σ) ≤ Cm L1−θ q (Σ,σ) · m q L (Σ, | ·−x

o|

,

d σ)

(4.5.22)

for some constant C = C(Σ, n, po, q, θ o ) ∈ (0, ∞). Moreover, m is a multiple of a H po -molecule on Σ, hence m is also a multiple of a H 1 -molecule on Σ. On the other hand, since m ∈ L q (Σ, σ) and we are currently assuming that q ∈ (1, ∞), we conclude from [133, (3.6.27)] that m ∈ H q (Σ, σ) and m H q (Σ,σ) ≤ Cm L q (Σ,σ) .

(4.5.23)

θ ∈ (0, 1) with the property that Given that 1 ∈ (po, q) we may select 1 = (1 − θ)/po + θ/q

(4.5.24)

and apply Proposition 4.2.2 (with p0 := po , q0 := q, and θ := θ) to obtain that m ∈ H 1 (Σ, σ) and



1−θ θ m H 1 (Σ,σ) ≤ Cm H p o (Σ,σ) · m H q (Σ,σ)



θ) o )(1−θ) ≤ Cm L(1−θ · m Lθoq(1− q (Σ,σ) (Σ, | ·−x



o|

θ) o +θo θ = Cm L1−θ · m Lθoq(1− q (Σ,σ) (Σ, | ·−x

o|

d σ)

· m Lθ q (Σ,σ)

,

(4.5.25)

d σ)

for some constant C = C(Σ, n, po, q, θ o ) ∈ (0, ∞). The second inequality in (4.5.25) is based on (4.5.22)-(4.5.23). Since (4.5.21) and (4.5.24) imply θ o (1 − θ) = θ

(4.5.26)

which further entails 1 − θ o + θ o θ) = 1 − θ, we may refashion (4.5.25) θ = 1 − θ o (1 − as θ m H 1 (Σ,σ) ≤ Cm L1−θ q (Σ,σ) · m q L (Σ, | ·−x

o|

,

d σ)

(4.5.27)

4.5 Molecules in Hardy Spaces

179

which goes to prove that estimate (4.5.10) also holds when p = 1. This finishes the treatment of Case II, and completes the proof of the first part in the statement of the lemma. In the opposite direction, assume the exponents p, q, ε, θ, d are as in (4.5.11), and suppose m is a (p, q, ε)-molecule centered near the ball B(xo, r) on Σ, for some finite r ∈ (0, diam Σ] which is not a constant function. Then, using notation and estimates from Definition 4.5.1, for some C ∈ (0, ∞) depending only on n, p, q, ε, and the Ahlfors regularity constants of Σ we may write q m L q (Σ,σ)

=

∞ ∫  k=0

A k (x o ,r)

|m| q dσ

≤ Cr (n−1)(1−q/p)

∞ 

2k(n−1)[1−(1+ε)q] = Cr (n−1)(1−q/p),

(4.5.28)

k=0

and q

m L q (Σ, | ·−x

o|

d σ)

=

∞ ∫  k=0

A k (x o ,r)

|m| q | · −xo | d dσ

≤ Cr d+(n−1)(1−q/p)

∞ 

2k



d+(n−1)[1−(1+ε)q]



k=0

= Cr

,

d+(n−1)(1−q/p)

(4.5.29)

where in the last step above we have used the fact that our choice of parameters in (4.5.11) guarantees that d + (n − 1)[1 − (1 + ε)q] < 0. From (4.5.28)-(4.5.29) and the definition of d in (4.5.11) we then readily see that (4.5.12) holds in this case. Finally, there remains to observe that in the case when Σ is bounded the constant  function given by m(x) := [σ(Σ)]−1/p for every x ∈ Σ also satisfies (4.5.12). Here is a useful consequence of the above lemma. Corollary 4.5.3 Let Σ ⊆ Rn be a closed set which is Ahlfors regular and abbreviate σ := H n−1 Σ. Also, fix some point xo ∈ Σ and consider exponents p, q, d such that    p ∈ n−1 q ∈ [1, ∞) with q > p, and d > (n − 1) q/p − 1). (4.5.30) n ,1 , Then



q



m ∈ L Σ, (1 + | · −xo | )σ : d

∫ Σ

m dσ = 0

 (4.5.31)

embeds continuously into the Hardy space H p (Σ, σ). Moreover, each function belonging to the space defined in (4.5.31) is a multiple of a H p -molecule on Σ (in the sense of Definition 4.5.1), and there exists a finite constant C > 0, independent of xo and m, with the property that

180

4 Hardy Spaces on Ahlfors Regular Sets θ m H p (Σ,σ) ≤ Cm L1−θ q (Σ,σ) · m q L (Σ, | ·−x

where θ := (n − 1)(q/p − 1)/d ∈ (0, 1).

o|

d σ)

(4.5.32)

Proof All claims in the statement are direct consequences of Lemma 4.5.2.



In relation to the space defined in (4.5.31), it is useful to point out that having fixed p, q as in (4.5.30) then a given (p, q, ε)-molecule m on Σ, in the sense of Definition 4.5.1, belongs to (4.5.31) for some d as in (4.5.30) provided p > 1/(1 + ε) (a condition automatically satisfied if ε ≥ 1/(n − 1)). The integral pairing between a molecule and a function of bounded mean oscillations is always absolutely convergent. More specifically, we have the following result. regular and abbreviate Lemma 4.5.4 Let Σ ⊆ Rn be a closed set which   is Ahlfors , 1 and q ∈ (1, ∞], and fix some σ := H n−1 Σ. Also, pick two exponents p ∈ n−1 n ε ∈ (0, ∞). Then ∫ | f ||m| dσ < +∞ whenever f ∈ BMO(Σ, σ) and m (4.5.33) Σ is a (p, q, ε)-molecule on Σ (cf. Definition 4.5.1). Proof Fix some function f ∈ BMO(Σ, σ) and suppose m is a (p, q, ε)-molecule centered near the ball B(xo, r) on Σ (where the point xo ∈ Σ and r > 0). Also, let the exponent q ∈ [1, ∞) be such that 1/q + 1/q = 1. Observe that [133, (7.4.122)] (considered with (X, ρ, μ) := (Σ, | · − · |, σ), p := q and (n − 1)ε in place of ε/p) presently gives ∞  k=1

2−k(n−1)ε

⨏ B(x o ,2k r)∩Σ

≤C

∞ 

  1/q  f − fB(x ,r)∩Σ q dσ o

−k(n−1)ε

2

k=1

≤ C f BMO(Σ,σ) .

⨏ B(x o ,2k r)

  1/q  f − fB(x ,2k r)∩Σ q dσ o (4.5.34)

To proceed, recall the set Ak (xo, r) defined for each k ∈ N0 as in (4.5.2) and estimate

4.5 Molecules in Hardy Spaces

∫ Σ\B(x o ,r)

181

   f − fB(x ,r)∩Σ  |m| dσ o ≤

∞ ∫  k=1

≤

A k (x o ,r)

  1/q  f − fB(x ,r)∩Σ q dσ o

⨏ ∞    1/q σ B(xo, 2k r) ∩ Σ k=1

B(x o ,2k r)∩Σ

∫ A k (x o ,r)

|m| q dσ

1/q

  1/q  f − fB(x ,r)∩Σ q dσ × o

  1/q−1/p × 2k(n−1)[1/q−1−ε] σ B(xo, r) ∩ Σ ∞   1−1/p  ≤ Cσ B(xo, r) ∩ Σ 2−k(n−1)ε ×

×

k=1

⨏ B(x o ,2k r)∩Σ

  1/q  f − fB(x ,r)∩Σ q dσ o

  1−1/p ≤ Cσ B(xo, r) ∩ Σ  f BMO(Σ,σ) < +∞.

(4.5.35)

Above, the first inequality is obtained by breaking up the domain of integration into mutually disjoint dyadic annuli and then using Hölder’s inequality. Next, the second inequality is a consequence of (4.5.1), the third inequality follows from the Ahlfors regularity of Σ, while the last inequality is simply (4.5.34).∫Since, as seen from (4.5.4), we also have m ∈ L 1 (Σ, σ), the reasoning so far gives that Σ\B(x ,r) | f ||m| dσ < +∞. o ∫ As such, there remains to show that Σ∩B(x ,r) | f ||m| dσ < +∞. This, however, is a o consequence of [133, (7.4.105)] and (4.5.4).  Here is a companion to Lemma 4.5.4, showing that the integral pairing between a molecule and a Hölder function of appropriate order is also absolutely convergent. regular and abbreviate Lemma 4.5.5 Let Σ ⊆ Rn be a closed set which   is Ahlfors , 1 and q ∈ (1, ∞], and fix some σ := H n−1 Σ. Also, pick two exponents p ∈ n−1 n ε ∈ (0, ∞). Then for each exponent α ∈ (0, 1) such that α < (n − 1)ε one has ∫ . | f ||m| dσ < +∞ whenever f ∈ C α (Σ) and m is a (4.5.36) Σ (p, q, ε)-molecule on Σ (cf. Definition 4.5.1).

.

Proof Fix some function f ∈ C α (Σ) and consider a (p, q, ε)-molecule m centered near the ball B(xo, r) on Σ. If q ∈ [1, ∞) is such that 1/q + 1/q = 1 we may reason as in (4.5.34) then invoke [133, (7.4.127)] to estimate

182

4 Hardy Spaces on Ahlfors Regular Sets ∞ 

2−k(n−1)ε

⨏ B(x o ,2k r)∩Σ

k=1

≤C

∞ 

  1/q  f − fB(x ,r)∩Σ q dσ o

2−k(n−1)ε

⨏

k=1

≤C

∞  k=1

B(x o ,2k r)

  1/q  f − fB(x ,2k r)∩Σ q dσ o

 α 2−k(n−1)ε 2k r  f C.α (Σ)

≤ Cr α  f C.α (Σ),

(4.5.37)

where the convergence of the geometric series uses the fact that α < (n − 1)ε. Next, if we proceed as in (4.5.35) and, this time, use (4.5.37) in the last inequality, we obtain ∫    f − fB(x ,r)∩Σ  |m| dσ o Σ\B(x o ,r)

  1−1/p ≤ Cσ B(xo, r) ∩ Σ × ×

∞ 

2−k(n−1)ε

k=1

⨏ B(x o ,2k r)∩Σ

  1/q  f − fB(x ,r)∩Σ q dσ o

  1−1/p ≤ Cr α σ B(xo, r) ∩ Σ  f C.α (Σ) < +∞.

(4.5.38)

∫ Given that m ∈ L 1 (Σ, σ) (cf. (4.5.4)), we conclude that Σ\B(x ,r) | f ||m| dσ < +∞. o Upon observing that we also have ∫

| f ||m| dσ ≤ sup | f (x)| m L 1 (Σ,σ) < +∞, (4.5.39) Σ∩B(x o ,r)

x ∈Σ∩B(x o ,r)

the proof is complete.



4.6 Duality in Hardy Spaces We have the following result (cf., e.g., [36, Theorem B, p. 593], [9, Theorem 7.20, p. 323]) describing the dual of the Hardy spaces from Definition 4.2.1 in the range p ≤ 1. Theorem 4.6.1 Given a closed set Σ ⊆ Rn which is Ahlfors regular, abbreviate σ := H n−1 Σ and fix an exponent p ∈ n−1 n , 1 . Then, in a quantitative fashion,

4.6 Duality in Hardy Spaces

183

 ∗ Σ unbounded =⇒ H p (Σ, σ) =

.    ⎧ ⎨ C (n−1)(1/p−1) (Σ) ∼ if p ∈ n−1 ⎪ n ,1 , ⎪ BMO(Σ, σ)  ∼ ⎩

if p = 1,

(4.6.1) (where the equivalence identifies functions which differ by a constant on Σ) and    (n−1)(1/p−1) (Σ) if p ∈ n−1 C  p ∗ n ,1 , Σ bounded =⇒ H (Σ, σ) = (4.6.2) BMO(Σ, σ) if p = 1. The actual nature of the identification of the dual of the Hardy spaces in (4.6.1) is as follows. Having fixed some q ∈ [1, ∞] satisfying q > p, consider the assignment taking any equivalence class [ f ] belonging to

.    ⎧ ⎨ C (n−1)(1/p−1) (Σ) ∼ if p ∈ n−1 ⎪ n ,1 , ⎪ BMO(Σ, σ)  ∼ if p = 1, ⎩

(4.6.3)

 ∗ into the functional Λ[ f ] ∈ H p (Σ, σ) defined as 

∫ N   Λ[ f ], g := lim λj f a j dσ, N →∞

whenever

j=1

Σ

g ∈ H p (Σ, σ), {λ j } j ∈N ∈  p (N), and {a j } j ∈N is a sequence of (p, q)-atoms on Σ, with the property that ! p g= ∞ j=1 λ j a j in H (Σ, σ).

(4.6.4)

(4.6.5)

Then Λ[ f ] is unambiguously defined, and the assignment [ f ] → Λ[ f ] is a bounded  ∗ linear isomorphism from the space in (4.6.3) onto H p (Σ, σ) , with bounded inverse. In particular, identifying Λ[ f ] with [ f ] yields (bearing in mind the format of the norm . on BMO(Σ, σ); cf. [133, (7.4.95)])      f BMO(Σ,σ) =  [ f ] BMO(Σ,σ)/∼ ≈ Λ[ f ] (H 1 (Σ,σ))∗     = sup  Λ[ f ], g  : g ∈ H 1 (Σ, σ) with g H 1 (Σ,σ) = 1

    = sup  [ f ], g  : g ∈ H 1 (Σ, σ) with g H 1 (Σ,σ) = 1 ,

(4.6.6)

uniformly for f ∈ BMO(Σ, σ) (compare with [133, Proposition 7.4.12]), and

184

 f C.(n−1)(1/p−1) (Σ)

4 Hardy Spaces on Ahlfors Regular Sets

  =  [ f ]  .(n−1)(1/p−1) C

(Σ)/∼

  ≈ Λ[ f ] (H p (Σ,σ))∗

    = sup  Λ[ f ], g  : g ∈ H p (Σ, σ) with g H p (Σ,σ) = 1     = sup  [ f ], g  : g ∈ H p (Σ, σ) with g H p (Σ,σ) = 1 , (4.6.7)

.   uniformly for f ∈ C (n−1)(1/p−1) (Σ) with p ∈ n−1 n , 1 (compare with [133, Proposition 7.4.8]). Also, (4.6.4) yields the duality formula ∫   ! [ f ], g = ∞ j=1 λ j Σ f a j dσ whenever [ f ] belongs to the space in (4.6.3), g ∈ H p (Σ, σ), {λ j } j ∈N ∈  p (N), and {a j } j ∈N ! (4.6.8) is a sequence of (p, q)-atoms on Σ, such that g = ∞ j=1 λ j a j in H p (Σ, σ). Another consequence is that there exists a finite constant C > 0, which depends only on the environment, having the property that     f C.(n−1)(1/p−1) (Σ) · g H p (Σ,σ) if p ∈ n−1   n ,1 ,  [ f ], g  ≤ C (4.6.9) if p = 1.  f BMO(Σ,σ) · g H 1 (Σ,σ) Finally, there is an analogous interpretation of the identification of the dual of the Hardy space in (4.6.2) (plus naturally accompanying results as in (4.6.6), (4.6.7), (4.6.8), (4.6.9)), this time involving inhomogeneous Hölder spaces and without having to consider equivalence classes of functions (modulo constants).   Hence, if Σ ⊆ Rn is a closed Ahlfors regular set, σ := H n−1 Σ, and p ∈ n−1 n ,∞ , we have: .    ⎧ C (n−1)(1/p−1) (Σ) ∼ if p ∈ n−1 ⎪ n , 1 and Σ unbounded, ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ C (n−1)(1/p−1) (Σ) if p ∈ n−1 ⎪ n , 1 and Σ bounded, ⎪ ⎨ ⎪  p ∗ ⎪  H (Σ, σ) = BMO(Σ, σ) ∼ if p = 1 and Σ unbounded, ⎪ ⎪ ⎪ ⎪ ⎪ BMO(Σ, σ) if p = 1 and Σ bounded, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L p/(p−1) (Σ, σ) if p ∈ (1, ∞). ⎩ (4.6.10) Continue to assume that Σ ⊆ Rn is a closed Ahlfors regular set, abbreviate   σ := H n−1 Σ. In general, the Hardy space H p (Σ, σ) with p ∈ n−1 n , 1 is not a dual space, but the case p = 1 is different. To elaborate, following [36] we shall4 denote by CMO(Σ, σ) the closure in BMO(Σ, σ) of C00 (Σ), the space of all continuous functions on Σ which vanish at infinity.

(4.6.11)

4 In [36] the authors employ the notation VMO(Σ, σ), in place of CMO(Σ, σ) as we do here (since the former already has a precise, typically distinct, meaning; see (3.1.1))

4.6 Duality in Hardy Spaces

185

Hence, by design, CMO(Σ, σ) is a closed linear subspace of BMO(Σ, σ).

(4.6.12)

Since Lipc (Σ) is a linear subspace of the space of continuous functions on Σ vanishing at infinity which is dense with respect to  ·  L ∞ (Σ,σ) , and since convergence in L ∞ (Σ, σ) implies convergence in BMO(Σ, σ), we conclude that (4.6.11) selfimproves to CMO(Σ, σ) is the closure of Lipc (Σ) in BMO(Σ, σ).

(4.6.13)

Let us also observe that f + c belongs to CMO(Σ, σ) whenever f ∈ CMO(Σ, σ) and c ∈ C, and CMO(Σ, σ) is the closure of Lipc (Σ) + C in BMO(Σ, σ).

(4.6.14)

Indeed, if Σ is bounded then constants belong to Lipc (Σ), whereas if Σ is unbounded this follows from the fact that  · BMO(Σ,σ) is insensitive to additive constants. Hence, constants are contained in CMO(Σ, σ) and . CMO(Σ, σ) := CMO(Σ, σ)/∼ is a closed linear . subspace of BMO(Σ, σ) := BMO(Σ, σ)/∼.

(4.6.15)

In addition, from (4.6.13) and (3.1.50) we see that if Σ ⊆ Rn is a compact Ahlfors regular set and σ := H n−1 Σ, then CMO(Σ, σ) = VMO(Σ, σ).

(4.6.16)

Also, according to [36, Theorem 4.1, p. 638], H 1 (Σ, σ) is the dual space of CMO(Σ, σ); more precisely, for each continuous linear functional Λ ∫on CMO(Σ, σ) there exists a unique f ∈ H 1 (Σ, σ) such that Λ(φ) = Σ f φ dσ for all continuous functions φ on Σ which vanish at infinity, and  f  H 1 (Σ,σ) is equivalent to the linear functional norm of Λ.

(4.6.17)

From (4.6.11), (4.6.17), Lemma4.6.5 (with p := 1), and Theorem 4.6.1 we see that  ∗ if a functional Λ ∈ CMO(Σ, σ) is identified with some f ∈ H 1 (Σ, σ) in the sense of (4.6.17), it follows that  H 1 (Σ,σ) f , [·]BMO(Σ,σ)/∼ if Σ unbounded, on CMO(Σ, σ). (4.6.18) Λ= if Σ bounded, H 1 (Σ,σ) f , ·BMO(Σ,σ) Let us take a separate look at the case when the set Σ is unbounded. In this scenario, denote by π : CMO(Σ, σ) → CMO(Σ, σ)/∼ the canonical projection  

∈ CMO(Σ, σ)/∼ ∗ it onto equivalence classes modulo constants. Then for each Λ  

◦ π belongs to CMO(Σ, σ) ∗ . Based on this follows that the composition Λ := Λ

186

4 Hardy Spaces on Ahlfors Regular Sets

and (4.6.18) we then conclude that there exists a unique function f ∈ H 1 (Σ, σ) such that

= Λ

H 1 (Σ,σ) f , ·BMO(Σ,σ)/∼

on CMO(Σ, σ)/∼ and

 f  H 1 (Σ,σ) is equivalent to the linear functional norm of Λ.

(4.6.19)

This proves that the assignment

f := H 1 (Σ, σ)  f → Λ

H 1 (Σ,σ) f , ·BMO(Σ,σ)/∼

 ∗ ∈ CMO(Σ, σ)/∼

(4.6.20)

is well defined, linear, continuous, and surjective. From (4.6.11), Lemma 4.6.5 (with p := 1), and [133, Corollary 3.7.3] we also see that the assignment (4.6.20) is injective. Ultimately, this discussion shows that 1 if  Σ is unbounded  ∗ then H (Σ, σ) may be identified with CMO(Σ, σ)/∼ via the isomorphism (4.6.20).

Moving on, we note that, collectively, (4.6.16) and (4.6.17) give  ∗ H 1 (Σ, σ) = VMO(Σ, σ) if Σ is bounded.

(4.6.21)

(4.6.22)

More precisely, whenever the set Σ is assumed to be bounded, from (4.6.16)-(4.6.18)  ∗ we see that for any functional Λ ∈ VMO(Σ, σ) there exists a unique function 1 f ∈ H (Σ, σ) with  f  H 1 (Σ,σ) equivalent to the linear functional norm of Λ and such that (4.6.23) Λ = H 1 (Σ,σ) f , ·BMO(Σ,σ) on VMO(Σ, σ). Another point of view on (4.6.17), stemming from the discussion in [36, p. 638] (and (4.6.13)), is as follows. If μ is a Borel measure on Σ with the property that there exists a constant C ∈ (0, ∞) such that ∫    (4.6.24)  φ dμ ≤ CφBMO(Σ,σ) for all φ ∈ Lipc (Σ), Σ

then μ p.

.  ∗ C α0 (Σ) ∩ L p1 (Σ, σ) → H p,q (Σ, σ)

Then

(4.7.1)

(4.7.2)

in a continuous and injective fashion. Moreover, (H p, q (Σ,σ))∗



f, g



∫ H p, q (Σ,σ)

=

Σ

+ whenever

f g1 dσ

(4.7.3) 

 [ f ], g0 H p0 (Σ,σ) if Σ is unbounded,   if Σ is bounded, C α0 (Σ) f , g0 H p0 (Σ,σ)

 . α C

0 (Σ)/∼

.  ∗ f ∈ C α0 (Σ) ∩ L p1 (Σ, σ) → H p,q (Σ, σ) and

g ∈ H p,q (Σ, σ) ⊆ H p0 (Σ, σ) + L p1 (Σ, σ) is written as5

(4.7.4)

p1

g = g0 + g1 with g0 ∈ H p0 (Σ, σ) and g1 ∈ L (Σ, σ).

.

Proof Pick an arbitrary function f ∈ C α0 (Σ)∩ L p1 (Σ, σ). In relation to this, consider the linear and continuous functionals p0 Λ(0) f : H (Σ, σ) −→ C

(4.7.5)

acting on each g ∈ H p0 (Σ, σ) according to (cf. Theorem 4.6.1)    . [ f ], g H p0 (Σ,σ) if Σ is unbounded, C α0 (Σ)/∼ (0) Λ f (g) :=   if Σ is bounded, C α0 (Σ) f , g H p0 (Σ,σ) and

p1 Λ(1) f : L (Σ, σ) −→ C,

Λ(1) f (g) :=

∫ Σ

f g dσ,



∀g ∈ L p1 (Σ, σ).

(4.7.6)

(4.7.7)

192

4 Hardy Spaces on Ahlfors Regular Sets

(1) Since Lemma 4.6.6 then ensures that Λ(0) f and Λ f are compatible with one another, we may unambiguously define the linear and bounded mapping

Λ f : H p0 (Σ, σ) + L p1 (Σ, σ) → C,

(1) Λ f (g) := Λ(0) f (g0 ) + Λ f (g1 ),

whenever g ∈ H p0 (Σ, σ) + L p1 (Σ, σ) is expressed as

(4.7.8)

p1

g = g0 + g1 with g0 ∈ H p0 (Σ, σ) and g1 ∈ L (Σ, σ). By design, this satisfies   Λf 

H p0 (Σ,σ)

  = Λ(0) and Λ  f f

L

p 1 (Σ,σ)

= Λ(1) f .

(4.7.9)

Thanks to Proposition 1.3.7, Theorem 4.3.1, and [133, (3.6.27)] to conclude that Λ f maps H p,q (Σ, σ) (itself, a subspace of H p0 (Σ, σ) + L p1 (Σ, σ)), equipped with the quasi-norm  ·  H p, q (Σ,σ) , boundedly into C. Thus, regarding this as a linear and bounded mapping Λ f : H p,q (Σ, σ) → C, the reasoning so far yields a well-defined and linear assignment

.  ∗ C α0 (Σ) ∩ L p1 (Σ, σ)  f −→ Λ f ∈ H p,q (Σ, σ) .

(4.7.10)

From (4.7.5)-(4.7.7), (4.6.9), Hölder’s inequality, and (1.3.64) it follows that this assignment is also continuous. As such, the claim in (4.7.2) follows (by identifying each f with Λ f ) as soon as we show that the assignment (4.7.10) is also injective. . To this end, suppose f ∈ C α0 (Σ) ∩ L p1 (Σ, σ) is such that Λ f = 0 as a functional  ∗ in H p,q (Σ, σ) . In such a scenario, (1.3.44) and (4.3.3) allow us to conclude that actually Λ f  H p0 (Σ,σ)∩L p1 (Σ,σ) ≡ 0. In particular, Λ f (a) = 0 for each atom a on Σ ∫which, on account of the very last property recorded in (1.3.66) and (4.7.7), implies f a dσ = 0 for each atom a on Σ. With this in hand, Lemma 4.6.9 applies and Σ ultimately proves that f = 0, as wanted. This finishes the proof of (4.7.2). Finally, whenever f , g are as in (4.7.4), by virtue of (4.7.10), and (4.7.9) we may write   (H p, q (Σ,σ))∗ f , g H p, q (Σ,σ) = Λ f (g) = Λ f (g0 + g1 ) (0) = Λ f (g0 ) + Λ f (g1 ) = Λ(0) f (g0 ) + Λ f (g1 ),

from which (4.7.3) follows on account of (4.7.5)-(4.7.6), and (4.7.7).

(4.7.11) 

Here is a companion result to Lemma 4.7.1, once again pointing to the fact that Lorentz-based Hardy spaces have reasonably rich dual spaces. Lemma 4.7.2 Suppose Σ ⊆ Rn is an unbounded closed  set which is also Ahlfors , 1 and q ∈ (0, ∞], then pick regular, and abbreviate σ := H n−1 Σ. Fix p ∈ n−1 n α0, α1 ∈ (0, 1) such that  p0 := 1 +

α0  −1 n−1

 < p < p1 := 1 +

α1  −1 . n−1

(4.7.12)

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

Then

.  .α   ∗ C 0 (Σ) ∩ C α1 (Σ) ∼ embeds into H p,q (Σ, σ) in a continuous and injective fashion,

193

(4.7.13)

and (H p, q (Σ,σ))∗



[ f ], g



H p, q (Σ,σ)

  = C.α0 (Σ)/∼ [ f ], g0 H p0 (Σ,σ)

  + C.α1 (Σ)/∼ [ f ], g1 H p1 (Σ,σ)

whenever

.

(4.7.14)

.

f belongs to the intersection C α0 (Σ) ∩ C α1 (Σ) and g ∈ H p,q (Σ, σ) ⊆ H p0 (Σ, σ) + H p1 (Σ, σ) is written as

(4.7.15)

g = g0 + g1 with g0 ∈ H p0 (Σ, σ) and g1 ∈ H p1 (Σ, σ). Furthermore, similar claims are valid in the case when Σ is bounded, this time using inhomogeneous Hölder spaces and no longer modding out constants. Proof The same type of argument as in the proof of Lemma 4.7.1 works in the present setting, with Lemma 4.6.8 now employed in place of Lemma 4.6.6. 

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings In concert, the Sequential Banach-Alaoglu Theorem recorded in [133, (3.6.22)] (whose present applicability is ensured by [133, (3.6.3)]) and Lebesgue’s Dominated Convergence Theorem give that if (X, μ) is a sigma-finite measure space, the measure μ is separable, and if { f j } j ∈N is a bounded sequence in L ∞ (X, μ) with the property that f (x) := lim f j (x) exists for μ-a.e. point x ∈ X, then f ∈ L ∞ (X, μ) j→∞  ∗ and lim f j = f weak-∗ in L ∞ (X, μ) = L 1 (X, μ) .

(4.8.1)

j→∞

Here is a version of this result in which the space of essentially bounded functions is replaced by the John-Nirenberg space of functions of bounded mean oscillations. Lemma 4.8.1 Consider a closed, unbounded set Σ ⊆ Rn , which is also Ahlfors 1 (Σ, σ) and suppose { f } regular, and abbreviate σ := H n−1 Σ. Pick f ∈ Lloc j j ∈N is a sequence of functions in BMO(Σ, σ) satisfying 1 sup  f j BMO(Σ,σ) < +∞ and f j → f in Lloc (Σ, σ) as j → ∞. j ∈N

(4.8.2)

194

4 Hardy Spaces on Ahlfors Regular Sets

Then f belongs to BMO(Σ, σ) and . lim [ f j ] = [ f ] weak-∗ in BMO(Σ, σ).

j→∞

(4.8.3)

That is, with ·, · denoting the duality bracket between the John-Nirenberg space of functions of bounded mean oscillations on Σ, modulo constants, and the Hardy space H 1 on Σ (cf. Theorem 4.6.1),     for each g ∈ H 1 (Σ, σ) one has [ f j ], g → [ f ], g as j → ∞. (4.8.4) In addition, a similar weak-∗ convergence result holds in the case when Σ is bounded, namely (4.8.5) lim f j = f weak-∗ in BMO(Σ, σ). j→∞

Proof Suppose (4.8.4) fails. Then there exists some function g ∈ H 1 (Σ, σ) along with some strictly increasing sequence { jk }k ∈N ⊆ N such that     does not converge to [ f ], g . (4.8.6) any sub-sequence of [ f jk ], g k ∈N

  On the other hand, the first condition in (4.8.2) implies that the sequence [ f jk ] k ∈N  ∗ . . is bounded in BMO(Σ, σ). Since BMO(Σ, σ) = H 1 (Σ, σ) (cf. (4.6.1) in Theorem 4.6.1), the Sequential Banach-Alaoglu Theorem recalled in [133, (3.6.22)] (whose applicability in the present case is ensured by the separability result from  Proposition 4.4.2) implies the existence of a sub-sequence, say [ f jk i ] i ∈N , along with some function f ∈ BMO(Σ, σ), such that . [ f jk i ] −→ [ f ] weak-∗ in BMO(Σ, σ), as i → ∞.

(4.8.7)

Then for each (1, ∞)-atom a on Σ we may write ∫ ∫   f a dσ = lim f jk i a dσ = lim [ f jk i ], a Σ

i→∞

Σ

  = [ f ], a =

i→∞

∫ Σ

f a dσ.

(4.8.8)

Above, the first equality is a consequence of the last property in (4.8.2), the second and fourth equalities are implied by (4.6.4), and the third equality follows from (4.8.7). Granted (4.8.8), we may invoke Lemma 4.6.9 to conclude that f− f is constant on Σ.

(4.8.9)

. In turn, from (4.8.7) and (4.8.9) we see that [ f jk i ] → [ f ] weak-∗ in BMO(Σ, σ) as i → ∞. Hence, in particular,     [ f jk i ], g −→ [ f ], g as i → ∞, (4.8.10)

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

195

which contradicts (4.8.6). This finishes the proof of (4.8.3). Finally, the weakconvergence result recorded in (4.8.5) in the case when Σ is bounded is proved similarly.  The formal convolution paring between the logarithm, viewed as a function in BMO, and a fixed function in the Hardy space H 1 , turns out to be continuous and bounded. This is made precise in the lemma below. Lemma 4.8.2 Suppose Σ is a closed Ahlfors regular set in Rn . Let σ := H n−1 Σ, fix some f ∈ H 1 (Σ, σ), and consider the function   ln |x − ·|, f if Σ is bounded, ∀x ∈ Rn, F(x) :=  (4.8.11)  [ln |x − ·|], f if Σ is unbounded, where ·, · stands for the duality bracket between the John-Nirenberg space of functions of bounded mean oscillations on Σ (modulo constants, if Σ is unbounded) and the Hardy space H 1 on Σ, described in Theorem 4.6.1. Then F is a well-defined continuous function in Rn . Moreover, the restriction of F to Rn \ Σ is of class C ∞ , and its partial derivatives in Rn \ Σ may be computed by differentiating directly under the integral pairing in (4.8.11). Finally, if Σ is unbounded then there exists a constant C = C(Σ) ∈ (0, ∞) with the property that (4.8.12) sup |F(x)| ≤ C f  H 1 (Σ,σ) . x ∈R n

Proof In a first stage, work under the assumption that Σ is unbounded. From [133, Lemma 7.4.13] and Theorem 4.6.1 we see that the function F is well defined, bounded, and that (4.8.12) holds. To establish the continuity of F, let {x j } j ∈N ⊆ Rn be a sequence convergent to a point x ∈ Rn . Define       g := ln |x − ·|  and g j := ln |x j − ·|  for each j ∈ N. (4.8.13) Σ

Σ

Then [133, Lemma 7.4.13] ensures that {g j } j ∈N is a bounded sequence in BMO(Σ, σ). Also, Vitali’s Convergence Theorem (cf., e.g., [130, Theorem 14.29, p. 559]) guar1 (Σ, σ) as j → ∞. Granted these properties, Lemma 4.8.1 antees that g j → g in Lloc     applies and gives that [g j ], f → [g], f as j → ∞. In turn, this readily implies that F is continuous in Rn . To prove that F is smooth on the complement of Σ, recall from Theorem 4.4.1 that there exist a numerical sequence {λ j } j ∈N ∈  p (N) and a sequence {a j } j ∈N of (1, ∞)-atoms on Σ such that f = lim fm in H 1 (Σ, σ) where, for each m ∈ N, m→∞

fm :=

m  j=1

If for each m ∈ N we now set

∞ λ j a j ∈ Lcomp (Σ, σ).

(4.8.14)

196

4 Hardy Spaces on Ahlfors Regular Sets



Fm (x) := [ln |x − ·|], fm ,

∀x ∈ Rn \ Σ,

(4.8.15)

then (4.6.8) permits us to express Fm (x) =

m  j=1

∫ λj

Σ

ln |x − y|a j (y) dσ(y)

∫ =

Σ

ln |x − y| fm (y) dσ(y),

∀x ∈ Rn \ Σ.

(4.8.16)

It is then clear from this representation that for each m ∈ N we have Fm ∈ C ∞ (Rn \ Σ) and, for each α ∈ N0n and x ∈ Rn \ Σ, ∫       α ∂xα ln |x − y| fm (y) dσ(y) = ∂xα ln |x − ·| , fm , (∂ Fm )(x) =

(4.8.17)

Σ

  where the last equality relies on (4.6.8) plus the fact that ∂xα ln |x − ·| belongs to L ∞ (Σ, σ) if |α| > 0 and to BMO(Σ, σ) if |α| = 0. If for each given α ∈ N0n and each given x ∈ Rn \ Σ we now introduce    α  ∂x ln |x − ·|  L ∞ (Σ,σ) if |α| > 0, Cα (x) := (4.8.18) ln |x − ·| BMO(Σ,σ) if |α| = 0, then based on (4.8.17), (4.6.9), (4.8.18), and (4.2.10) we may estimate          ∂xα ln |x − ·| , f − (∂ α Fm )(x)            =  ∂xα ln |x − ·| , f − ∂xα ln |x − ·| , fm         =  ∂xα ln |x − ·| , f − fm  ≤ Cα (x) f − fm  H 1 (Σ,σ),

(4.8.19)

for each α ∈ N0n , m ∈ N, and x ∈ Rn \ Σ. For each xo ∈ Rn \ Σ and α ∈ N0n fixed, [133, Lemma 7.4.13] and (4.8.18) imply that sup

x ∈B(x o ,r)

Cα (x) < +∞ if 0 < r < dist(xo, Σ).

Together with (4.8.14) and (4.8.19), this proves that   for each multi-index α ∈ N0n the sequence (∂ α Fm )(x) m∈N converges     to ∂xα ln |x − ·| , f uniformly for x in compact subsets of Rn \ Σ.

(4.8.20)

(4.8.21)

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

197

Based this and the differentiation theorem for sequences of functions we then readily conclude that the restriction of F to Rn \ Σ is of class C ∞ , and its partial derivatives may be computed by differentiating directly under the integral pairing. Finally, the case when Σ is bounded is treated very similarly.  Going back to the result described in Lemma 4.8.1, it turns out that truncating (in the range) any given BMO function, in the most basic fashion, yields a sequence of bounded functions which is weak-∗ convergent in BMO to the original function. Lemma 4.8.3 Let Σ ⊆ Rn be a closed, unbounded set, which is Ahlfors regular, and abbreviate σ := H n−1 Σ. Given a real-valued function f ∈ BMO(Σ, σ), for each M ∈ N define ⎧ M if f (x) > M ⎪ ⎪ ⎨ ⎪ fM (x) := f (x) if − M ≤ f (x) ≤ M, ⎪ ⎪ ⎪ −M if f (x) < −M, ⎩ Then

∀x ∈ Σ.

. lim [ fM ] = [ f ] weak-∗ in BMO(Σ, σ).

M→∞

(4.8.22)

(4.8.23)

That is, with ·, · denoting the duality bracket between the John-Nirenberg space of functions of bounded mean oscillations on Σ, modulo constants, and the Hardy space H 1 on Σ (cf. Theorem 4.6.1),     for each g ∈ H 1 (Σ, σ) one has [ fM ], g → [ f ], g as M → ∞. (4.8.24) In addition, a similar weak-∗ convergence result holds in the case when Σ is bounded, namely lim fM = f weak-∗ in BMO(Σ, σ). (4.8.25) M→∞

Proof In view of [133, (7.4.104), (7.4.102)], this is a consequence of Lemma 4.8.1. We aim to replicate the results in Lemma 4.8.1 and Lemma 4.8.3 working with Hölder functions. In this regard, the analogue of Lemma 4.8.1 reads as follows. Lemma 4.8.4 Let Σ ⊆ Rn be a closed unbounded Ahlfors regular set. Abbreviate   σ := H n−1 Σ, pick α ∈ (0, 1), and introduce p := (n − 1)/(n − 1 + α) ∈ n−1 n ,1 . .α Consider a sequence { f j } j ∈N ⊆ C (Σ) of functions satisfying sup  f j C.α (Σ) < +∞ and f (x) := lim f j (x) exists for each x ∈ Σ. j→∞

j ∈N

(4.8.26)

.

Then the function f belongs to C α (Σ) and

.  lim [ f j ] = [ f ] weak-∗ in C α (Σ) ∼ .

j→∞

(4.8.27)

198

4 Hardy Spaces on Ahlfors Regular Sets

Specifically, with ·, · denoting the duality bracket between functions satisfying a homogeneous Hölder condition of order α on Σ, modulo constants, and the Hardy space H p on Σ (cf. Theorem 4.6.1),     for each g ∈ H p (Σ, σ) one has [ f j ], g → [ f ], g as j → ∞. (4.8.28) Moreover, a similar weak-∗ convergence result holds in the case when Σ is bounded, namely starting with a bounded sequence { f j } j ∈N in C α (Σ) which converges pointwise to a function f and concluding that lim f j = f weak-∗ in C α (Σ).

(4.8.29)

j→∞

Proof If M ∈ [0, ∞) denotes the supremum in (4.8.26), it follows that for each j ∈ N we have | f j (x) − f j (y)| ≤ M |x − y| α for every x, y ∈ Σ. Passing to limit j → ∞ then . yields | f (x) − f (y)| ≤ M |x − y| α for each x, y ∈ Σ, which shows that f ∈ C α (Σ). Next, seeking a contradiction, assume the claim in (4.8.28) fails. Then there exists a distribution g ∈ H p (Σ, σ) together with a strictly increasing sequence { jk }k ∈N of natural numbers such that     any sub-sequence of [ f jk ], g does not converge to [ f ], g . (4.8.30) k ∈N

To proceed, observe that the first condition in (4.8.26) implies that the sequence . .      ∗ [ f jk ] k ∈N is bounded in C α (Σ) ∼. Given that C α (Σ) ∼ = H p (Σ, σ) (cf. (4.6.1) in Theorem 4.6.1), the separability result from Proposition 4.4.2 together with the Sequential Banach-Alaoglu Theorem from [133, (3.6.22)] (which covers the class of separable quasi-Banach spaces) imply the existence of a sub-sequence, .   f ∈ C α (Σ), such that say [ f jk i ] i ∈N , along with some function

.  [ f jk i ] −→ [ f ] weak-∗ in C α (Σ) ∼, as i → ∞.

(4.8.31)

Having fixed an arbitrary point xo ∈ Σ, for each (p, ∞)-atom a on Σ we then write ∫ ∫   f (x) − f (xo ) a(x) dσ(x) f a dσ = Σ

Σ

= lim

i→∞

∫ Σ



 f jk i (x) − f jk i (xo ) a(x) dσ(x)

    = lim [ f jk i ], a = [ f ], a = i→∞

∫ Σ

f a dσ.

(4.8.32)

Above, the first equality is a consequence of the cancellation property of the atom, the second equality is implied by Lebesgue’s Dominated Convergence Theorem (with ∞ (Σ, σ)), the third the uniform domination provided by M |x − xo | α · |a(x)| ∈ Lcomp and final equalities comes from (4.6.4), while the fourth equality is guaranteed by (4.8.31). With (4.8.32) in hand, we may rely on Lemma 4.6.9 to conclude that

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

f− f is constant on Σ.

199

(4.8.33)

In turn, from (4.8.31) and (4.8.33) we conclude that [ f jk i ] → [ f ] weak-∗ in .  C α (Σ) ∼ as i → ∞. Consequently,     [ f jk i ], g −→ [ f ], g as i → ∞, (4.8.34) which contradicts (4.8.30). This finishes the proof of (4.8.27). Finally, the weakconvergence result recorded in (4.8.29) in the case when Σ is bounded is proved similarly.  Here is a companion result for Lemma 4.8.3, valid for Hölder functions. Lemma 4.8.5 Let Σ ⊆ Rn be a closed, unbounded set,  which is Ahlfors regintroduce ular, and abbreviate σ := H n−1 Σ. Also, pick p ∈ n−1 n , 1 and .α  1 α := (n − 1) p − 1 ∈ (0, 1). Given a real-valued function f ∈ C (Σ), define     fM := min max{ f , −M }, M = max min{ f , M }, −M for each M ∈ N. Then

.  lim [ fM ] = [ f ] weak-∗ in C α (Σ) ∼ .

M→∞

(4.8.35)

Specifically, with ·, · denoting the duality bracket between functions satisfying a homogeneous Hölder condition of order α on Σ, modulo constants, and the Hardy space H p on Σ (cf. Theorem 4.6.1),     for each g ∈ H p (Σ, σ) one has [ fM ], g → [ f ], g as M → ∞. (4.8.36) Moreover, a similar weak-∗ convergence result holds in the case when Σ is bounded, namely (4.8.37) lim fM = f weak-∗ in C α (Σ). M→∞

Proof This is a consequence of Lemma 4.8.4, bearing in mind [133, (7.3.13)(7.3.14)].  The next proposition further refines the compatibility result from Lemma 4.6.5. Proposition 4.8.6 Let Σ ⊆ Rn be a closed, unbounded set, which is Ahlfors regular, and abbreviate σ := H n−1 Σ. In this setting, suppose ∫ | f ||g| dσ < +∞. (4.8.38) f ∈ BMO(Σ, σ) and g ∈ H 1 (Σ, σ) are such that Σ

Then, with ·, · denoting the duality bracket between the John-Nirenberg space of functions of bounded mean oscillations on Σ, modulo constants, and the Hardy space H 1 on Σ (cf. Theorem 4.6.1), one has ∫   f g dσ. (4.8.39) [ f ], g = Σ

200

4 Hardy Spaces on Ahlfors Regular Sets

In particular, 

 [ f ], g =

∫ Σ

f g dσ for each f ∈ L ∞ (Σ, σ) and g ∈ H 1 (Σ, σ).

(4.8.40)

Moreover, similar formulas are valid in the case when Σ is bounded, this time without having to consider equivalence classes of functions (modulo constants). Proof If for each integer M ∈ N we define the function fM as in (4.8.22), then from [133, (7.4.102)], Lemma 4.8.3, and Lebesgue’s Dominated Convergence Theorem we have fM ∈ L ∞ (Σ, σ) ⊂ BMO(Σ, σ), . σ), lim [ fM ] = [ f ] weak-∗ in BMO(Σ,

M→∞

lim fM g = f g in

M→∞

(4.8.41)

L 1 (Σ, σ).

Having fixed some q ∈ (1, ∞), Theorem 4.4.1 guarantees the existence of a numerical sequence {λ j } j ∈N ∈  1 (N) along with a sequence {a j } j ∈N of (p, q)-atoms on Σ with the property that g N :=

N 

λ j a j converges to g in H 1 (Σ, σ) as N → ∞.

(4.8.42)

j=1

We may then write       [ f ], g = lim [ fM ], g = lim lim [ fM ], g N M→∞

= lim

M→∞

∫

lim

N →∞

∫ =

Σ

Σ

M→∞ N →∞

∫

fM g N dσ = lim fM g dσ M→∞

f g dσ.

Σ

(4.8.43)

The first equality in (4.8.43) is a consequence of the second property in (4.8.41), while the second equality in (4.8.43) is implied by (4.8.42), and the third equality in (4.8.43) follows from (4.6.8). The fourth equality in (4.8.43) may be justified based on the first property in (4.8.41) and the fact that, since H 1 (Σ, σ) embeds continuously into L 1 (Σ, σ), from (4.8.42) we also have that lim g N = g in L 1 (Σ, σ). The last N →∞

equality in (4.8.43) is simply the last formula in (4.8.41). This establishes (4.8.39). Finally, the version of the formula (4.8.39) in the case when Σ is bounded is proved in a similar fashion.  In the next proposition we refine the compatibility result from Lemma 4.6.6. Proposition 4.8.7 Let Σ ⊆ Rn be a closed, unbounded set, which is Ahlfors regular,  , and abbreviate σ := H n−1 Σ. Also, pick an arbitrary p ∈ n−1 n 1 and introduce   α := (n − 1) p1 − 1 ∈ (0, 1). In this setting, suppose

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

.

f ∈ C α (Σ) and g ∈ H p (Σ, σ) ∩ H 1 (Σ, σ) are such that

201

∫ Σ

| f ||g| dσ < +∞.

(4.8.44) Then, with ·, · denoting the duality bracket between functions satisfying a homogeneous Hölder condition of order α on Σ, modulo constants, and the Hardy space H p on Σ (cf. Theorem 4.6.1), one has ∫   f g dσ. (4.8.45) [ f ], g = Σ

In addition, a similar formula is valid in the case when Σ is bounded, this time working with inhomogeneous Hölder spaces and without having to consider equivalence classes of functions (modulo constants). Moreover, in the case when Σ is bounded there exists a finite constant C > 0, which depends only on the environment, with the property that   | f , g | ≤ C f C.α (Σ) · g H p (Σ,σ) for each f ∈ C α (Σ) ∫ (4.8.46) p 1 g dσ = 0. and g ∈ H (Σ, σ) ∩ H (Σ, σ) such that Σ

The reader is alerted to the fact that the inequality in (4.8.46) is finer than the generic manner in which the paring between a given function f ∈ C α (Σ) with an arbitrary g ∈ H p (Σ, σ) may be estimated according to (the last part of) Theorem 4.6.1 when Σ is bounded, since the estimate in (4.8.46) only involves the . semi-norm associated with the homogeneous Hölder space C α (Σ) (as opposed to the larger norm in C α (Σ)). Proof of Proposition 4.8.7 The argument is similar to that in the  proof of Proposi tion 4.8.6. To get started, for each M ∈ N consider fM := min max{ f , −M }, M . Then from [133, (7.3.13)], Lemma 4.8.5, and Lebesgue’s Dominated Convergence Theorem we have

.

fM ∈ L ∞ (Σ, σ) ∩ C α (Σ) = C α (Σ),

.  lim [ fM ] = [ f ] weak-∗ in C α (Σ) ∼,

M→∞

(4.8.47)

lim fM g = f g in L 1 (Σ, σ).

M→∞

Next, Theorem 4.4.3 ensures the existence of a numerical sequence {λ j } j ∈N ∈  1 (N) along with a sequence {a j } j ∈N of (p, 2)-atoms on Σ with the property that g N :=

N 

λ j a j converges to g as N → ∞ both in H p (Σ, σ) and in H 1 (Σ, σ).

j=1

(4.8.48) In particular, since H 1 (Σ, σ) embeds continuously into L 1 (Σ, σ), we also have that g N converges to g as N → ∞ in L 1 (Σ, σ).

(4.8.49)

202

4 Hardy Spaces on Ahlfors Regular Sets

We may now write       [ f ], g = lim [ fM ], g = lim lim [ fM ], g N M→∞

= lim

M→∞

M→∞ N →∞

∫

lim

N →∞

Σ

∫

∫



fM g N dσ = lim

M→∞

Σ

fM g dσ =

Σ

f g dσ. (4.8.50)

The first equality in (4.8.50) is a consequence of the second property in (4.8.47), while the second equality in (4.8.50) is implied by (4.8.48), and the third equality in (4.8.50) follows from (4.6.8). The fourth equality in (4.8.50) follows from (4.8.49) and the first property in (4.8.47). The last equality in (4.8.50) is simply the last formula in (4.8.47). This establishes (4.8.45). p 1 Consider next the case ∫ when Σ is bounded. If g ∈ H (Σ, σ) ∩ H (Σ, σ) and α f ∈ C (Σ) are such that Σ | f ||g| dσ < ∞, then the fact that 

 f, g =

∫

Σ

f g dσ

(4.8.51)

may be justified in a very similar fashion to the proof of formula (4.8.45). As regards the claim∫ in (4.8.46), suppose now that f ∈ C α (Σ) and g ∈ H p (Σ, σ) ∩ H 1 (Σ, σ) satisfies Σ g dσ = 0. As before, invoke Theorem 4.4.3 to produce a numerical sequence {λ j } j ∈N ∈  1 (N) satisfying ∞ 

|λ j | p

1/p

≤ Cg H p (Σ,σ)

(4.8.52)

j=1

for some finite constant C > 0 which depends only on the ambient, along with a ∫sequence {a j }∫j ∈N of (p, 2)-atoms on Σ such that (4.8.48)-(4.8.49) hold. In particular, g dσ → Σ g dσ = 0 as N → ∞. Even though the constant function a ≡ σ(Σ)−1 Σ N is now considered an atom, this convergence ensures that none of the atoms a j is this special atom. Consequently, each a j has vanishing moment which, in turn, permits us to estimate  ∫   f a j dσ  ≤ C f C.α (Σ) for each j ∈ N, (4.8.53)  Σ

where the constant C ∈ (0, ∞) depends only on the ambient. At this stage, we may write ∫ ∞ ∞   ∫          f , g  =  λj f a j dσ  ≤ |λ j |  f a j dσ  j=1

≤C

∞  j=1

Σ

Σ

j=1

∞

1/p

 |λ j |  f C.α (Σ) ≤ C |λ j | p  f C.α (Σ)

≤ C f C.α (Σ) g H p (Σ,σ),

j=1

(4.8.54)

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

203

using the last part of Theorem 4.6.1, (4.8.53), the fact that p < 1, and (4.8.52). This establishes (4.8.46), and finishes the proof of Proposition 4.8.7.  A useful version of Proposition 4.8.7, in which the membership of g to H 1 (Σ, σ) 1 (Σ, σ) (at the (cf. (4.8.44)) is replaced by the less restrictive membership to Lloc expense of strengthening the assumptions on f ) is discussed next. Proposition 4.8.8 Let Σ ⊆ Rn be a closed, unbounded set, which is Ahlfors regular,  , and abbreviate σ := H n−1 Σ. Also, pick an arbitrary p ∈ n−1 n 1 and introduce 1  α := (n − 1) p − 1 ∈ (0, 1). In this setting, suppose

.

1 (Σ, σ) f ∈ C α (Σ) ∩ Lip(Σ) and g ∈ H p (Σ, σ) ∩ Lloc ∫ are such that | f ||g| dσ < +∞.

(4.8.55)

Σ

Then, with ·, · denoting the duality bracket between functions satisfying a homogeneous Hölder condition of order α on Σ, modulo constants, and the Hardy space H p on Σ (cf. Theorem 4.6.1), one has ∫   f g dσ. (4.8.56) [ f ], g = Σ

In addition, a similar formula is valid when Σ is bounded (in which case, the last property in (4.8.55) becomes redundant), this time working with inhomogeneous Hölder spaces and without having to consider equivalence classes of functions (modulo constants). In relation to the hypotheses made on f in (4.8.55) it is relevant to observe that (compare with [133, (7.3.15)]) " if f ∈ Lip(Σ) and sup | f | < +∞ then f ∈ C α (Σ). (4.8.57) Σ

0 p.

(4.8.61)

∫ H p, q (Σ,σ)

=

Σ

f g dσ

(4.8.62)

provided (with the inclusion considered in the sense of (4.7.2))

.  ∗ f ∈ Lip(Σ) ∩ C α0 (Σ) ∩ L p1 (Σ, σ) → H p,q (Σ, σ) ∫ p,q 1 and g ∈ H (Σ, σ) ∩ Lloc (Σ, σ) with | f ||g| dσ < +∞.

(4.8.63)

Σ

Proof Let f , g be as in (4.8.63) and decompose g = g0 +g1 , with g0 ∈ H p0 (Σ, σ) and g1 ∈ L p1 (Σ, σ) (recall that H p,q (Σ, σ) embeds into H p0 (Σ, σ) + L p1 (Σ, σ)). Given 1 (Σ, σ), it follows that actually that by assumption f ∈ L p1 (Σ, σ) and g ∈ Lloc ∫ p0 1 | f ||g0 | dσ < +∞ (4.8.64) g0 ∈ H (Σ, σ) ∩ Lloc (Σ, σ) and Σ

since

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

∫

∫ Σ

| f ||g0 | dσ ≤

Σ

205

  | f | |g| + |g1 | dσ

∫ ≤

Σ

| f ||g| dσ +  f  L p1 (Σ,σ) g1  L p1 (Σ,σ) < +∞.

(4.8.65)

Based on (4.8.64), (4.7.3), and Proposition 4.8.8 applied here   with α, g, p replaced, respectively, by α0 , g0 , p0 (note that we have p0 ∈ n−1 , 1 by design; cf. (4.8.61)) n we may then write ∫ ∫ ∫   f g1 dσ + f g0 dσ = f g dσ, (4.8.66) (H p, q (Σ,σ))∗ f , g H p, q (Σ,σ) = Σ

Σ

Σ



as wanted.

Let us formally record the fact that the duality pairing between any BMO function and any H 1 -molecule agrees with the integral pairing of said functions. Corollary 4.8.10 Let Σ ⊆ Rn be a closed, unbounded set, which is Ahlfors regular, and abbreviate σ := H n−1 Σ. Then for any function f ∈ BMO(Σ, σ) and any H 1 -molecule m on Σ (cf. Definition 4.5.1) one has ∫   f m dσ, (4.8.67) [ f ], m = Σ

where ·, · denotes the duality bracket between the John-Nirenberg space of functions of bounded mean oscillations on Σ, modulo constants, and the Hardy space H 1 on Σ (cf. Theorem 4.6.1). As a consequence, ∫   ! one has [ f ], g = lim N j=1 λ j Σ f m j dσ whenever N →∞

f ∈ BMO(Σ, σ), g ∈ H 1 (Σ, σ), {λ j } j ∈N ∈  1 (N), and {m j }!j ∈N is a sequence of H 1 -molecules on Σ, such that 1 g= ∞ j=1 λ j m j in H (Σ, σ).

(4.8.68)

Moreover, similar results are valid in the case when Σ is bounded, this time without having to consider equivalence classes of functions (modulo constants). Proof This is an immediate consequence of Proposition 4.8.6 and Lemma 4.5.4.  There is also a version of Corollary 4.8.10 for Hölder functions and H p Hardy spaces with p < 1, of the sort described below. Corollary 4.8.11 Let Σ ⊆ Rn be a closed, unbounded set, which  is Ahlfors regular, and abbreviate σ := H n−1 Σ. Also, pick an arbitrary p ∈ n−1 n , 1 and introduce   α := (n − 1) p1 − 1 ∈ (0, 1). Finally, choose q ∈ (1, ∞) along with ε > α/(n − 1).

.

Then for any function f ∈ C α (Σ) and any (p, q, ε)-molecule m on Σ (cf. Definition 4.5.1) one has ∫   [ f ], m = f m dσ, (4.8.69) Σ

206

4 Hardy Spaces on Ahlfors Regular Sets

where ·, · stands for the duality bracket between functions satisfying a homogeneous Hölder condition of order α on Σ, modulo constants, and the Hardy space H p on Σ (cf. Theorem 4.6.1). As a consequence, ∫   ! one has [ f ], g = lim N j=1 λ j Σ f m j dσ whenev-

.

N →∞

er f ∈ C α (Σ), g ∈ H p (Σ, σ), {λ j } j ∈N ∈  1 (N), and {m j } j ∈N is a sequence of (p, q, ε)-molecules on Σ, with ! p the property that g = ∞ j=1 λ j m j in H (Σ, σ).

(4.8.70)

In addition, similar results are valid in the case when Σ is bounded, this time working with the inhomogeneous Hölder space and without having to consider equivalence classes of functions (modulo constants). Proof All claims are readily justified with the help of Proposition 4.8.7 and Lemma 4.5.5, bearing in mind that any (p, q, ε)-molecule on Σ is a scalar multiple of a (1, q, ε)-molecule on Σ and, hence, belongs to the Hardy space H 1 (Σ, σ).  Lemma 4.8.12, discussed below, is a weak-∗ density result which is relevant when studying Hardy-based Sobolev spaces, introduced later. Lemma 4.8.12 Suppose Σ is a closed set in Rn which is Ahlfors regular and let n−1 ∈ n−1 σ := H n−1 Σ. Fix α ∈ (0, 1) and define p = n−1+α n , 1 . Then for each .α ∞ n g ∈ C (Σ) there exists a sequence {φ j } j ∈N ⊆ Cc (R ) with the following property. .     If Σ is unbounded, then φ j Σ , viewed as an element in C α (Σ) ∼, converges to [g] .α  weak-∗ in C (Σ) ∼ as j → ∞, i.e.,     φ j Σ , h −→ [g], h as j → ∞ for every h ∈ H p (Σ, σ), (4.8.71) where the pairings are understood in the sense of Theorem 4.6.1. Also, if Σ is bounded then, with a similar interpretation,    φ j Σ, h −→ g, h as j → ∞ for every h ∈ H p (Σ, σ). (4.8.72)  . Proof Fix some G ∈ C α (Rn ) such that GΣ = g (such an extension always exists; see the discussion in [138]). Also consider a cut-off function ξ ∈ Cc∞ (Rn ) with supp ξ ⊆ B(0, 2), 0 ≤ ξ ≤ 1, ξ ≡ 1 on B(0, 1). Then, having picked some x0 ∈ Σ, for each ε ∈ (0, ∞) define ξε (x) := ξ ε(x − x0 ) for x ∈ Rn . Fix ε ∈ (0, 1). We claim that there exists some constant C ∈ (0, ∞), independent of ε (actually depending only on inf Σ |g| and GC.α (Rn ) ) such that   (ξε G)(x) − (ξε G)(x) ≤ C|x − y| α for every x, y ∈ Rn . We split the proof of (4.8.73) into four cases. Case I: Assume x ∈ B(x0, 2/ε) and y ∈ B(x0, 4/ε). Then

(4.8.73)

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

207

       (ξε G)(x) − (ξε G)(x) ≤ ξε (x)G(x) − G(y) + |G(y) − G(x0 )| ξε (x) − ξε (y)   + |g(x0 )| ξε (x) − ξε (y). (4.8.74) Observe that, the properties of ξ, the definition of ξε , and the current assumptions on x and y, imply     ξε (x) − ξε (y) ≤ |x − y| α |x − y| 1−α sup ∇ξε  ≤ C|x − y| α ε α−1 · Cε Rn

= C|x − y| α ε α .

(4.8.75)

.

Making use of (4.8.75), the fact that G ∈ C α (Σ), that 0 ≤ ξε ≤ 1, that |y − x0 | < 4/ε, and that |ε| ≤ 1, we may now return to (4.8.74) and further estimate   (ξε G)(x) − (ξε G)(x) ≤ C|x − y| α + C|y − x0 | α |x − y| α ε α + |g(x0 )| · C|x − y| α ε α ≤ C|x − y| α,

(4.8.76)

for some constant C ∈ (0, ∞) independent of ε, x, and y. This proves (4.8.73) in the current case. Case II: Assume x ∈ B(x0, 2/ε) and y ∈ Rn \ B(x0, 4/ε). Then |x − y| ≥ 2/ε and ξε (y) = 0. As such,     (ξε G)(x) − (ξε G)(x) = (ξε G)(x) ≤ |G(x)| ≤ |G(x) − G(x0 )| + |g(x0 )| ≤ C|x − x0 | α + |g(x0 )|ε α ·

1 C ≤ α ≤ C|x − y| α, (4.8.77) εα ε

for some constant C ∈ (0, ∞) independent of ε, x, and y. Hence, (4.8.73) holds in the current case as well. Case III: Assume y ∈ B(x0, 2/ε) and x ∈ Rn arbitrary. Since the estimate in (4.8.73) is symmetric in x and y, the desired conclusion follows from what we proved in Case I and Case II. Case IV: Assume x, y ∈ Rn \ B(x0, 2/ε). In this scenario ξε (x) = 0 = ξε (y), so (4.8.73) is trivial. Having completed the proof of (4.8.73), bring in ∫a mollifier of the following sort. Pick a non-negative function θ ∈ Cc∞ (Rn ) with Rn θ dL n = 1 and, for each ε ∈ (0, 1), set θ ε (x) := ε −n θ(x/ε) for every x ∈ Rn . Finally, define   φε := ξε G ∗ θ ε for each ε ∈ (0, 1). (4.8.78) Then each φε belongs to Cc∞ (Rn ). Moreover, for every x, y ∈ Rn we may estimate, thanks to (4.8.73) and the qualities of θ,

208

4 Hardy Spaces on Ahlfors Regular Sets

  φε (x) − φε (y) ≤

∫ Rn

  (ξε G)(x − z) − (ξε G)(y − z)θ ε (z) dz

≤ C|x − y| α, for each ε ∈ (0, 1),

(4.8.79)

where the constant C ∈ (0, ∞) is as in (4.8.73) (hence, independent of ε). Thus, . φε ∈ C α (Rn ) and (4.8.79) shows that, for some constant C ∈ (0, ∞) independent of ε, we have      φε   .α ≤ φε  .α n ≤ C, for each ε ∈ (0, 1). (4.8.80) Σ C (Σ) C (R ) Also, given any compact set K ⊂ Σ, observe that ξε becomes identically 1 near K when ε ∈ (0, 1) is sufficiently small. Together with (4.8.73), this also permits us to conclude that there exists some small εK ∈ (0, 1) such that ∫     φε (x) − g(x) ≤ (ξε G)(x − z) − (ξε G)(x)θ ε (z) dz Rn

≤ Cε α whenever ε ∈ (0, εK ).  φε Σ converges uniformly on arbitrary bounded subsets of Σ to the function g, as ε −→ 0+ .

As a result,

(4.8.81)

(4.8.82)

Suppose Then (4.8.80) implies that the sequence    next that Σ is unbounded. .   ∗ φε Σ ε ∈(0,1) is bounded in C α (Σ) ∼= H p (Σ, σ) . Granted this and (4.8.82), Lemma 4.8.4 applies and yields (4.8.71). Finally, when Σ is bounded, the end-game in the proof is similar. This time, in addition to (4.8.80), we also have    (4.8.83) sup φε Σ  ≤ sup |G| < +∞ for each ε ∈ (0, 1), 

Σ

U

where Σ, y ∈ supp θ, |t| ≤ 1} is a compact set. Thus,    U := {x + t y : x ∈ ∗ φε Σ ε ∈(0,1) is bounded in C α (Σ) = H p (Σ, σ) . From this point on, the same type of argument as before, based on Lemma 4.8.4, then justifies (4.8.72).  It is also useful to have a weak-∗ convergence result involving duals of Lorentzbased Hardy spaces, of the sort described in our next lemma. Lemma 4.8.13 Let Σ ⊆ Rn be a closed set which is Ahlfors regular, and  abbreviate  σ := H n−1 Σ. Pick α0 ∈ (0, 1), introduce p0 := (n − 1)/(n − 1 + α0 ) ∈ n−1 n , 1 , and select p1 ∈ (1, ∞). Also, fix some p ∈ (p0, p1 ) along . with some q ∈ (0, ∞). In this setting, consider a bounded sequence { f j } j ∈N ⊆ C α0 (Σ) ∩ L p1 (Σ, σ), i.e., satisfying sup  f j C.α0 (Σ) < +∞ and sup  f j  L p1 (Σ,σ) < +∞, j ∈N

and make the assumption that

j ∈N

(4.8.84)

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings

209

f (x) := lim f j (x) exists for each x ∈ Σ. j→∞

(4.8.85)

.

the function f belongs to C α0 (Σ) ∩ L p1 (Σ, σ) and, when viewed in  Then ∗ p,q is the weak-∗ limit of { f j } j ∈N , itself regarded H (Σ, σ) (cf. Lemma 4.7.1), ∗ as a sequence in H p,q (Σ, σ) (again, in the sense of Lemma 4.7.1). Specifically,

(H p, q (Σ,σ))∗



for each g ∈ H p,q (Σ, σ) one has   f j , g H p, q (Σ,σ) → (H p, q (Σ,σ))∗ f , g H p, q (Σ,σ) as j → ∞. 

(4.8.86)

Proof Let M ∈ [0, ∞) denote the first supremum in (4.8.84). Consequently, for each j ∈ N we have | f j (x) − f j (y)| ≤ M |x − y| α0 for all x, y ∈ Σ. Passing to limit j → ∞ . then yields | f (x)− f (y)| ≤ M |x − y| α0 for all x, y ∈ Σ, which proves that f ∈ C α0 (Σ). Next, we make the claim that f belongs to L p1 (Σ, σ) and is the weak-∗ limit of the sequence { f j } j ∈N in L p1 (Σ, σ).

(4.8.87)



To justify this, fix g ∈ L p1 (Σ, σ) where p1 ∈ (1, ∞) is such that 1/p1 + 1/p1 = 1 and . Since from (4.8.84) we know consider an arbitrary sub-sequence { f jk }k ∈N  of { f j } j ∈N ∗ that { f jk }k ∈N is bounded in L p1 (Σ, σ) = L p1 (Σ, σ) , and since [133, (3.6.27)] en sures that L p1 (Σ, σ) is a separable Banach space, the Sequential Banach-Alaoglu Theorem (recalled in [133, (3.6.22)]) permits us to extract a sub-sub-sequence { f jk i }i ∈N which converges weak-∗ in L p1 (Σ, σ) to some function h ∈ L p1 (Σ, σ). As such, for each φ ∈ Cc∞ (Rn ) we may write ∫ ∫ ∫ h φ dσ = lim f jk i φ dσ = f φ dσ, (4.8.88) i→∞

Σ

Σ

Σ

where the last equality is a consequence of Lebesgue’s Dominated Convergence Theorem, whose applicability is presently guaranteed by (4.8.85) and the first property in (4.8.84), as they together imply that | f j (x)| ≤ M ·

sup

y ∈suppφ∩Σ

|y − x0 | α0 + sup | fk (x0 )| ≤ C < +∞, k ∈N

(4.8.89)

for each x ∈ suppφ ∩ Σ and each j ∈ N, where x0 ∈ Σ is some fixed reference point. In turn, from (4.8.88) and [133, (3.7.23)] we conclude that h = f on Σ. This has two consequences of interest. First, it implies that f ∈ L p1 (Σ, σ). Second, it allows us to write (for the function g selected earlier) ∫ ∫ f jk i g dσ = f φ dσ. (4.8.90) lim i→∞

Σ

Σ

Let us summarize our progress The  ∫ at this stage.   ∫argumentpresented so far shows that for any sub-sequence Σ f jk g dσ of Σ f j g dσ there exists a subk ∈N

j ∈N

210

4 Hardy Spaces on Ahlfors Regular Sets

∫



sub-sequence Σ f jk i g dσ with the property that (4.8.90) holds. Ultimately, i ∈N from this we conclude that actually ∫ ∫ f j g dσ = f g dσ, (4.8.91) lim j→∞

Σ

Σ



and since g ∈ L p1 (Σ, σ) has been chosen arbitrarily it follows that the sequence { f j } j ∈N is weak-∗ convergent in L p1 (Σ, σ) to the function f . The proof of the claims made in (4.8.87) is therefore complete. Going further, consider again an arbitrary sub-sequence { f jk }k ∈N of { f j } j ∈N and pick some arbitrary g ∈ H p,q (Σ, σ) (which is going to come into play a little later; see . p1 in the space C α0 (Σ) (4.8.94)). Since { f j } j ∈N is assumed to be bounded  ∩ L (Σ, σ)  . p (equipped with the natural quasi-norm max  · C α0 (Σ),  ·  L 1 (Σ,σ) ; cf. (1.3.3)),  p,q ∗ and since the latter space is continuously embedded into  p,q  ∗ H (Σ, σ) (cf. (4.7.2)), we conclude that { f j } j ∈N is bounded in H (Σ, σ) . Given that we know from Proposition 4.4.5 that the Lorentz-based Hardy space H p,q (Σ, σ) is a separable quasiBanach space, we may once more rely on the Sequential Banach-Alaoglu Theorem (cf. [133, (3.6.22)]) to extract a sub-sub-sequence { f jk i }i ∈N which converges weak-∗ ∗  ∗  in H p,q (Σ, σ) to some functional Λ ∈ H p,q (Σ, σ) . For any (p0, p1 )-atom a on Σ we may then write     (H p, q (Σ,σ))∗ Λ, a H p, q (Σ,σ) = lim (H p, q (Σ,σ))∗ f jk i , a H p, q (Σ,σ) i→∞

= lim

i→∞

= lim

i→∞

∫ =

Σ

(H p0 (Σ,σ))∗



f jk i , a



H p0 (Σ,σ)

∫ Σ

f jk i a dσ

  f a dσ = (H p, q (Σ,σ))∗ f , a H p, q (Σ,σ) .

(4.8.92)

The first equality above is implied by the fact that { f jk i }i ∈N converges weak-∗ ∗  ∗  in H p,q (Σ, σ) to Λ ∈ H p,q (Σ, σ) , and the membership of a to the space H p0 (Σ, σ) ∩ L p1 (Σ, σ) ⊆ H p,q (Σ, σ) (cf. (4.3.145)). The second equality in (4.8.92) is a consequence of (4.7.3)-(4.7.4). The third equality in (4.8.92) is seen from (4.6.4)(4.6.5). The fourth equality in (4.8.92) may be justified using Lebesgue’s Dominated Convergence Theorem, whose applicability is currently ensured by (4.8.85) and (4.8.89). Finally, the fifth equality in (4.8.92) may be justified much as the second equality in (4.8.92). Having proved (4.8.92), bring on the family F used in the proof of Proposition 4.4.2, constructed as in (4.4.15) for an exponent r ∈ (p1, ∞). From (4.8.92) and the fact that any function in F may be written as a finite linear combination of (p0, p1 )-atoms on Σ we conclude that

4.8 Weak-∗ Convergence and More on the Compatibility of Pairings (H p, q (Σ,σ))∗



Λ, h



H p, q (Σ,σ)



= (H p, q (Σ,σ))∗ f , h



H p, q (Σ,σ)

211

for each h ∈ F . (4.8.93)

In concert with (4.4.122), this further gives (with g as above)     (H p, q (Σ,σ))∗ Λ, g H p, q (Σ,σ) = (H p, q (Σ,σ))∗ f , g H p, q (Σ,σ) .

(4.8.94)

 ∗ Thus, necessarily, Λ = f in Hp,q (Σ, σ) . In summary, the proof so far shows   of the sequence that for any given sub-sequence (H p, q (Σ,σ))∗ f jk , g H p, q (Σ,σ) k ∈N     , wre are able to find a sub-sub-sequence (H p, q (Σ,σ))∗ f j , g H p, q (Σ,σ) j∈N    with the property that (H p, q (Σ,σ))∗ f jk i , g H p, q (Σ,σ) lim

i→∞

(H p, q (Σ,σ))∗



f jk i , g



i ∈N

H p, q (Σ,σ)

  = (H p, q (Σ,σ))∗ f , g H p, q (Σ,σ) .

From this we eventually conclude that, in fact,     lim (H p, q (Σ,σ))∗ f j , g H p, q (Σ,σ) = (H p, q (Σ,σ))∗ f , g H p, q (Σ,σ) j→∞

(4.8.95)

(4.8.96)

we finally see that the sequence and since g ∈ H p,q (Σ, σ) has been  chosen arbitrarily ∗  { f j } j ∈N is weak-∗ convergent in H p,q (Σ, σ) to the function f . It is also useful to have a result pertaining to the compatibility of the paring of Lorentz-based Hardy spaces and their duals, on the one hand, and the distributional pairing, on the other hand, of the sort described in the lemma below. Lemma 4.8.14 Let Σ be a closed set in Rn which is Ahlfors regular and abbreviate   , ∞ and some q ∈ (0, ∞). σ := H n−1 Σ. Fix some integrability exponent p ∈ n−1 n  p,q (Σ, σ) → Lip (Σ) and each given Then for each given distribution f ∈ H c  ∗ function g ∈ Lipc (Σ) → H p,q (Σ, σ) (with the inclusion considered in the sense of Lemma 4.7.1; cf. (4.7.2)) one has     (4.8.97) (Lip c (Σ)) f , g Lip c (Σ) = H p, q (Σ,σ) f , g (H p, q (Σ,σ))∗ .   Proof Consider first the case when we have p ∈ n−1 n , 1 . In such a scenario, r ∈ (1, ∞) then bring back the set F first used in the proof of Proposition 4.4.2 (cf. (4.4.15)). Thanks to (4.4.122) and (4.2.29), it suffices to show that (4.8.97) holds if f ∈ F . However, whenever this is the case we may write ∫     f , g = f g dσ = H p, q (Σ,σ) f , g (H p, q (Σ,σ))∗ (4.8.98) (Lip c (Σ)) Lip c (Σ) Σ

thanks to [133, (4.1.47)]and (4.7.3)-(4.7.4). Suppose now that p ∈ (1, ∞), and pick p0 ∈ (1, p) along with p1 ∈ (p, ∞). We run the same argument as before, with one adjustment. Specifically, in place of the family F this time we consider L p0 (Σ, σ) ∩ L p1 (Σ, σ) which, according to [133,  (6.2.51)] and (4.2.30), continues to be a dense subset of H p,q (Σ, σ).

212

4 Hardy Spaces on Ahlfors Regular Sets

4.9 More on H p Versus L p : The Filtering Operator On a given closed Ahlfors regular set in Rn , it turns out that the identity map between the Hardy scale H p and the Lebesgue scale L p when p ∈ (1, ∞) (cf. [133, (3.6.27)]) may  be further extended uniquely to a linear and bounded mapping in the the identity map thus extended as an operator range p ∈ n−1 n , 1 . We shall refer to  n−1 p p from H into L with p ∈ n , ∞ as the L p -filtering operator. Details are contained in the theorem below. Theorem 4.9.1 Suppose Σ ⊆ Rn is a closed Ahlfors regular set. Let σ := H n−1  Σ and consider a family of kernels St : Σ × Σ → R indexed by t ∈ 0, diam Σ and satisfying (4.2.14). Then the limit6   (H f )(x) := lim+ (H p (Σ,σ))∗ St (x, ·), f H p (Σ,σ) t→0 (4.9.1) for each f ∈ H p (Σ, σ) with n−1 n < p < ∞ exists for σ-a.e. point x ∈ Σ and is unambiguously defined7. Moreover, the assignment f → H f induces a well-defined linear and bounded operator   H : H p (Σ, σ) −→ L p (Σ, σ) for each p ∈ n−1 (4.9.2) n ,∞ ,   and the operators H associated with various values of p in n−1 n , ∞ are compatible with one another. Also,   1 (Σ, σ) with p ∈ n−1 , ∞ , hence H f = f whenever f ∈ H p (Σ, σ) ∩ Lloc n (4.9.3) in particular for each function f ∈ H p (Σ, σ) with 1 ≤ p < ∞, properties (4.9.2)-(4.9.3) determine uniquely the operator H, and   p p if p ∈ n−1 n , 1 then the operator H : H (Σ, σ) −→ L (Σ, σ) acts p on any given f ∈ H (Σ, σ) according to H f = lim f j in L p (Σ, σ) j→∞

p,q

provided f = lim f j in H p (Σ, σ) for some { f j } j ∈N ⊆ Hfin (Σ, σ) with q ∈ (1, ∞).

(4.9.4)

j→∞

Finally, the operator (4.9.2) further induces well-defined linear and bounded mappings   H : H p,q (Σ, σ) −→ L p,q (Σ, σ) for p ∈ n−1 (4.9.5) n , ∞ and q ∈ (0, ∞], (henceforth referred to as L p,q -filtering operators) according to 6 with the convention that St (x, ·) in (4.9.1) by [St (x, ·)], its class modulo constants, in   is replaced the case when Σ is unbounded and p ∈ n−1 , 1 n   7 in the sense that it is not affected by the particular index p ∈ n−1 n , ∞ labeling the Hardy space to which f happen to belong to

4.9 More on H p Versus L p : The Filtering Operator



(H f )(x) = lim+ Lipc (Σ) St (x, ·), f t→0

for each f ∈ H p,q (Σ, σ) with



213

at σ-a.e. x ∈ Σ,

(Lip c (Σ))

n−1 n

< p < ∞ and q ∈ (0, ∞]

(4.9.6)

which are compatible with one another as well as with the operator in (4.9.2), and 1 (Σ, σ) H f = f if f ∈ H p,q (Σ, σ) ∩ Lloc  n−1  with p ∈ n , ∞ and q ∈ (0, ∞].

(4.9.7)

We wish to note that while H in (4.9.5) becomes the identification of H p,q (Σ, σ) with L p,q (Σ, σ) when 1 < p < ∞ and 0 < q ≤ ∞ (cf. (4.2.30)), this operator is not injective in the range p ≤ 1. To substantiate this  claim, recall from (4.2.34) that for each point y ∈ Σ the Dirac distribution δy ∈ Lipc (Σ) belongs to H 1,∞ (Σ, σ). Then, on account of (4.9.6) and (4.2.14), we may write   (Hδy )(x) = lim+ Lipc (Σ) St (x, ·), δy (Lipc (Σ)) = lim+ St (x, y) = 0 (4.9.8) t→0

t→0

first for σ-a.e. x ∈ Σ \ {y}, for σ-a.e. x ∈ Σ if n ≥ 2.  hence  ultimately p -filtering operator (4.9.1)-(4.9.2) fails to be Likewise, when p ∈ n−1 , 1 , the L n injective. Indeed, according to (4.2.17), for each two  distinct points x0, x1 ∈ Σ we , 1 and (4.9.8) implies (assuming have δx0 − δx1 ∈ H p (Σ, σ) for every p ∈ n−1 n n ≥ 2) that (4.9.9) H(δx0 − δx1 ) = 0 at σ-a.e. point on Σ. After this digression, we now turn to the proof of Theorem 4.9.1. 1 (Σ, σ) with p ∈ Proof of Theorem 4.9.1 Suppose f ∈ H p (Σ, σ) ∩ Lloc Then for σ-a.e. x ∈ Σ we have   ∫      lim+ St (x, y) f (y) dσ(y) − f (x)   t→0 Σ

 n−1 n

 ,∞ .

∫ ≤ lim sup t→0+

Σ

St (x, y)| f (y) − f (x)| dσ(y)

⨏

≤ C lim sup t→0+

| f (y) − f (x)| dσ(y) = 0,

(4.9.10)

B(x,Ct)∩Σ

thanks to (4.2.14), the current assumptions on Σ, and [133, Proposition 7.4.4] (whose applicability in the present setting is ensured by [133, (3.6.26)]). We may then rely on (4.6.10), (4.2.14), part (a) in Lemma 4.6.4, and [133, Proposition 4.1.4] to write     lim+ (H p (Σ,σ))∗ St (x, ·), f H p (Σ,σ) = lim+ Lipc (Σ) St (x, ·), f (Lipc (Σ)) t→0

t→0

= lim+ t→0

∫ Σ

St (x, y) f (y) dσ(y)

= f (x) at σ-a.e. x ∈ Σ,

(4.9.11)

214

4 Hardy Spaces on Ahlfors Regular Sets

where the last equality follows from (4.9.10). From (4.9.11) (and also keeping in mind [133, (3.6.27)], (4.2.10)) the claims in (4.9.3) follow. ∞), consider Having shown that H acts as the identity on H p (Σ, σ) when p ∈ [1,  n−1  1 

next the case when p ∈ n , 1 . Having fixed some γ ∈ (n − 1) p − 1 , 1 , we may rely on Lemma 4.6.4, (4.1.7), and (4.2.14) to conclude that there is a constant C ∈ (0, ∞) with the property that for every f ∈ H p (Σ, σ) we have       sup (H p (Σ,σ))∗ St (x, ·), f H p (Σ,σ)  ≤ C fγ (x) at each x ∈ Σ. (4.9.12) t>0

In concert with (4.2.6), this implies that there exists some C ∈ (0, ∞) such that8         ≤ C f  H p (Σ,σ) (4.9.13)  sup (H p (Σ,σ))∗ St (x, ·), f H p (Σ,σ)  p t>0

L x (Σ,σ)

for all f ∈ H p (Σ, σ). At this stage in the proof, the idea is to invoke [133, Proposition 6.2.11] in the following concrete context: X := Σ × 0, diam Σ viewed as a topological space with the topology inherited from Rn ×R, the set X := Σ×{0} ⊆ X , the measure μ := according to [133, Lemma 3.6.4],  σ on  X = Σ × {0} ≡ Σ which,  is complete, Γ (x, 0) := {x} × (0, diam Σ ⊆ X \ X for each (x, 0) ∈ X (so that the condition in [133, (6.2.71)] is actually satisfied at every point), the space Y := H p (Σ, σ) equipped with the quasi-norm  · Y :=  ·  H p (Σ,σ) , the linear operator T mapping mapping vectors from Y into functions defined on X \ X according  p ∗ St (x, ·), f H p (Σ,σ) for each f ∈ Y = H (Σ, σ) and each to (T f )(x, t) := (H p (Σ,σ))   p,q (x, t) ∈ X \ X = Σ× 0, diam Σ , and with V := Hfin (Σ, σ) for some fixed q ∈ (1, ∞) which, according to (4.4.114), is a dense linear subspace of Y . For these choices, (4.9.13) implies that the maximal operator T associated with T as in [133, (6.2.73)] satisfies [133, (6.2.74)]. Also, (4.9.11) used with p := q guarantees that the condition in [133, (6.2.75)] holds for every f ∈ V. Granted these, we may rely on [133, Proposition 6.2.11] to ultimately conclude that the limit in (4.9.1) exists at σ-a.e. point x ∈ Σ. In turn, the existence of this limit together with (4.9.13) proves that the operator (4.9.1)-(4.9.2) is well-defined, linear, and bounded.   Furthermore, that the operators H associated with various values of p ∈ n−1 n ,∞ are compatible with one another is a consequence of (4.9.1) and the compatibility results from Lemmas 4.6.4-4.6.8.  Next, recall from (4.9.3) that H f = f for each  p,q , 1 and q ∈ (1, ∞). In concert with the continuity f ∈ Hfin (Σ, σ) with p ∈ n−1 n of (4.9.2), this then yields (4.9.4). Also, that properties (4.9.2)-(4.9.3) determine H uniquely is a consequence of the density result in (4.4.114). At this stage, the claims about (4.9.5) follow from what we have proved so far and real-interpolation (cf. (4.3.3), [133, (6.2.48)], as well as Proposition 1.3.7). That the operator (4.9.5) acts according to (4.9.6) is a consequence of (4.9.1), Lemma 4.6.4, and (1.3.39). Finally, (4.9.7) may be justified much as (4.9.3).    n−1 Given a closed Ahlfors regular set Σ ⊆ Rn along with some exponent p ∈ n , 1 , in general there is no set-theoretic relationship between the Hardy scale H p (Σ, σ) 8 with  ·  L xp (Σ, σ) indicating that the L p quasi-norm on Σ is taken in the variable x

4.9 More on H p Versus L p : The Filtering Operator

215

and the Lebesgue space L p (Σ, σ) (where, as usual, σ := H n−1 Σ). This being said, Theorem 4.9.1 permits us to canonically associate to any linear and bounded

from H p (Σ, σ) operator T : H p (Σ, σ) → H p (Σ, σ) a linear and bounded operator T

:= H ◦ T. into L p (Σ, σ) via T

Chapter 5

Banach Function Spaces, Extrapolation, and Orlicz Spaces

In this chapter we begin by providing a definition which is more inclusive than that of a “standard” Banach function space (as traditionally used in the literature; cf. e.g., [14]), and indicate that a significant portion of the classical theory goes through for this more general brand, which we dub Generalized Banach Function Spaces. This is done in §5.1. The relevance of this extension is that a variety of scales of spaces of interest, such as Morrey spaces, block spaces, as well as Beurling algebras and their pre-duals, now fit naturally into this more accommodating label. Most significantly, in §5.2 we develop powerful and versatile extrapolation results serving as portal, allowing us to pass from estimates on Muckenhoupt weighted Lebesgue spaces (for a fixed integrability exponent and arbitrary weights) to estimates on the brand of Generalized Banach Function Spaces introduced earlier, on which the Hardy-Littlewood maximal operator happens to be bounded. Finally, in §5.3 we focus on Orlicz spaces which, in particular, are natural examples of classical Banach function spaces for which the machinery developed so far applies.

5.1 Generalized Banach Function Spaces The notion of Banach function space is a well-established concept in Functional and Harmonic Analysis, and there is a multitude of accounts where the theory of “standard” Banach function space is discussed at length (see, e.g., [14] and the references therein). Alas, this classical construct is not inclusive enough, as certain scales of spaces we are interested in (such as Morrey and block spaces, as well as Beurling algebras) fail to be Banach function spaces in a traditional sense. The goal in this section is to develop a more inclusive brand of Banach function space, which we dub Generalized Banach Function Spaces, in which the aforementioned scales of spaces now fit naturally. We emphasize that, throughout this section, we follow [14] very closely (with necessary alterations to accommodate our more general setting). To get started, we make the following definition: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1_5

217

218

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

Definition 5.1.1 Let (X, M, μ) be an arbitrary measure space1 and denote   M (X, μ) : = f : X → R : f is μ-measurable ,   M+ (X, μ) : = f ∈ M (X, μ) : f ≥ 0 .

(5.1.1) (5.1.2)

A mapping ρ : M+ (X, μ) → [0, ∞] is called a function norm provided the following properties are satisfied for all f , g ∈ M+ (X, μ): (P1) ρ( f ) = 0 if and only if f = 0 at μ-a.e. point in X, for each λ ∈ [0, ∞) one has ρ(λ f ) = λρ( f ), and ρ( f + g) ≤ ρ( f ) + ρ(g); (P2) if f ≤ g at μ-a.e. point in X then ρ( f ) ≤ ρ(g); (P3) if { fk }k ∈N ⊂ M+ (X, μ) is such that fk increases to f pointwise μ-a.e. in X as k → ∞, then ρ( fk ) increases to ρ( f ) as k → ∞; ∞  (P4) there exists a collection of sets {Yj } j ∈N ⊆ M with the property that X = Yj j=1   and ρ 1Yj < ∞ for all j ∈ N; ∞  (P5) there exists a collection of sets {Z j } j ∈N ⊆ M with the property that X = Zj j=1

and ∫ such that for every j ∈ N there exists some constant c j ∈ (0, ∞) satisfying f dμ ≤ c j ρ( f ). Z j

Note that (P2) implies that, for any two functions f , g ∈ M+ (X, μ), ρ( f ) = ρ(g) whenever f = g at μ-a.e. point in X.

(5.1.3)

Remark 5.1.2 Without loss of generality, we may assume that the sets in (P4)-(P5) are such that Yj = Z j for all j ∈ N. Indeed, if (P4)-(P5) are true as stated, we may consider the family {Wi j }(i, j)∈N2 defined by Wi j := Yi ∩ Z j , for (i, j) ∈ N2 . Then after re-denoting this collection as {Wk }k ∈N it is immediate that the latter satisfies the properties listed in (P4) and (P5). Furthermore, we may assume that {Wk }k ∈N just defined is an increasing nested k }k ∈N sequence of sets. This may be achieved by replacing it with the sequence {W k k :=  W j , for each k ∈ N. In summary, it is possible to combine properties where W j=1

(P4) and (P5) in Definition 5.1.1 as the following “locality” property: in the context of Definition 5.1.1, properties (P4)-(P5) are equivalent with the existence of a collection of sets {W N } N ∈N ⊆ M with the property  that W N X as N → ∞, and for each N ∈ N one has  < ∞ and there exists some constant c N ∈ (0, ∞) such that ρ 1 W N ∫ f dμ ≤ c N ρ( f ) for every f ∈ M+ (X, μ). W

(5.1.4)

N

Let us now compare our notion of Generalized Banach Function Space introduced in Definition 5.1.1 with the classical concept of Banach function space. 1 we emphasize that we do not require that the measure μ is complete, or sigma-finite

5.1 Generalized Banach Function Spaces

219

Remark 5.1.3 The class of function norms defined in Definition 5.1.1 is a more general than the class of Banach function norms defined as in [14, Definition 1.1, p. 2]. The difference is that [14, Definition 1.1, p. 2] also makes the assumption that μ is sigma-finite and properties (P4)-(P5) above are replaced by the following more restrictive axioms:   (P4’) for every μ-measurable set E ⊆ X with μ(E) < ∞ one has ρ 1E < ∞; (P5’) for each μ-measurable set E ⊆ X with μ(E) < ∞ there exists CE ∈ (0, ∞) with ∫ the property E f dμ ≤ CE ρ( f ) for every f ∈ M+ (X, μ). Observe that if the measure μ is sigma-finite then there exists some  countable collection {X j } j ∈N of μ-measurable sets with the property that X = ∞ j=1 X j and μ(X j ) < ∞ for all j ∈ N, and properties (P4’)-(P5’) being satisfied imply that (P4)-(P5) hold with Yj := X j and Z j := X j for each j ∈ N. Even in a standard metric-measure theoretic setting, such as the Euclidean space Rn equipped with the Lebesgue measure, the classical scales of Morrey spaces and block spaces (see §6.2) fail to satisfy properties (P4’)-(P5’) above, but they do satisfy (P4)-(P5) in Definition 5.1.1 (see Proposition 6.2.17). Moving on, we make the following definition, formally introducing the concept of Generalized Banach Function Space. Definition 5.1.4 Let (X, M, μ) be an arbitrary measure space and let ρ be a function norm as in Definition 5.1.1. Then the set   X := f ∈ M (X, μ) : ρ(| f |) < ∞ (5.1.5) is called a Generalized Banach Function Space (GBFS) on X. In such a scenario, for each f ∈ M (X, μ) define ρ(| f |) if f ∈ X, (5.1.6)  f X := ρ(| f |) = +∞ if f  X. Remark 5.1.5 Whenever (X, M, μ) is a sigma-finite measure space, and ρ is a function norm in a “classical” sense, as defined in [14, Definition 1.1, p. 2] (i.e., with properties (P4)-(P5) in our Definition 5.1.1 replaced by properties (P4’)-(P5’) from Remark 5.1.3), we shall refer to the X defined in (5.1.5) as being a (classical) Banach function space. Going back to Definition 5.1.4, in view of (5.1.6) one may then re-cast (5.1.5) simply as   X = f ∈ M (X, μ) :  f X < ∞ . (5.1.7) Also, from (5.1.6) and (P1) we see that  f X = 0 if and only if f = 0 at μ-a.e. point in X. (5.1.8)   Ultimately, we conclude that X,  · X is a normed vector space. In addition, (P5) simply says that, for each j ∈ N,

220

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

the restriction operator X f → f Z j ∈ L 1 (Z j , μ) is well defined and bounded, with norm ≤ c j .

(5.1.9)

For future reference, we observe that this implies if f ∈ X then | f | < ∞ at μ-a.e. point in X.

(5.1.10)

Indeed, given any f ∈ X, from (5.1.9) we see that f | Z j ∈ L 1 (Z j , μ) for each j ∈ N, hence | f | < ∞ at μ-a.e. point on Z j for each j ∈ N. Now (5.1.10) follows since ∞  Zj. X= j=1

It is also useful to remark that, as seen from (5.1.3) and (5.1.6), for any two given functions f , g ∈ M (X, μ) we have  f X = gX whenever f = g at μ-a.e. point in X.

(5.1.11)

Finally, for later use we observe here that if f , g : X → R are two μ-measurable functions such that |g| ≤ | f | at μ-a.e. point on X and f ∈ X, then g also belongs to the space X and gX ≤  f X ,

(5.1.12)

the operator of pointwise multiplication by some given function b in L ∞ (X, μ) is a bounded mapping from the space X into itself, with operator norm ≤ b L ∞ (X,μ) .

(5.1.13)

and

Proposition 5.1.6 Let (X, M, μ) be an arbitrary measure space, let ρ be a function norm as in Definition 5.1.1, and let X and  · X be as in Definition 5.1.4. Then  X,  · X is a normed vector space. In addition, if f ∈ X and { fn }n∈N ⊂ X is a sequence with the property that  fn − f X → 0 as n → ∞, then there exists a sub-sequence { fnk }k ∈N of { fn }n∈N that converges to f pointwise μ-a.e. in X as k → ∞. Proof As seen in (5.1.10), every f ∈ X is finite μ-a.e. on X. This and the properties in (P1)-(P2) further imply that X,  · X is a normed vector space. Next, suppose { fn }n∈N ⊂ X and f ∈ X are such that  fn − f X → 0 as n → ∞. Recall the sets {W N } N ∈N from (5.1.4). In particular, for each N ∈ N there exists some CN ∈ (0, ∞) with the property that for each g ∈ X we have   g|W  1 (5.1.14) N L (WN ,μ) ≤ C N gX . Hence, if we write (5.1.14) for N := 1 and with  g := fn − f , n ∈ N, and then use the current assumptions, we obtain that fn |W1 n∈N converges to f |W1 in L 1 (W1, μ) as n → ∞. As a consequence, there exists a sub-sequence { fnk }k ∈N that converges to f at μ-a.e. point in W1 as k → ∞. Next, iterate this process. That is, we write (5.1.14) for N := 2 and g := fnk − f for each k ∈ N, to obtain a sub-sub-sequence { fnk j } j ∈N

5.1 Generalized Banach Function Spaces

221

that converges to f at μ-a.e. point in W2 as j → ∞. Inductively, we do this for each N ∈ N. The desired final sub-sequence is then selected via a Cantor diagonalization argument. That this final sub-sequence converges to f pointwise μ-a.e. on X is then  seen from its construction and the fact that W N X as N ∞ (cf. (5.1.4)). Any Generalized Banach Function Space enjoys suitable versions of Lebesgue’s Monotone Convergence Theorem and Fatou’s Lemma, as indicated in the next lemma.   Lemma 5.1.7 Let X,  · X be a Generalized Banach Function Space associated as in Definition 5.1.4 with a measure space (X, M, μ) and a function norm ρ. Then for any given sequence { fk }k ∈N in X the following properties are true. (i) Suppose that for each k ∈ N one has 0 ≤ fk (x) ≤ fk+1 (x) for μ-a.e. point x ∈ X, and define f (x) := lim fk (x) at μ-a.e. point x ∈ X. Redefine2 f on a μ-nullset k→∞

as to make it a non-negative μ-measurable (see [133, (3.1.30)]). Then either f  X and  fk X ∞ as k → ∞, or f ∈ X and  fk X  f X as k → ∞.

(5.1.15)

Simply put, with the convention made in (5.1.6), in all cases one has  fk X  f X as k → ∞.

(5.1.16)

(ii) Assume lim inf  fk X < ∞ and suppose f (x) := lim fk (x) exists at μ-a.e. point k→∞

k→∞

x ∈ X. Redefine3 the function f on a μ-nullset as to make it a non-negative μ-measurable (cf. [133, (3.1.30)]). Then f ∈ X and  f X ≤ lim inf  fk X . k→∞

(5.1.17)

Proof The statement in (i) is a consequence of (P3), bearing in mind that f redefined as indicated in the statement belongs to M+ (X, μ). To deal with the statement in (ii), note that f redefined belongs to M+ (X, μ) and there exists some set A ∈ M of μ-measure zero such that sequence { fk (x)}k ∈N converges pointwise to f (x) at each point x ∈ X \ A. To proceed, consider hk (x) := inf | fm (x)| for each x ∈ X \ A and m≥k

every k ∈ N. This is a non-negative, increasing sequence of functions, and   lim hk (x) = sup hk (x) = sup inf | fm (x)| = lim inf | fk (x)| k→∞

k ∈N

k ∈N

m≥k

k→∞

= lim | fn (x)| = | f (x)| at each x ∈ X \ A. k→∞

(5.1.18)

Thus, 0 ≤ hk | f | pointwise on X \ A as k → ∞. Since as a consequence of (P2), for each k ∈ N we have ρ(hk ) ≤ ρ(| fm |) for all m ≥ k, we may apply (P3) to write 2 this step is not necessary if the measure space (X, M, μ) is complete 3 again, this is no longer necessary if the measure μ is complete

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

 f X = ρ(| f |) = lim ρ(hk ) ≤ lim k→∞

k→∞



inf ρ(| fm |)



m≥k

= lim inf ρ(| fk |) = lim inf  fk X < +∞. k→∞

k→∞

(5.1.19) 

With this in hand, (5.1.17) follows, so the proof of (ii) is complete.

The point of the next proposition is that any Generalized Function Space also enjoys the Riesz-Fischer property (which renders said space Banach).   Proposition 5.1.8 Let X, ·X be a Generalized Banach Function Space associated as in Definition 5.1.4 with a measure space (X, M, μ). Suppose fk ∈ X, with k ∈ N, is a family of functions satisfying ∞

 fk X < ∞.

(5.1.20)

k=1

Then

∞

k=1

fk converges in X to a function f ∈ X and  f X ≤

∞

 fk X .

(5.1.21)

k=1

  As a consequence, the space X,  · X is complete, hence Banach.

n Proof For each n ∈ N, consider the function tn := k=1 | fk | ∈ M+ (X, μ). Invoking (P1) and (5.1.20) we may write tn X ≤

n k=1

 fk X ≤

∞

 fk X < ∞ for all n ∈ N.

(5.1.22)

k=1

Thus, tn ∈ X for all n ∈ N, and if we set t :=

∞

k=1

| fk |, then 0 ≤ tn t as n → ∞.

Since this convergence happens at every point in X, it follows that t is a μ-measurable function. In light of (5.1.22), by part (i) in Lemma 5.1.7 we have t ∈ X. The latter ∞

| fk (x)| < ∞ for μ-a.e. x ∈ X which, in turn, implies that and (5.1.10) ensure that ∞

k=1

k=1

fk (x) is absolutely convergent for μ-a.e. x ∈ X. From [133, (3.1.30)] we know

that there exists a μ-measurable function f on X with the property that f = at μ-a.e. point in X. In particular, if we define sn :=

n

k=1

∞

k=1

fk

fk for each n ∈ N, then

sn ∈ X for each n ∈ N and sn → f at μ-a.e. point in X as n → ∞. Moreover, for each m ∈ N, we have sn − sm → f − sm at μ-a.e. point in X as n → ∞ and

5.1 Generalized Banach Function Spaces ∞

lim inf sn − sm X ≤ n→∞

223

 fk X ≤

k=m+1

∞

 fk X < ∞.

(5.1.23)

k=1

We may therefore invoke part (ii) in Lemma 5.1.7 to conclude that f − sm ∈ X for each m ∈ N and  f − sm X ≤ lim inf sn − sm X ≤ n→∞

∞

 fk X .

(5.1.24)

k=m+1

Consequently, f = ( f − sm ) + sm ∈ X and if we pass to the limit as m → ∞ in (5.1.24) while relying on assumption (5.1.20), we obtain lim  f − sm X = 0. This m→∞ proves that sm → f in X as m → ∞. Furthermore, for each m ∈ N, we may write  f X ≤  f − sm X + sm X ≤  f − sm X +

m

 fk X .

(5.1.25)

k=1

The estimate in (5.1.21) now follows from (5.1.25) by passing to limit m → ∞. To show that X is complete, pick an arbitrary Cauchy sequence {gn }n∈N in X.  Then for each k ∈ N there exists nk ∈ N such that gnk+1 −gnk X < 2−k . In particular, the sequence fk := gnk+1 − gnk , indexed by k ∈ N, is contained in X and satisfies (5.1.20). Invoking what we have proved so far, we obtain that there exists some f ∈ X such that N f = lim fk = lim gn N +1 − gn1 in X. (5.1.26) N →∞

k=1

N →∞

This shows that the sub-sequence {gnk }k ∈N converges in X to f + gn1 , which further implies that the {gn }n∈N converges in X to f + gn1 . Ultimately, we   entire sequence  conclude that X,  · X is complete. To each given function norm one can associate a new function norm of the sort described below. Definition 5.1.9 Let (X, M, μ) be a measure space and let ρ be a function norm as in Definition 5.1.1. Its associated norm is the function ρ : M+ (X, μ) → [0, ∞] defined by ∫  f g dμ : f ∈ M+ (X, μ), ρ( f ) ≤ 1 for all g ∈ M+ (X, μ). ρ(g) := sup X

(5.1.27)

Our next proposition clarifies the fact that ρ defined above is indeed a function norm. Proposition 5.1.10 If (X, M, μ) is a measure space and ρ is a function norm as in Definition 5.1.1, then its associated norm ρ defined in Definition 5.1.9 is a function norm.

224

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

Proof Let g ∈ M+ (X, μ) be such that ρ(g) = 0. Then ∫ f g dμ = 0 for all f ∈ M+ (X, μ) with ρ( f ) ≤ 1.

(5.1.28)

X

Since the goal is to show that g = 0 at μ-a.e. point in X, without loss of generality we may assume that μ(X) > 0. By (P4) in Definition 5.1.1, there exists a collection ∞    of sets {Yj } j ∈N ⊆ M with the property that X = Yj and ρ 1Yj < ∞ for all j ∈ N. j=1

Let Then 



J := { j ∈ N : μ(Yj ) = 0}.

(5.1.29)

  ρ 1Yj > 0 for each j ∈ N \ J.

(5.1.30)

Indeed, otherwise ρ 1Yj = 0 would imply 1Yj = 0 at μ-a.e. point in X, thus μ(Yj ) = 0, contradicting the fact that j  J. In light of (5.1.30), for each integer j ∈ N \ J, we may set f j := μ(11Y ) 1Yj which j

is μ-measurable and ρ( f j ) = 1 (the latter a consequence of property (P1) for ρ with λ := 1/ρ(1Yj )). Applying (5.1.28), we have ∫ ∫ f j g dμ = g dμ, for all j ∈ N \ J. (5.1.31) 0= X

Yj

 Yj . Since by the definition of J we have Hence, g = 0 at μ-a.e. point in j ∈N\J   μ Yj = 0, we may conclude that g = 0 at μ-a.e. point in X. j ∈J

Conversely, assume g = 0 at μ-a.e. point in X and pick an arbitrary f ∈ M+ (X, μ) with ρ( f ) ≤ 1. Then f ∈ X and | f | < ∞ ∫at μ-a.e. point in X (recall (5.1.10)) which implies f g = 0 at μ-a.e. point in X, thus X f g dμ = 0. Hence, ρ(g) = 0 as wanted. That ρ also satisfies the rest of the properties listed in (P1)-(P2) in Definition 5.1.1 follows from (5.1.27) and standard properties of integrals. We also observe that, with the sets {Yj } j ∈N ⊆ M and J as above, if f ∈ M+ (X, μ) then ∫ ∫ ∫   1Y f dμ = f 1Yk dμ = ρ 1Yk f ·  k  dμ ρ 1Yk Yk X X    ≤ ρ 1Yk ρ ( f ) (5.1.32) ∫ for each k ∈ N \ J, and Y f dμ = 0 if k ∈ J. Hence, ρ satisfies (P5) in Definik   tion 5.1.1 with Z j := Yj for each j ∈ N, and with ck := ρ 1Yk if k ∈ N \ J and ck := 0 if k ∈ J. To show that ρ satisfies (P3), let g ∈ M+ (X, μ) and {gk }k ∈N ⊂ M+ (X, μ) be such that gk increases to g pointwise μ-a.e. in X as k → ∞. Since ρ satisfies (P2), it follows that the sequence {ρ(gk )}k ∈N is increasing and that ρ(gk ) ≤ ρ(g) for all k ∈ N. To finish proving that ρ satisfies (P3), we have to show that ρ(gk ) ρ(g)

5.1 Generalized Banach Function Spaces

225

as k → ∞. Without loss of generality we can assume ρ(gk ) < ∞ for all k ∈ N.  If c ∈ R satisfies c < ρ(g), then the definition ∫ of ρ implies the existence of some f ∈ M+ (X, μ) with ρ( f ) ≤ 1 and such that X f g dμ > c. Since gk f increases to g f pointwise μ-a.e. ∫ in X as k ∫→ ∞, by the Monotone Convergence Theorem we necessarily have X g fk dμ X f g dμ as k → ∞. Hence, there exists some N ∈ N ∫ with the property that X g fk dμ > c for all k ≥ N. The arbitrariness of c satisfying the condition c < ρ(g) now allows us to conclude that, as wanted, ρ(gk ) increases to ρ(g) as k → ∞. Finally, the fact that ρ satisfies (P5) in Definition 5.1.1, implies the existence of ∞  Z j and such that for every a collection {Z j } j ∈N ⊆ M with the property that X = j=1 ∫ k ∈ N there exists some constant ck ∈ (0, ∞) satisfying Z f dμ ≤ ck ρ( f ) for each k f ∈ M+ (X, μ). Then, for each k ∈ N, ∫    f dμ : f ∈ M+ (X, μ), ρ( f ) ≤ 1 ≤ ck < ∞. (5.1.33) ρ 1 Zk = sup Zk

This shows that ρ satisfies (P4) in Definition 5.1.1 with Yj := Z j , for each j ∈ N, and completes the proof of the proposition.  We are now in a position to define X, the associated space of a given Generalized Banach Function Space X. Definition 5.1.11 Let (X, M, μ) be a measure space, ρ be a function norm as in Definition 5.1.1, and ρ its associated norm as in Definition 5.1.9. Then the Generalized Banach Function Space on (X, μ) defined in relation to ρ:   X := g ∈ M (X, μ) : ρ(|g|) < ∞ (5.1.34) is called the associated space of X (defined in relation to ρ as in Definition 5.1.4). According to the recipe in (5.1.6) implemented for ρ in place of ρ, the formula given in (5.1.27), and (5.1.5), for every g ∈ M (X, μ) we have ∫  gX = ρ(|g|) = sup f |g| dμ : f ∈ M+ (X, μ), ρ( f ) ≤ 1 = sup

∫ X

X

 | f g| dμ : f ∈ X,  f X ≤ 1 ,

and the Generalized Banach Function Space X may be characterized as   X = g ∈ M (X, μ) : gX < ∞ .

(5.1.35)

(5.1.36)

There is a very useful (generalized) Hölder inequality accompanying the pair X, X, where X is a given Generalized Banach Function Space and X is its associated space.

226

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

Proposition 5.1.12 Let X be a Generalized Banach Function Space on a measure space (X, M, μ) with associated space X. Then for every f ∈ X and every g ∈ X the function f g is absolutely integrable on X and ∫ | f g| dμ ≤  f X gX . (5.1.37) X

Proof Let f ∈ X and g ∈ X be arbitrary. According to (5.1.10) (used for X in place of X), we have |g| < ∞ at μ-a.e. point in X. If  f X = 0, then f = 0 at μ-a.e. point in X, so f g = 0 at μ-a.e. point in X. As such, both sides in the inequality in (5.1.37) are zero. If  f X > 0 then f / f X ∈ X has norm equal to 1 and, due to (5.1.35), we have ∫



f

g dμ ≤ gX (5.1.38)

 f  X X 

which in turn yields (5.1.37).

The lemma below plays a role in the proof of Proposition 5.1.14, given a little later. Lemma 5.1.13 Let X be a Generalized Banach Function Space on a measure space (X, M, μ) with associated space X. Then a μ-measurable function f on X belongs to X if and only if f g is absolutely integrable on X for every g ∈ X. Proof The left-to-right implication is an immediate consequence of Proposition 5.1.12. For the opposite implication we reason by contradiction. Suppose f ∈ M (X, μ) is such that f g ∈ L 1 (X, μ) for every g ∈ X and ρ(| f |) = ∞. Recalling (5.1.35), the latter implies the existence of a sequence {gn }n∈N ⊂ M (X, μ) satisfying ∫ |gn f | dμ > n3 for every n ∈ N. (5.1.39) gn X ≤ 1 and X

Invoking Proposition 5.1.8, it follows that

∞

n=1

n−2 gn converges in X to some function

g. However, by the second inequality in (5.1.39) we have ∫ ∫ |g f | dμ ≥ n−2 |gn f | dμ > n for every n ∈ N, X

(5.1.40)

X

which contradicts the assumption that f g is absolutely integrable on X.



Here is a natural version of the classical Lorentz-Luxemburg theorem corresponding to Generalized Banach Function Spaces, which generalizes (and corrects4) [14, Theorem 2.7, p. 10]. Proposition 5.1.14 Let X be a Generalized Banach Function Space on a measure space (X, M, μ). Assume the measure μ is sigma-finite. Then X = X isometrically, 4 note that [14, (2.12), p. 11] forces γ to be non-negative, which is not necessarily the case

5.1 Generalized Banach Function Spaces

227

i.e., X coincides with its second associated space X := (X) and  f X =  f X for each f ∈ X. In fact, one has  f X =  f X for each f ∈ M (X, μ).

(5.1.41)

Proof Let f ∈ X be arbitrary. Applying Proposition 5.1.12, it follows that f g is in L 1 (X, μ) for every g ∈ X. Hence, Lemma 5.1.13 applies (with X in place of X) and gives that f belongs to the associated space X. This establishes the inclusion X ⊆ X. In addition, (5.1.35) and (5.1.37) imply ∫   f X = sup | f g| dμ : g ∈ X, gX ≤ 1 ≤  f X . (5.1.42) X

To finish the proof of the fact that X coincides with its second associated space X = (X) in an isometric fashion, it remains to show that X ⊆ X and that  f X ≤  f X for all f ∈ X .

(5.1.43)

Pick f ∈ X arbitrary, let {W N } N ∈N be as in (5.1.4), and fix N ∈ N. Define the μ-measurable function   (5.1.44) f N := min | f |, N 1WN , and observe that since 1WN ∈ X and 0 ≤ f N ≤ N1WN , by (P2) we have f N ∈ X. Hence, 0 ≤ f N | f | at every point in X as N → ∞, (5.1.45) and f N ∈ X ⊆ X for each N ∈ N. We may now invoke (ii) in Lemma 5.1.7 (with X in place of X) to conclude f ∈ X and  f N X  f X as N → ∞.

(5.1.46)

Moreover, a second application of (ii) in Lemma 5.1.7 (as stated for X) implies either f  X and  f N X ∞,

(5.1.47)

or f ∈ X and  f N X  f X .

(5.1.48)

At this stage we make the claim that  f N X ≤  f N X for all N ∈ N.

(5.1.49)

Assuming (5.1.49) to be true, we may then send N → ∞ in the inequality in (5.1.49) and (keeping in mind (5.1.46)) obtain lim  f N X ≤ lim  f N X =  f X < ∞.

N →∞

N →∞

(5.1.50)

In light of (5.1.47)-(5.1.48), this further implies f ∈ X and  f X ≤  f X , as wanted.

228

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

Let us now return to the task of justifying (5.1.49). Henceforth fix N ∈ N. To set the stage, we introduce a couple of symbols and make some preliminary observations. First, we define   L 1N := f ∈ L 1 (X, μ) : f = 0 μ-a.e. on X \ W N (5.1.51) which is a closed subspace of L 1 (X, μ). Indeed, if {gn }n∈N ⊂ L 1N converges in L 1 (X, μ) to some g ∈ L 1 (X, μ), then there exists a sub-sequence {gnk }k ∈N that converges pointwise at μ-a.e. point in X to g (recall that μ is complete). In turn, this forces g = 0 at μ-a.e. point in X \ W N (given that gnk = 0 at μ-a.e. point in X \ W N ), so g ∈ L 1N , as wanted. As a consequence, 

 L 1N ,  ·  L 1 (X,μ) is a Banach space.

(5.1.52)

Second, we consider a subset of L 1N defined by   U N := L 1N ∩ S where S := g ∈ X : gX ≤ 1 .

(5.1.53)

It is clear that the set U N is convex and we claim that U N is also closed in L 1N . To see why the latter is true, let {hn }n∈N ⊂ U N be a sequence that converges in L 1 (X, μ) to some h ∈ L 1 (X, μ). Then there exists a sub-sequence {hnk }k ∈N that converges pointwise at μ-a.e. point in X to h. Since also hnk X ≤ 1 for all k ∈ N, we may apply item (ii) in Lemma 5.1.7 to obtain hX ≤ 1. Thus, h ∈ L 1N . Third, we have   1 ∗  L N = φ ∈ L ∞ (X, μ) : φ = 0 μ-a.e. on X \ W N . (5.1.54) To establish (5.1.54) we introduce the mapping    ∗ Φ : φ ∈ L ∞ (X, μ) : φ = 0 at μ-a.e. point in X \ W N −→ L 1N

Φ(φ) := Λφ for each φ ∈ L ∞ (X, μ) vanishing at μ-a.e. point in X \ W N , ∫ where Λφ ( f ) := φ f dμ for each function f ∈ L 1N . X

(5.1.55) This is a well-defined map and (5.1.54) will follow as soon as we prove that Φ is a bijection. Suppose φ ∈ L ∞ (X, μ) is such that φ = 0 at μ-a.e. point in X \ W N and ∫ φ f dμ = 0 for all f ∈ L 1N . Corresponding to f := (sgn φ)1WN ∈ L 1N , the latter X ∫ gives 0 = W |φ| dμ, which further implies φ = 0 at μ-a.e. point in W N , hence N φ = 0 at μ-a.e. point in X. This proves that Φ is injective. ∗ To show that Φ is also surjective, let Λ ∈ L 1N be arbitrary. The Hahn-Banach  ∈ (L 1 (X, μ))∗ of Λ. Given that theorem guarantees the existence of an extension Λ 1 ∗ we have the isometric identification (L (X, μ)) = L ∞ (X, μ) since the measure μ is sigma-finite (cf., e.g., [55, Theorem 6.15, p. 190]), it follows that there exists some function η ∈ L ∞ (X, μ) with the property that

5.1 Generalized Banach Function Spaces

     f · 1WN = Λ f · 1WN = Λ

∫ X

229

  f η · 1WN dμ for all f ∈ L 1 (X, μ).

(5.1.56)

If we now define φ0 := η · 1WN , then φ0 = 0 on X \ W N and (5.1.56)  implies ∗ Λ f = Λφ0 ( f ) for all f ∈ L 1N . Thus, Λ = Λφ0 as functionals in L 1N . This establishes that Φ is also surjective, so the proof of (5.1.54) is finished. Returning to the claim in (5.1.49), we remark that without loss of generality we may assume  f N X > 0. With this a standing assumption, we may then consider the function f N / f N X which belongs to U N . On account of this and (P1), for any dilation factor λ ∈ (1, ∞) we have gλ :=

λ f N ∈ L 1N and gλ  U N .  f N X

(5.1.57)

Fix λ > 1 and invoke the Hahn-Banach Separation Theorem (cf., e.g., [46, Corollary 12, p. 418], [165, Theorem 3.4, p. 59]) in the context of the Banach space L 1N , the closed and convex subset U N of L 1N , and the function gλ  U N . This guarantees  ∗ the existence of a number γ ∈ R and a functional Λ ∈ L 1N such that Re Λ(h) < γ < Re Λ(gλ ) for all h ∈ U N .

(5.1.58)

Upon recalling (5.1.54), we know that there exists φ ∈ L ∞ (X, μ) satisfying φ = 0 ∫ at μ-a.e. point in X \ W N and Λ(h) = X φh dμ for all h ∈ L 1N . Then (5.1.58) is equivalent with having ∫  ∫  Re φh dμ < γ < Re φgλ dμ for all h ∈ U N . (5.1.59) WN

WN

In turn, (5.1.59) implies  ∫ ∫ φh dμ ≤ Re WN

WN

|φgλ | dμ for all h ∈ U N .

(5.1.60)

Let us now consider the function ψ : X → R defined by ⎧ φ(x) ⎪ ⎪ if φ(x)  0, ⎨ ⎪ |φ(x)| ψ(x) := at μ-a.e. x ∈ X. ⎪ ⎪ ⎪1 if φ(x) = 0, ⎩

(5.1.61)

Then ψ is μ-measurable, |ψ| = 1 and φ = |φ|ψ at μ-a.e. point in X,

(5.1.62)

and for every h ∈ L 1N , h ∈ U N ⇐⇒ ψh ∈ U N ⇐⇒ ψ −1 |h| ∈ U N ⇐⇒ |h| ∈ U N .

(5.1.63)

230

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

(consequences of the fact that hX = ψ −1 hX = ψhX for all h ∈ L 1N ). Together, (5.1.63) and (5.1.60) imply ∫ ∫ ∫ |φh| dμ = Re |φh| dμ ≤ |φgλ | dμ for all h ∈ U N , (5.1.64) WN

WN

WN

∫

∫

thus

|φh| dμ ≤

sup

h ∈U N

WN

WN

|φgλ | dμ.

(5.1.65)

Next, we will show that (5.1.65) remains valid if the supremum is taken over h ∈ S. To see why this is true, for each h ∈ S introduce the sequence of functions hn := min{|h|, n} · 1WN for all n ∈ N.

(5.1.66)

Then 0 ≤ hn ≤ |h| for each n ∈ N. From this, (P2), and the fact that hX ≤ 1 we conclude that hn X ≤ 1 for each n ∈ N. Moreover, in light of (5.1.4), for each n ∈ N we have hn ∈ L 1N , thus hn ∈ U N for each n ∈ N. In addition, hn |h|1WN as n → ∞ at every point in X, so an application of Lebesgue’s Monotone Convergence Theorem gives ∫ ∫ |φhn | dμ

|φh| dμ as n → ∞, for each h ∈ S. WN

(5.1.67)

(5.1.68)

WN

This ultimately implies ∫ h ∈U N

∫ |φh| dμ = sup

sup

h ∈S

WN

|φh| dμ.

(5.1.69)

WN

Recalling that φ = 0 at μ-a.e. point in X \ W N , then using (5.1.69), then (5.1.65) and finally, recalling the definition of gλ , we arrive at ∫ ∫ ∫ φX = sup |φh| dμ = sup |φh| dμ = sup |φh| dμ h ∈S

X

∫ ≤ WN

|φgλ | dμ =

h ∈S

WN

λ  f N X

∫ WN

h ∈U N

|φ f N | dμ.

WN

(5.1.70)

After multiplying the resulting inequality in (5.1.70) by  f N X /φX and invoking Proposition 5.1.12 we obtain ∫



φ

f N dμ ≤ λ f N X . (5.1.71)  f N X ≤ λ

X φX Now (5.1.49) follows by passing to the limit with λ  1 in (5.1.71). This finishes the proof of the fact that X coincides with its second associated space X = (X) in an isometric fashion.

5.1 Generalized Banach Function Spaces

231

In turn, this shows that (5.1.41) holds whenever f ∈ X. If f  X then f  (X) so  f X = ∞ by (5.1.6) and  f (X ) = ∞ by (5.1.36) written for X in place of X. Hence, (5.1.41) holds in all cases.  A simple yet useful consequence of Proposition 5.1.14 is the following characterization of the norm in given Generalized Banach Function Space, in terms of its associated space. Corollary 5.1.15 Let X be a Generalized Banach Function Space on a measure space (X, M, μ). Assume that the measure μ is sigma-finite. Then for each f ∈ M (X, μ) one has ∫  | f g| dμ : g ∈ X, gX ≤ 1 . (5.1.72)  f X = sup X

Proof This is a direct consequence of (5.1.35) and Proposition 5.1.14.



Other equivalent descriptions of the norms in a given Generalized Banach Function Space and its associated space are provided below.   Proposition 5.1.16 Let X,  · X be a Generalized Banach Function Space associated as in Definition 5.1.4 with a measure space (X, M, μ). Consider its associated space X from Definition 5.1.11 with norm defined in (5.1.35). Then

 ∫ 



f g dμ : f ∈ X,  f X ≤ 1 for all g ∈ X (5.1.73) gX = sup

X

and if the measure μ is sigma-finite one also has

 ∫ 



f g dμ : g ∈ X, gX ≤ 1 for all f ∈ X.  f X = sup

(5.1.74)

X

Proof Fix an arbitrary g ∈ X. That the supremum in the right-hand

∫ side of (5.1.73) ∫



is ≤ gX follows from (5.1.35) and the fact that X f g dμ ≤ X | f g| dμ for every f ∈ X. It remains to prove the opposite inequality, that is, ∫

∫



(5.1.75) sup | f g| dμ ≤ sup

f g dμ

f ∈S



f ∈S

X

X



where S := f ∈ X :  f X ≤ 1 . To proceed, set E := {x ∈ X : g(x)  0} and define φ : X → R by φ(x) := g(x)/|g(x)| if x ∈ E and φ(x) := 0 if x ∈ X \ E. In particular, φ is μ-measurable. If f ∈ S is arbitrary, then the function h := | f |φ is μ-measurable and satisfies |h| ≤ | f |. Hence hX ≤  f X ≤ 1, which also implies h ∈ X. Ultimately, h ∈ S. This together with the fact that |g| = φg then allow us to write ∫ ∫ ∫

∫

∫







| f g| dμ = | f |φg dμ = hg dμ ≤

hg dμ ≤ sup

ψg dμ . (5.1.76) X

X

X

X

ψ ∈S

X

Now (5.1.75) follows by taking the supremum over f ∈ S in (5.1.76). This completes the proof of (5.1.73).

232

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

Finally, if the measure μ is sigma-finite, Proposition 5.1.14 is applicable. Bearing this in mind, that the equality claimed in (5.1.74) holds for each f ∈ X = (X) is  seen from (5.1.41) and (5.1.73) written for X in place of X. We conclude with result to the effect that set theoretic inclusions for Generalized Banach Function Spaces are always continuous injections.   Proposition 5.1.17 Let X,  · X be a Generalized Banach Function Space associated as in Definition 5.1.4 with a measure space (X, M, μ). Suppose V ⊆ M (X, μ) is a linear space such that X ⊆ V and for which there exists a mapping ·V : V → [0, ∞) satisfying the following properties: (i) | f | ∈ V and  | f | V =  f V for every f ∈ V; (ii) λ f V = λ f V for all f ∈ V and all λ ∈ [0, ∞); (iii) if f , g ∈ V are such that 0 ≤ f ≤ g at μ-a.e. point in X, then  f V ≤ gV . Then there exists C ∈ (0, ∞) such that  f V ≤ C f X for all f ∈ X.

(5.1.77)

Proof First we observe that if f ∈ V is such that f = 0 at μ-a.e. point in X then  f V = 0. Indeed, if f ∈ V satisfies f = 0 at μ-a.e. point in X, then | f | ∈ V by (i), and 2| f | = | f | at μ-a.e. point in X. This and (iii) imply  f V = 2 f V , hence  f V = 2 f V by (ii), which in the end gives  f V = 0 as wanted. Suppose (5.1.77) is false. Then for each n ∈ N there exists gn ∈ X such that gn V > n3 gn X .

(5.1.78)

If some gn X = 0, then gn = 0 at μ-a.e. point in X, which by the earlier observation gives gn V = 0 contradicting (5.1.78). This shows that we necessarily have that ∞ > gn X  0 for every n ∈ N and we can define fn := g|gnn|X for each n ∈ N. This sequence of μ-measurable functions satisfies fn ≥ 0,  fn X = 1, and  fn V > n3 for all n ∈ N. Consequently,

∞

n=1

(5.1.79)

n−2 fn X < ∞ and Proposition 5.1.8 gives that

verges in X to some f ∈ X. Proposition 5.1.6 then ensures that

∞

n=1

∞

n=1

n−2 fn con-

n−2 fn also con-

verges pointwise μ-a.e. to f . In particular, for every n ∈ N, we have f ≥ n−2 fn ≥ 0 at μ-a.e. point in X. Invoking (iii), (ii), and the last inequality in (5.1.79) it follows that (5.1.80)  f V ≥ n−2 fn V = n−2  fn V > n for each n ∈ N. which contradicts the fact that  f V is finite. This proves that (5.1.77) holds for some constant C ∈ (0, ∞). 

5.2 Extrapolation Theory

233

5.2 Extrapolation Theory The proof of Rubio de Francia’s extrapolation theorem, formulated in [133, Proposition 7.7.6], lends itself to a considerably more versatile and powerful result to the effect that estimates on Muckenhoupt weighted Lebesgue spaces (for a fixed integrability exponent and arbitrary weights) yield estimates on Generalized Banach Function Spaces (on which the Hardy-Littlewood maximal operator turns out to be bounded). Our main result to this effect (refining work in [37], [123]) is the theorem below, itself a generalization of [133, Proposition 7.7.6]. Theorem 5.2.1 Let (X, d, μ) be a space of homogeneous type with the property that the quasi-distance d : X × X → [0, ∞) is continuous5 in the product topology τd ×τd . Also, denote by M X the Hardy-Littlewood maximal operator on (X, d, μ) and, given a Generalized Banach Function Space X on (X, μ), denote by X its associated space (cf. Definition 5.1.4 and Definition 5.1.11). Then there exists some CX ∈ (0, ∞), depending only on the quasi-distance d (via d defined as in (A.0.19)-(A.0.20)) and the doubling charter of μ, the constants Cd, C such that the following statements are true: (1) Fix some integrability exponent p0 ∈ (1, ∞) and, assuming M X : X → X and M X : X → X are well-defined bounded mappings, introduce

(5.2.1)

p −1

p

0 WX, p0 := CX0 · M X X→X · M X X →X

p −1

p

and WX, p0 := CX0 · M X X0 →X · M X X→X .

(5.2.2)

Also, suppose f , g are two μ-measurable real-valued functions defined on X with the property that for each Muckenhoupt weight w ∈ Ap0 (X, d, μ) with [w] A p0 ≤ WX, p0 one has  f  L p0 (X,w) ≤ Cw g L p0 (X,w)

(5.2.3)

for some constant Cw ∈ (0, ∞) which depends only on the ambient (via d, μ, p0 ), and w. Then    f X ≤ 22−1/p0 ·

sup

w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

Cw gX .

(5.2.4)

0

Moreover, if (5.2.3) holds for each Muckenhoupt weight w ∈ Ap0 (X, d, μ) with [w] A p0 ≤ WX, p0 then

5 The result proved in [133, Theorem 7.1.2] guarantees that any quasi-metric space has an equivalent quasi-distance satisfying this property

234

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

  f X ≤ 2

2−1/p0

·

 sup

w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

Cw gX .

(5.2.5)

0

(2) Assume

M X : X → X is a well-defined bounded mapping,

(5.2.6)

WX,1 := CX · M X X →X .

(5.2.7)

and define In addition, suppose f , g are two μ-measurable real-valued functions defined on X with the property that for each Muckenhoupt weight w ∈ A1 (X, d, μ) with [w] A1 ≤ WX,1 one has  f  L 1 (X,w) ≤ Cw g L 1 (X,w)

(5.2.8)

for some constant Cw ∈ (0, ∞) which depends only on the ambient (via d, μ), and w. Then    f X ≤ 2 ·

sup

w ∈ A1 (X,d,μ) [w] A1 ≤WX,1

Cw gX .

(5.2.9)

(3) Assume M X : X → X is a well-defined bounded mapping,

(5.2.10)

and define WX,1 := CX · M X X→X .

(5.2.11)

In addition, suppose f , g are two μ-measurable real-valued functions defined on X with the property that for each Muckenhoupt weight w ∈ A1 (X, d, μ) with [w] A1 ≤ WX,1 one has  f  L 1 (X,w) ≤ Cw g L 1 (X,w)

(5.2.12)

for some constant Cw ∈ (0, ∞) which depends only on the ambient (via d, μ), and w. Then    f X ≤ 2 ·

sup

w ∈ A1 (X,d,μ) [w] A1 ≤WX,1

Cw gX .

(5.2.13)

Proof To prove the claims made in item (1), assume p0 ∈ (1, ∞) and work under the assumptions made in (5.2.1). Then M X X→X < ∞ and M X X →X < ∞. We claim that we also have M X X→X > 0. To justify this, bring in the family {W j } j ∈N of μ-measurable subsets of X considered in (5.1.4). Then 1Wj ∈ X for each j ∈ N there exists some jo ∈ N such and μ(W j ) μ(X) > 0 as j → ∞. In particular,   that μ(W jo ) > 0. If M X X→X = 0 then M X 1Wj o = 0 in X which, by (P1) in Definition 5.1.1 forces M X 1Wj o = 0 at μ-a.e. point in X hence, further, 1Wj o = 0 at μ-a.e. point in X, in contradiction with the choice of jo . Likewise, M X X →X > 0.

5.2 Extrapolation Theory

235

These considerations allow us to implement a version of Rubio de Francia’s iterative algorithm in the present setting, i.e., define T : X −→ X by setting T φ :=

∞ j=0

j

MX φ 2 j M

j X X→X

for each function φ ∈ X,

(5.2.14)

j

where M X0 φ := |φ| for each φ ∈ X, and M X := M X ◦ · · · ◦ M X is the j-fold composition of M X : X → X with itself, for each j ∈ N, as well as T  : X −→ X by setting 

T ψ :=

∞

MX ψ

j=0

2 j M X X →X

j

j

for each function ψ ∈ X,

(5.2.15)

where M X0 ψ := |ψ| for each ψ ∈ X, and M X := M X ◦ · · · ◦ M X is the j-fold composition of the operator M X : X → X with itself, for each j ∈ N. Then, thanks to item (1) in [133, Lemma 7.7.1], T, T  are well-defined bounded sublinear operators with   T X→X := sup T φX : φX = 1 ≤ 2, (5.2.16)   T  X →X := sup T  ψX : ψX = 1 ≤ 2. j

Bearing in mind the nature of the first term in (5.2.14) it follows that for each function φ ∈ X one has |φ| ≤ T φ and M X (T φ) ≤ 2M X X→X T φ.

(5.2.17)

The above properties further imply 0 < T φ < ∞ at μ-a.e. point on X whenever 0  φ ∈ X.

(5.2.18)

In a similar fashion, for each function ψ ∈ X one has |ψ| ≤ T ψ and M X (T ψ) ≤ 2M X X →X T ψ,

(5.2.19)

plus 0 < T ψ < ∞ at μ-a.e. point on X if ψ  0. Based on the properties recorded in (5.2.17)-(5.2.18) and item (4) in [133, Lemma 7.7.1] we then conclude that if φ ∈ X is not identically zero (μ-a.e.), then T φ ∈ A1 (X, μ) and [T φ] A1 ≤ CM X X→X

(5.2.20)

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

d where C ∈ (0, ∞) depends only on the quasi-distance d (via the constants Cd, C defined as in (A.0.19)-(A.0.20)), and the doubling charter of μ. Likewise, from (5.2.19) and item (4) in [133, Lemma 7.7.1] we see that if ψ ∈ X is not identically zero (μ-a.e.), then T ψ ∈ A1 (X, μ) and [T ψ] A1 ≤ CM X X →X .

(5.2.21)

Let us focus on (5.2.4). Note that this is trivially true if f = 0 at μ-a.e. point in X, or if gX is either ∞ or 0 (since in the latter case (5.2.3) with, say, w ≡ 1, forces f = 0 at μ-a.e. point in X). As far as (5.2.4) is concerned, we are therefore left with considering the situation when f is not identically zero (μ-a.e.), and 0 < gX < ∞.

(5.2.22)

Hence,  f X > 0 and 0  g ∈ X. In particular, it is meaningful to define  g :=

g , gX

(5.2.23)

which is a function satisfying  g ∈ X and  g X = 1.

(5.2.24)

The remainder of the proof of (5.2.4) is divided into several steps, starting with: Step I. Proof of (5.2.4) under the additional assumption f ∈ X. In particular, 0 <  f X < ∞, and Proposition 5.1.14 together with Definition 5.1.9 ensure that ∫  f X =  f X = sup | f ||h| dμ. (5.2.25) h ∈X h X =1

X

Fix an arbitrary number θ ∈ (0, 1). Then the above considerations guarantee the existence of a function ∫ hθ ∈ X with hθ X = 1 and θ  f X ≤ | f ||hθ | dμ. (5.2.26) X

Granted (5.2.24) and the first two properties in (5.2.26), we conclude from (5.2.20) and item (2) in [133, Lemma 7.7.1] that there exists CX ∈ (0, ∞) such that T g ∈ A1 (X, μ) and T  hθ ∈ A1 (X, μ) with [T g ] A1 ≤ CX M X X→X and [T  hθ ] A1 ≤ CX M X X →X .

(5.2.27)

In turn, based on these properties and item (3) in [133, Lemma 7.7.1] we see that if wθ := (T g )1−p0 (T  hθ ) then

(5.2.28)

5.2 Extrapolation Theory

237

wθ ∈ Ap0 (X, μ) and [wθ ] A p0 ≤ WX, p0

(5.2.29)

with the inequality in (5.2.29) implied by (5.2.2). With q0 ∈ (1, ∞) denoting the Hölder conjugate exponent of p0 , may now write ∫ ∫ ∫  θ  f X ≤ | f ||hθ | dμ ≤ | f |(T hθ ) dμ = | f |(T g ) p0 −1 wθ dμ ∫

X

= X

X

| f |(T g ) p0 −1 dwθ ≤

=  f  L p0 (X,wθ ) =  f  L p0 (X,wθ )

∫ X

∫ 

X

∫ X

X

| f | p0 dwθ

 1/p0  ∫ X

(T g )(p0 −1)q0 dwθ

(T g )(p0 −1)q0 (T g )1−p0 (T  hθ ) dμ (T g )(T  hθ ) dμ

g X T  hθ X ≤  f  L p0 (X,wθ ) T

 1/q0

 1/q0

 1/q0

 1/q0

≤ 41/q0  f  L p0 (X,wθ ) .

(5.2.30)

Above, the first two inequalities come from (5.2.26) and (5.2.19), respectively. In the next two equalities we have used the definition of the function wθ given in (5.2.28), and the fact that dwθ = wθ dμ. Going further, we have employed Hölder’s inequality and once again (5.2.28), then relied on the identity (p0 −1)(q0 −1) = 1 and invoked Proposition 5.1.12. The final inequality in (5.2.30) is implied by (5.2.16). In summary, (5.2.30) proves that θ  f X ≤ 41/q0  f  L p0 (X,wθ ) .

(5.2.31)

We next estimate g. To get started, recall from (5.2.18) and (5.2.24) that 0 < T g < ∞ at μ-a.e. point in X.

(5.2.32)

Also, use (5.2.23) and the first property in (5.2.17) to write |g| = gX  g ≤ gXT g.

(5.2.33)

Bearing (5.2.32) in mind, this implies |g|(T g )−1 ≤ gX at μ-a.e. point in X.

(5.2.34)

After rising both sides of (5.2.34) to the power p0 − 1 > 0 and multiplying by |g|, we arrive (again, bearing (5.2.32) in mind) at the conclusion that   1−p0 p −1 g ≤ gX0 |g| at μ-a.e. point in X. |g| p0 T

(5.2.35)

Based on the definition of the weight wθ given in (5.2.28), (5.2.35), Proposition 5.1.12, (5.2.16), and (5.2.26) we therefore obtain

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

∫ g L p0 (X,wθ ) =

X

  1−p0  |g| p0 T (T hθ ) dμ g p0 −1 p0

∫

≤ gX

X

p0 −1 p0

≤ gX





|g| (T hθ ) dμ

gX T  hθ X

 1/p0

 1/p0

 1/p0

= gX T  hθ X

1/p0

≤ 21/p0 gX,

(5.2.36)

hence g L p0 (X,wθ ) ≤ 21/p0 gX .

(5.2.37)

Combining (5.2.31), (5.2.3), (5.2.37) (in this order) and recalling (5.2.29) yields   θ  f X ≤ 22−1/p0 ·

sup

w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

Cw gX .

(5.2.38)

0

Upon sending θ 1 then justifies (5.2.4) in the scenario in which f ∈ X. Step II. The end-game in the proof of (5.2.4). From Step I we know that (5.2.4) is true if f ∈ X. To eliminate the latter additional assumption, bring in the family {Wk }k ∈N of μ-measurable subsets of X considered in (5.1.4). Then 1Wk ∈ X for each k ∈ N and Wk X as k → ∞. If we now define fk := min{| f |, k} · 1Wk for each k ∈ N,

(5.2.39)

then each fk is a μ-measurable function on X satisfying 0 ≤ fk ≤ k · 1Wk . As such, (P2) in Definition 5.1.1 together with Definition 5.1.4 imply that each fk belongs to X.

(5.2.40)

Also, 0 ≤ fk | f | as k → ∞. In particular, (5.2.3) implies that for every Muckenhoupt weight w ∈ Ap0 (X, d, μ) with [w] A p0 ≤ WX, p0 we have  fk  L p0 (X,w) ≤  f  L p0 (X,w) ≤ Cw g L p0 (X,w) .

(5.2.41)

On account of this and (5.2.40), we conclude from (5.2.41) and Step I that    fk X ≤ 22−1/p0 ·

sup

w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

Cw gX for each k ∈ N.

(5.2.42)

0

Passing to limit k → ∞ and relying on item (ii) in Lemma 5.1.7 (while bearing in mind (5.2.22)) we then arrive at the conclusion that (5.2.4) holds as stated.

5.2 Extrapolation Theory

239

Step III. The proof of (5.2.5). In view of Proposition 5.1.14, the estimate claimed in (5.2.5) is a consequence of (5.2.4) written for the Generalized Banach Function Space X. At this stage, all claims in item (1) have been justified, and we turn our attention to item (2). The goal is to establish (5.2.9), now working under the assumption made in (5.2.6) in place of (5.2.1), and with (5.2.8) in place of (5.2.3). To this end, we shall largely follow the same approach as in the treatment of item (1) in which we now formally take p0 := 1, so we will only focus on the novel aspects. As before, suppose first that f ∈ X. In this scenario, in place of (5.2.28) we now simply define wθ := T  hθ

(5.2.43)

which in view of (5.2.27) implies wθ ∈ A1 (X, μ) and [wθ ] A1 ≤ CX M X X →X = WX,1

(5.2.44)

where the final equality in (5.2.44) is a consequence of (5.2.7). Much as in (5.2.30), we then have ∫ ∫ θ  f X ≤ | f ||hθ | dμ ≤ | f |(T  hθ ) dμ ∫

X

= X

X

| f | dwθ =  f  L 1 (X,wθ ),

(5.2.45)

hence θ  f X ≤  f  L 1 (X,wθ ) .

(5.2.46)

As far as the function g is concerned, (5.2.43), Proposition 5.1.12, (5.2.16), and (5.2.26) yield ∫ g L 1 (X,wθ ) = |g| (T  hθ ) dμ ≤ gX T  hθ X ≤ 2gX . (5.2.47) X

Collectively, (5.2.46), (5.2.8), (5.2.47), and (5.2.44) permit us to write   θ  f X ≤ 2 ·

sup

w ∈ A1 (X,d,μ) [w] A1 ≤WX,1

Cw gX .

(5.2.48)

The estimate claimed in (5.2.9) is then obtained from this, in the case when f ∈ X, after sending θ 1. Lastly, the case when f is arbitrary is dealt using the same type of reasoning as the one employed in Step II above, with p0 := 1. This completes the proof of item (2). Finally, item (3) is a consequence of the result in item (2) written for the Generalized Banach Function Space X in place of X, bearing in mind the fact that X = X (cf. Proposition 5.1.14). 

240

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

A byproduct of the proof of the extrapolation result in Theorem 5.2.1 is the (abstract) embedding result of Generalized Banach Function Spaces into Muckenhoupt weighted Lebesgue spaces, described in the corollary below. Corollary 5.2.2 Assume (X, d, μ) is a space of homogeneous type with the property that the quasi-distance d : X × X → [0, ∞) is continuous6 in the product topology τd × τd . Let M X be the Hardy-Littlewood maximal operator on (X, d, μ). Finally, suppose X is a Generalized Banach Function Space on (X, μ) and denote by X its associated space (cf. Definition 5.1.4 and Definition 5.1.11). Then there exists some CX ∈ (0, ∞), depending only on the quasi-distance d (via d defined as in (A.0.19)-(A.0.20)) and the doubling charter of μ, the constants Cd, C such that  M X : X → X boundedly =⇒ X ⊆ L 1 (X, w μ), (5.2.49) w ∈ A1 (X,d,μ) [w] A1 ≤WX,1

where WX,1 := CX · M X X →X

(5.2.50)

and 

M X : X → X boundedly =⇒ X ⊆

L 1 (X, w μ)

(5.2.51)

w ∈ A1 (X,d,μ) [w] A1 ≤WX,1

where WX,1 := CX · M X X→X . Finally, if

(5.2.52)

M X : X → X and M X : X → X are well-defined bounded mappings,

then for each p0 ∈ (1, ∞) one has  X⊆ L p0 (X, w μ) and X ⊆ w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

L p0 (X, w μ)

(5.2.54)

w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

0

where



(5.2.53)

0

p −1

p

0 · M X X →X WX, p0 := CX0 · M X X→X

p

p −1

and WX, p0 := CX0 · M X X0 →X · M X X→X .

(5.2.55)

Proof All desired conclusions are implicit in the proof of Theorem 5.2.1. To be specific, assume first that M X is bounded on X and fix an arbitrary function h ∈ X with hX = 1. For this choice of h used in place of hθ , the estimates in (5.2.47) 6 The result proved in [133, Theorem 7.1.2] guarantees that any quasi-metric space has an equivalent quasi-distance satisfying this property

5.2 Extrapolation Theory

241

prove that any function g ∈ X belongs to the space L 1 (X, w μ) for some choice of a weight w ∈ A1 (X, μ) with [w] A1 ≤ WX,1 . This justifies the claim in (5.2.49). The implication in (5.2.51) is then a consequence of (5.2.49) and Proposition 5.1.14. To deal with (5.2.54), assume (5.2.53) and pick an exponent p0 ∈ (1, ∞). Given any nontrivial function g ∈ X, run the argument which has produced (5.2.37) (with h as above). This shows that there exists some weight w ∈ Ap0 (X, μ) satisfying [w] A p0 ≤ WX, p0 and with the property that g ∈ L p0 (X, w), thus establishing the first inclusion in (5.2.54). Finally, the second inclusion in (5.2.54) becomes a consequence  of the first, keeping in mind the fact that X = X (cf. Proposition 5.1.14). For certain applications, it is useful to formulate an extrapolation result in the spirit of Theorem 5.2.1 for operators (which are not necessarily linear), of the sort described in the next corollary. Corollary 5.2.3 Assume (X, d, μ) is a space of homogeneous type with the property that the quasi-distance d : X × X → [0, ∞) is continuous7 in the product topology τd × τd . Let M X be the Hardy-Littlewood maximal operator on (X, d, μ). Finally, suppose X is a Generalized Banach Function Space on (X, μ) and denote by X its associated space (cf. Definition 5.1.4 and Definition 5.1.11). Fix some integrability exponent p0 ∈ [1, ∞) and assume    p0 L (X, w) f −→ Θ( f ) ∈ M (X, μ) (5.2.56) w ∈ A p0 (X,μ)

constitutes an assignment8 with the property that for any given Muckenhoupt weight w ∈ Ap0 (X, d, μ) one has Θ( f ) L p0 (X,w) ≤ Cw  f  L p0 (X,w) for each f ∈ L p0 (X, w),

(5.2.57)

where the constant9 Cw ∈ (0, ∞] depends only on Θ, d, μ, p0 , and w. Finally, denote by M X the Hardy-Littlewood maximal operator on (X, d, μ) and, given a Generalized Banach Function Space X on (X, μ), denote by X its associated space (cf. Definition 5.1.4 and Definition 5.1.11). Then there exists some CX ∈ (0, ∞), depending only on the quasi-distance d (via d defined as in (A.0.19)-(A.0.20)) and the doubling charter of μ, the constants Cd, C such that the following statements are true: (1) If p0 ∈ (1, ∞) and M X : X → X and M X : X → X are well-defined bounded mappings,

(5.2.58)

7 from [133, Theorem 7.1.2] it is known that any quasi-metric space has an equivalent quasi-distance satisfying this property 8 not necessarily linear 9 note that Cw = ∞ is allowed, and we make the convention that ∞ · 0 = ∞

242

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

then, with

p −1

p

0 · M X X →X WX, p0 := CX0 · M X X→X

p

p −1

and WX, p0 := CX0 · M X X0 →X · M X X→X,

(5.2.59)

it follows that for each f ∈ X the function Θ( f ) is well defined and μ-measurable on X (see the first inclusion in (5.2.54)) and   Θ( f )X ≤ 22−1/p0 ·

sup

w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

Cw  f X,

(5.2.60)

0

while for each f ∈ X the function Θ( f ) is well defined and μ-measurable on X (see the second inclusion in (5.2.54)) and   Θ( f )X ≤ 22−1/p0 ·

sup

w ∈ A p0 (X,d,μ) [w] A p ≤WX, p0

Cw  f X .

(5.2.61)

0

(2) If p0 = 1 and M X : X → X is a well-defined bounded mapping,

(5.2.62)

then, with WX,1 := CX · M X X →X,

(5.2.63)

it follows that for each f ∈ X the function Θ( f ) is well defined and μ-measurable on X (see (5.2.49)) and   Θ( f )X ≤ 2 ·

sup

w ∈ A1 (X,d,μ) [w] A1 ≤WX,1

Cw  f X .

(5.2.64)

(3) If p0 = 1 and M X : X → X is a well-defined bounded mapping,

(5.2.65)

then, with WX,1 := CX · M X X→X,

(5.2.66)

it follows that for each f ∈ X the function Θ( f ) is well defined and μ-measurable on X (see (5.2.51)) and   Θ( f )X ≤ 2 ·

sup

w ∈ A1 (X,d,μ) [w] A1 ≤WX,1

Cw  f X .

(5.2.67)

5.2 Extrapolation Theory

243

Before presenting the proof of this corollary, we make two comments. First, the fact that the constant Cw appearing in (5.2.57) is allowed to be ∞ permits us to obtain a version of Corollary 5.2.3 in which the demand in (5.2.57) is imposed for a restrictive class of weights, namely only for those w ∈ Ap0 (X, d, μ) satisfying [w] A p0 ≤ W where W ∈ (0, ∞] is some (a priori) fixed threshold. Specifically, in the scenario in which (5.2.57) is known to hold only for this smaller category of weights, we define Cw for each w ∈ Ap0 (X, μ) with [w] A p0 ≤ W, w := (5.2.68) C ∞ for each w ∈ Ap0 (X, μ) with [w] A p0 > W, and note that (5.2.57) is satisfied for any weight w ∈ Ap0 (X, d, μ) provided Cw is w (given that the latter is larger than the former). Wit this adjustment, replaced by C Corollary 5.2.3 applies and yields bounds like (5.2.61), (5.2.64), (5.2.67) written w in place of Cw . However, since the characteristic of the weight is restricted with C in all aforementioned estimates, we may return to the original constant Cw in said bounds provided the threshold W is sufficiently large, to begin with. The second comment we wish to make in relation to Corollary 5.2.3 is that in  the case when the space of homogeneous type (X, d, μ) is actually Σ, | · − · |, σ where Σ is a closed Ahlfors regular subset of Rn (for some n ∈ N with n ≥ 2) and σ := H n−1 Σ, then (5.2.56) naturally takes effect if we start from the premise that the operator Θ acts at the level  L 1 Σ,

σ(x)  f −→ Θ( f ) ∈ M (Σ, σ). 1 + |x| n−1

(5.2.69)

See [133, (7.7.104)] in this regard. Here is the proof of Corollary 5.2.3. Proof of Corollary 5.2.3 This is a direct consequence of the extrapolation results from Theorem 5.2.1 and the embeddings established in Corollary 5.2.2.  The results so far in this section highlight the importance of the boundedness of the Hardy-Littlewood maximal operator on a given Generalized Banach Function Space X and/or its associate space X. Our next proposition sheds light on the delicate balance between these properties.   Proposition 5.2.4 Let X, d, μ be a space of homogeneous type with the property that the quasi-distance d : X × X → [0, ∞) is continuous10 in the product topology τd × τd . Fix an arbitrary exponent s ∈ (0, 1) and denote by M X and M X,s , respectively, the Hardy-Littlewood maximal operator on X (cf. (A.0.71)) and its L s -based version (defined as in (A.0.69)). Finally, suppose X is a Generalized Banach Function Space on (X, μ) and denote by X its associated space (cf. Definition 5.1.4 and Definition 5.1.11). In this setting, 10 The result proved in [133, Theorem 7.1.2] guarantees that any quasi-metric space has an equivalent quasi-distance satisfying this property

244

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

if M X : X → X is a well-defined bounded mapping then M X,s : X → X is a well-defined bounded mapping,

(5.2.70)

and if M X : X → X is a well-defined bounded mapping then M X,s : X → X is a well-defined bounded mapping.

(5.2.71)

It is straightforward to check that for any sigma-finite measure space (X, M, μ) both L 1 (X, μ) and L ∞ (X, μ) are Generalized Banach Function spaces for which  1    L (X, μ) = L ∞ (X, μ) and L ∞ (X, μ) = L 1 (X, μ) (5.2.72)   (see, e.g., [14, Theorem 2.5, p. 10]). Whenever X, d, μ happens to be a space of homogeneous type with the property that the quasi-distance d : X × X → [0, ∞) is continuous, we know that M X is a well-defined bounded mapping on L ∞ (X, μ) but (in general) M X,s maps L 1 (X, μ) boundedly into itself only if s < 1. This explains the restriction of the exponent s to the interval (0, 1) and highlights the optimality of Proposition 5.2.4. Proof of Proposition 5.2.4 Assume first that M X is bounded on X, and pick an arbitrary function f ∈ X. In addition, suppose the functions g ∈ X has gX ≤ 1. Then M X,s f : X → [0, ∞] is a well-defined μ-measurable function, and the version of Fefferman-Stein’s maximal inequality from [133, Proposition 7.6.7] ensures the existence of a constant C = C(μ, d, s) ∈ (0, ∞) such that ∫ ∫     M X,s f |g| dμ ≤ C | f | M X g dμ ≤ C f X M X gX X

X

≤ CM X X →X  f X gX ≤ C f X,

(5.2.73)

where we have also employed the Hölder type inequality established in Proposition 5.1.12. From this, (5.1.72), and Definition 5.1.4 we then conclude that M X,s f ∈ X and M X,s f X ≤ C f X . In view of the arbitrariness of f , this shows that M X,s induces a well-defined sublinear bounded mapping from X into itself. The claim in (5.2.70) is therefore established. Finally, (5.2.71) follows from (5.2.70) written for X in place of X, keeping in mind the fact that X = X (see Proposition 5.1.14).  It is also of interest to note that if a Generalized Banach Function Space, considered over a space of homogeneous type, is invariant under the action of the Hardy-Littlewood maximal operator then the Generalized Banach Function Space enjoys some useful embedding properties, of the sort discussed in the lemma below.

5.2 Extrapolation Theory

245

  Lemma 5.2.5 Assume X, d, μ is a space of homogeneous type with the property that the quasi-distance d : X × X → [0, ∞) is continuous11 in the product topology 1 (X, μ) the space of μ-measurable functions on X which are τd × τd . Denote by Lloc ∞ (X, μ) the space of μabsolutely integrable on any d-ball in X, and denote by Lcomp measurable functions on X which are essentially bounded and vanish μ-a.e. outside of some d-ball in X. Also, bring in the Hardy-Littlewood maximal operator M X on X (cf. (A.0.71)). Finally, suppose X is a Generalized Banach Function Space on (X, μ) and denote by X its associated space (cf. Definition 5.1.4 and Definition 5.1.11). In relation to these, make the assumption that either M X (X) ⊆ X, or M X (X) ⊆ X .

(5.2.74)

Then ∞ 1 ∞ 1 Lcomp (X, μ) ⊆ X ⊆ Lloc (X, μ) and Lcomp (X, μ) ⊆ X ⊆ Lloc (X, μ).

(5.2.75)

In addition12, whenever μ(X) < ∞ one has the continuous embeddings L ∞ (X, μ) → X → L 1 (X, μ) and L ∞ (X, μ) → X → L 1 (X, μ).

(5.2.76)

Proof To fix ideas, assume first that M X (X) ⊆ X. Pick an arbitrary f ∈ X. Then M X f ∈ X, so (5.1.10) implies that there ⨏ exists xo ∈ X such that (M X f )(xo ) < ∞. In view of (A.0.71), this further entails B (x ,r) | f | dμ < ∞ for each r ∈ (0, ∞), which d

o

1 (X, μ). The inclusion X ⊆ L 1 (X, μ) has therefore ultimately shows that f ∈ Lloc loc been established. Moving on, recall from Definition 5.1.1 and Definition 5.1.4 that there exists a collection {Yj } j ∈N of μ-measurable subsets sets of X with the property that we have ∞  Yj and 1Yj ∈ X for each j ∈ N. Pick j∗ ∈ N such that μ(Yj∗ ) > 0. Fix an X = j=1

arbitrary point x∗ ∈ X. Then there exists r∗ ∈ (0, ∞) such that   0 < μ Yj∗ ∩ Bd (x∗, r∗ ) < ∞.

(5.2.77)

Fix an arbitrary real number R ≥ r∗ . Then there exists a constant C = C(d) ∈ (0, ∞) for which we have Bd (x, CR) ⊆ Bd (x∗, R) for each x ∈ Bd (x∗, R).

(5.2.78)

Consequently, with Cμ ∈ (0, ∞) and Dμ ∈ [0, ∞) denoting the doubling constant and doubling order of μ (see [133, (7.7.4)]), for each x ∈ Bd (x∗, R) we may estimate 11 from [133, Theorem 7.1.2] it is known that any quasi-metric space has an equivalent quasidistance satisfying this property   12 recall that for a space of homogeneous type X, d, μ one has μ(X) < ∞ if and only if X is d-bounded; see, e.g., [9, Proposition 2.12(7), pp. 49-50]

246

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces



  M X 1Yj∗ (x) ≥

1  μ Bd (x, CR) 

∫ B d (x,C R)

1Yj∗ dμ

∫ 1  1Y dμ ≥  μ Bd (x, CR) B d (x∗,R) j∗     μ Bd (x∗, R)  · μ Yj∗ ∩ Bd (x∗, R) =  μ Bd (x, CR) ≥

  1  R  Dμ · μ Yj∗ ∩ Bd (x∗, r∗ ) Cμ CR

≥ c :=

  1  1  Dμ · μ Yj∗ ∩ Bd (x∗, r∗ ) > 0. Cμ C

(5.2.79)

  This shows that M X 1Yj∗ ≥ c · 1B d (x∗,R) with c ∈ (0, ∞), hence 1B d (x∗,R) ∈ X for each real number R ≥ r∗ , by the lattice property enjoyed by X (cf. (5.1.12)). Since for any R ∈ (0, r∗ ) we have 1B d (x∗,r∗ ) ≥ 1B d (x∗,R) , the lattice property just mentioned also implies that 1B d (x∗,R) ∈ X for each R ∈ (0, r∗ ). All together, 1B d (x∗,R) ∈ X for ∞ (X, μ) ⊆ X each x∗ ∈ X and each R ∈ (0, ∞), from which we then conclude that Lcomp by one last application of the lattice property for the vector space X. Moving on, given any g ∈ X, Proposition 5.1.12 guarantees that f g is absolutely integrable on X for each f ∈ X. In particular, in view of what we have just proved, 1B d (x,R) · g is an absolutely integrable function on X for each x ∈ X and each 1 (X, μ), which goes to show that X  ⊆ L 1 (X, μ). To R ∈ (0, ∞). Thus, g ∈ Lloc loc finish the proof of (5.2.75) in the current case, there remains to notice that (5.1.35)1 (X, μ) guarantee that we also have the inclusion (5.1.36) and the fact that X ⊆ Lloc ∞  Lcomp (X, μ) ⊆ X . Going further, if in (5.2.74) we now assume that M X (X) ⊆ X, then all desired conclusions in (5.2.75) follow from what we have proved so far applied to the Generalized Banach Function Space X (in place of X) and Proposition 5.1.14. Finally, to deal with (5.2.76), assume μ(X) < ∞ and recall (see., e.g., [9]) that this is equivalent to having X bounded (relative to the quasi-distance d). Bearing this in mind, (5.2.75) presently implies X ⊆ L 1 (X, μ) and X ⊆ L 1 (X, μ). In view of Proposition 5.1.17, these set-theoretic inclusions self-improve to continuous embeddings X → L 1 (X, μ) and X → L 1 (X, μ). Having established these, we may now rely on (5.1.7) together with Corollary 5.1.15 to conclude that L ∞ (X, μ) → X, and  invoke (5.1.35)-(5.1.36) to also obtain that L ∞ (X, μ) → X. Our next definition brings forth a distinguished linear subspace of a Generalized Banach Function Space, which is going to play a significant role in future endeavors. Indeed, this offers a natural functional analytic environment in which the HardyLittlewood maximal operator as well as singular integral operators are well behaved.

5.2 Extrapolation Theory

247

  Definition 5.2.6 Suppose X, d, μ is a space of homogeneous type with the property that the quasi-distance d : X × X → [0, ∞) is continuous13 in the product topology τd × τd . Denote by M X the Hardy-Littlewood maximal operator on X (cf. (A.0.71)). Also, assume X is a Generalized Banach Function Space on (X, μ) with the property that (5.2.80) either M X (X) ⊆ X, or M X (X) ⊆ X, where X is its associated space (cf. Definition 5.1.4 and Definition 5.1.11). In this ˚ as being the closure setting, recall that (5.2.75) holds, soit is meaningful to define X  ∞ of Lcomp (X, μ) in the Banach space X,  · X , i.e., ∞ (X, μ)  · X . ˚ := Lcomp X

(5.2.81)

Thus, by design,   ˚ is a closed linear subspace of X, hence X, ˚  · X becomes a Banach X ∞ (X, μ) is a dense subspace. space in which Lcomp

(5.2.82)

In addition, from (5.2.75) we see that ∞ ˚ ⊆ X ⊆ L 1 (X, μ), (X, μ) ⊆ X Lcomp loc

(5.2.83)

while from (5.2.82) and (5.2.76) we conclude that  · X ∞ ˚ if μ(X) < ∞ , hence L ∞ (X, μ) is a dense  then X = L ∞(X, μ) ˚ → X → L 1 (X, μ) contin˚  · X and L (X, μ) → X subspace of X, uously in this case.

(5.2.84)

Finally, from (5.1.13), (5.2.81), and Lemma 1.2.20 we see that the operator of pointwise multiplication by some given function b in ˚ into itself, with L ∞ (X, μ) is a bounded mapping from the space X operator norm ≤ b L ∞ (X,μ) ,

(5.2.85)

˚ to the effect that which, in turn, readily implies the lattice property for X, if f , g : X → R are two μ-measurable functions such that ˚ then g also |g| ≤ | f | at μ-a.e. point on X and f ∈ X, ˚ belongs to the space X.

(5.2.86)

In the case when the underlying space of homogeneous type is an Ahlfors regular set equipped with its “surface” measure (and the standard Euclidean distance), we may combine the abstract embedding results from Corollary 5.2.2 with the embedding results involving weighted Lebesgue spaces from [133, Lemma 7.7.13] to

13 from [133, Theorem 7.1.2] it is known that any quasi-metric space has an equivalent quasidistance satisfying this property

248

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

establish useful embedding properties, like the ones formulated in the proposition below. Proposition 5.2.7 Let Σ ⊆ Rn (n ∈ N with n ≥ 2) be a closed set which is Ahlfors regular and abbreviate σ := H n−1 Σ. Also, denote by M Σ the Hardy-Littlewood   maximal operator associated with the space of homogeneous type Σ, | · − · |, σ . Finally, consider a Generalized Banach Function Space X on (Σ, σ) and denote by X its associated space (cf. Definition 5.1.4 and Definition 5.1.11). Then, with M (Σ, σ) standing for the space of σ-measurable functions on Σ, whenever M Σ (X) ⊆ X one has the continuous embeddings   1+ |x | n−1 f ∈ M (Σ, σ) : ess-supx ∈Σ 1+log | f (x)| < ∞ → X |x | +     (1+log+ |x |)σ(x)  1 → L 1 Σ, 1+σ(x) , as well as X → L Σ, 1+ |x | n−1 |x | n−1

(5.2.87)

whenever M Σ (X) ⊆ X one has the continuous embeddings   1+ |x | n−1 g ∈ M (Σ, σ) : ess-supx ∈Σ 1+log |g(x)| < ∞ → X + |x |     (1+log |x |)σ(x) → L 1 Σ, 1+σ(x) . as well as X → L 1 Σ, 1+ +|x | n−1 n−1 |x |

(5.2.88)

and

Furthermore, if

M Σ : X → X and M Σ : X → X are well-defined bounded mappings

(5.2.89)

then one can find an exponent p∗ = p∗ (Σ, X) ∈ (1, ∞) with the property that for each q ∈ (0, p∗ ) there exists ε = ε(Σ, X, q) ∈ (0, 1) such that     σ(x) σ(x)  q and X Σ, . X → L q Σ, → L 1 + |x| n−1−ε 1 + |x| n−1−ε

(5.2.90)

Moreover, under the assumptions made in (5.2.89) and with p∗ = p∗ (Σ, X) ∈ (1, ∞) as above, whenever q ∈ (1, p∗ ) and ε = ε(Σ, X, q) ∈ (0, 1) is defined as before, if q  := (1 − 1/q)−1 ∈ (1, ∞) is the Hölder conjugate exponent of q, it follows that one also has     L q Σ, (1 + |x| n−1−ε )q −1 σ(x) → X and   (5.2.91)   L q Σ, (1 + |x| n−1−ε )q −1 σ(x) → X . As a corollary, whenever the Ahlfors regular set Σ is compact and (5.2.89) holds,  there exists q ∈ (1, 2) such that L q (Σ, σ) → X → L q (Σ, σ) and  L q (Σ, σ) → X → L q (Σ, σ) continuously, with q  := (1 − 1/q)−1 .

(5.2.92)

5.2 Extrapolation Theory

249

Proof To deal with (5.2.87), assume M Σ (X) ⊆ X. Having fixed some x0 ∈ Σ, Lemma 5.2.5 then guarantees that 1Δ(x0,1) ∈ X, hence also M Σ (1Δ(x0,1) ) ∈ X and, further, (M Σ ◦ M Σ )(1Δ(x0,1) ) ∈ X. Then, on account of [133, (7.6.69)] and Proposition 5.1.12, for each function g ∈ X we may estimate ∫ ∫ 1 + log+ |x| 1 + log+ |x − x0 | |g(x)| dσ(x) ≈ |g(x)| dσ(x) n−1 1 + |x| 1 + |x − x0 | n−1 Σ Σ ∫   ≈ |g(x)| (M Σ ◦ M Σ ) 1Δ(x0,1) (x) dσ(x) Σ

  ≤ CΣ,x0 (M Σ ◦ M Σ )(1Δ(x0,1) )X gX = CgX where

  C := CΣ,x0 · (M Σ ◦ M Σ )(1Δ(x0,1) )X ∈ (0, ∞).

(5.2.93) (5.2.94)

This establishes the second embedding in (5.2.87). With this in hand, the first embedding in (5.2.87) then follows on account of Corollary 5.1.15. At this stage, (5.2.87) has been fully justified and (5.2.88) follows from it and Proposition 5.1.14. Moving on, work under the assumptions made in (5.2.89). From these, the first inclusion in (5.2.54) (written for some arbitrary fixed exponent p0 ∈ (1, ∞)), and [133, (7.7.103)] we then conclude that, with q and ε as in the statement, we have the set theoretic inclusion   σ(x) . (5.2.95) X ⊆ L q Σ, 1 + |x| n−1−ε Granted this, we may invoke  Proposition  5.1.17 with (V,  · ) taken to be the weighted Lebesgue space L q Σ, 1+ |xσ(x) equipped with its canonical norm and, on | n−1−ε account of (5.1.77), conclude that we have the first continuous embedding specified in (5.2.90). Parenthetically, we note that while (5.1.77) only ensures that the operator norm of the inclusion (5.2.90) is finite, we can actually estimate said operator norm more concretely. To be specific, fix some point x0 ∈ Σ and recall from Lemma 5.2.5 that we presently have 1Δ(x0,1) ∈ X. Pick an arbitrary f ∈ X and, with q and ε as before, consider the functions  1/q ∫ | f (x)| q dσ(x) · 1Δ(x0,1) and G := f . (5.2.96) F := n−1−ε Σ 1 + |x| Then, thanks to [133, (7.7.113)] (written with ε in place of θ), the extrapolation result established in Theorem 5.2.1 applies to the pair (F, G) and ultimately provides a constant C ∈ (0, ∞) which depends only on X and Σ for which  1/q  ∫  | f (x)| q 1Δ(x ,1)  ≤ C f X . dσ(x) · (5.2.97) 0 X n−1−ε Σ 1 + |x|

250

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

−1  Hence the operator norm of the first inclusion in (5.2.90) is ≤ C 1Δ(x0,1) X . Finally, the second embedding claimed in (5.2.90) is a consequence of the first embedding specified in (5.2.90) written for the Generalized Banach Function Space  X (in place of X) and Proposition 5.1.14. ˚ introduced in relation to a given Generalized Other basic properties of the space X, Banach Function Space X as in Definition 5.2.6, are described in the next lemma. Lemma 5.2.8 Let Σ ⊆ Rn (n ∈ N with n ≥ 2) be a closed set which is Ahlfors regular and abbreviate σ := H n−1 Σ. Also, denote by M Σ the Hardy-Littlewood   maximal operator associated with the space of homogeneous type Σ, | · − · |, σ . Consider a Generalized Banach Function Space X on (Σ, σ) with the property that M Σ : X → X and M Σ : X → X

(5.2.98)

are well-defined bounded mappings,

where X is its associated space (cf. Definition 5.1.4 and Definition 5.1.11). Finally, bring in the exponent p∗ = p∗ (Σ, X) ∈ (1, ∞) from Proposition 5.2.7. Then for each q ∈ (1, p∗ ), if q := (1 − 1/q )−1 ∈ (1, ∞) is its Hölder conjugate ˚ originally introduced in Definition 5.2.6 (with X := Σ and exponent it follows that X q μ := σ)  may bealternatively described as the closure of Lcomp (Σ, σ) in the Banach space X,  · X , i.e., 

q ˚ = Lcomp (Σ, σ) X

 · X

,





(5.2.99)

as well as the closure of Lipc (Σ) in the Banach space X,  · X , i.e., ˚ = Lipc (Σ)  · X . X

(5.2.100)

In particular, (5.2.99), (5.2.100), and (5.2.92) imply that whenever the Ahlfors regular set Σ is compact and (5.2.89) holds, it ˚ is the closure of L q (Σ, σ) in the Banach space X,  · X , follows that X  ˚ → X → L q (Σ, σ) continuously, and Lipc (Σ) ⊆ X ˚ (5.2.101) L q (Σ, σ) → X densely. Furthermore,

˚ −→ X ˚ MΣ : X is a well-defined bounded mapping.

(5.2.102)

q

∞ (Σ, σ) ⊆ L Proof Staring with Lcomp comp (Σ, σ) and taking closures in X yields, on account of (5.2.81), 

q ˚ ⊆ Lcomp (Σ, σ) X

We next claim that



 · X

q ˚ Lcomp (Σ, σ) ⊆ X.

.

(5.2.103) (5.2.104)

5.2 Extrapolation Theory

251

To prove this, let the number ε = ε(Σ, X, q) ∈ (0, 1) be as in Proposition 5.2.7 and q pick an arbitrary function f ∈ Lcomp (Σ, σ). Then there exists a sequence {s j } j ∈N of  simple functions which converges to f in L q (Σ, σ) (see [133, (3.1.11)]), and matters may be arranged so that each s j vanishes outside of the support of f . Since f is compactly supported, {s j } j ∈N actually converges to  we conclude that the sequence  



f in the space L q Σ, (1 + |x| n−1−ε )q −1 σ(x) . Together with (5.2.91), this proves ∞ (Σ, σ), we conclude that {s j } j ∈N converges to f in X. Since each s j belongs to Lcomp ˚ In view of the arbitrariness of f , this establishes from this and (5.2.81) that f ∈ X. (5.2.104). ˚ is a closed subspace of X (cf. (5.2.82)), In turn, from (5.2.104) and the fact that X we deduce that  · X q ˚ Lcomp (Σ, σ) ⊆ X. (5.2.105)

At this stage, (5.2.99) follows from (5.2.103) and (5.2.105). Also, (5.2.100) is a consequence of (5.2.99), [133, (3.7.22)], and the continuity of the first embedding in (5.2.91). Let us now turn our attention to (5.2.102). Since M Σ maps L ∞ (Σ, σ) into itself, it follows that   ∞ (Σ, σ) ⊆ L ∞ (Σ, σ). (5.2.106) M Σ Lcomp ∞ (Σ, σ). Then there exists Fix a point x0 ∈ Σ, and pick an arbitrary function f ∈ Lcomp some R ∈ (0, ∞) such that | f | ≤ R · 1Δ(x0,R) at σ-a.e. point in Σ. In concert with [133, (7.6.66)], this permits us to write

     0 ≤ M Σ f (x) ≤ R · M Σ 1Δ(x0,R) (x) ≤ CΣ R · ≤

Rn (R + |x − x0 |)n−1

CΣ,x0,R at σ-a.e. point x ∈ Σ. 1 + |x − x0 | n−1

(5.2.107)

Also, (5.2.106) implies that ∞ g j := 1Δ(x0, j) · M Σ f ∈ Lcomp (Σ, σ) for each j ∈ N.

For each large j ∈ N now estimate

(5.2.108)

252

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

    M Σ f − g j  = 1Σ\Δ(x , j) · M Σ f  0 X X   ≤ C 1Σ\Δ(x0, j) ·

 1   1 + | · −x0 | n−1 X

1Σ\Δ(x0, j) 1 + log+ | · −x0 |  ·  1 + log+ | · −x0 | 1 + | · −x0 | n−1 X C  1 + log+ | · −x0 |  ≤   ln j 1 + | · −x0 | n−1 X  C   ≤ (M Σ ◦ M Σ )(1Δ(x0,1) ) X ln j

  = C

≤

2   C  M Σ X→X 1Δ(x0,1) X ln j

(5.2.109)

where the first equality comes from (5.2.108), the subsequent inequality is implied by (5.2.107) (bearing in mind property (P2) in Definition 5.1.1), the second equality is obvious, the penultimate inequality uses [133, (7.6.69)] and once again the monotonicity of the norm in X, and the final inequality is a consequence of (5.2.98). In  view of the fact that M Σ X→X < ∞ thanks to (5.2.98), and that 1Δ(x0,1) X < ∞ thanks to (5.2.75), this proves that g j → M Σ f in X as j → ∞.

(5.2.110)

˚ Hence, In concert with (5.2.108) and (5.2.81), this ultimately shows that M Σ f ∈ X.  ∞  ˚ M Σ Lcomp (Σ, σ) ⊆ X. (5.2.111) From (5.2.111), (5.2.98), and item (1) in Lemma 1.2.20 we then conclude that the claim made in (5.2.102) is true. 

5.3 Young Functions and Orlicz Spaces Orlicz spaces are natural examples of classical Banach function spaces (cf. Remark 5.1.5). Moreover, it is possible to discern when the Hardy-Littlewood operator is bounded on an Orlicz space constructed in relation to a given space of homogenous type, based on the nature of the dilation indices associated with the corresponding Young function. Here we elaborate on these aspects. This ties up with the material from §5.1 and §5.2. See Proposition 5.3.15 for a concrete result of such flavor. For the benefit of the reader, we begin by reviewing the class of Young functions.

5.3 Young Functions and Orlicz Spaces

253

Call Φ a Young function14 provided Φ : [0, ∞) → [0, ∞) is convex, does not vanish on (0, ∞), and satisfies lim+ Φ(t)/t = 0 as well as lim Φ(t)/t = ∞.

(5.3.1)

t→∞

t→0

There are other qualities implicit in (5.3.1). For one thing, Φ(λt) ≤ λΦ(t) for each λ ∈ [0, 1] and t ∈ [0, ∞), which may be justified by writing   Φ(λt) = Φ λt + (1 − λ)0 ≤ λΦ(t) + (1 − λ)Φ(0) = λΦ(t) for each λ ∈ [0, 1] and t ∈ [0, ∞).

(5.3.2)

(5.3.3)

In turn, (5.3.2) has several notable consequences. First, we may use (5.3.3) and the fact that Φ is non-negative on [0, ∞) to obtain 0 ≤ Φ(λ) ≤ λΦ(1) for each λ ∈ [0, 1]. This implies that Φ vanishes continuously at 0. Upon recalling that any convex function on an open interval is continuous (cf., e.g., [164, Theorem 3.2, p. 61]), we then conclude that Φ is continuous on [0, ∞). We may also use (5.3.3) and the fact that Φ is strictly positive on (0, ∞) to conclude that Φ(λt) < Φ(t) for each λ ∈ (0, 1) and t ∈ (0, ∞). In particular, Φ is strictly increasing on [0, ∞). Another consequence of (5.3.3) is that Φ(λt)/(λt) ≤ Φ(t)/t for each λ ∈ (0, 1] and t ∈ (0, ∞), which readily shows that the assignment (0, ∞) t →

Φ(t) ∈ (0, ∞) is non-decreasing. t

(5.3.4)

As such, the limits lim+ Φ(t)/t and lim Φ(t)/t are guaranteed to exist (in the interval t→0

t→∞

[0, ∞]). Having lim+ Φ(t)/t = 0 implies, since Φ(0) = 0, that Φ is differentiable from t→0

the right at 0, and Φ(0) = 0. Being a convex function makes Φ differentiable15 at L 1 -a.e. point in (0, ∞) (cf., e.g., [160, Theorem 25.5, p. 246]). Bearing this in mind, we conclude from (5.3.4) that tΦ(t) ≥ 1 at any differentiability point t ∈ (0, ∞) for Φ, Φ(t) hence at L 1 -a.e. point t in the interval (0, ∞).

(5.3.5)

It also turns out that Φ is super-additive, i.e., Φ(t1 ) + Φ(t2 ) ≤ Φ(t1 + t2 ) for each t1, t2 ∈ [0, ∞). Indeed, this is clear if either t1 = 0 or t2 = 0, and in the case when t1, t2 ∈ (0, ∞) we employ (5.3.2) to write 14 in the literature, the demands in the second line of (5.3.1) are not always imposed at full strength, but even when they are replaced by weaker hypotheses the corresponding Orlicz spaces (cf. (5.3.27)) continue to enjoy a significant number of properties deduced in this section 15 In fact, any convex function is locally Lipschitz (cf., e.g., [158]), hence a.e. differentiable by the classical Rademacher theorem. Aleksandov’s theorem also guarantees that Φ has a second-order derivative at L 1 -a.e. point in (0, ∞) (see, e.g., [49, Theorem 1, p. 242]).

254

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

  t   t1 2 (t1 + t2 ) + Φ (t1 + t2 ) t1 + t2 t1 + t2  t   t  1 2 Φ(t1 + t2 ) + Φ(t1 + t2 ) = Φ(t1 + t2 ), ≤ t1 + t2 t1 + t2

Φ(t1 ) + Φ(t2 ) = Φ





(5.3.6)

as wanted. Finally, having the property lim Φ(t)/t = ∞ forces16 lim Φ(t) = ∞, t→∞ t→∞ and since the function Φ has already been shown to be strictly increasing, it follows that the function Φ : [0, ∞) → [0, ∞) is a bijection. Granted this, it makes sense to consider its inverse Φ−1 : [0, ∞) → [0, ∞), itself a continuous strictly increasing function, which vanishes at 0, and has lim Φ−1 (t) = ∞. t→∞

Call a Young function Φ doubling, and indicate this by simply writing Φ ∈ Δ2 , if there exists a constant CΔ ∈ (0, ∞) with the property that Φ(2t) ≤ CΔ Φ(t) for each t ∈ [0, ∞).

(5.3.7)

For future use we note that Φ being strictly increasing forces CΔ > 1, hence # := log2 CΔ ∈ (0, ∞).

(5.3.8)

For the Young function Φ, the doubling property is intimately related with the behavior of the function hΦ (t) := sup s>0

Φ(st) ∈ (0, ∞] for each t ∈ (0, ∞) Φ(s)

(5.3.9)

at infinity. Let us elaborate on this. First, it is immediate from its definition that the function hΦ is non-decreasing (given that Φ is increasing), submultiplicative (i.e., hΦ (t1 t2 ) ≤ hΦ (t1 )hΦ (t2 ) for all t1, t2 > 0), and (5.3.10) hΦ (1) = 1; also, (5.3.2) implies hΦ (t) ≤ t for each t ∈ (0, 1].  m As a consequence, hΦ (t) ≤ hΦ (t 1/m ) for each t ∈ (0, ∞) and m ∈ N. Bearing in mind that hΦ is non-decreasing, this further implies that if hΦ (t∗ ) = ∞ for some t∗ ∈ (1, ∞) then hΦ is identically equal to +∞ on the interval (1, ∞).

(5.3.11)

In relation to (5.3.11) we claim that, for any Young function Φ, hΦ is not identically equal to +∞ on the interval (1, ∞) if and only if Φ ∈ Δ2 if and only if there exists CΔ ∈ (1, ∞) such that hΦ (t) ≤ CΔ t # for each t ∈ (1, ∞) where # := log2 CΔ ∈ (0, ∞).

(5.3.12)

To justify this, assume first that Φ ∈ Δ2 . Pick some arbitrary t ∈ (1, ∞). Then there exists a unique N ∈ N with 2 N −1 < t ≤ 2 N , which also entails N − 1 < log2 t ≤ N. 16 henceforth, we shall tacitly assume that Φ is extended to [0, ∞] with Φ(∞) = ∞

5.3 Young Functions and Orlicz Spaces

255

As such, given any s ∈ (0, ∞) we can iterate (5.3.7) and make use of (5.3.8) to write Φ(ts) ≤ Φ(2 N s) ≤ (CΔ ) N Φ(s) = CΔ (CΔ ) N −1 Φ(s) < CΔ (CΔ )log2 t Φ(s) = CΔ (2# )log2 t Φ(s) = CΔ t # Φ(s).

(5.3.13)

# Hence, hΦ (t) = sups>0 Φ(ts) Φ(s) ≤ CΔ t for each t ∈ (1, ∞), as wanted. In the opposite direction, if hΦ is not identically equal to +∞ on the interval (1, ∞) it follows that hΦ is finite at any point in (1, ∞) (cf. (5.3.11)). As such, (5.3.7) holds with CΔ := hΦ (2) < ∞, so Φ is doubling. This finishes the proof of (5.3.12). The growth of a Young function Φ is encoded in its lower and upper dilation indices defined, respectively, as

ln hΦ (t) ln hΦ (t) = lim+ , t→0 ln t ln t 0 0 : 0

One can also see directly from definitions that for each f ∈ M (X, μ) we have ∫   Φ | f (x)|/ f  L Φ (X,μ) dμ(x) ≤ 1 whenever 0 <  f  L Φ (X,μ) < ∞ (5.3.25) X

∫

and  f  L Φ (X,μ) ≤ 1 if and only if

  Φ | f (x)| dμ(x) ≤ 1.

(5.3.26)

X

The Luxemburg norm is a genuine norm on the Orlicz space L Φ (X, μ), defined as   L Φ (X, μ) := f ∈ M (X, μ) :  f  L Φ (X,μ) < ∞ .

(5.3.27)

It turns out that the assignment ρΦ : M+ (X, μ) → [0, +∞] defined in (5.3.22) is a function norm, in the sense of Definition 5.1.1. In fact, it may be easily checked from (5.3.22) that for each E ∈ M with μ(E) < ∞ we have  −1   1 Φ 1/μ(E) if μ(E) > 0, Φ (5.3.28) ρ (1E ) = 0 if μ(E) = 0. In particular, ρΦ (1E ) < ∞ whenever E ∈ M has μ(E) < ∞, hence

5.3 Young Functions and Orlicz Spaces

1E ∈ L Φ (X, μ) for each E ∈ M with μ(E) < ∞.

257

(5.3.29)

Also,∫ for each set E ∈ M with μ(E) < ∞ there exists a constant CE ∈ (0, ∞) such that E f dμ ≤ CE ρΦ ( f ) for each function f ∈ M+ (X, μ). As a consequence, for any given set E ∈ M with μ(E) < ∞, the restriction operator L Φ (X, μ) f −→ f E ∈ L 1 (E, μ) is well defined and bounded.

(5.3.30)

For all these and related matters see, e.g., [14, pp. 268–271]. In particular, the Orlicz space L Φ (X, μ) associated with any sigma-finite measure space (X, M, μ) and any Young function Φ is a classical Banach function space (as introduced in [14, Definition 1.3, p. 3]; see Remark 5.1.5), hence also a Generalized Banach Function Space, in the sense of our Definition 5.1.4.

(5.3.31)

If the sigma-finite measure space (X, M, μ) is non-atomic, recall that a (classic) Banach function space X on (X, μ) is called rearrangement invariant if for any two function f , g ∈ M (X, μ) one has  f X = gX provided     μ {x ∈ X : | f (x)| > λ} = μ {x ∈ X : |g(x)| > λ} for each λ ≥ 0.

(5.3.32)

See, e.g., [14, Definition 4.1, p. 59]. It is then apparent from this definition, (5.3.31), and (5.3.24) that if (X, M, μ) is a non-atomic sigma-finite measure space then the Orlicz space L Φ (X, μ) associated with any Young function Φ is a rearrange- (5.3.33) ment invariant Banach function space. Let us also record here the well known fact that given a sigma-finite measure space (X, M, μ) along with a Young function Φ, the Orlicz space L Φ (X, μ) is reflexive if one has 1 < i(Φ) ≤ I(Φ) < ∞.

(5.3.34)

Our next lemma contributes to the understanding of the Luxemburg norm of an Orlicz space. Lemma 5.3.1 Assume (X, M, μ) is a sigma-finite measure space, and pick some Young function17 Φ ∈ Δ2 . In addition, suppose α, β ∈ R are two numbers with the property that 0 < α < i(Φ) ≤ I(Φ) < β < ∞. Then there exist constants cα,β ∈ (0, ∞) and Cα,β ∈ (0, ∞) such that for each function f ∈ M (X, μ) one has

17 the reader is reminded of the convention that Φ(∞) := ∞

258

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

 ∫    β cα,β · min  f  LαΦ (X,μ),  f  L Φ (X,μ) ≤ Φ | f (x)| dμ(x) X

  β ≤ Cα,β · max  f  LαΦ (X,μ),  f  L Φ (X,μ) . (5.3.35) As a corollary, for each function f ∈ M (X, μ) one has ∫   f ∈ L Φ (X, μ) ⇐⇒ Φ | f (x)| dμ(x) < +∞,

(5.3.36)

X

hence Φ

∫



L (X, μ) = f ∈ M (X, μ) :

   Φ | f (x)| dμ(x) < +∞ .

(5.3.37)

X

In view of this result, given a sigma-finite measure space (X, M, μ) and some Young function Φ, for each f ∈ M (X, μ) we henceforth agree to abbreviate ∫   Φ | f (x)| dμ(x) ∈ [0, +∞], (5.3.38) N Φ ( f ) := X

and refer to it as the modular size of the function f . In terms of this symbol, we may then recast (5.3.37) simply as   L Φ (X, μ) = f ∈ M (X, μ) : N Φ ( f ) < +∞ . (5.3.39) So, while the assignment M (X, μ) f → N Φ ( f ) ∈ [0, +∞] does not give rise to a conventional norm (as it lacks homogeneity and the triangle inequality), if Φ ∈ Δ2 it does satisfy weaker substitutes. For example, it is clear from (5.3.20) and (5.3.21) that, given any Young function Φ ∈ Δ2 , whenever α, β ∈ R satisfy 0 < α < i(Φ) ≤ I(Φ) < β < ∞ it follows that there exist two constants, cα,β ∈ (0, 1] and Cα,β ∈ [1, ∞), such that cα,β · min{λα, λ β } · N Φ ( f ) ≤ N Φ (λ f ) ≤ Cα,β · max{λα, λ β } · N Φ ( f )

(5.3.40)

for each number λ ∈ (0, ∞) and each function f ∈ M (X, μ). Let us now present the proof of Lemma 5.3.1. Proof of Lemma 5.3.1 Fix an arbitrary function f ∈ M (X, μ). Let us justify the second inequality in (5.3.35). This is clear if  f  L Φ (X,μ) = 0, and if  f  L Φ (X,μ) = ∞ there is nothing to prove. Therefore assume 0 <  f  L Φ (X,μ) < ∞, in which case (5.3.25) and (5.3.21) permit us to estimate

5.3 Young Functions and Orlicz Spaces

cα,β

259

 ∫    −β · min  f  L−αΦ (X,μ),  f  L Φ (X,μ) · Φ | f (x)| dμ(x) X

∫

≤ X

  Φ | f (x)|/ f  L Φ (X,μ) dμ(x) ≤ 1. (5.3.41)

The second inequality in (5.3.35) now readily follows from this. Turning our attention to ∫  the first inequality in (5.3.35), we may assume that 0 <  f  L Φ (X,μ) and X Φ | f (x)| dμ(x) < ∞, since otherwise there is nothing to prove. Since in view of (5.3.2) we may bound ∫ ∫     Φ λ| f (x)| dμ(x) ≤ λ Φ | f (x)| dμ(x) for each λ ∈ (0, 1), (5.3.42) X

X

the assumed finiteness ∫ property   ensures that there exists some small number λ ∈ (0, 1) such that X Φ λ| f (x)| dμ(x) ≤ 1. Together with (5.3.26), this implies −1 λ f  L Φ (X,μ) ≤ 1 from which  we further conclude that  f  L Φ (X,μ) ≤ λ < ∞. Then for each ε ∈ 0,  f  L Φ (X,μ) we may rely on (5.3.26) to write ∫     f | f (x)|   dμ(x) > 1, (5.3.43) > 1 =⇒ Φ  Φ   f  L Φ (X,μ) − ε L (X,μ)  f  L Φ (X,μ) − ε X   −1 and then use (5.3.20) with λ :=  f  L Φ (X,μ) −ε to deduce from the latter inequality that ∫   −α   −β    · Cα,β · max  f  L Φ (X,μ) − ε ,  f  L Φ (X,μ) − ε Φ | f (x)| dμ(x) > 1. X

(5.3.44) After sending ε  0 we eventually arrive at the first inequality in (5.3.35) after some simple algebra.  In turn, the above lemma is an ingredient in the proof of the following useful density result. Lemma 5.3.2 Suppose (X, M, μ) is a sigma-finite measure space, and select a Young function Φ ∈ Δ2 . Then the space of simple functions Sfin (X, μ) defined in (A.0.107) is dense in the Orlicz space L Φ (X, μ). To offer an example, consider Φ : [0, ∞) → [0, ∞), Φ(t) := t ln(1 + t) for each t ∈ [0, ∞).

(5.3.45)

It may be verified without difficulty that this is a Young function satisfying lim

t→0+

Φ(2t) Φ(2t) = 2 and lim = 1, t→∞ Φ(t) Φ(t)

(5.3.46)

260

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

hence Φ ∈ Δ2 (in fact i(Φ) = 1 and I(Φ) = 2; see (5.3.103)). Consequently, given any sigma-finite measure space (X, M, μ), the corresponding Orlicz space L Φ (X, μ), aka the Zygmund space ∫   | f (x)|ln(1 + | f (x)|) dμ(x) < ∞ , (5.3.47) L log L(X, μ) := f ∈ M (X, μ) : X

has Sfin (X, μ) as a dense subspace. Proof of Lemma 5.3.2 From (5.3.29) and (A.0.107) we see that Sfin (X, μ) is a subspace of L Φ (X, μ). To show that this subspace is actually dense in the space L Φ (X, μ), pick two exponents α, β such that 0 < α < i(Φ) ≤ I(Φ) < β < ∞. Given an arbitrary function f ∈ L Φ (X, μ), by decomposing f = f+ − f− where f± := 12 (| f | ± f ) matters are reduced to the case when f is non-negative to begin with. Assuming the latter, bring in the sequence {s j } j ∈N ⊂ S(X, μ) associated with the non-negative, μ-measurable, function f on X as in [133, (3.1.12)]. In particular, each simple func N ai 1 Ai , where N ∈ N, the ai ’s are strictly positive real tion s j is of the form i=1 numbers, and the Ai ’s are mutually disjoint sets in M. Then, for each i, the fact that 0 ≤ ai 1 Ai ≤  f at every point in X implies, in view of the monotonicity of Φ, that Φ ai 1 Ai (x) ≤ Φ f (x) for each x ∈ X. As a consequence of this and the second inequality in (5.3.35), for each i we may write ∫ ∫     Φ ai 1 Ai (x) dμ(x) ≤ Φ f (x) dμ(x) Φ(ai ) · μ(Ai ) = X

X

 β ≤ Cα,β · max  f  LαΦ (X,μ),  f  L Φ (X,μ) < ∞, 

(5.3.48)

with the last inequality above a consequence of the fact that f ∈ L Φ (X, μ). Since Φ(ai ) ∈ (0, ∞), this forces μ(Ai ) < ∞. Ultimately, this shows that each function in the sequence {s j } j ∈N actually belongs the space Sfin (X, μ) defined in (A.0.107). Going further, [133, (3.1.12)] implies 0 ≤ f (x) − s j (x) ≤ f (x) for each x ∈ X and each j ∈ N which, in viewof the monotonicity of the function Φ, leads to the   conclusion that Φ f (x) − s j (x) ≤ Φ f (x) for each point x ∈ X and each integer j ∈ N. The function X x → Φ f (x) is, by the second inequality in (5.3.35), in the space L 1 (X, μ).Also, thanks to [133, (3.1.12)] and the continuity of the function Φ, we have lim Φ f (x) − s j (x) = Φ(0) = 0 at every point x ∈ X. Granted these j→∞

properties, Lebesgue’s Dominated Convergence Theorem applies and gives ∫   lim Φ f (x) − s j (x) dμ(x) = 0. (5.3.49) j→∞

X

In concert with the first inequality in (5.3.35), this proves   β lim min  f − s j  LαΦ (X,μ),  f − s j  L Φ (X,μ) = 0, j→∞

(5.3.50)

5.3 Young Functions and Orlicz Spaces

261

which ultimately forces lim  f − s j  L Φ (X,μ) = 0. Now the desired conclusion folj→∞



lows. Here is another noteworthy consequence of Lemma 5.3.1.

Lemma 5.3.3 Let (X, M, μ) be a sigma-finite measure space, and fix a Young function Φ ∈ Δ2 . Suppose T is a homogeneous mapping18 from the Orlicz space L Φ (X, μ) into M (X, μ). Then T is bounded on L Φ (X, μ), i.e., there exists a constant C ∈ (0, ∞) such that T f  L Φ (X,μ) ≤ C f  L Φ (X,μ) for each f ∈ L Φ (X, μ),

(5.3.51)

if and only if whenever α, β are such that 0 < α < i(Φ) ≤ I(Φ) < β < ∞ there exists a constant Cα,β ∈ (0, ∞) with the property that for each function f ∈ L Φ (X, μ) one has ∫   Φ |T f | dμ (5.3.52) X

≤ Cα,β · max

 ∫

  Φ | f | dμ

α/β

X

,

∫

  Φ | f | dμ

β/α 

.

X

As a corollary, T is bounded on L Φ (X, μ) whenever the following modular inequality ∫ ∫     Φ |(T f )(x)| dμ(x) ≤ C Φ | f (x)| dμ(x) (5.3.53) X

holds for each function f ∈ of f .

X

L Φ (X,

μ), where C ∈ (0, ∞) is a constant independent

Proof If (5.3.52) holds, then (5.3.35) shows that T maps the unit ball of L Φ (X, μ) into a bounded set in L Φ (X, μ). In view of the fact that T is homogeneous, this proves (5.3.51). Conversely, if (5.3.51) holds, then (5.3.35) readily yields (5.3.52).  Pressing on, fix a Young function Φ. The complementary function of Φ is the function  : [0, ∞) −→ [0, ∞) given by Φ     := sup{st − Φ(s)} = sup s t − Φ(s) for each t ≥ 0. Φ(t) s s>0

(5.3.54)

s>0

The last equality (5.3.54) together with the second line in (5.3.1) show (also bearing in mind that Φ is continuous) that the supremum is attained and is a positive number  for each t > 0. Also, it is clear from the first equality (5.3.54) that Φ(0) = 0, In  is itself addition, as the supremum of a family of linear (hence convex) functions, Φ  is itself a a convex function. It actually turns out that the complementary function Φ Young function, and 18 not necessarily linear

262

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

  = Φ for each Young function Φ. Φ

(5.3.55)

See [160, Theorem 12.2, p. 104] for a more general result of this flavor, involving generic convex functions in Rn . An immediate consequence of the definition given in (5.3.54) is Young’s inequality, to the effect that  for all s, t ∈ [0, ∞). st ≤ Φ(s) + Φ(t)

(5.3.56)

 −1 (t) ≤ 2t for each t ∈ [0, ∞), t ≤ Φ−1 (t)Φ

(5.3.57)

It is also known that

which further implies, after replacing t by Φ(t) and applying the strictly increasing  function Φ,    Φ(t)/t ≤ Φ(t) for each t ∈ (0, ∞). Φ (5.3.58) With “prime” denoting Hölder conjugation, the lower and upper dilation indices satisfy (cf. [38, p. 71], [54], and the references therein)  and I(Φ) = i(Φ).  i(Φ) = I(Φ)

(5.3.59)

In view of (5.3.19), this further implies  ∈ Δ2 if and only if i(Φ) > 1. Φ

(5.3.60)

Thus, as a consequence of (5.3.19) and (5.3.60),  ∈ Δ2 if and only if 1 < i(Φ) ≤ I(Φ) < ∞. Φ, Φ

(5.3.61)

Suppose (X, M, μ) is a sigma-finite measure space and fix a Young function Φ. It has already been noted in (5.3.31) that the Orlicz space L Φ (X, μ) a classical Banach function space (cf. Remark 5.1.5) hence, in particular, a Generalized Banach Function Space. Then, it turns out that the associated space X of the Generalized Banach Function Space X := L Φ (X, μ) (in the sense of Definition 5.1.11) is the Orlicz   space L Φ (X, μ) defined as in (5.3.27) in relation to the complementary function Φ of the original Young function Φ. See [14, Corollary 8.15, p. 275]. In view of this, Proposition 5.1.14, and (5.3.55), we therefore have 

if X := L Φ (X, μ) then X = L Φ (X, μ), 

and if X := L Φ (X, μ) then X = L Φ (X, μ).

(5.3.62)

Moreover, as a classical Banach function space (cf. Remark 5.1.5), X contains the family of all essentially bounded functions vanishing outside sets of finite measure, ˚ the closure of the latter family in X (compare with Defiand we agree to denote by X nition 5.2.6 in the case of a Generalized Banach Function Space). Then Lemma 5.3.2 tells us that

5.3 Young Functions and Orlicz Spaces

263

˚ = L Φ (X, μ) provided Φ ∈ Δ2 . if X := L Φ (X, μ) then X

(5.3.63)

Time to shift perspectives, and make the following definition. Definition 5.3.4 (i) Call a function φ : (0, ∞) → R quasi-increasing if there exists C ∈ (0, ∞) such that φ(t1 ) ≤ Cφ(t2 ) whenever 0 < t1 < t2 . Also, call a given function φ : (0, ∞) → R quasi-decreasing if there exists C ∈ (0, ∞) such that φ(t2 ) ≤ Cφ(t1 ) whenever 0 < t1 < t2 . (ii) Call a function φ : (0, ∞) → R pseudo-increasing if there exist two constants C1, C2 ∈ (0, ∞) such that φ(t1 ) ≤ C1 φ(C2 t2 ) whenever 0 < t1 < t2 . Finally, call a function φ : (0, ∞) → R pseudo-decreasing if there exist two constants C1, C2 ∈ (0, ∞) such that φ(t2 ) ≤ C1 φ(C2 t1 ) whenever 0 < t1 < t2 . The next lemma sheds some light on the interplay between quasi/pseudo monotonicity properties of power-modulated Young functions and their complementary functions. Lemma 5.3.5 Let p, p ∈ (1, ∞) be such that 1/p + 1/p = 1, and consider a Young  Then t −p Φ(t) is pseudofunction Φ, along with its complementary function Φ.  −p  increasing if and only if t Φ(t) is pseudo-decreasing, and t −p Φ(t) is pseudo  is pseudo-increasing. decreasing if and only if t −p Φ(t) Proof Suppose t −p Φ(t) is pseudo-increasing, i.e., there exist C1, C2 ∈ (0, ∞) such that −p (5.3.64) t1 Φ(t1 ) ≤ C1 (C2 t2 )−p Φ(C2 t2 ) whenever 0 < t1 < t2 .  p −1  Fix 0 < t1 < t2 . Since 0 < s < t2 /t1 · s, we see from (5.3.64) that   p   t  p −1  2 p t2 · s > C1−1 C2 · Φ(s) > 0. Φ C2 t1 t1

(5.3.65)

Based on this and (5.3.54) we may then write !  t  p −1   t  p −1  2 2  2 ) = sup{st2 − Φ(s)} = sup C2 · st2 − Φ C2 ·s Φ(t t1 t1 s>0 s>0 ≤ sup C2 s>0

=

p C1−1 C2

·

= C1−1 C2 · p

 t  p 2

t1  t  p 2

t1  t  p 2

t1

·

p st1 − C1−1 C2

 t  p 2

t1

! · Φ(s) !

· sup s>0

1−p C1 C2

· st1 − Φ(s)

   C1 C 1−p · t1 . ·Φ 2

(5.3.66)

 In view of Definition 5.3.4, this tells us that the function t −p Φ(t) is pseudodecreasing. 

264

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

Assume next that t −p Φ(t) is pseudo-decreasing, i.e., there exist C1, C2 ∈ (0, ∞) such that −p

t2 Φ(t2 ) ≤ C1 (C2 t1 )−p Φ(C2 t1 ) whenever 0 < t1 < t2 .

(5.3.67)

  p −1 Consider 0 < t1 < t2 . Given that 0 < t1 /t2 · s < 2, we conclude from (5.3.67) that   t  p −1  −p    p −1  1 · s Φ C2 t1 /t2 ·s . (5.3.68) 0 < s−p Φ(s) ≤ C1−1 C2 t2 Use this and (5.3.54) to write  1 ) = sup{st1 − Φ(s)} = sup C2 Φ(t s>0

s>0

≤ sup C2 s>0

=

p C1−1 C2

·

= C1−1 C2 · p

 t  p 1

t2  t  p 1

t2  t  p 1

t2

·

 t  p −1 1

t2

p st2 − C1−1 C2

 t  p 1

t2



· st1 − Φ C2

s>0

1

t2

·s



!

! · Φ(s) !

· sup

 t  p −1

1−p C1 C2

· st2 − Φ(s)

   C1 C 1−p · t2 ·Φ 2

(5.3.69)

  which, in light of Definition 5.3.4, implies that function t −p Φ(t) is pseudoincreasing. At this stage, the equivalences claimed in the statement become consequences of what we have proved and (5.3.55). 

The relevance of Definition 5.3.4 is most apparent in the context of the lemma below, containing useful characterizations of the dilation indices of a Young function. Lemma 5.3.6 For each Young function Φ one has   i(Φ) = sup p ∈ (0, ∞) : t −p Φ(t) is quasi-increasing   = sup p ∈ (0, ∞) : t −p Φ(t) is pseudo-increasing ,

(5.3.70)

and   I(Φ) = inf p ∈ (1, ∞) : t −p Φ(t) is pseudo-decreasing   = inf p ∈ (1, ∞) : t −p Φ(t) is quasi-decreasing , with the usual convention inf  = ∞ in place.

(5.3.71)

5.3 Young Functions and Orlicz Spaces

265

Proof Recall from the first inequality in (5.3.16) that 1 ≤ i(Φ), and pick a strictly positive number p < i(Φ). In light of (5.3.14) this enures the existence of a number t∗ ∈ (0, 1) with the property that ln hΦ (t) > p for each t ∈ (0, t∗ ). ln t

(5.3.72)

Since ln < 0 on (0, 1), this implies ln hΦ (t) < p ln t = ln(t p ). Hence, on the one hand, hΦ (t) < t p for each t ∈ (0, t∗ ). On the other hand, for each t ∈ (t∗, 1) we may −p write hΦ (t) ≤ 1 < Ct p with C := t∗ ∈ (1, ∞). On account of (5.3.9), this analysis shows that Φ(st) < Ct p Φ(s) for each t ∈ (0, 1) and s ∈ (0, ∞).

(5.3.73)

Suppose next that 0 < t1 < t2 . Since any Young function is strictly increasing, (5.3.73) used with t := t1 /t2 ∈ (0, 1) and s := t2 ∈ (0, ∞) gives   Φ(t1 ) = Φ (t1 /t2 )t2 ≤ C(t1 /t2 ) p Φ(t2 ), (5.3.74) −p

−p

from which we deduce that t1 Φ(t1 ) ≤ Ct2 Φ(t2 ). According to Definition 5.3.4, the function t −p Φ(t) is therefore quasi-increasing, hence p is less than, or equal to, the first supremum in (5.3.70). Upon letting p i(Φ) we conclude that   i(Φ) ≤ sup p ∈ (0, ∞) : t −p Φ(t) is quasi-increasing   ≤ sup p ∈ (0, ∞) : t −p Φ(t) is pseudo-increasing , (5.3.75) where the last inequality is a simple consequence of the fact that any quasi-increasing is also pseudo-increasing (cf. Definition 5.3.4). Going further, from what we have proved earlier we know that any strictly positive   number p < i(Φ) belongs to the set p ∈ (0, ∞) : t −p Φ(t) is pseudo-increasing . As such, the set in question is nonempty. Let us now pick an element in said set, i.e., suppose p ∈ (0, ∞) is such that t −p Φ(t) is pseudo-increasing. Definition 5.3.4 guarantees the existence of two constants C1, C2 ∈ (0, ∞) with the property that −p t1 Φ(t1 ) ≤ C1 (C2 t2 )−p Φ(C2 t2 ) whenever 0 < t1 < t2 . To proceed, fix an arbitrary t ∈ (0, 1). Then for each s ∈ (0, ∞) we have 0 < st < s so (st)−p Φ(st) < C1 (C2 s)−p Φ(C2 s) for each s ∈ (0, ∞).

(5.3.76)

In turn, this implies Φ(st) < C1 (C2 )−p t p for each s ∈ (0, ∞) Φ(C2 s) which, after re-naming C2 s simply as s, become equivalent to   Φ s(t/C2 ) < C1 (t/C2 ) p for each s ∈ (0, ∞). Φ(s)

(5.3.77)

(5.3.78)

266

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

If we also re-name t/C2 ∈ (0, 1/C2 ) as t, we arrive at the conclusion that Φ(st) < C1 t p for each t ∈ (0, 1/C2 ) and each s ∈ (0, ∞). Φ(s)

(5.3.79)

From this and (5.3.9) we then obtain hΦ (t) = sup s>0

Φ(st) ≤ C1 t p for each t ∈ (0, 1/C2 ). Φ(s)

(5.3.80)

As such, ln hΦ (t) ≤ ln C1 + p ln t for all t > 0 small which, in concert with (5.3.9), permits us to write i(Φ) = lim+ t→0

ln hΦ (t) C1 ≥ p + lim+ = p. t→0 ln t ln t

(5.3.81)

Taking the supremum over all p’s satisfying the current working assumptions then proves that   i(Φ) ≥ sup p ∈ (0, ∞) : t −p Φ(t) is pseudo-increasing . (5.3.82) At this stage, (5.3.70) follows from (5.3.75) and (5.3.82). Turing to the task of proving (5.3.71), first consider the question whether   I(Φ) ≤ inf p ∈ (1, ∞) : t −p Φ(t) is pseudo-decreasing . (5.3.83) Since inf  = ∞, this is trivially true when there is no p ∈ (1, ∞) such that t −p Φ(t) is quasi-decreasing. To treat the remaining case, suppose that p ∈ (1, ∞) is such   is quasithat t −p Φ(t) is quasi-decreasing. Then Lemma (5.3.5) implies that t −p Φ(t)   is the increasing, where p ∈ (1, ∞) is the Hölder conjugate exponent of p and Φ  ≥ p, complementary function of Φ. As such, from (5.3.70) we conclude that i(Φ)   = I(Φ), with the last equality a consequence of (5.3.59) and (5.3.55). hence p ≥ i(Φ) With this in hand, (5.3.83) follows. Given that any quasi-decreasing function is also pseudo-decreasing, we then have   I(Φ) ≤ inf p ∈ (1, ∞) : t −p Φ(t) is pseudo-decreasing   (5.3.84) ≤ inf p ∈ (1, ∞) : t −p Φ(t) is quasi-decreasing . Consider the issue whether   I(Φ) ≥ inf p ∈ (1, ∞) : t −p Φ(t) is quasi-decreasing .

(5.3.85)

If I(Φ) = ∞ there is nothing to prove, so assume I(Φ) < ∞. Recall that we always have I(Φ) ≥ 1 (cf. (5.3.16)). Pick a number p > I(Φ). In light of (5.3.15), this enures the existence of some t∗ ∈ (1, ∞) such that ln hΦ (t) < p for each t ∈ (t∗, ∞), ln t

(5.3.86)

5.3 Young Functions and Orlicz Spaces

267

which implies that for each t ∈ (t∗, ∞) we have ln hΦ (t) < p ln t = ln(t p ), hence hΦ (t) < t p for each t ∈ (t∗, ∞). Since hΦ is non-decreasing, therefore there exists some C ∈ (0, ∞) such that hΦ (t) ≤ Ct p for each t ∈ (1, ∞). On account of (5.3.9), this forces Φ(st) ≤ Ct p Φ(s) for each t ∈ (1, ∞) and each s ∈ (0, ∞).

(5.3.87)

Suppose next that 0 < t1 < t2 . Keeping in mind that any Young function is strictly increasing, (5.3.87) used with t := t2 /t1 ∈ (1, ∞) and s := t1 ∈ (0, ∞) yields   Φ(t2 ) = Φ (t2 /t1 )t1 ≤ C(t2 /t1 ) p Φ(t1 ), (5.3.88) −p

−p

from which we conclude that t2 Φ(t2 ) ≤ Ct1 Φ(t1 ). The function t −p Φ(t) is therefore quasi-decreasing. Since we also know that p ∈ (1, ∞), we conclude that p is greater than, or equal to, the infimum in (5.3.85). Sending p  I(Φ) then proves (5.3.85).  The characterizations of the dilation indices of a Young function given in Lemma 5.3.6 are the key ingredient in the estimates contained in our next lemma, itself a basic tool in the proof of the version of the Marcinkiewicz interpolation result established a little later, in Theorem 5.3.16. Lemma 5.3.7 Consider a Young function Φ. Then whenever 0 < p < i(Φ) one has lim+

t→0

Φ(t) =0 tp

(5.3.89)

and there exists a constant Cp ∈ (0, ∞) such that ∫ 0

t

Φ(s) Φ(t) ds ≤ Cp p for each t ∈ (0, ∞). p+1 t s

(5.3.90)

Moreover, whenever I(Φ) < q < ∞ one has lim

t→∞

Φ(t) = 0, tq

and there exists a constant Cq ∈ (0, ∞) such that ∫ ∞ Φ(s) Φ(t) ds ≤ Cq q for each t ∈ (0, ∞). q+1 t s t

(5.3.91)

(5.3.92)

Proof If 0 < p < i(Φ) then (5.3.70) guarantees the existence of some p∗ > p with the property that the function t −p∗ Φ(t) is quasi-increasing. Using this, for each t ∈ (0, ∞) we may estimate

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

∫

t

0

Φ(s) ds = s p+1

∫

t

0

≤C

1 s1−(p∗ −p)

Φ(t) t p∗

∫

0

Φ(s) ds s p∗

t

1 s1−(p∗ −p)

ds = C

Φ(t) , tp

(5.3.93)

proving (5.3.90). In addition, lim+

t→0

 Φ(t) Φ(t) = lim + t p∗ −p p = 0, p (0,1) t→0 t t ∗

(5.3.94)

since p∗ − p > 0 and 0 < Φ(t) t p∗ ≤ CΦ(1) < ∞ for each t ∈ (0, 1). Next, if I(Φ) < q < ∞ then (5.3.71) ensures that there exists q∗ ∈ (1, q) such that t −q Φ(t) is quasi-decreasing. Keeping this in mind, we may now write ∫ ∞ ∫ ∞ Φ(s) 1 Φ(s) ds = ds q+1 1+(q−q ) ∗ s q∗ s s t t ∫ 1 Φ(t) Φ(t) ∞ ds = C q , (5.3.95) ≤C q 1+(q−q ) ∗ ∗ t t s t which establishes (5.3.92). Finally,  1 Φ(t) Φ(t) = 0, = lim q t→∞ t (1,∞) t→∞ t q−q∗ t q∗ lim

since q − q∗ > 0 and 0
0 Φ(t)

 ∈ Δ2 if and only if 1 < inf Φ, Φ



(t) ≤ sup tΦ Φ(t) < ∞.

(5.3.97)

t>0

Moreover, given any Young function Φ of class C 1 on (0, ∞), it has been noted in [54, Theorem 1.3, p. 435] that  tΦ(t) tΦ(t)  , lim inf , i(Φ) ≥ min lim inf t→∞ t→0+ Φ(t) Φ(t)  tΦ(t) tΦ(t)  i(Φ) ≤ min lim sup , lim sup , Φ(t) t→∞ Φ(t) t→0+ and

(5.3.98) (5.3.99)

5.3 Young Functions and Orlicz Spaces

269

 tΦ(t) tΦ(t)  , lim sup , I(Φ) ≤ max lim sup Φ(t) t→∞ Φ(t) t→0+

(5.3.100)

 tΦ(t) tΦ(t)  I(Φ) ≥ max lim inf , lim inf . t→∞ t→0+ Φ(t) Φ(t)

(5.3.101)

In particular, if the limits r0 := lim+ t→0

tΦ(t) tΦ(t) and r∞ := lim exist t→∞ Φ(t) Φ(t)

(5.3.102)

then i(Φ) = min(r0, r∞ ) and I(Φ) = max(r0, r∞ ).

(5.3.103)

The differentiability condition imposed above on a Young function is not essential. To elaborate on this aspect, let us introduce an equivalence relation in the class of non-decreasing non-negative functions defined on [0, ∞) and which vanish at the origin, by setting Φ ∼ Ψ if and only if there exists c ∈ (0, 1] such that Ψ(c t) ≤ Φ(t) ≤ Ψ(t/c) for each t ≥ 0.

(5.3.104)

Then, for one thing, for two Young functions Φ, Ψ, having Φ ∼ Ψ implies that i(Φ) = i(Ψ),  in addition, Φ ∈ Δ2 if and only if Ψ ∈ Δ2 ,  ∼ Ψ; I(Φ) = I(Ψ), and Φ  ∈ Δ2 ; also, CΦ ∼ Φ for any constant (5.3.105)  ∈ Δ2 if and only if Ψ while Φ C ∈ (0, ∞) if Φ ∈ Δ2 (cf. (5.3.20)-(5.3.21)). For another thing, from the discussion in (5.3.1)-(5.3.4) we see that if Φ is a Young function then ∫ t Φ(s) ds for each t ∈ [0, ∞) (5.3.106) Θ(t) := s 0 is a Young function which is continuously differentiable on the interval (0, ∞), and is equivalent to Φ (i.e., Θ ∼ Φ), specifically Φ(t/2) ≤ Θ(t) ≤ Φ(t) for each t ≥ 0.

(5.3.107)

In addition, Θ is doubling if Φ is doubling, and has the property that its comple is continuously differentiable on (0, ∞) and equivalent to Φ  (i.e., mentary function Θ   Θ ∼ Φ). As a consequence of (5.3.105), we also have i(Θ) = i(Φ) plus I(Θ) = I(Φ),  ∈ Δ2 if Φ  ∈ Δ2 , while Φ  ∈ Δ2 if and only if Φ  ∈ Δ2 . Of course, this smoothing and Θ procedure can be iterated, yielding successively more regular (equivalent) Young functions. In applications, given a function φ which may fail to be globally convex, it is of interest to associate a genuine Young function Φ having roughly the same size at φ. This is made precise in our next lemma.

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

Lemma 5.3.8 Suppose φ : [0, ∞) → [0, ∞) is a function satisfying the following properties:   φ(0) = 0, φ > 0 on (0, ∞), φ ∈ C 0 (0, ∞) , near zero and near infinity (i.e., outside of a compact subinterval of (0, ∞)) the function φ is of class C 2 and φ  > 0, and one has lim+ φ (t) = 0, t→0

lim φ (t) = ∞,

t→∞

lim inf t→∞

tφ (t) φ(t)

(5.3.108)

> 1.

Then there exists a Young function Φ which is continuously differentiable on (0, ∞) and a constant C ∈ (0, ∞) such that Φ(t) ≤ φ(t) ≤ Φ(Ct) for each t ∈ (0, ∞), and Φ coincides with φ near zero and near infinity

(5.3.109)

(i.e., outside of a compact subinterval of (0, ∞)). Moreover, if one also assumes that lim sup t→0+

tφ (t) tφ (t) < ∞ and lim sup < ∞, φ(t) φ(t) t→∞

(5.3.110)

then Φ ∈ Δ2 and, for any given sigma-finite measure space (X, M, μ), the Orlicz space originally defined in relation to Φ as in (5.3.27) may alternatively described (compare with (5.3.37)) as ∫     φ | f (x)| dμ(x) < +∞ . (5.3.111) L Φ (X, μ) = f ∈ M (X, μ) : X

 ∈ Δ2 , if one Finally, the complementary function of Φ is also doubling, i.e., Φ also assumes that tφ (t) > 1. (5.3.112) lim inf t→0+ φ(t) In relation to the situation described in the above lemma, we shall henceforth adopt the following: Convention 5.3.9 Whenever φ : [0, ∞) → [0, ∞) is a function as described in (5.3.108), we shall write Φ(t) ≈ φ(t) (or, simply, Φ ≈ φ) to indicate that Φ is a Young function continuously differentiable on (0, ∞) which is related to φ as in (5.3.109). Proof of Lemma 5.3.8 We begin by noting that for each number t > 0 sufficiently small there exists some ξt ∈ (0, t) such that 0
0 there). In particular,

5.3 Young Functions and Orlicz Spaces

271

tφ (t) > 1 for each t > 0 sufficiently small. φ(t)

(5.3.114)

We claim that the properties listed in (5.3.108) also imply lim

t→0+

φ(t) φ(t) = 0 and lim = ∞. t→∞ t t

(5.3.115)

Indeed, the first formula in (5.3.115) follows from (5.3.113), the Squeeze Theorem, and the fact that lim+ φ (t) = 0. Consider next the second formula in (5.3.115). The t→0

fact that φ is convex near infinity implies that in that regime the graph of φ lies above any of its tangent lines, i.e., there exists some T ∈ (0, ∞) with the property that for each t∗ ≥ T we have φ(t) ≥ φ (t∗ )(t − t∗ ) + φ(t∗ ) at every t ≥ T. This forces   lim inf φ(t) t ≥ φ (t∗ ), and since lim φ (t∗ ) = ∞ the second formula in (5.3.115) is t∗ →∞

t→∞

proved. In particular, (5.3.115) shows that lim φ(t) = 0 and lim φ(t) = ∞.

t→0+

t→∞

(5.3.116)

Going further, let ε ∈ (0, 1) be small enough such that the function φ is of class C 2 and φ  > 0 on (0, ε) ∪ (ε −1, ∞). We may also assume that ε ∈ (0, 1) is sufficiently that φ is continuous on (0, ∞), small so that φ (t) > 0 for each t ∈ [ε −1, ∞). Given  it follows that m := min φ(t) : ε ≤ t ≤ ε −1 is a well-defined strictly positive number, Moreover, since lim+ φ (t) = 0 = lim+ φ(t) there exists some small number

t0 ∈ (0, ε) such that

t→0

t→0

m > φ (t0 )(ε −1 − t0 ) + φ(t0 ) and φ (ε −1 ) > φ (t0 ). For this choice, define Φ0 : [0, ∞) → [0, ∞) by setting, for each t ≥ 0, φ(t) if t ∈ (0, t0 ], Φ0 (t) := φ (t0 )(t − t0 ) + φ(t0 ) if t ∈ [t0, ∞).

(5.3.117)

(5.3.118)

Then Φ0 convex and strictly increasing on [0, ∞), Φ0 > 0 on (0, ∞),   Φ0 ∈ C 1 (0, ∞) , Φ0 (t) ≤ φ(t) for each t ∈ [0, ∞), and Φ0 = φ on [0, t0 ]. (5.3.119) Next, since the very last property in (5.3.108) entails    φ(t)  φ(t)  lim inf t −  = lim inf t 1 −  = ∞, t→∞ t→∞ φ (t) tφ (t)

(5.3.120)

we may find some large number t1 ∈ (ε −1, ∞) such that t1 −

φ(t1 ) > ε −1 . φ (t1 )

(5.3.121)

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

For this choice, define Φ1 : [0, ∞) → [0, ∞) by setting, for each t ≥ 0, 1) ⎧ 0 if t ≤ t1 − φφ(t ⎪  (t ) , ⎪ 1 ⎪ ⎨ ⎪ φ(t ) 1  Φ1 (t) := φ (t1 )(t − t1 ) + φ(t1 ) if t1 − φ (t ) ≤ t ≤ t1, 1 ⎪ ⎪ ⎪ ⎪ φ(t) if t ≥ t1 . ⎩

Then

 Φ1 is convex on [0, ∞), C 1 and strictly increasing on t1 −

φ(t1 ) φ (t1 ) , ∞

Φ1 (t) ≤ φ(t) for each t ∈ [0, ∞), and Φ1 = φ on [t1, ∞).

(5.3.122)



,

(5.3.123)

If we now define Ψ : [0, ∞) → [0, ∞) by setting Ψ(t) := max{Φ0 (t), Φ1 (t)} for each t ≥ 0,

(5.3.124)

then the function Ψ is convex and strictly increasing on [0, ∞), such that Ψ(t) ≤ φ(t) for each t ∈ [0, ∞) and which actually coincides with φ outside the compact interval [t0, t1 ]. In addition, the function Ψ is continuously differentiable everywhere on (0, ∞) except at the point t# :=

t1 φ (t1 ) − t0 φ (t0 ) − φ(t1 ) + φ(t0 )  −1  ∈ ε ,∞ φ (t1 ) − φ (t0 )

(5.3.125)

where the two slanted lines from (5.3.118) and (5.3.122)  actually meet. Finally,  slightly round off the graph of Ψ near the corner t#, Ψ(t# ) (using a small convex arc tangent to both lines) to yield Φ : [0, ∞) → [0, ∞) with Ψ(t) ≤ Φ(t) ≤ φ(t) for each t ∈ [0, ∞), such that Φ is strictly increasing, convex, and of class C 1 on the entire interval (0, ∞), while continues to coincide with φ outside the compact interval [t0, t1 ]. In particular, Φ(0) = φ(0) = 0 and lim Φ(t) = ∞ (cf. (5.3.116)). By t→∞

design, Φ(t) ≥ Φ0 (t) > 0 for each t ∈ (0, ∞). We also claim that there exists some C ∈ [1, ∞) such that Φ(Ct) ≥ φ(t) for each t ∈ [0, ∞).

(5.3.126)

Indeed, having C ≥ 1 implies, since Φ is increasing everywhere and agrees with φ on [0, t0 ] ∪ [t1, ∞), that Φ(Ct) ≥ Φ(t) = φ(t) for each t ∈ [0, t0 ] ∪ [t1, ∞). To deal with the interval [t0, t1 ], define M := max{φ(t) : t0 ≤ t ≤ t1 }, and notice that M ∈ (0, ∞). To finish the proof of (5.3.126), it suffices to show that Φ(Ct) ≥ M for each t ∈ [t0, t1 ] which, in view of the monotonicity of Φ, is implied by Φ(Ct0 ) ≥ M. However, since lim Φ(t) = ∞, this may always be arranged by taking C ∈ [1, ∞) t→∞ sufficiently large. With (5.3.126) in hand, all properties claimed in (5.3.109) have been justified. In light of these and (5.3.115) we then readily conclude that lim

t→0+

Φ(t) Φ(t) = 0 and lim = ∞. t→∞ t t

(5.3.127)

5.3 Young Functions and Orlicz Spaces

273

Hence, Φ is actually a Young function (cf. (5.3.1)). For the remainder of the proof work under the additional assumption made in (5.3.110). Based on this, (5.3.100), and the fact that Φ coincides with φ near zero and near infinity we then estimate  tΦ(t) tΦ(t)  , lim sup , I(Φ) ≤ max lim sup Φ(t) t→∞ Φ(t) t→0+  tφ (t) tφ (t)  = max lim sup , lim sup < ∞. φ(t) t→∞ φ(t) t→0+

(5.3.128)

On account of (5.3.19), this permits us to conclude that Φ ∈ Δ2 . The description of the Orlicz space given in (5.3.111) is a consequence of the double inequality in the first line of (5.3.109), and (5.3.37) (bearing in mind that Φ is doubling). Finally, the additional assumption in (5.3.112) ensures that  tΦ(t) tΦ(t)  , lim inf , i(Φ) ≥ min lim inf t→∞ t→0+ Φ(t) Φ(t)  tφ (t) tφ (t)  = min lim inf , lim inf > 1, + t→0 φ(t) t→∞ φ(t)

(5.3.129)

thanks to (5.3.98), the fact that Φ coincides with φ near zero and near infinity, and the last property in (5.3.108). Once this has been established, (5.3.60) tells us that  ∈ Δ2 .  the complementary function of Φ is doubling, i.e., Φ The usefulness of Lemma 5.3.8 becomes apparent in the discussion below, where a certain variety of Orlicz spaces, called Zygmund spaces, are introduced. Example 5.3.10 (The Zygmund Spaces L p (log L)α ) Fix an exponent p ∈ (1, ∞) along with a power α ∈ R and define the function φ : [0, ∞) → [0, ∞) by setting φ(t) := t p [ln(e + t)]α for each t ∈ [0, ∞).

(5.3.130)

Then φ(0) = 0, the function φ is strictly positive and of class C ∞ on (0, ∞), and for each t ∈ (0, ∞) we have   t   1 , φ (t) = t p−1 [ln(e + t)]α p + α e + t ln(e + t) hence

 t  tφ (t) 1 = p+α . φ(t) e + t ln(e + t)

(5.3.131)

(5.3.132)

From these we conclude that lim φ (t) = 0 and lim φ (t) = ∞,

t→0+

and

t→∞

(5.3.133)

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

lim+

t→0

tφ (t) tφ (t) = lim = p ∈ (1, ∞). t→∞ φ(t) φ(t)

In addition, for each t ∈ (0, ∞) we have φ (t) = t p−2 [ln(e + t)]α p(p − 1) + 2αp

 t  1 e + t ln(e + t)

(5.3.134)

(5.3.135)

!  t 2  t 2 1 1 + α(α − 1) −α . e + t [ln(e + t)]2 e + t ln(e + t) Hence

  φ (t) = t p−2 [ln(e + t)]α p(p − 1) + o(1) as either t → 0+ or t → ∞, (5.3.136)

from which we see that φ  > 0 near zero and near infinity19. As such, all properties demanded in (5.3.108) and (5.3.110) are satisfied. Granted these, Lemma 5.3.8 then guarantees the existence of a Young function Φ ∈ Δ2 as in (5.3.109), which is continuously differentiable on (0, ∞), and whose complementary function is doubling  ∈ Δ2 . In fact, i(Φ) = I(Φ) = p ∈ (1, ∞) (see (5.3.61)). Also, in as well, i.e., Φ accordance with Convention 5.3.9, we may write Φ(t) ≈ t p [ln(e + t)]α,

(5.3.137)

 the complementary function of Φ, has the property that and it turns out that Φ,  ) p α(1−p  Φ(t) ≈ t [ln(e + t)] where p ∈ (1, ∞) is the Hölder conjugate exponent of p (cf. [38, p. 72]). Given a sigma-finite measure space (X, M, μ), we agree to denote the Orlicz space L Φ (X, μ), defined in relation to Φ as in (5.3.27), by L p (log L)α (X, μ). This is typically referred to as Zygmund’s space (cf., e.g., [14, Example 8.3(e), p. 266]). According to (5.3.111), we then have the following description of the Zygmund space just introduced: ∫    α | f (x)| p ln(e + | f (x)|) dμ(x) < +∞ . L p (log L)α (X, μ) = f ∈ M (X, μ) : X

(5.3.138) In fact, with Convention 5.3.9 in place, similar considerations are valid in relation to

   β Φ(t) ≈ t p [ln(e + t)]α ln ln(ee + t) with p ∈ (1, ∞) and α, β ∈ R,

(5.3.139)

or, more generally, given any p ∈ (1, ∞), m ∈ N, and α1, . . . , αm ∈ R, in relation to

19 if α ≥ 2 − 2p then actually φ > 0 everywhere on (0, ∞), hence in this case φ is globally convex and, ultimately, a Young function itself

5.3 Young Functions and Orlicz Spaces

275



 α  Φ(t) ≈ t p [ln(e + t)]α1 ln ln(ee + t) 2 × · · · ×  # αm "  e    ..  e. × ln ln · · · ln ln(e + t) · · · ,

(5.3.140)

where the last expression involves m logarithms and e is m-fold exponentiated. A related construction is described in the next example. Example 5.3.11 (The Space L p exp (logθ L)) Pick an exponent p ∈ (1, ∞) together with a number θ ∈ [0, 1) and define the function φ : [0, ∞) → [0, ∞) by setting   φ(t) := t p exp [ln(e + t)]θ for each t ∈ [0, ∞). (5.3.141) Then φ(0) = 0, the function φ is strictly positive and of class C ∞ on (0, ∞), and for each t ∈ (0, ∞) we have  t     1 , φ (t) = t p−1 exp [ln(e + t)]θ p + θ e + t [ln(e + t)]1−θ

(5.3.142)

which further implies  t  tφ (t) 1 = p+θ . φ(t) e + t [ln(e + t)]1−θ

(5.3.143)

It is then apparent from these formulas that lim φ (t) = 0 and lim φ (t) = ∞,

t→0+

and lim+

t→0

t→∞

tφ (t) tφ (t) = lim = p ∈ (1, ∞). t→∞ φ(t) φ(t)

(5.3.144)

(5.3.145)

Moreover, for each t ∈ (0, ∞) we have  t   1  p−2 θ φ (t) = t exp [ln(e + t)] p(p − 1) + 2pθ e + t [ln(e + t)]1−θ  t 2 1 · e+t [ln(e + t)]1−θ  t 2 1 + θ(θ − 1) · e+t [ln(e + t)]2−θ !  t 2 1 + θ2 · . e+t [ln(e + t)]2(1−θ) −θ

Thus,

(5.3.146)

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5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

   φ (t) = t p−2 exp [ln(e + t)]θ p(p − 1) + o(1) as either t → 0+ or t → ∞, (5.3.147) so φ  > 0 near zero and near infinity. Consequently, all properties specified in (5.3.108) and (5.3.110) are presently satisfied. Lemma 5.3.8 then guarantees the existence of a Young function   Φ(t) ≈ t p exp [ln(e + t)]θ (5.3.148)  ∈ Δ2 . In addition, i(Φ) = I(Φ) = p ∈ (1, ∞) (see (5.3.61)). such that Φ, Φ Given a sigma-finite measure space (X, M, μ), it is natural to denote the Orlicz space L Φ (X, μ), defined in relation to Φ as in (5.3.27), by L p exp (logθ L)(X, μ). In light of (5.3.111), we then have the following description of the space just introduced: L p exp (logθ L)(X, μ)

(5.3.149)

∫   θ  | f (x)| p exp ln(e + | f (x)|) dμ(x) < ∞ . = f ∈ M (X, μ) : X

In our last example we note that intersections and sums of Lebesgue spaces may be construed as Orlicz spaces. Example 5.3.12 Given any two Young functions Φ1, Φ2 , we see from definitions that max{Φ1, Φ2 } and Φ1 + Φ2 are also Young functions, satisfying max{Φ1, Φ2 } ≤ Φ1 + Φ2 ≤ 2 max{Φ1, Φ2 }.

(5.3.150)

Moreover, the Young functions max{Φ1, Φ2 } and Φ1 + Φ2 are doubling and actually equivalent (in the sense of (5.3.104)) if Φ1, Φ2 ∈ Δ2 . In such a scenario, for any sigma-finite measure space (X, M, μ) it is apparent from (5.3.37) and (5.3.150) that L max{Φ1,Φ2 } (X, μ) = L Φ1 +Φ2 (X, μ) = L Φ1 (X, μ) ∩ L Φ1 (X, μ).

(5.3.151)

For example, having fixed two exponents p, q ∈ (1, ∞), then for the Young function defined as Φ(t) := max{t p, t q } for each t ∈ [0, ∞) (5.3.152) we have

i(Φ) = min{p, q}, I(Φ) = max{p, q}, and L Φ (X, μ) = L p (X, μ) ∩ L q (X, μ),

(5.3.153)

with the first two formulas implied by (5.3.103). In this vein, let us also observe that  of Φ in (5.3.152) as a we may use (5.3.54) to identify the complementary function Φ Young function satisfying  ≈ min{t p, t q } for each t ∈ [0, ∞), Φ(t) where p, q  ∈ (1, ∞) are the Hölder conjugates of p, q.

(5.3.154)

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277

In general, for a Young function Ψ(t) ≈ min{t p, t q } with p, q ∈ (1, ∞) we have i(Ψ) = min{p, q}, I(Ψ) = max{p, q}, and L Ψ (X, μ) = L p (X, μ) + L q (X, μ).

(5.3.155)

To justify the last equality above, suppose 1 < p∫≤ q < ∞. Then for  f in M (X, μ) arbitrary we have f ∈ L Ψ (X, μ) if and only if X min | f | p, | f | q dμ < ∞ if and ∫ ∫ ∫ only if | f | 1 then T maps L (log L)α (X, μ) bound- (5.3.170) edly into L 1 (X, μ); in fact, if μ is merely sigma-finite then T maps L (log L)α (X, μ) boundedly into L 1 (X, μ) + L ∞ (X, μ). In particular, from Lemma  5.3.1  and either (5.3.168), or (5.3.170) and [133, (7.6.18)], it may be seen that if X, d, μ is a space of homogeneous type such that the quasidistance d : X × X → [0, ∞) is continuous in the product topology τd × τd , then the Hardy-Littlewood maximal operator M X : L log L(X, μ) −→ L 1 (X, μ) is well defined and bounded if μ(X) < ∞. (5.3.171) The proof of the classical Marcinkiewicz interpolation theorem naturally extends to produce modular estimates in the context described in the theorem below.   Theorem 5.3.16 (Modular Interpolation Theorem) Let X, μ be a non-atomic sigma-finite measure space. Consider a doubling Young function Φ (i.e., a Young function whose upper and lower dilation indices satisfy 1 ≤ i(Φ) ≤ I(Φ) < ∞; cf. (5.3.19)), and pick two integrability exponents p, q ∈ (0, ∞] such that

Then

0 < p < i(Φ) ≤ I(Φ) < q ≤ ∞.

(5.3.172)

L Φ (X, μ) ⊆ L p (X, μ) + L q (X, μ).

(5.3.173)

Also, for each quasi-subadditive operator26 T : L p (X, μ) + L q (X, μ) −→ M (X, μ)

(5.3.174)

which is both of weak-type (p, p) and of weak-type (q, q), i.e., there exist two constants Mp, Mq ∈ (0, ∞) such that T f  L p,∞ (X,μ) ≤ Mp  f  L p (X,μ) for every f ∈ L p (X, μ),

(5.3.175)

26 i.e., there exists some  constant C  ∈ (0, ∞) such that for all functions f , g in the domain of T one has |T ( f + g)| ≤ C |T f | + |T g | at μ-a.e. point

5.3 Young Functions and Orlicz Spaces

281

and27 T f  L q,∞ (X,μ) ≤ Mq  f  L q (X,μ) for every f ∈ L q (X, μ),

(5.3.176)

it follows that there exists C ∈ (0, ∞), depending only on p, q, i(Φ), I(Φ), and the quasi-subadditivity constant of T, such that for each function f ∈ L Φ (X, μ) one has ∫ ∫     Φ |(T f )(x)| dμ(x) ≤ C · CT Φ | f (x)| dμ(x), (5.3.177) X

for

X

CT :=

p

q

Mp + Mq Cα,β ·

p Mp

· max



α, M∞

β M∞

if q < ∞, if q = ∞,

(5.3.178)

where α, β ∈ R are any numbers satisfying 0 < α < i(Φ) ≤ I(Φ) < β < ∞ and Cα,β ∈ [1, ∞) depends only on α, β, and the doubling character of Φ. As a corollary of this modular estimate and the last property in the statement of Lemma 5.3.3, T is bounded on the Orlicz space L Φ (X, μ). Proof Choose another exponent p0 ∈ (p, ∞) such that 1 ≤ p0 < i(Φ). Then L Φ (X, μ) ⊆ L p0 (X, μ) + L q (X, μ) ⊆ L p (X, μ) + L q (X, μ)

(5.3.179)

with the first inclusion implied by (5.3.161) (keeping in mind (5.3.165)), and the second inclusion implied by the fact that L p0 (X, μ) ⊆ L p (X, μ) + L q (X, μ) since p < p0 < q. This establishes (5.3.173). Next, assume T is a quasi-sublinear operator as in (5.3.174) which is both of weak-type (p, p) and of weak-type (q, q). The goal is to establish (5.3.177). To this end, we first note that, as seen from the discussion surrounding (5.3.106), there is no loss of generality in assuming that the Young function Φ is also continuously differentiable on (0, ∞). Given any number t ∈ (0, ∞), we may then integrate by parts ∫ ∫ t Φ(s) 1 t −p  ds = − (s ) Φ(s) ds (5.3.180) p+1 p 0 0 s ! ∫ t  ∫ Φ (s) 1 Φ(t) 1 t Φ(s) 1 Φ(s)

t − ds = − + ds, =−

p sp 0 sp p tp p 0 sp 0 where the last equality uses (5.3.89) and the fact that 0 < p < i(Φ). From (5.3.180) and (5.3.90) we then conclude that there exists a constant C ∈ (0, ∞) such that ∫ t  Φ (s) Φ(t) ds ≤ C p for each t ∈ (0, ∞). (5.3.181) p s t 0 To proceed, consider the case when q < ∞. Then for any t ∈ (0, ∞) we may integrate by parts 27 with the convention that L ∞,∞ (X, μ) := L ∞ (X, μ)

282

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

∫

∞

t

Φ(s) 1 ds = − q s q+1

∫

∞

(s−q )Φ(s) ds

(5.3.182)

t

! ∫ ∞  ∫ Φ (s) 1 Φ(t) 1 ∞ Φ(s) 1 Φ(s)

∞ ds = + ds, =−

− q sp t sq q tq q t sq t

where the last equality uses (5.3.91) and the fact that I(Φ) < q < ∞. In concert with (5.3.92), this implies that there exists a constant C ∈ (0, ∞) such that ∫ ∞  Φ (s) Φ(t) ds ≤ C q for each t ∈ (0, ∞). (5.3.183) q s t t Going further, pick α, β ∈ R satisfying 0 < α < i(Φ) ≤ I(Φ) < β < ∞. Also, select an arbitrary function f ∈ L p (X, μ) + L q (X, μ). In particular, | f (x)| < ∞ for μ-a.e. point x ∈ X. Finally, fix some constant γ ∈ (0, ∞), to be determined later. For each t ∈ (0, ∞) decompose f = gt + ht where gt := f 1 { | f |>t/γ } and ht := f 1 {0< | f | ≤t/γ } .

(5.3.184)

Then for each t ∈ (0, ∞) we have |gt | ≤ | f | and |ht | ≤ | f |, which places gt , ht in L p (X, μ)+ L q (X, μ) (since this is a lattice). In view of this and the quasi-subadditivity property of T, for each t ∈ (0, ∞) we may estimate   |(T f )(x)| ≤ C |(T gt )(x)| + |(T ht )(x)| at μ-a.e. point x ∈ X. (5.3.185) Based on this and upon recalling that (X, μ) is sigma-finite, we may estimate (cf. [133, (8.4.102)]) ∫ ∞  ∫     Φ |(T f )(x)| dμ(x) = μ x ∈ X : |(T f )(x)| > t Φ(t) dt X

0

∫

∞

≤

μ



0

∫

+

∞

0

μ

 x ∈ X : |(T gt )(x)| > t/(2C) Φ(t) dt



 x ∈ X : |(T ht )(x)| > t/(2C) Φ(t) dt

=: I + II. The fact that T is of weak-type (p, p) with p ∈ (0, ∞) allows us to estimate

(5.3.186)

5.3 Young Functions and Orlicz Spaces

∫

283

 t  −p p gt  L p (X,μ) Φ(t) dt 2C 0 ∫ ∞ ∫  p p −p = (2C) Mp t | f (x)| p dμ(x) Φ(t) dt p

I ≤ Mp

∞

{x ∈X: | f (x) |>t/γ }

0

∫

∫

Φ(t)  dt dμ(x) tp 0 X ∫  ∫ | f (x) |γ Φ(t)  p = (2C) p Mp | f (x)| p dt dμ(x) tp {x ∈X: ∞> | f (x) |>0} 0 ∫ Φ(| f (x)|γ) p | f (x)| p dμ(x) ≤ (2C) p Mp | f (x)| p {x ∈X: ∞> | f (x) |>0} ∫ p = (2C) p Mp Φ(| f (x)|γ) dμ(x) = (2C)

p

p Mp

| f (x)|

∫

| f (x) |γ

{x ∈X: ∞> | f (x) |>0}

p

= (2C) p Mp

p

Φ(| f (x)|γ) dμ(x) X

≤ CMp · max{γ α, γ β } ·

∫

p

Φ(| f (x)|) dμ(x),

(5.3.187)

X

also using Fubini’s theorem (recall that (X, μ) is sigma-finite), (5.3.184), (5.3.181), and (5.3.40). In addition, the fact that T is of weak-type (q, q) with q ∈ (1, ∞) permits us to estimate

284

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

∫

 t  −q q ht  L p (X,μ) Φ(t) dt 2C 0 ∫ ∞ ∫ q q −q = (2C) Mq t q

II ≤ Mq

∞

{x ∈X: 0< | f (x) | ≤t/γ }

0

∫ q

= (2C)q Mq

{x ∈X: ∞> | f (x) |>0}

| f (x)| q

∫ q

≤ (2C)q Mq

∫

{x ∈X: ∞> | f (x) |>0}

q

= (2C)q Mq

∫ = (2C) ≤

q

q CMq

q Mq

{x ∈X: ∞> | f (x) |>0}

| f (x)| q

∫

 | f (x)| q dμ(x) Φ(t) dt ∞ | f (x) |γ

Φ(t)  dt dμ(x) tp

Φ(| f (x)|γ) dμ(x) | f (x)| q

Φ(| f (x)|γ) dμ(x)

Φ(| f (x)|γ) dμ(x) X α

β

∫

· max{γ , γ } ·

Φ(| f (x)|) dμ(x),

(5.3.188)

X

after also availing ourselves of Fubini’s theorem, (5.3.184), (5.3.183), and (5.3.40). From (5.3.186), (5.3.187), (5.3.188), and (5.3.173) we see that the modular estimate claimed in (5.3.177) holds if q < ∞ (in this scenario, the actual value of the parameter γ ∈ (0, ∞) is inconsequential; e.g., we make take γ := 1). In the case when q = ∞, for each t ∈ (0, ∞) we see from the fact that T is bounded on L ∞ (X, μ) and (5.3.184) that |(T ht )(x)| ≤ T ht  L ∞ (X,μ) ≤ M∞ ht  L ∞ (X,μ) ≤ M∞ · (t/γ) < t/(2C) at μ-a.e. point x ∈ X,

(5.3.189)

with the very last inequality valid provided we choose, e.g., γ := 4CM∞ to begin with28. For such a choice,   μ x ∈ X : |(T ht )(x)| > t/(2C) = 0 for each t ∈ (0, ∞).

(5.3.190)

(5.3.191)

hence II = 0. Based on this, (5.3.186), and (5.3.187) we then once again conclude that the modular estimate claimed in (5.3.177) is true in the case when q = ∞ as well.  In turn, the modular interpolation result established Theorem 5.3.16 readily yields weighted modular estimates for the Hardy-Littlewood maximal operator, of the 28 assuming T is not identically zero, in which case everything is trivially true

5.3 Young Functions and Orlicz Spaces

285

sort described in the corollary below. In particular, they allow us to recapture the boundedness properties of the Hardy-Littlewood maximal operator on Orlicz spaces already established in (5.3.167).   Corollary 5.3.17 Let X, d, μ be a space of homogeneous type such that the measure μ does not charge singletons29 and the quasi-distance d : X × X → [0, ∞) is continuous30 in the product topology τd ×τd . Bring in the Hardy-Littlewood maximal operator M X on X (cf. (A.0.71)). Finally, consider a Young function Φ whose upper and lower dilation indices satisfy 1 < i(Φ) ≤ I(Φ) < ∞.

(5.3.192)

Then for each Muckenhoupt weight w ∈ Ai(Φ) (X, μ) there exists some constant C ∈ (0, ∞), which depends only on the ambient (X, d, μ), the dilation indices of Φ, and [w] Ai(Φ) , with the property that ∫ ∫     Φ |(M X f )(x)| w(x) dμ(x) ≤ C Φ | f (x)| w(x) dμ(x) (5.3.193) X X for each function f belonging to the weighted Orlicz space L Φ (X, w μ). In particular, corresponding to the case when w ≡ 1, one has the “plain” modular estimate ∫ ∫     Φ |(M X f )(x)| dμ(x) ≤ C Φ | f (x)| dμ(x) for each f ∈ L Φ (X, μ). X

X

(5.3.194) Finally, as a corollary of (5.3.194), (5.3.59), and the last property in the statement of Lemma 5.3.3, the Hardy-Littlewood maximal operator M X induces well-defined sublinear bounded mappings both from the Orlicz space L Φ (X, μ) into itself, and   is the complementary function from the Orlicz space L Φ (X, μ) into itself, where Φ of Φ.   Proof Given any weight w ∈ Ai(Φ) (X, μ) there exists ε ∈ 0, i(Φ) − 1 with the property that (5.3.195) w ∈ Ai(Φ)−ε (X, μ) ⊆ AI(Φ)+ε (X, μ). See items (9) and (5) in [133, Lemma 7.7.1] which also show that ε as well as [w] Ai(Φ)−ε and [w] A I (Φ)+ε are controlled in terms the ambient (X, d, μ), the lower dilation index i(Φ), and [w] Ai(Φ) . Then the claim made in (5.3.193) is seen from   Theorem 5.3.16,presently  used for the choices T := M X , p := i(Φ) − ε ∈ 1, i(Φ) , q := I(Φ) + ε ∈ I(Φ), ∞ , and with μ replaced by the measure w μ, bearing in mind item (1) in [133, Lemma 7.7.1], [133, Lemma 7.4.2], and [133, Lemma 7.4.3].  We conclude by presenting an extrapolation result from Muckenhoupt weighted Lebesgue space estimates to modular estimates on Orlicz spaces. This refines and 29 i.e., μ({x }) = 0 for each x ∈ X 30 it is known from [133, Theorem 7.1.2] that any quasi-metric space has an equivalent quasidistance satisfying this property

286

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

extends [38, Theorem 4.15, p. 78]. For example, in contrast to the argument given on [38, pp. 80-83] in the entire Euclidean space, our proof is carried out in spaces of homogeneous type and does not rely on the classical Rubio de Francia extrapolation theorem (see [133, Proposition 7.7.6]). Also, we provide a more transparent formula for the constant involved in the main estimate. Theorem 5.3.18 (Modular Extrapolation Theorem) Let (X, d, μ) be a space of homogeneous type such that the measure μ does not charge singletons31 and the quasi-distance d : X × X → [0, ∞) is continuous32 in the product topology τd × τd . Let M X stand for the Hardy-Littlewood maximal operator on (X, d, μ). Also, pick a Young function Φ whose upper and lower dilation indices satisfy 1 < i(Φ) ≤ I(Φ) < ∞,

(5.3.196)

 Then the following statements are true. and denote its complementary function by Φ.  are doubling, and the operator M X is bounded both on (1) The functions Φ, Φ  L Φ (X, μ) and on L Φ (X, μ). In particular, with the piece of notation introduced in (5.3.38), N Φ (M X f ) N Φ [M X ] := sup (5.3.197) N Φ( f ) f ∈L Φ (X,μ) f 0

and 

N Φ [M X ] :=



sup 

f ∈L Φ (X,μ) f 0

N Φ (M X f )  NΦ (f)

(5.3.198)

are well-defined real numbers, belonging to the interval (0, ∞). (2) There exists some CX,Φ ∈ (0, ∞), depending only on the quasi-distance d (via d defined as in (A.0.19)-(A.0.20)) the doubling charter of μ, the constants Cd, C as well as the dilation indices of Φ, with the following significance. Fix some integrability exponent p0 ∈ (1, ∞) and define p   WΦ, p0 := CX,Φ 0 · N Φ [M X ] p0 −1 · N Φ [M X ].

(5.3.199)

Also, suppose the number CW ∈ (0, ∞) and the μ-measurable real-valued functions f , g defined on X are such that for any given Muckenhoupt weight w ∈ Ap0 (X, d, μ) with [w] A p0 ≤ WΦ, p0 one has  f  L p0 (X,w) ≤ CW g L p0 (X,w) .

(5.3.200)

Then whenever 0 < α < i(Φ) ≤ I(Φ) < β < ∞ there exists some constant CΦ,α,β ∈ (0, ∞), which depends only on p0 , the dilation indices of Φ, as well as 31 i.e., μ({x }) = 0 for each x ∈ X 32 The result proved in [133, Theorem 7.1.2] guarantees that any quasi-metric space has an equivalent quasi-distance satisfying this property

5.3 Young Functions and Orlicz Spaces

287

α and β, with the property that ∫ ∫       Φ | f (x)| dμ(x) ≤ CΦ,α,β · max (CW )α, (CW )β · Φ |g(x)| dμ(x). X

X

(5.3.201)

 ∈ Δ2 . Also, from Proof Based on (5.3.61) and (5.3.196) we conclude that Φ, Φ (5.3.196) and (5.3.167) (or the very last property in the statement of Corollary 5.3.17)  we see that M X is bounded both on L Φ (X, μ) and on L Φ (X, μ). To proceed, from  Φ Φ (5.3.193)-(5.3.194) we know that N [M X ] and N [M X ] defined in (5.3.197)(5.3.198) are are well-defined numbers in the interval [0, ∞), with the property that (5.3.202) N Φ (M X f ) ≤ N Φ [M X ] · N Φ ( f ) for each f ∈ L Φ (X, μ), and









N Φ (M X f ) ≤ N Φ [M X ] · N Φ ( f ) for each f ∈ L Φ (X, μ).

(5.3.203)

 N Φ [M

To get strictly positive lower bounds for N Φ [M X ], X ], fix a point xo ∈ X along with a radius r ∈ (0, ∞), and consider the d-ball B := Bd (xo, r) in X. Note that ∫     N Φ 1B = Φ 1B (x) dμ(x) = Φ(1)μ(B). (5.3.204) X

Also, it may be seen from (A.0.71) that there exists some CX ∈ (1, ∞), which depends only on the doubling constant of μ and the quasi-distance d, with the property that33   CX · M X 1B ≥ 1B pointwise on X. (5.3.205) In concert with (5.3.204), (5.3.202), and (5.3.20) (with, say α := 1 and β := 2I(Φ)), this permits us to estimate (also bearing in mind that Φ is increasing, as well as doubling)      Φ(1)μ(B) = N Φ 1B ≤ N Φ CX · M X 1B    ≤ CΦ · (CX )2I(Φ) · N Φ M X 1B   ≤ CΦ · (CX )2I(Φ) · N Φ [M X ]N Φ 1B = CΦ · (CX )2I(Φ) · N Φ [M X ]Φ(1)μ(B),

(5.3.206)

for some CΦ ∈ (1, ∞) depending only on I(Φ). Ultimately, this proves that

33 if the measure μ is Borel-semiregular on (X, τρ ), then Lebesgue’s Differentiation Theorem (see (1) ⇔ (3) in [133, Proposition 7.4.4]) ensures that (5.3.205) holds with C X := 1

288

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

If we now define

N Φ [M X ] ≥ CΦ · (CX )−2I(Φ) .

(5.3.207)

  −1 ΛΦ := 2CΦ · (CX )2I(Φ) · N Φ [M X ]

(5.3.208)

then ΛΦ ∈ (0, 1/2] and

∞

j

(1 − ΛΦ )ΛΦ = 1.

(5.3.209)

j=0

These considerations allow us to implement a version of Rubio de Francia’s iterative algorithm in the present setting. Specifically, define T : L Φ (X, μ) −→ M+ (X, μ) by setting T φ :=

∞

ΛΦ M X φ for each function φ ∈ L Φ (X, μ), j

j

(5.3.210)

j=0

where M X0 φ := |φ| for each φ ∈ L Φ (X, μ), and M X := M X ◦ · · · ◦ M X is the j-fold composition of M X : L Φ (X, μ) → L Φ (X, μ) with itself. In view of the nature of the first term in (5.3.210), we have j

|φ| ≤ T φ for each function φ ∈ L Φ (X, μ).

(5.3.211)

By also taking into account to the sublinearity of M X on M+ (X, μ) and (5.3.208), it follows that for each function φ ∈ L Φ (X, μ) we may write M X (T φ) ≤

∞ j=0

ΛΦ M X φ = Λ−1 Φ j

j+1

∞

ΛΦ M X φ ≤ Λ−1 Φ Tφ j+1

j+1

j=0

= 2CΦ · (CX )2I(Φ) · N Φ [M X ] · T φ.

(5.3.212)

Next, given any function φ ∈ L Φ (X, μ), we make use of (5.3.38), (5.3.210), the fact that Φ is convex, (5.3.209), iterations of (5.3.202), and the monotonicity of N Φ to write

5.3 Young Functions and Orlicz Spaces

N Φ (T φ) =

∫ Φ X

∫ Φ X

∞ 

  j  j (1 − ΛΦ )ΛΦ M X (φ/(1 − ΛΦ )) (x) dμ(x)

j=0

∫ ∞    j j (1 − ΛΦ )ΛΦ Φ M X (φ/(1 − ΛΦ )) (x) dμ(x) X

j=0

≤

 j  j  ΛΦ M X φ (x) dμ(x)

j=0

=

≤

∞ 

289

∞

  j j  ΛΦ N Φ M X φ/(1 − ΛΦ )

j=0

≤

∞

  j ΛΦ N Φ [M X ] j N Φ φ/(1 − ΛΦ )

j=0 ∞  j   = N Φ φ/(1 − ΛΦ ) ΛΦ · N Φ [M X ] j=0

=

 Φ



N φ/(1 − ΛΦ ) ≤ 2N Φ (2φ) 1 − ΛΦ · N Φ [M X ]

≤ CΦ · 22I(Φ)+1 · N Φ (φ),

(5.3.213)

where in the penultimate line we have also used that  −1  ΛΦ · N Φ [M X ] = 2CΦ · (CX )2I(Φ) ∈ (0, 1/2),

(5.3.214)

itself a consequence of (5.3.208) and the fact that CΦ, CX ∈ (1, ∞), while the very last line in (5.3.213) employs (5.3.40) (with λ := 2 and, say, α := 1, β := 2I(Φ)) and (5.3.8). Hence, N Φ (T φ) ≤ CΦ · 22I(Φ)+1 · N Φ (φ) for each φ ∈ L Φ (X, μ).

(5.3.215)

From (5.3.215) and Lemma 5.3.3 we deduce that T actually maps the Orlicz space L Φ (X, μ) boundedly into itself, thus the operator T : L Φ (X, μ) −→ L Φ (X, μ) is well-defined, sublinear, and bounded.

(5.3.216)

Together with (5.3.211) and (5.3.212), this further implies 0 < T φ < ∞ at μ-a.e. point on X whenever 0  φ ∈ L Φ (X, μ).

(5.3.217)

290

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

From (5.3.217), (5.3.212), and item (4) in [133, Lemma 7.7.1] we then conclude that if φ ∈ L Φ (X, μ) is not identically zero (μ-a.e.), then T φ ∈ A1 (X, μ) and [T φ] A1 ≤ CX,Φ · N Φ [M X ]

(5.3.218)

d where CX,Φ ∈ (0, ∞) depends only on the quasi-distance d (via the constants Cd, C defined as in (A.0.19)-(A.0.20)), the doubling charter of μ, and upper dilation index of Φ. In a similar fashion, we define   : LΦ (X, μ) −→ M+ (X, μ) by setting T

 := Tψ

∞



Λ M X ψ for each function φ ∈ L Φ (X, μ), j Φ

j=0

j

(5.3.219)

 where ΛΦ  ∈ (0, 1/2] is defined in relation to Φ the same way ΛΦ has been introduced  in reference to Φ in (5.3.209), i.e., for some CΦ  ∈ (1, ∞) depending only on I(Φ),   −1   −1    2I(Φ) 2i(Φ) · N Φ [M X ] = 2CΦ · N Φ [M X ] . (5.3.220) ΛΦ  := 2CΦ  · (CX )  · (CX ) Then, much as above,    : LΦ the operator T (X, μ) −→ L Φ (X, μ)

is well-defined, sublinear, bounded,  for each ψ ∈ and |ψ| ≤ Tψ

 L Φ (X,

(5.3.221)

μ).

Also,      2I(Φ)+1 2i(Φ)+1 · N Φ N Φ (Tψ) ≤ CΦ · N Φ (ψ) = CΦ (ψ)  ·2  ·2 

for each function ψ ∈ L Φ (X, μ), and

(5.3.222)



if ψ ∈ L Φ (X, μ) is not identically zero (μ-a.e.), then   ∈ A1 (X, μ) and [Tψ]  A1 ≤ C X,Φ · N Φ Tψ [M X ]

(5.3.223)

X,Φ ∈ (0, ∞) depends only on the quasi-distance d (via the constants Cd, C d where C defined as in (A.0.19)-(A.0.20)), the doubling charter of μ, and lower dilation index of Φ. Let us turn our attention to (5.3.201). Note that this is trivially true if f = 0 at μ-a.e. point in X, or if N Φ (g) is either ∞ or 0 (since in the latter case (5.3.200) with, say, w ≡ 1, forces f = 0 at μ-a.e. point in X). As far as (5.3.201) is concerned, we are therefore left with considering the situation when

5.3 Young Functions and Orlicz Spaces

291

f is not identically zero (μ-a.e.), and 0 < N Φ (g) < ∞. Define ψ : X → [0, ∞) by setting, for each x ∈ X,   Φ | f (x)| | f (x)| if f (x)  0, ψ(x) := 0 if f (x)  0.

(5.3.224)

(5.3.225)

In particular, since Φ(0) = 0 and Φ > 0 on (0, ∞),

and

ψ is not identically zero (μ-a.e.) on X,

(5.3.226)

  | f (x)|ψ(x) = Φ | f (x)| for each x ∈ X.

(5.3.227)

 = 0 to conclude that We may also use (5.3.225), (5.3.58), and the fact that Φ(0)    ψ(x) ≤ Φ(| f (x)|) for each x ∈ X. Φ (5.3.228) As a consequence of this, (5.3.226), and (5.3.38), we have 

0 < N Φ (ψ) ≤ N Φ ( f ).

(5.3.229)

The remainder of the proof of (5.3.201) is divided into two steps. Step I. Proof of (5.3.201) under the additional assumption f ∈ L Φ (X, μ). In particular, thanks to (5.3.39) and (5.3.229), 

0 < N Φ (ψ) ≤ N Φ ( f ) < ∞,

(5.3.230)

which further entails (cf. (5.3.39)) 

0  ψ ∈ L Φ (X, μ).

(5.3.231)

For some δ ∈ (0, 1/2) to be chosen later, define φ := δ| f | + (1 − δ)|g| ∈ M+ (X, μ).

(5.3.232)

Upon recalling that Φ is convex, we have N Φ (φ) ≤ δN Φ ( f ) + (1 − δ)N Φ (g) < ∞,

(5.3.233)

with the last inequality a consequence of (5.3.224) and the current working assumption. In view of (5.3.39) and (5.3.224), this implies that 0  φ ∈ L Φ (X, μ).

(5.3.234)

From the properties listed in (5.3.231), (5.3.218), (5.3.234), (5.3.223), and item (3) in [133, Lemma 7.7.1] we see that if

292

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

 w := (T φ)1−p0 (Tψ)

(5.3.235)

then w is a weight in the Muckenhoupt class Ap0 (X, μ), and   p −1    X,Φ · N Φ [w] A p0 ≤ CX,Φ · N Φ [M X ] 0 · C [M X ] ≤ WΦ, p0

(5.3.236)

with the final inequality in (5.3.236) coming from (5.3.199). With q0 ∈ (1, ∞) denoting the Hölder conjugate exponent of p0 , we are now in a position to write (with justifications provided below) ∫ ∫ ∫    dμ Φ | f (x)| dμ(x) = | f |ψ dμ ≤ | f |(Tψ) N Φ( f ) = X

X

∫

X

 dμ | f |(T φ)−1/q0 (T φ)1/q0 (Tψ)

= X

≤ =

∫ ∫

 dμ | f | p0 (T φ)1−p0 (Tψ)

X

| f | p0 dw

 1/p0

X

≤ CW = CW

= 2 p0 CW

 dμ (T φ)(Tψ)

 1/q0

X

∫

 dμ (T φ)(Tψ)

 1/q0

X

∫

∫

 1/p0  ∫

|g| p0 dw

 1/p0  ∫

X

X

|g| (T φ) p0

1−p0

 dμ (Tψ)

X

∫

 dμ (T φ)(Tψ)

 1/q0

 1/p0  ∫

 1/p0  ∫

X

∫ = 2 p0 CW ·

 dμ (T φ)(Tψ)

 dμ (T φ)(Tψ)

 1/q0

X

 dμ (T φ)(Tψ)

 1/q0

X

 dμ. (T φ)(Tψ)

(5.3.237)

X

Above, the first two equalities come from (5.3.38) and (5.3.227), the subsequent inequality is a consequence of (5.3.221), and the third equality is clear from (5.3.217)  dμ, then use and (5.3.234). We next use Hölder’s inequality for the measure (Tψ) the definition of the weight w (cf. (5.3.235)). The third inequality is implied by the hypothesis made in (5.3.200) plus (5.3.236), after which we unpack the definition the weight w (cf. (5.3.235)). Next, bearing in mind that δ ∈ (0, 1/2), from (5.3.232) together with (5.3.211) and (5.3.234) we see that 2−1 |g| ≤ (1 − δ)|g| ≤ φ ≤ T φ pointwise on X,

(5.3.238)

so |g| p0 ≤ 2 p0 (T φ) p0 . Based on this we conclude that the equality in the penultimate line of (5.3.237) is true. Finally, we use the fact that 1/p0 + 1/q0 = 1 in the last equality of (5.3.237).

5.3 Young Functions and Orlicz Spaces

293

Going further, for each ε ∈ (0, 1) we may estimate ∫ ∫       dμ = (ε −1T φ)(εTψ)  dμ ≤ N Φ ε −1T φ + N Φ (T φ)(Tψ) εTψ X

X   ≤ CΦ · ε −2I(Φ) · N Φ (T φ) + ε · N Φ (Tψ) 

2i(Φ)+1 ε · N Φ (ψ) ≤ (CΦ )2 · 22I(Φ)+1 ε −2I(Φ) · N Φ (φ) + CΦ  ·2   ≤ (CΦ )2 · 22I(Φ)+1 ε −2I(Φ) · δN Φ ( f ) + (1 − δ)N Φ (g) 2i(Φ)+1 ε · N Φ( f ) + CΦ  ·2   2i(Φ) ≤ 2 (CΦ )2 · 22I(Φ) ε −2I(Φ) · δ + CΦ ε · N Φ( f )  ·2

+ (CΦ )2 · 22I(Φ) ε −2I(Φ) · N Φ (g),

(5.3.239)

where we have used Young’s inequality (recorded in (5.3.56)) and (5.3.38) in the first inequality, (5.3.40) (with, say, α := 1 and β := 2I(Φ)) as well as (5.3.2) (written  in place of Φ and for λ := ε ∈ (0, 1)) in the second inequality, (5.3.215) and for Φ (5.3.222) in the third inequality, (5.3.230) and (5.3.233) in the fourth inequality, and the fact that δ ∈ (0, 1/2) in the final inequality. If we further restrict ε to (0, 1/2) and take (5.3.240) δ := ε 1+2I(Φ) ∈ (0, 1/2), we see that (5.3.239) becomes ∫    dμ ≤ 2ε · (CΦ )2 · 22I(Φ) + C · 22i(Φ) · N Φ ( f ) (T φ)(Tψ) Φ X

+ (CΦ )2 · 22I(Φ) ε −2I(Φ) · N Φ (g).

(5.3.241)

At this stage, use (5.3.241) back into (5.3.237) for the choice    −1  2i(Φ) ε := min 2−1, 2−(p0 +2) (CΦ )2 · 22I(Φ) + CΦ (CW )−1 ∈ (0, 1/2]. (5.3.242)  ·2 After absorbing the term N Φ ( f ) from the right side, whose coefficient is ≤ 1/2, into the left side we obtain   N Φ ( f ) ≤ CΦ · CW · max 1, CΦ · (CW )2I(Φ) · N Φ (g), (5.3.243) for some constant CΦ ∈ (0, ∞) which depends only on p0 and the dilation indices of Φ. Having proved (5.3.243), running the same argument as above for the function CW · g in place of g produces the version of (5.3.243) with CW = 1 and g replaced by CW · g, i.e.,

294

5 Banach Function Spaces, Extrapolation, and Orlicz Spaces

  N Φ ( f ) ≤ CΦ · N Φ CW · g ,

(5.3.244)

for some CΦ ∈ (0, ∞) depending only on p0 and the dilation indices of Φ. At this stage, whenever 0 < α < i(Φ) ≤ I(Φ) < β < ∞ we conclude from (5.3.244) and (5.3.40) that (5.3.201) holds. Step II. The end-game in the proof of (5.3.201). From Step I we know that (5.3.201) is true if f ∈ L Φ (X, μ). The goal here is to eliminate the latter additional assumption. To this end, fix a reference point xo ∈ X and define f j := min{| f |, j} · 1B d (xo, j) for each j ∈ N.

(5.3.245)

Then each f j is a μ-measurable function on X satisfying 0 ≤ f j ≤ j · 1B d (xo, j) . From the monotonicity of Φ, (5.3.29) and (5.3.37) we see that each f j belongs to L Φ (X, μ).

(5.3.246)

Also, 0 ≤ f j | f | as j → ∞ in a pointwise fashion on X. In particular, (5.3.200) implies that for every Muckenhoupt weight w ∈ Ap0 (X, d, μ) with [w] A p0 ≤ WΦ, p0 we have (5.3.247)  f j  L p0 (X,w) ≤  f  L p0 (X,w) ≤ CW g L p0 (X,w) . On account of this and (5.3.246), we conclude from (5.3.247) and Step I that whenever we have 0 < α < i(Φ) ≤ I(Φ) < β < ∞ there exists a constant CΦ,α,β ∈ (0, ∞), depending only on p0 , the dilation indices of Φ, as well as α and β, with the property that for each j ∈ N we have ∫ ∫       Φ | f j (x)| dμ(x) ≤ CΦ,α,β · max (CW )α, (CW )β · Φ |g(x)| dμ(x). X

X

(5.3.248) Passing to limit j → ∞ and relying on Lebesgue’s Monotone Convergence Theorem (while bearing in mind the monotonicity of Φ) we then arrive at the conclusion that (5.3.201) holds as stated. 

Chapter 6

Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals on Ahlfors Regular Sets

General information pertaining to the classical Morrey and Morrey-Campanato spaces in the entire Euclidean ambient may be found in a multitude of sources (cf., e.g., [2], [6], [63], [117], [155], [157], [184] and the references therein). The goal here is to develop a theory for these scales of spaces, which is comparable in scope and power to its Euclidean counterpart, in more general geometric settings.

6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets To set the stage, throughout we let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. For each p ∈ (1, ∞) and . λ ∈ (0, n − 1) define the homogeneous Morrey-Campanato space L p,λ (Σ, σ) as   . 1 L p,λ (Σ, σ) := f ∈ Lloc (Σ, σ) :  f L. p, λ (Σ,σ) < +∞ (6.1.1) ⨏ 1 (Σ, σ) and with f where, for each f ∈ Lloc f dσ for each R > 0 Δ(x,R) := Σ∩B(x,R) and x ∈ Σ, we have set    p1 n−1−λ  ⨏  p  f (y) − fΔ(x,R)  dσ(y) sup  f L. p, λ (Σ,σ) := R p . x ∈Σ and 0 0. Hence, m ∈ L 1 (Σ, σ) and m L 1 (Σ,σ) ≤ CR(n−1)/p+λ(1/q−1) . Together with the first property in (6.1.33) this also proves that  any H q,λ,θ -dome on Σ belongs to L r (Σ, σ).

(6.1.35)

(6.1.36)

1≤r ≤q

A function m ∈ L q (Σ, σ) is said to be a H

q,λ,θ -molecule

on Σ provided

∫ m is a H

q,λ,θ

-dome on Σ and

Σ

m dσ = 0.

(6.1.37)

In particular, (6.1.34) ensures that the last condition in (6.1.37) is meaningful. Our next lemma extends an Euclidean result found in [10, Proposition 17.4.2, p. 312].

302

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

Lemma 6.1.2 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set, and abbreviate σ := H n−1 Σ. Also, fix q ∈ (1, ∞) along with λ ∈ (0, n − 1), then pick θ > (n − 1)(q − 1). Then there exists C = C(Σ, q, λ, θ) ∈ (0, ∞) such that any H q,λ,θ -molecule m on Σ belongs to the space H q,λ (Σ, σ) and has mH q, λ (Σ,σ) ≤ C.

(6.1.38)

Proof If cΣ, CΣ denote the lower and upper Ahlfors regularity constants of Σ, then 0 < cΣ ≤ CΣ < ∞. Throughout, fix a number

1/(n−1) . 2∗ > CΣ /cΣ

(6.1.39)

Let us first treat the case when Σ is unbounded. In such a scanario, the choice made in (6.1.39) ensures that we have

σ Δ(x, ρ) \ Δ(x, ρ/2∗ ) ≈ ρn−1, (6.1.40) uniformly for x ∈ Σ and ρ ∈ (0, ∞). Given a H q,λ,θ -molecule m on Σ, let xo ∈ Σ and R ∈ (0, ∞) be as in (6.1.33). From (6.1.35) we know that m ∈ L 1 (Σ, σ). j For each j ∈ N0 abbreviate Δ j := Σ ∩ B(xo, 2∗ R) and set Δ−1 := . For each j ∈ N0 let us also introduce A j := Δ j \ Δ j−1 and observe that thanks to (6.1.40) we have j (6.1.41) σ(A j ) ≈ (2∗ R)n−1 uniformly for j ∈ N0 . ⨏ In particular, it is meaningful to define m A j := A m dσ for each j ∈ N0 . Next, fix j some N ∈ N and write N 

mAj 1Aj =

j=0

∫ N  1Aj j=0

=

=

σ(A j )

Δ j \Δ j−1

∫ N  1Aj  j=0

σ(A j )

N −1 

 1A j+1

j=0

(6.1.42) ∫

Σ\Δ j−1

σ(A j+1 )

m dσ

−

m dσ −

Σ\Δ j

∫ 1Aj  σ(A j )



Σ\Δ j

m dσ

m dσ −

1AN σ(A N )

∫ Σ\Δ N

m dσ.

The last equality above uses the summation by parts formula, to the effect that for any a0, a1, . . . , a N ∈ C and b−1, b0, . . . , b N ∈ C we have N  j=0

a j (b j−1 − b j ) =

N −1 

(a j+1 − a j )b j − a N b N + a0 b−1 .

j=0

In the context of (6.1.42), at any point x ∈ Σ this is used with

(6.1.43)

6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets

303

a j := 1 A j (x)/σ(A j ) for 0 ≤ j ≤ N and ∫ b j :=

Σ\Δ j

(6.1.44)

m dσ for −1 ≤ j ≤ N.

The vanishing moment property of the molecule m translates into b−1 = 0, so the very last term in (6.1.43) presently drops out. This explains the structure of the expression in the final line of (6.1.42). Going further, express N 

N N N   

m1 A j − m A j 1 A j = m1Δ N − mAj 1Aj . m − mAj 1Aj =

j=0

j=0

j=0

(6.1.45)

j=0

Hence, if for each j ∈ N0 we introduce ∫  1A 1Aj 

j+1 α j := (m − m A j 1 A j and β j := − m dσ, σ(A j+1 ) σ(A j ) Σ\Δ j

(6.1.46)

then combining (6.1.42) and (6.1.45)-(6.1.46) we arrive at the formula m1Δ N =

N 

αj +

N −1 

j=0

1AN σ(A N )

βj −

j=0

∫

Note that for each j ∈ N0 we have supp α j ⊆ Δ j , ∫ ∫ ∫ ∫

m − m A j dσ = α j dσ = m dσ − Σ

Aj

Aj

m dσ.

(6.1.47)

m dσ = 0.

(6.1.48)

Σ\Δ N

Aj

Also, if j ≥ 1 α j  L q (Σ,σ) ≤ 2

∫

|m| q dσ

 1/q

=2

∫

Aj

|m(x)| q |x − xo | θ |x − xo | −θ dσ(x)

Aj

≤ C(2∗ R)−θ/q j

∫ Σ

|m(x)| q |x − xo | θ dσ(x) −jθ/q

≤ C(2∗ R)−θ/q Rλ(1/q−1)+θ/q = C2∗ j

−j[θ/q+λ(1/q−1)]

= C2∗

 1/q

 1/q

Rλ(1/q−1)

(2∗ R)λ(1/q−1), j

(6.1.49)

while if j = 0 α0  L q (Σ,σ) ≤ 2

∫

|m| q dσ A0

Thus, for each j ∈ N0 ,

 1/q

≤2

∫ Σ

|m| q dσ

 1/q

≤ 2Rλ(1/q−1) .

(6.1.50)

304

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . . −j[θ/q+λ(1/q−1)]

if λ j := C2∗

then  α j := α j /λ j is a H

q,λ

-atom on Σ.

(6.1.51)

Also, for each j ∈ N0 we have supp β j ⊆ Δ j+1 ,   ∫ ∫ ∫  ∫ 1A 1Aj j+1 dσ − dσ = 0, β j dσ = m dσ Σ Σ\Δ j Σ σ(A j+1 ) Σ σ(A j ) and  β j  L q (Σ,σ) ≤ =

∫ Σ\Δ j

∫ Σ\Δ j

    1 A  j+1   |m| dσ  σ(A j+1 ) 

L q (Σ,σ)

 L q (Σ,σ)

 |m(x)||x − xo | θ/q |x − xo | −θ/q dσ(x) ×



× σ(A j+1 )1/q−1 + σ(A j )1/q−1 ≤ C(2∗ R)(n−1)(1/q−1) j

×

   1Aj    + σ(A j ) 

(6.1.52)

∫ Σ\Δ j

∫ Σ



|m(x)| q |x − xo | θ dσ(x)

 1/q

×

dσ(x)  1/p |x − xo | pθ/q

≤ C(2∗ R)(n−1)(1/q−1) Rλ(1/q−1)+θ/q (2∗ R)(n−1)/p−θ/q j

j

−jθ/q

= C2∗

−j[θ/q+λ(1/q−1)]

Rλ(1/q−1) = C2∗

(2∗ R)λ(1/q−1) .

(6.1.53)

-atom on Σ.

(6.1.54)

m dσ,

(6.1.55)

j

Hence, for each j ∈ N0 , −j[θ/q+λ(1/q−1)]

if η j := C2∗

then βj := β j /η j is a H

q,λ

We may then re-write (6.1.47) as m1Δ N =

N  j=0

λj αj +

N −1 

η j βj −

j=0

1AN σ(A N )

∫ Σ\Δ N

where  α j and βj are H and

∞  j=0

-atoms on Σ for each j ∈ N0

(6.1.56)

∞ 

−j[θ/q+λ(1/q−1)] |λ j | + |η j | ≤ C 2∗ < +∞,

(6.1.57)

q,λ

j=0

given that θ/q + λ(1/q − 1) > (n − 1)(q − 1)/q + λ(1/q − 1) = (n − 1 − λ)(1 − 1/q) > 0 since θ > (n − 1)(q − 1), q ∈ (1, ∞), and λ ∈ (0, n − 1). Since m ∈ L 1 (Σ, σ) (cf.

6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets

305

∫ 1A (6.1.35)), it follows that both m1Δ N → m and σ(ANN ) Σ\Δ m dσ → 0 in L 1 (Σ, σ) N as N → ∞. Consequently, from (6.1.55) we deduce that ∞ 

m=

λj αj +

j=0

∞ 



 η j βj in Lipc (Σ) ,

(6.1.58)

j=0

while (6.1.57) ensures that C :=

∞ 

|λ j | + |η j | < +∞ is independent of m.

(6.1.59)

j=0

In view of (6.1.56) and (6.1.17), the claims recorded in (6.1.38) now follow from (6.1.58) and (6.1.59). This finishes the proof of the lemma in the case when Σ is unbounded. The argument in the case when Σ is bounded follows along similar lines, with some important alterations, indicated below. For starters, when Σ is bounded the choice made in (6.1.39) only guarantees that

σ Δ(x, ρ) \ Δ(x, ρ/2∗ ) ≈ ρn−1 uniformly for (6.1.60)

x ∈ Σ and ρ ∈ 0, 2 diam(Σ) .

To proceed, given a H q,λ,θ -molecule m on Σ, let xo ∈ Σ and R ∈ 0, 2 diam(Σ) be as in (6.1.33). Once again, set Δ−1 :=  and, for each j ∈ N0 , define the surface balls j Δ j := Σ ∩ B(xo, 2∗ R) and the annuli A j := Δ j \ Δ j−1 as before. If we now introduce

j NR := min j ∈ N : 2∗ R ≥ 2 diam(Σ) ,

(6.1.61)

then thanks to (6.1.40) and the definition of NR we have (compare with (6.1.41)) j

σ(A j ) ≈ (2∗ R)n−1 uniformly for j ≤ NR − 1,

(6.1.62)

A j =  whenever j ≥ NR + 1.

(6.1.63)

and Moreover, corresponding to j = NR , Δ N R = Σ and A N R = Σ \ Δ N R −1

(6.1.64)

which, in particular, implies that ∫ Σ\Δ N R −1

m dσ = 0 if σ(A N R ) = 0.

Glancing back at the definition of NR given in (6.1.61) we see that

(6.1.65)

306

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

2∗N R R ≥ 2 diam(Σ) and 2∗N R −1 R < 2 diam(Σ) hence 2∗N R R ≈ diam(Σ) uniformly with respect to R.

(6.1.66)

We now fix an arbitrary point x ∈ Σ and write the summation by parts formula (6.1.43) for N := NR and the following choices: for each j ∈ {0, 1, . . . , NR } take ⎧ 1 A j (x)/σ(A j ) if j ≤ NR − 1, ⎪ ⎪ ⎪ ⎨ ⎪ a j := 1 A N R (x)/σ(A N R ) if j = NR and σ(A N R ) > 0, ⎪ ⎪ ⎪ ⎪0 if j = NR and σ(A N R ) = 0, ⎩ while for each j ∈ {−1, 0, . . . , NR } take ∫ b j :=

Σ\Δ j

m dσ.

(6.1.67)

(6.1.68)

As a consequence of (6.1.68), (6.1.64), and the vanishing moment condition for m, b N R = 0 and b−1 = 0.

(6.1.69)

For these choices, (6.1.43) gives (bearing in mind (6.1.64)-(6.1.65)) m=

NR  j=0

where, if for each j ∈ {0, 1, . . . , NR },

αj +

N R −1

βj

(6.1.70)

j=0

⨏ ⎧ m dσ if j ≤ NR − 1, ⎪ ⎪ A ⎪ ⎨⨏ j ⎪

α j := (m − m A j 1 A j with m A j := m dσ if j = NR and σ(A N R ) > 0, ANR ⎪ ⎪ ⎪ ⎪0 if j = NR and σ(A N R ) = 0, ⎩ (6.1.71) and, for each j ∈ {0, 1, . . . , NR − 1},  1A ∫ 1Aj ⎧ j+1 ⎪ − m dσ if j ≤ NR − 2, ⎪ σ(A ) σ(A ) ⎪ Σ\Δ j j j+1 ⎪ ⎨ ⎪ 1 A N −1  ∫ 1AN β j := R R ⎪ σ(A N R ) − σ(A N R −1 ) Σ\Δ N R −1 m dσ if j = NR − 1 and σ(A N R ) > 0, ⎪ ⎪ ⎪ ⎪0 if j = NR − 1 and σ(A N R ) = 0. ⎩ (6.1.72) Much as in (6.1.48), (6.1.52), these formulas imply that ∫ α j dσ = 0 for each j ∈ {0, 1, . . . , NR }, (6.1.73) Σ

and

6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets

307

∫ Σ

β j dσ = 0 for each j ∈ {0, 1, . . . , NR − 1}.

(6.1.74)

Also, for each j ∈ {0, 1, . . . , NR } we have supp α j ⊆ Δ j and the same type of estimates as in (6.1.49)-(6.1.50) give that −j[θ/q+λ(1/q−1)]

α j  L q (Σ,σ) ≤ C2∗

(2∗ R)λ(1/q−1) j

for j ∈ {0, 1, . . . , NR }. Thus,

−j[θ/q+λ(1/q−1)]

if λ j := C2∗ is a H

q,λ -atom

then  α j := α j /λ j

on Σ, for each {0, 1, . . . , NR }.

(6.1.75)

(6.1.76)

Going further, for each j ∈ {0, 1, . . . , NR − 1} we have supp β j ⊆ Δ j+1 and the same type of estimate as in (6.1.53) shows that −j[θ/q+λ(1/q−1)]

 β j  L q (Σ,σ) ≤ C2∗

(2∗ R)λ(1/q−1) j

for j ∈ {0, 1, . . . , NR − 2}.

(6.1.77)

As far as the case j = NR − 1 is concerned, observe first that   ∫   1A N R −1   −N [θ/q+λ(1/q−1)] N R m dσ  ≤ C2∗ R (2∗ R)λ(1/q−1)   q  σ(A N R −1 ) Σ\Δ N −1 R L (Σ,σ)

≤ C(2∗N R R)λ(1/q−1) .

(6.1.78)

Indeed, the first estimate above is justified by the same type of argument used in (6.1.53) (thanks to (6.1.62) for j = NR − 1), while the second estimate is a simple consequence of the fact that 2∗ > 1 and θ/q + λ(1/q − 1) > 0. We complement the estimate established in (6.1.78) by showing that there exists a constant C = C(Σ, q, λ, θ) ∈ (0, ∞) (in particular, independent of x0, R, m) with the property that in the case when σ(A N R ) > 0 we also have   ∫   1A NR   m dσ  ≤ C(2∗N R R)λ(1/q−1) . (6.1.79)   q  σ(A N R ) Σ\Δ N −1 R

L (Σ,σ)

(The main difference between (6.1.78) and (6.1.79) is the absence of a concrete lower bound for σ(A N R ) of the sort appearing in (6.1.62).) To justify (6.1.79), assume σ(A N R ) > 0 and first estimate

308

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

 ∫  1    σ(A N R ) Σ\Δ N ≤

R

   m dσ   −1

1 σ(A N R )

∫ Σ\Δ N R −1

|m(x)||x − xo | θ/q |x − xo | −θ/q dσ(x)

∫ 1 |m(x)||x − xo | θ/q dσ(x) σ(A N R ) Σ\Δ N R −1 ⨏ NR −θ/q |m(x)||x − xo | θ/q dσ(x) = C(2∗ R) ≤ C(2∗N R R)−θ/q

ANR

≤ C(2∗N R R)−θ/q ≤

C(2∗N R R)−θ/q

≤ C(2∗N R R)−θ/q ≤ C(2∗N R R)−θ/q ≤ C(2∗N R R)−θ/q =

C σ(A N R )1/q

⨏

ANR

|m(x)| q |x − xo | θ dσ(x) ∫

1 σ(A N R )1/q 1 σ(A N R )1/q

Σ

σ(A N R

)1/q

|m(x)| q |x − xo | θ dσ(x)

 1/q

Rλ(1/q−1)+θ/q

1

 1/q

2 diam(Σ)

λ(1/q−1)+θ/q

1 (2∗N R R)λ(1/q−1)+θ/q σ(A N R )1/q

(2∗N R R)λ(1/q−1),

(6.1.80)

where C = C(Σ, q, λ, θ) ∈ (0, ∞) is a constant independent of xo, R, m. Above, the first two inequalities are clear, the first equality is based on (6.1.64) and, subsequently, we have used Hölder’s inequality. Next, we unpack the integral average and extend the domain of integration,

then we rely on the second inequality in (6.1.33). The fact that R ∈ 0, 2 diam(Σ) and λ(1/q − 1) + θ/q > 0 permit us to justify the penultimate inequality. The last inequality is implied by (6.1.66). The final equality in (6.1.80) is obvious. Having established (6.1.80), the claim made in (6.1.79) follows by writing     ∫ ∫      1A 1 NR      m dσ  = m dσ  1 A N R  q   q  σ(A N R ) Σ\Δ N −1   σ(A N R ) Σ\Δ N −1 L (Σ,σ) R

L (Σ,σ)

R

≤

C σ(A N R )1/q

(2∗N R R)λ(1/q−1) σ(A N R )1/q

= C(2∗N R R)λ(1/q−1) .

(6.1.81)

6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets

309

Collectively, (6.1.72) (with j := NR − 1), (6.1.78), (6.1.79), (6.1.66), and (6.1.64) prove that

λ(1/q−1) and supp β N R −1 ⊆ Δ N R = Σ.  β N R −1  L q (Σ,σ) ≤ C diam(Σ)

(6.1.82)

As such, from (6.1.77), (6.1.82), (6.1.74), the fact that supp β j ⊆ Δ j+1 for each j ∈ {0, 1, . . . , NR − 1}, and (6.1.64) we conclude that if  −j[θ/q+λ(1/q−1)] C2∗ if j ∈ {0, 1, . . . , NR − 2} η j := C if j = NR − 1,

(6.1.83)

(6.1.84)

then βj := β j /η j is a H

q,λ

-atom on Σ for each j ∈ {0, 1, . . . , NR − 1}.

(6.1.85)

Additionally, it is apparent from definitions that NR  j=0

|λ j | +

N R −1 j=0

|η j | ≤ C

∞ 

−j[θ/q+λ(1/q−1)]

2∗



+ C =: C(Σ, q, λ, θ) < +∞, (6.1.86)

j=0

since θ/q + λ(1/q − 1) > 0. From (6.1.70), (6.1.76), (6.1.84), (6.1.86), and (6.1.16) we see that the function m=

NR  j=0

λj αj +

N R −1

η j βj

(6.1.87)

j=0

belongs to the space H q,λ (Σ, σ) and satisfies mH q, λ (Σ,σ) ≤ C(Σ, q, λ, θ). This establishes the desired conclusions in the case when Σ is bounded, so the proof of Lemma 6.1.2 is complete.  It is also useful to observe (from our earlier definitions) that, given q ∈ (1, ∞) and λ ∈ (0, n − 1), any H q,λ -atom on Σ is also a H q,λ,θ -molecule on Σ for any choice of the parameter θ in the interval (n − 1)(q − 1), ∞ .

(6.1.88)

In the lemma below we address the fundamental issue of correlating the integral pairing of a function belonging to the homogeneous Morrey-Campanato space (defined on a given Ahlfors regular set) with a molecule belonging to the pre-dual of this space. This compatibility result is essential in establishing boundedness properties for singular integral operators (of double layer type) acting on Morrey-Campanato spaces, a topic considered a little later in this section.

310

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

Lemma 6.1.3 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed set which is Ahlfors regular and abbreviate σ := H n−1 Σ. Also, pick two exponents p, q ∈ (1, ∞) such that 1/p + 1/q = 1, along with . two parameters λ ∈ (0, n − 1) and θ > (n − 1)(q − 1). Then for any function f ∈ L p,λ (Σ, σ) and any H q,λ,θ -molecule m on Σ (cf. (6.1.37)) one has   ∫ ∫ [ f ], m if Σ is unbounded,  | f ||m| dσ < +∞ and f m dσ =  (6.1.89) f, m if Σ is bounded, Σ Σ where ·, · denotes the duality bracket between the homogeneous Morrey-Campanato space and its pre-dual (cf. (6.1.25)). . Furthermore, for each function f ∈ L p,λ (Σ, σ) ⊆ L p,λ (Σ, σ) and each function g ∈ H q,λ (Σ, σ) with the property that ∫ | f ||g| dσ < +∞ (6.1.90) Σ

one has



∫ Σ

f g dσ =

[ f ], g   f, g



if Σ is unbounded, if Σ is bounded,

(6.1.91)

with ·, · denoting, as before, the duality bracket between the inhomogeneous Morrey-Campanato space and its pre-dual. Proof To fix ideas, suppose first that the set Σ is unbounded. ⨏ For each x ∈ Σ and R > 0 abbreviate Δ(x, R) := Σ ∩ B(x, R) and fΔ(x,R) := Δ(x,R) f dσ. Fix a point x ∈ Σ along with a scale R ∈ (0, ∞), then use [133, (7.4.121)] with X := Σ, μ := σ, ρ := | · − · |, q := p, x0 := x, and r := R to write for each j ∈ N0 ⨏  1/p    f − fΔ(x,R)  p dσ Δ(x,2 j+1 R)

 j  λ−n+1 n−1−λ  ⨏ +1 ≤C (2 R) p (2+1 R) p

Δ(x,2+1 R)

=0

≤ C f L p, λ (Σ,σ)

j 

(2+1 R)

λ−n+1 p

 1/p    f − fΔ(x,2+1 R)  p dσ

≤ C f L p, λ (Σ,σ) R

λ−n+1 p ,



(6.1.92)

=0

since (λ − n + 1)/p < 0 ensures that

j 

(2+1 )

=0

λ−n+1 p

and R ∈ (0, ∞) be as in (6.1.33) for the given H then estimate:

< +∞. To proceed, let xo ∈ Σ

q,λ,θ -molecule

m on Σ. We may

6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets

∫ Σ\Δ(xo ,R)

≤

   f − fΔ(x ,R)  |m| dσ o

∞ ∫  Δ(x o ,2k R)\Δ(x o ,2k−1 R)

k=1

×

∫

 1/p    f − fΔ(x ,R)  p dσ × o

|m| q dσ

Δ(x o ,2k R)\Δ(x o ,2k−1 R)

⨏ ∞ 

1/p  k = σ Δ(xo, 2 R)

 1/q  1/p    f − fΔ(x ,R)  p dσ × o

k=1

Δ(x o ,2k R)

×

|m(x)| q |x − xo | θ |x − xo | −θ dσ(x)

∫ Δ(x o ,2k R)\Δ(x o ,2k−1 R)

≤ C f L p, λ (Σ,σ) R ×

311

∫ Σ

λ−n+1 p

∞ 

 1/q

 (2k R)(n−1)/p−θ/q ×

k=1

|m(x)| q |x − xo | θ dσ(x)

≤ C f L p, λ (Σ,σ) R

λ−n+1 p

∞ 

 1/q

 (2k R)(n−1)/p−θ/q Rλ(1/q−1)+θ/q

k=1



= C f L p, λ (Σ,σ) · R



θ λ−n+1 n−1 θ 1 p + p − q +λ q −1 + q

∞ 

2−k[θ/q−(n−1)/p]



k=1

= C f L p, λ (Σ,σ) .

(6.1.93)

The first inequality above is obtained by breaking up the domain of integration into mutually disjoint dyadic annuli and then using Hölder’s inequality (keeping in mind that 1/p + 1/q = 1). The subsequent equality is clear, and the second inequality is a consequence of (6.1.92) and the upper Ahlfors regularity of Σ. Next, the third inequality follows from (6.1.33), while the ensuing equality is just algebra. Finally, the last equality uses the fact that the series in the curly brackets converges, since θ/q − (n − 1)/p = [θ − (n − 1)q/p]/q = [θ − (n − 1)(q − 1)]/q ∫ > 0. Since from (6.1.36) we know that m ∈ L 1 (Σ, σ), the argument so far gives Σ\B(x ,R) | f ||m| dσ < +∞. o ∫ As such, there remains to show that Σ∩B(x ,R) | f ||m| dσ < +∞. This, however, is o a consequence of (6.1.3) and (6.1.36). The first property in (6.1.89) has now been established if Σ is unbounded. In the case when Σ is bounded, the first property in (6.1.89) follows by observing that we now have f ∈ L p (Σ, σ) (cf. (6.1.13)) and recalling from (6.1.36) that m ∈ L q (Σ, σ). To prove the second property in (6.1.89), suppose first that Σ is unbounded. Note that, in light of the nature of the conclusion we seek, there is no loss of generality

312

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

in assuming that the given function f is real-valued. For each M ∈ N we may then define fM := min max{ f , −M }, M . Lemma 6.1.1 ensures that

.

{ fM } M ∈N ⊆ L ∞ (Σ, σ) ∩ L p,λ (Σ, σ) and .  lim [ fM ] = [ f ] weak-∗ in L p,λ (Σ, σ) ∼ .

(6.1.94)

M→∞

As such, since m belongs to H q,λ (Σ, σ) (cf. (6.1.38)), we may write     [ f ], m = lim [ fM ], m .

(6.1.95)

M→∞

To proceed, express m ∈ H q,λ (Σ, σ) as a series m = H q,λ (Σ, σ), where {λ j } j ∈N ∈  1 (N) and each a j is a H each M ∈ N fixed, (6.1.26)-(6.1.27) permit us to write 

∫



[ fM ], m = lim

N →∞

Σ

fM

N 

∞

j=1 λ j a j convergent in q,λ -atom on Σ. Then for

 λ j a j dσ.

(6.1.96)

j=1



Let xo ∈ Σ and R ∈ 0, 2 diam(Σ) be such that m satisfies the properties listed in (6.1.33). Also, recall the number 2∗ from (6.1.39). For each j ∈ N0 let us now define j Δ j := Σ ∩ B(xo, 2∗ R), set Δ−1 := , and introduce A j := Δ j \ Δ j−1 . Then from (6.1.55)-(6.1.57) it follows that matters may be arranged so that for each N ∈ N we have ∫ N  1AN λ j a j = m1Δ N + m dσ. (6.1.97) σ(A N ) Σ\Δ N j=1 The availability of such an explicit formula for the partial sums of an atomic decomposition series for m is a crucial element in the present proof. For this allows us to compute, on the one hand, ∫ lim

N →∞

Σ

fM

∫ ∫    1AN λ j a j dσ = lim fM m1Δ N + m dσ dσ N →∞ Σ σ(A N ) Σ\Δ N j=1

N 

∫ = lim

N →∞

+ lim

ΔN

N →∞

fM m dσ

∫ Σ\Δ N

 m dσ

1 σ(A N )

∫ AN

 fM dσ . (6.1.98)

On the other hand, the fact that m ∈ L 1 (Σ, σ) and fM ∈ L ∞ (Σ, σ) implies

6.1 Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets

∫

∫ lim

N →∞

ΔN

fM m dσ =

  and lim sup  N →∞

313

∫ fM m dσ,

Σ

1 σ(A N )

∫

lim

N →∞

Σ\Δ N

m dσ = 0,

  fM dσ  ≤  fM  L ∞ (Σ,σ) .

AN

All together, from (6.1.96), (6.1.98), and (6.1.99) we deduce that ∫   [ fM ], m = fM m dσ, ∀M ∈ N. Σ

(6.1.99)

(6.1.100)

on Σ as M → ∞, that | fM | ≤ | f | for each M ∈ N Given that fM → f pointwise ∫ and that, as already noted, Σ | f ||m| dσ < +∞, Lebesgue’s Dominated Convergence Theorem applies and gives that ∫ ∫ fM m dσ = f m dσ. (6.1.101) lim M→∞

Σ

Σ

At this stage, the second property claimed in (6.1.89) follows in the case when Σ is unbounded by combining (6.1.95), (6.1.100), and (6.1.101). To justify the second property in (6.1.89) in the case when Σ is bounded, recall . from (6.1.13) and (6.1.11) that f ∈ L p,λ (Σ, σ) = L p,λ (Σ, σ) ⊆ L p (Σ, σ). Also, in the current setting, ⨏  ⨏  m dσ + m dσ (6.1.102) m= m− Σ

Σ

constitutes an atomic decomposition of the function m ∈ L q (Σ, σ) ⊆ H q,λ (Σ, σ) (in the sense of (6.1.16)), so the discussion regarding the sense in which (6.1.25) is to be understood implies that ⨏ ∫ ⨏ ∫ ∫    m− m dσ f dσ + m dσ f dσ = f m dσ, (6.1.103)  f , m = Σ

Σ

Σ

Σ

Σ

as wanted. Let us now turn to the task of proving (6.1.91) for any given f ∈ L p,λ (Σ, σ) and g ∈ H q,λ (Σ, σ) satisfying (6.1.90). As above, the crux of the matter is establishing that for each fixed M ∈ N we have ∫ lim

N →∞

Σ

fM

N 

∫  λ j a j dσ = fM g dσ,

j=1

Σ

 q,λ (Σ, σ), where {λ } 1 if g = ∞ j j ∈N ∈  (N) and each a j is a H j=1 λ j a j in H on Σ. To this end, we first recall from (6.1.22) that lim

N →∞

N  j=1

λ j a j = g in L r (Σ, σ) where r :=

q(n − 1) . n − 1 + λ(q − 1)

(6.1.104) q,λ -atom

(6.1.105)

314

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

Note that 1/p + 1/r = 1/p + 1/q + [λ(q − 1)]/[q(n − 1)] > 1. Hence, if r  ∈ (1, ∞) is such that 1/r + 1/r  = 1, then necessarily r  ∈ (p, ∞). In particular, for each  M ∈ N the function fM belongs to L ∞ (Σ, σ) ∩ L p (Σ, σ) ⊆ L r (Σ, σ). Granted this membership, we may now conclude from (6.1.105) that (6.1.104) is indeed valid. The desired conclusion now follows by reasoning as before. 

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets We begin by discussing the scale of Morrey spaces on Ahlfors regular sets. As in the past, we assume that Σ ⊆ Rn (where n ∈ N with n ≥ 2) is a closed Ahlfors regular set, and that σ := H n−1 Σ. Given p ∈ (0, ∞) and λ ∈ (0, n − 1), we then define the Morrey space M p,λ (Σ, σ) as   M p,λ (Σ, σ) := f : Σ → C : f is σ-measurable and  f  M p, λ (Σ,σ) < +∞ (6.2.1) where, for each σ-measurable function f on Σ, we have set    p1 n−1−λ  ⨏ sup | f | p dσ  f  M p, λ (Σ,σ) := R p . (6.2.2) x ∈Σ and 0 n − 1 − (n−1−λ)τ , estimate p

318

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

∫ Σ

| f (x)|τ dσ(x) ≤ C 1 + |x| a ≤C

∫ Σ

∫

| f (x)|τ dσ(x) 1 + |x − x0 | a

B(x0,1)∩Σ

+C

| f (x)|τ dσ(x) 1 + |x − x0 | a

∞ ∫  j=1

[B(x0,2 j+1 )\B(x0,2 j )]∩Σ

∫ ≤C

≤C

τ

B(x0,1)∩Σ ∞ 

| f | dσ + C

2−j(a−n+1)

∞ 

(2−j )

−j a

2

⨏ B(x0,2 j+1 )∩Σ

a−n+1+

(n−1−λ)τ p



∫ B(x0,2 j+1 )∩Σ

j=1

j=0

≤C

∞ 

| f (x)|τ dσ(x) 1 + |x − x0 | a

| f | p dσ

| f |τ dσ

 τ/p

 f  M p, λ (Σ,σ)

j=0

= C f  M p, λ (Σ,σ),

(6.2.24)

from which the desired conclusion follows. In particular, from (6.2.15) and (6.2.23) (used with τ := p and τ := 1) we see that we have continuous embeddings   σ(x)  σ(x)  ∩ L 1 Σ, M˚ p,λ (Σ, σ) → M p,λ (Σ, σ) → L p Σ, a 1 + |x| 1 + |x| n−1  p → Lloc (Σ, σ) ∩ L 1 Σ,

σ(x)  1 + |x| n−1

if p ∈ (1, ∞), λ ∈ (0, n − 1), and a > λ.

(6.2.25)

As it turns out, there is a natural description of Morrey spaces in terms of the fractional Hardy-Littlewood maximal operator (cf. (A.0.68) with X := Σ), as indicated in the lemma below. Lemma 6.2.2 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set, and abbreviate σ := H n−1 Σ. Also, fix p ∈ (0, ∞) along with λ ∈ (0, n − 1), and p recall from (A.0.68)  1 that L -based fractional Hardy-Littlewood maximal operator of order α ∈ 0, p on Σ acts on each σ-measurable function f on Σ as  ⨏

α  M Σ, p,α f (x) := sup σ B(x, r) ∩ Σ r >0

Then, if α :=

| f | p dσ

 p1 

,

∀x ∈ Σ. (6.2.26)

B(x,r)∩Σ

n−1−λ , (n − 1)p

(6.2.27)

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

one has

  M Σ, p,α f 

L ∞ (Σ,σ)

≈  f  M p, λ (Σ,σ),

uniformly for σ-measurable functions f on Σ, and

319

(6.2.28)

  M p,λ (Σ, σ) = f : Σ → C : f is σ-measurable and M Σ, p,α f ∈ L ∞ (Σ, σ) . (6.2.29)

The description of the Morrey space in (6.2.29) offers a natural interpretation of the embedding L s (Σ, σ) → M p,λ (Σ, σ) with s := p(n−1) n−1−λ (cf. (6.2.7)) in view of the fact that the fractional Hardy-Littlewood maximal operator M Σ, p,α with α as in (6.2.27), i.e., α = 1/s, maps L s (Σ, σ) into L ∞ (Σ, σ) (see [133, (7.6.6)]). Proof of Lemma 6.2.2 As in (4.4.42), we find it convenient to work with a version of M Σ, p,α which exhibits better measurability properties than the original fractional Hardy-Littlewood maximal operator. Specifically, if for each σ-measurable function f : Σ → C we define ∫   1/p 1 p Σ, p,α f (x) := sup | f | dσ M (n−1)(1−αp) B(x,R)∩Σ R ∈(0,∞) R  1 ∫  1/p p = sup | f | dσ, ∀x ∈ Σ, (6.2.30) λ B(x,R)∩Σ R ∈(0,∞) R then, in a similar fashion to (4.4.43), we have Σ, p,α f : Σ −→ [0, +∞] is a lower-semicontinuous function M for each given each σ-measurable function f : Σ → C.

(6.2.31)

The fact that Σ is an Ahlfors regular set ensures that there exist 0 < co ≤ Co < +∞ with the property that for each σ-measurable function f : Σ → C we have Σ, p,α f ≤ Co · M Σ, p,α f everywhere on Σ. co · M Σ, p,α f ≤ M

(6.2.32)

As such, for each σ-measurable function f on Σ we may write     Σ, p,α f  ∞ Σ, p,α f (x) M Σ, p,α f  ∞ ≈ M = sup M L (Σ,σ)

L (Σ,σ)

x ∈Σ

≈ sup M Σ, p,α f (x) ≈  f  M p, λ (Σ,σ), x ∈Σ

(6.2.33)

with proportionality constants independent of f . The first two equivalences above are implied by (6.2.32), the equality is a consequence of (6.2.31), and the last equivalence is clear from (6.2.26), (6.2.27), and (6.2.2). This proves (6.2.28) which, in turn, justifies (6.2.29). 

320

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

The Morrey spaces from (6.2.1)-(6.2.2) interface tightly with the MorreyCampanato spaces discussed earlier. For starters let us observe that, as is apparent from [133, (7.4.58)] and definitions, if p ∈ (1, ∞) and λ ∈ (0, n − 1) we have

.

M p,λ (Σ, σ) ⊆ L p,λ (Σ, σ) and there exists a constant C ∈ (0, ∞) such that  f L. p, λ (Σ,σ) ≤ C f  M p, λ (Σ,σ) for all functions f ∈ M p,λ (Σ, σ).

(6.2.34)

To be able to further elaborate on this relationship, we first establish the following result. Lemma 6.2.3 Suppose Σ ⊆ Rn (where n ∈ N with n ≥ 2) is an unbounded closed n−1 set which is Ahlfors regular, and abbreviate . p,λσ := H Σ. Also, fix p ∈ (1, ∞) and λ ∈ (0, n − 1). Then for each function f ∈ L (Σ, σ) and each point x0 ∈ Σ, the limit ⨏ f dσ (6.2.35) fΣ := lim R→∞

B(x0,R)∩Σ

exists and is actually independent of the point x0 . In addition, there exists a constant C ∈ (0, ∞) with the property that   n−1−λ  ⨏   p  p1  f − fΣ  dσ R p  f − fΣ  M p, λ (Σ,σ) = sup x ∈Σ and R ∈(0,∞)

Σ∩B(x,R)

≤ C f L. p, λ (Σ,σ) .

(6.2.36)

It is worth pointing out that, as seen with the help of Hölder’s inequality, whenever f ∈ L q (Σ, σ) for some q ∈ [1, ∞) the limit in (6.2.35) is zero for each point x0 ∈ Σ.

(6.2.37)

Proof of Lemma 6.2.3⨏ As in the past, for each point x ∈ Σ and each radius R > 0, abbreviate fΔ(x,R) := B(x,R)∩Σ f dσ. Let us now fix some x ∈ Σ. Then, for each r, R > 0 such that R > r, based on [133, Lemma 7.4.15], (A.0.83), and (6.1.2) we may estimate    fΔ(x,r) − fΔ(x,R)  ≤ C

∫ r

2R

osc p ( f ; t) ∫

≤ C f L. p, λ (Σ,σ) ≤ C f L. p, λ (Σ,σ) r

dt t

∞

t

−1−

n−1−λ p

dt

r −

n−1−λ p

= o(1) as r → ∞.

(6.2.38)



This proves that the sequence fΔ(x,r) r >0 is Cauchy, hence convergent to some complex number which we shall call fΣ,x . Let us now fix two arbitrary points x, y ∈ Σ. Then for each r > |x − y| we have B(y, r) ⊆ B(x, 2r) so we may invoke [133, (7.4.56)] (with p := 1), Hölder’s

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

321

inequality, the Ahlfors regularity of Σ, (6.1.2), and what we have just proved in order to write        fΔ(y,r) − fΣ,x  ≤  fΔ(y,r) − fΔ(x,2r)  +  fΣ,x − fΔ(x,2r)  ⨏   ≤ | f − fΔ(y,r) | dσ +  fΣ,x − fΔ(x,2r)  Δ(x,2r)



⨏  p1 σ Σ ∩ B(x, 2r) 

≤ 1+ | f − fΔ(x,2r) | p dσ σ Σ ∩ B(y, r) Δ(x,2r)   +  fΣ,x − fΔ(x,2r)  

≤ C f L. p, λ (Σ,σ) r

−

n−1−λ p

  +  fΣ,x − fΔ(x,2r) 

= o(1) as r → ∞.

(6.2.39)

Ultimately, this shows that fΣ,y = fΣ,x which establishes the independence of fΣ,x on the point x. As regards (6.2.36), recall from (6.2.38) that for each for each x ∈ Σ and each r, R > 0 such that R > r we have n−1−λ   − p  fΔ(x,r) − fΔ(x,R)  ≤ C f  . r . L p, λ (Σ,σ)

(6.2.40)

Upon passing to limit as R → ∞ we then obtain, on account of (6.2.35), that n−1−λ   − p  fΔ(x,r) − fΣ  ≤ C f  . r . p, λ L (Σ,σ)

(6.2.41)

Hence, n−1−λ r p

⨏ Σ∩B(x,r)

1 n−1−λ     f − fΣ  p dσ p ≤ r p +r

⨏ Σ∩B(x,r)

n−1−λ p |f Σ

1    f − fΔ(x,r)  p dσ p

− fΔ(x,r) |

≤ C f L. p, λ (Σ,σ) .

(6.2.42)

Taking the supremum over all x ∈ Σ and r ∈ (0, ∞) then yields (6.2.36). This finishes the proof of the lemma.  We are now prepared to fully clarify the relationship between the Morrey spaces and the Morrey-Campanato spaces in the case when the underlying Ahlfors regular is unbounded. Proposition 6.2.4 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be an unbounded closed set which is Ahlfors regular, and abbreviate σ := H n−1 Σ. Also, fix p ∈ (1, ∞) and

322

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

λ ∈ (0, n − 1). Then, with the piece of notation introduced in (6.2.35), the assignment . 

 L p,λ (Σ, σ) ∼ ,  · L. p, λ (Σ,σ)  ∼  [ f ] −→ f − fΣ ∈ M p,λ (Σ, σ),  ·  M p, λ (Σ,σ) (6.2.43) is a well-defined linear and bounded isomorphism, whose inverse is the mapping .  p,λ

 M (Σ, σ),  ·  M p, λ (Σ,σ)  f −→ [ f ] ∈ L p,λ (Σ, σ) ∼ ,  · L. p, λ (Σ,σ)  ∼ . (6.2.44)

.

Moreover,

for each f ∈ L p,λ (Σ, σ) one has

(6.2.45)

f ∈ M p,λ (Σ, σ) ⇐⇒ fΣ = 0.

We wish to note that, as a consequence of (6.2.45), (6.2.37), and (6.2.34), in the context of the above proposition we have

.

L q (Σ, σ) ∩ L p,λ (Σ, σ) = L q (Σ, σ) ∩ M p,λ (Σ, σ) for each q ∈ [1, ∞).

(6.2.46)

.

Proof of Proposition 6.2.4 Observe that for each f ∈ M p,λ (Σ, σ) ⊆ L p,λ (Σ, σ) we have ⨏ ⨏  1/p | f | dσ ≤ lim sup | f | p dσ | fΣ | ≤ lim sup R→∞

≤ lim sup R R→∞

B(x0,R)∩Σ

−

n−1−λ p

R→∞

B(x0,R)∩Σ

 f  M p, λ (Σ,σ) = 0,

(6.2.47)

hence fΣ = 0 for each f ∈ M p,λ (Σ, σ).

(6.2.48)

. p,λ

In the opposite direction, if f ∈ L (Σ, σ) has fΣ = 0 then (6.2.36) implies that f ∈ M p,λ (Σ, σ). This finishes the proof of (6.2.45). Next, the fact the assignment considered in (6.2.43) is well defined, linear, and bounded is clear from Lemma 6.2.3 and (6.2.1)-(6.2.2). Its injectivity is obvious, since f − fΣ = 0 in M p,λ (Σ, σ) forces f to be a constant, hence [ f ] = 0. To prove that the assignment in (6.2.43) is also surjective, pick f ∈ M p,λ (Σ, σ). Then (6.2.34) . implies f ∈ L p,λ (Σ, σ), and [ f ] is mapped into f − fΣ = f , by (6.2.48). This reasoning also shows that the inverse of (6.2.43) is the mapping in (6.2.44).  Here is a companion result to Proposition 6.2.4, establishing the coincidence of Morrey and Morrey-Campanato spaces in the case when the underlying Ahlfors regular is bounded. Proposition 6.2.5 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a bounded closed set which is Ahlfors regular, and abbreviate σ := H n−1 Σ. Also, fix p ∈ (1, ∞) and λ ∈ (0, n − 1). Then

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

323

.

L p,λ (Σ, σ) = L p,λ (Σ, σ) = M p,λ (Σ, σ) and

 f L p, λ (Σ,σ) ≈  f  . p, λ L

(6.2.49)

∫    + f dσ   ≈  f  M p, λ (Σ,σ), (Σ,σ)

(6.2.50)

Σ

uniformly for f ∈ L 1 (Σ, σ). Proof The first equality in (6.2.49) has been already noted in (6.1.13), and from . (6.2.34) we know that M p,λ (Σ, σ) ⊆ L p,λ (Σ, σ). To prove the opposite inclusion, fix . a function f ∈ L p,λ (Σ, σ), some point x ∈ Σ, and take R := 4 diam(Σ). Then for each r ∈ 0, 2 diam(Σ) we may rely on [133, Lemma 7.4.15], (A.0.83), and (6.1.2) to estimate ∫ 8diam(Σ) ∫     dt    −1  f dσ  = fΔ(x,r) − fΔ(x,R) ≤ C osc p ( f ; t)  fΔ(x,r) − σ(Σ) t r Σ ∫ 8diam(Σ) n−1−λ −1− p ≤ C f L. p, λ (Σ,σ) t dt r

≤ C f L. p, λ (Σ,σ) r Consequently, ⨏ Σ∩B(x,r)

| f | dσ p

 p1

≤

⨏ Σ∩B(x,r)

−

n−1−λ p .

(6.2.51)

1      f − fΔ(x,r)  p dσ p +  fΔ(x,r) 

≤ C f L. p, λ (Σ,σ) r

−

n−1−λ p

∫     +  fΔ(x,r) − σ(Σ)−1 f dσ  Σ

∫    + σ(Σ)−1  f dσ  Σ

≤ C f L. p, λ (Σ,σ) r

−

n−1−λ p

∫    + σ(Σ)−1  f dσ . Σ

(6.2.52)

n−1−λ

p Multiplying the most extreme sides by r x ∈ Σ and all r ∈ 0, 2 diam(Σ) then yields

, then taking the supremum over all

∫     f  M p, λ (Σ,σ) ≤ C f L. p, λ (Σ,σ) + C  f dσ . Σ

(6.2.53)

This shows that f ∈ M p,λ (Σ, σ), which finishes the proof of the inclusion . L p,λ (Σ, σ) ⊆ M p,λ (Σ, σ). At this stage, (6.2.49) is fully justified and the equivalences in (6.2.50) are implicit in the above argument. The proof of the proposition is therefore complete. 

324

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

In Lemma 6.2.7 we shall take a closer look at the nature of H q,λ,θ -domes on compact Ahlfors regular sets Σ ⊆ Rn . One useful technical result in this regard is isolated in the lemma below. Lemma 6.2.6 Suppose Σ ⊆ Rn (where n ∈ N with n ≥ 2) is a bounded closed set which is Ahlfors regular, and abbreviate σ := H n−1 Σ. Also, fix p, q ∈ (1, ∞) such that 1/p + 1/q = 1, along with some λ ∈ (0, n − 1). Then L q (Σ, σ) → H

q,λ

(Σ, σ) continuously,

(6.2.54)

and there exists C = C(Σ,

q, λ) ∈ (0, ∞) such that for any x0 ∈ Σ and any R ∈ 0, 2 diam(Σ) the function f := R(n−1)(1/p−1)+λ(1/q−1) 1B(x0,R)∩Σ belongs to the space H q,λ (Σ, σ) and satisfies  f H q, λ (Σ,σ) ≤ C.

(6.2.55)

For further reference let us observe that as a consequence of (6.2.54) and (6.1.90)(6.1.91), in the context of Lemma 6.2.6, we have that ∫   f g dσ for each f ∈ L p,λ (Σ, σ) and g ∈ L q (Σ, σ), (6.2.56) f, g = Σ

where ·, · denotes the duality bracket between the inhomogeneous MorreyCampanato space and its pre-dual (regarding g in H q,λ (Σ, σ)). In particular, given any θ > (n − 1)(q − 1), we have ∫   f, m = f m dσ for each f ∈ L p,λ (Σ, σ) and each H q,λ,θ -dome m on Σ. Σ

(6.2.57)

Proof of Lemma 6.2.6 Any given function f ∈ L q (Σ, σ) may ∫ ∫ be decomposed as −1 f = g + c, where g := f − σ(Σ) Σ f dσ and c := σ(Σ)−1 Σ f dσ. Since g may be regarded as a scalar multiple of a H q,λ -atom on Σ with “large” support (i.e., a function satisfying (6.1.15) with R := diam(Σ)) and g is a constant hence, by definition, also a multiple of a H q,λ -atom on Σ (given that Σ is bounded), the inclusion in (6.2.54) follows. Its continuity is implicit

in the above reasoning. Next, fix x0 ∈ Σ along with R ∈ 0, 2 diam(Σ) , and define f as in (6.2.55). Thanks to (6.2.54) we have f ∈ H q,λ (Σ, σ). To estimate the norm of f in the space H q,λ (Σ, σ) we write

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

 f H q, λ (Σ,σ) =

≤

≤

sup

|Λ( f )| ≤

sup

∫    f g dσ  

Λ∈(H q, λ (Σ,σ))∗ Λ (H q, λ (Σ, σ))∗ ≤1

g ∈M p, λ (Σ,σ) g  M p, λ (Σ, σ) ≤C

sup

g ∈M p, λ (Σ,σ) g  M p, λ (Σ, σ) ≤C

=C

Σ

R

sup

g ∈M p, λ (Σ,σ) g  M p, λ (Σ, σ) ≤C

sup

sup

g ∈L p, λ (Σ,σ) g L p, λ (Σ, σ) ≤C

(n−1)/p+λ(1/q−1)

g ∈M p, λ (Σ,σ) g  M p, λ (Σ, σ) ≤C

⨏ B(x0,R)∩Σ

 ≤C

325

   g, f  

R

(n−1−λ)/p

⨏

|g| dσ

|g| dσ p

B(x0,R)∩Σ

 1/p

g M p, λ (Σ,σ) ≤ C.



(6.2.58)

Above, the first step is one of the basic consequences of the Hahn-Banach Theorem, the second step is implied by (6.1.25), the third step originates in (6.1.91) in Lemma 6.1.3 and (6.2.49) in Proposition 6.2.5, the fourth step takes into account the specific format of the function f , the fifth step is based on Hölder’s inequality and the fact that 1/p + 1/q = 1, the sixth step is seen from (6.2.2), and the final step is tautological.  Remarkably, there is a companion result for Lemma 6.1.2 valid for domes (in lieu of molecules), provided the Ahlfors regular set is now assumed to be bounded. Lemma 6.2.7 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a bounded set which is closed and Ahlfors regular, and abbreviate σ := H n−1 Σ. Also, fix q ∈ (1, ∞), λ ∈ (0, n − 1), and θ > (n − 1)(q − 1). Then there exists C = C(Σ, q, λ, θ) ∈ (0, ∞) such that any H q,λ,θ -dome m on Σ belongs to the space H q,λ (Σ, σ) and satisfies mH q, λ (Σ,σ) ≤ C.

(6.2.59)

As a consequence, whenever Σ is compact, the operator of pointwise multiplication by some given function belonging to L ∞ (Σ, σ) is a bounded (6.2.60) mapping from the space H q,λ (Σ, σ) into itself.

Proof Fix a H q,λ,θ -dome m on Σ and pick x0 ∈ Σ along with R ∈ 0, 2 diam(Σ) such that the estimates in (6.1.33) are true. We perform the same decomposition procedure as in the proof of Lemma 6.1.2 with one basic difference. Specifically, in the process of writing (6.1.42) (presently with N := NR ) by once again making use of (6.1.43) for the same choices of the a j ’s and b j ’s as in (6.1.67)-(6.1.67), we

326

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

no longer have b−1 = 0 as the ∫ dome m may be lacking any cancellation property. Instead, we now have b−1 = Σ m dσ so we currently have an additional term, namely ∫

−1 a0 b−1 = σ B(x0, R) ∩ Σ 1B(x0,R)∩Σ Σ m dσ, in the last line of (6.1.42), hence also in the right-hand side of (6.1.70) and the right-hand side of (6.1.87). Thus, in place of (6.1.87) we now arrive at m=

NR 

λj αj +

N R −1

j=0

with m0 :=

η j βj + m0 pointwise on Σ,

j=0

1B(x0,R)∩Σ



σ B(x0, R) ∩ Σ

(6.2.61)

∫ Σ

m dσ,

α j ’s and βj ’s where the numerical coefficients λ j , η j satisfy (6.1.86) and where the  are H q,λ -atoms on Σ. Denote by p ∈ (1, ∞) the Hölder conjugate exponent of q. Since from (6.1.34) and the Ahlfors regularity of Σ we obtain ∫ 1B(x0,R)∩Σ

|m0 | ≤ |m| dσ ≤ CR(n−1)/p+λ(1/q−1) · R−(n−1) 1B(x0,R)∩Σ σ B(x0, R) ∩ Σ Σ = CR(n−1)(1/p−1)+λ(1/q−1) 1B(x0,R)∩Σ pointwise on Σ,

(6.2.62)

it follows that (6.2.55) applies to the function m0 defined in (6.2.61). Hence, m0 belongs to the space H q,λ (Σ, σ) and satisfies m0 H q, λ (Σ,σ) ≤ C for some constant C ∈ (0, ∞) independent of m, x0, R. In concert with (6.1.17), this proves the claim made in (6.2.59). Finally, the claim in (6.2.60) is a consequence of (6.1.17), (6.1.22), (6.2.59), and the fact that the product between a bounded function and a H q,λ -dome on Σ is a  multiple of a H q,λ -dome on Σ. The topic addressed next pertains to the pre-duals of Morrey spaces. To set the stage, given an integrability exponent q ∈ (1, ∞) and a parameter λ ∈ (0, n − 1), a function b ∈ L q (Σ, σ) is said to be a B q,λ -block on Σ (or, simply, a block) provided there exist some point xo ∈ Σ and some radius R ∈ 0, 2 diam(Σ) such that

supp b ⊆ B(xo, R) ∩ Σ and b L q (Σ,σ) ≤ R

1 λ q −1

.

(6.2.63)

We then define the block space 

 B q,λ (Σ, σ) := f ∈ Lipc (Σ) : there exist a sequence {λ j } j ∈N ∈  1 (N) and a family {b j } j ∈N of B q,λ -blocks on Σ such that ∞ 

 f = λ j b j with convergence in Lipc (Σ) , j=1

(6.2.64)

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

327

and for each f ∈ B q,λ (Σ, σ) define  f  B q, λ (Σ,σ) := inf

∞  j=1

|λ j | : f =

∞ 



 λ j b j in Lipc (Σ) with

(6.2.65)

j=1

 {λ j } j ∈N ∈  1 (N) and each b j a B q,λ -block on Σ .

In particular, it is clear from (6.2.65) that each B q,λ -block b on Σ belongs to

(6.2.66)

B q,λ (Σ, σ) and b B q, λ (Σ,σ) ≤ 1, and that if f ∈ B q,λ (Σ, σ) is expanded as f =

∞  j=1



 λ j b j in Lipc (Σ) for some

sequence {λ j } j ∈N ∈  1 (N) and with each b j a B q,λ -block on Σ, then ∞  λ j b j also converges to f in B q,λ (Σ, σ). the series

(6.2.67)

j=1

Additionally, it is of significance to note that (6.2.66) implies that if f ∈ L q (Σ, σ) has supp f ⊆ Δ(x0 , R) for some point x0 ∈ Σ and some radius R ∈ 0, 2 diam(Σ) , then f ∈ B q,λ (Σ, σ) and 1

λ −1 the estimate  f  B q, λ (Σ,σ) ≤ R q  f  L q (Σ,σ) holds.

(6.2.68)

As a consequence of (6.2.67) and (6.2.68) we have that q

Lcomp (Σ, σ) → B q,λ (Σ, σ) continuously and densely

(6.2.69)

which, thanks to [133, (3.7.22)] and (6.2.66), further self-improves to3: Lipc (Σ) is a dense linear subspace of B q,λ (Σ, σ).

(6.2.70)

Also, as in the case of the space introduced in (6.1.16), we see that   B q,λ (Σ, σ),  ·  B q, λ (Σ,σ) is a separable Banach space, and B q,λ (Σ, σ) → L r (Σ, σ) with r :=

q(n−1) n−1+λ(q−1)

∈ (1, q)

(6.2.71)

is a well-defined, continuous embedding, with dense range. In particular, (6.2.71) and [133, (7.7.106)] imply that  B q,λ (Σ, σ) → L 1 Σ,

σ(x)  continuously. 1 + |x| n−1

(6.2.72)

3 another proof of (6.2.70) may be given based on the Hahn-Banach theorem and Proposition 6.2.8

328

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

In addition, if f ∈ B q,λ (Σ, σ) is expressed as f =

∞  j=1



 λ j b j in Lipc (Σ) for a

sequence {λ j } j ∈N ∈  1 (N) and with each b j a B q,λ -block on Σ, then ∞  q(n−1) λ j b j converges to f L r (Σ, σ) if we take r := n−1+λ(q−1) ,

(6.2.73)

j=1

and the operator of pointwise multiplication by a given function h ∈ L ∞ (Σ, σ) is a bounded mapping from B q,λ (Σ, σ) into itself, with operator norm ≤ h L ∞ (Σ,σ) ,

(6.2.74)

since such a mapping does not distort dull atoms by more than a fixed multiplicative constant, which is ≤ h L ∞ (Σ,σ) . Note that the latter property further implies that if f , g : Σ −→ C are two σ-measurable functions such that |g| ≤ | f | at σ-a.e. point on Σ and f ∈ B q,λ (Σ, σ), then g is in B q,λ (Σ, σ) and one has g B q, λ (Σ,σ) ≤  f  B q, λ (Σ,σ) , since

 g = h f with h :=

g/ f if f  0, 0

if f = 0,

satisfying h L ∞ (Σ,σ) ≤ 1.

(6.2.75)

(6.2.76)

It is instructive to compare the space B q,λ (Σ, σ) introduced in (6.2.64) with H q,λ (Σ, σ) from (6.1.16) (the latter being the pre-dual of a Morrey-Campanato space; cf. (6.1.25)). In this regard, we wish to remark that H H

q,λ (Σ, σ)

q,λ (Σ, σ)

⊆ B q,λ (Σ, σ), and actually

= B q,λ (Σ, σ) whenever Σ is bounded.

(6.2.77)

Indeed, the inclusion in the first line of (6.2.77) is clear from definitions. Also, if Σ is bounded then any B q,λ -block on Σ is a H q,λ,θ -dome on Σ (in the sense of (6.1.33)) for any θ > (n − 1)(q − 1), so Lemma 6.2.7 now provides the opposite inclusion B q,λ (Σ, σ) ⊆ H q,λ (Σ, σ) from which the equality in the second line of (6.2.77) follows. This being said, our primary interest in the space (6.2.64) stems from the fact that this turns out to be the pre-dual of a Morrey space. Specifically, we have the following result. Proposition 6.2.8 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Fix two integrability exponents p, q ∈ (1, ∞) satisfying 1/p + 1/q = 1, along with a parameter λ ∈ (0, n − 1). Then there exists a constant C ∈ (0, ∞) with the property that

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

329

∫ Σ

| f ||g| dσ ≤ C f  M p, λ (Σ,σ) g B q, λ (Σ,σ)

(6.2.78)

for all f ∈ M p,λ (Σ, σ) and g ∈ B q,λ (Σ, σ). Moreover, the mapping

∗ M p,λ (Σ, σ)  f −→ Λ f ∈ B q,λ (Σ, σ) given by ∫ Λ f (g) := f g dσ for each g ∈ B q,λ (Σ, σ)

(6.2.79)

Σ

is a well-defined, linear, bounded isomorphism, with bounded inverse. Simply put, the integral paring yields the quantitative identification ∗  (6.2.80) B q,λ (Σ, σ) = M p,λ (Σ, σ).

 As a corollary, whenever Λ ∈ Lipc (Σ) is a distribution with the property that there exists some constant C ∈ (0, ∞) such that    Λ, φ  ≤ Cφ B q, λ (Σ,σ) for every φ ∈ Lip (Σ), (6.2.81) c there exists a unique function f ∈ M p,λ (Σ, σ) such that ∫ f φ dσ for every φ ∈ Lipc (Σ). Λ, φ = Σ

(6.2.82)

In addition, with C denoting the constant appearing in (6.2.81), the function f satisfies  f  M p, λ (Σ,σ) ≤ c C, where c ∈ (0, ∞) is a fixed number which depends only the ambient. It is useful to note that, in the context of Proposition 6.2.8, for each function f ∈ M p,λ (Σ, σ) we may write  f  M p, λ (Σ,σ) ≈ Λ f (B q, λ (Σ,σ))∗ =

=

sup

g ∈B q, λ (Σ,σ) g B q, λ (Σ, σ) ≤1

sup

g ∈B q, λ (Σ,σ) g B q, λ (Σ, σ) ≤1

∫    f g dσ ,  Σ

|Λ f (g)|

(6.2.83)

thanks to the fact that (6.2.79) is an isomorphism. A closely related result is the fact that for each σ-measurable function f : Σ → R we have ∫ f ∈ M p,λ (Σ, σ) if and only if sup | f ||g| dσ < +∞, (6.2.84) g ∈B q, λ (Σ,σ) g B q, λ (Σ, σ) ≤1

Σ

330

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

∫

and  f  M p, λ (Σ,σ) ≈

sup

| f ||g| dσ.

Σ

g ∈B q, λ (Σ,σ) g B q, λ (Σ, σ) ≤1

(6.2.85)

Indeed, the right-pointing implication in (6.2.84) is seen from (6.2.78), while the left-pointing implication can be proved using formula (6.2.80). To elaborate, assume f : Σ → R is a σ-measurable function for which the supremum in (6.2.84) is finite. 1 Thanks to (6.2.69), this ensures ∫ that f ∈ Lloc (Σ, σ). Next, observe that the assignment B q,λ (Σ, σ)  g → Λ(g) := Σ f g dσ is a continuous linear functional on B q,λ (Σ, σ),  ∗ hence Λ ∈ B q,λ (Σ, σ) = M p,λ (Σ, σ) by (6.2.80). The latter identification implies ∫ that there exists  f ∈ M p,λ (Σ, σ) such that Λ(g) = Σ  f g dσ for each g ∈ B q,λ (Σ, σ). ∫ ∫ Hence, Σ f g dσ = Σ  f g dσ for each g ∈ B q,λ (Σ, σ). In light of (6.2.70) and [133, Proposition 3.7.2], this ultimately permits us to conclude that f =  f ∈ M p,λ (Σ, σ), as wanted. Parenthetically, we wish to remark that the left-pointing implication in (6.2.84) may also be justified by pairing the given function f with individual blocks. Specifically, if C f ∈ [0, ∞) denotes the supremum appearing in (6.2.84) then, having fixed some point xo ∈ ∫Σ and some radius R ∈ 0, 2 diam(Σ) , for each function b as in (6.2.63) we have Σ | f ||b| dσ ≤ C f . By Riesz’s Representation Theorem, this places 

f Δ(xo,R) in L p Δ(xo, R), σ with ∫

| f | dσ p

Δ(x o ,R)

 p1



≤ Cf ·

1 λ 1− q R

In turn, from this and (6.2.2) we conclude that  n−1−λ  ⨏ sup R p  f  M p, λ (Σ,σ) = x o ∈Σ and 0 0. Take Wn,λ := Cθ and CΣ, p := C 1/p with C = C(Σ, p) ∈ (0, ∞) as in (6.2.132). For ( f , g) ∈ F arbitrary we may then estimate ∫ ∫ | f | p dσ = | f | p 1B(x,R)∩Σ dσ Σ

B(x,R)∩Σ

∫ ≤

Σ

∫ | f | p w dσ ≤ (Cw ) p

Σ

|g| p w dσ

≤ (CΣ, p,λ ) p · Rλ g M p, λ (Σ,σ) . p

(6.2.133)

340

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

Above, the first inequality uses the top line in (6.2.129), the second inequality is based on (6.2.120) and the bottom line in (6.2.129), while the final

inequality is provided by (6.2.132) and (6.2.121). Dividing by σ B(x, R) ∩ Σ , raising to the n−1−λ

power 1/p, then multiplying by R p and, finally, taking the supremum over all x ∈ Σ and R ∈ 0, 2 diam(Σ) then yields (6.2.122).  It turns out that the Hardy-Littlewood maximal operator is bounded on Morrey spaces defined on Ahlfors regular sets. The full-space Euclidean version of the result below first appeared in [27]. Corollary 6.2.13 Suppose Σ ⊆ Rn (where n ∈ N with n ≥ 2) is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Then, given any exponent p ∈ (1, ∞) and any number λ ∈ (0, n − 1), the Hardy-Littlewood maximal operator on Σ induces a well-defined, sub-linear and bounded mapping M Σ : M p,λ (Σ, σ) −→ M p,λ (Σ, σ).

(6.2.134)

As a consequence of this, (6.2.14), and [133, (7.6.18)], one also has that M Σ : M˚ p,λ (Σ, σ) −→ M˚ p,λ (Σ, σ)

(6.2.135)

is a well-defined, sub-linear and bounded mapping. Proof This is a direct consequence of Proposition 6.2.12 and items (1), (5) in [133, Lemma 7.7.1]. It is also possible to provide an alternative proof of (6.2.134) based on the boundedness result from Corollary 6.2.11, the duality result from Proposition 6.2.8, and Fefferman-Stein’s maximal inequality in the version recorded in [133, Proposition 7.6.7], which presently gives that for each s ∈ (1, ∞) ∫ C = C(Σ, s) ∈ (0, ∞) with ∫ there exists a constant

s

M Σ f g dσ ≤ C f s M Σ g dσ for any two the property that Σ

Σ

(6.2.136)

non-negative σ-measurable functions f , g defined on Σ. Specifically, pick some number θ ∈ (1, p) and denote by (p/θ) ∈ (1, ∞) the Hölder conjugate exponent of p/θ ∈ (1, ∞). Then, given any function f ∈ M p,λ (Σ, σ), we may write:

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

  M Σ f 

341

M p, λ (Σ,σ)

 θ 1/θ  = M Σ f   p/θ, λ M

≤C

(Σ,σ)

sup 

g ∈H (p/θ ) , λ (Σ,σ) g H (p/θ ), λ (Σ, σ) ≤1

≤C·

≤C·

≤C·

≤C·

 θ 1/θ  = M Σ f   (p/θ ), λ (H

∫  1/θ θ    M Σ f  g dσ  

sup 

g ∈H (p/θ ) , λ (Σ,σ) g H (p/θ ), λ (Σ, σ) ≤1

Σ

∫ Σ

∫ sup 

g ∈H (p/θ ) , λ (Σ,σ) g H (p/θ ), λ (Σ, σ) ≤1

sup 

g ∈H (p/θ ) , λ (Σ,σ) g H (p/θ ), λ (Σ, σ) ≤1

sup 

g ∈H (p/θ ) , λ (Σ,σ) g H (p/θ ), λ (Σ, σ) ≤1

(Σ,σ))∗

Σ

 1/θ

θ M Σ | f | |g| dσ



 1/θ | f | θ M Σ g dσ

  | f |θ 

M p/θ, λ (Σ,σ)

  M Σ g 

 1/θ

  | f |θ 

M p/θ, λ (Σ,σ)

gH (p/θ ), λ (Σ,σ)

H

1/θ  ≤ C  | f | θ  M p/θ, λ (Σ,σ) = C f  M p, λ (Σ,σ),

(p/θ ), λ (Σ,σ)

 1/θ

(6.2.137)

thanks to (6.2.4), Proposition 6.2.8, (6.2.136), and Corollary 6.2.11. This establishes the boundedness of the Hardy-Littlewood maximal operator in the context of (6.2.134).  Corollary 6.2.13 is a key ingredient the proof of the following version of the Fractional Integration Theorem in the context of Morrey spaces defined on Ahlfors regular sets, generalizing the classical result in [1]. Proposition 6.2.14 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Fix p ∈ (1, ∞) along with λ ∈ (0, n − 1) and suppose 0

n−1 λ and b > . p p

(6.2.142)

Next, fix a function f ∈ M p,λ (Σ, σ) along with a point x ∈ Σ. For some r > 0 to be specified later, use [133, (7.8.20)] to estimate ∫ | f (y)| dσ(y) ≤ Cr α (M Σ f )(x), (6.2.143) |x − y| n−1−α B(x,r)∩Σ for some constant C ∈ (0, ∞) which is independent of x, r, f . Also, based on (6.2.142), Hölder’s inequality, [133, (7.2.5)], and (6.2.2) we may write ∫ | f (y)| dσ(y) |x − y| n−1−α Σ\B(x,r) ∫  1/p  ∫ | f (y)| p dσ(y)  1/p ≤ dσ(y) ap bp Σ\B(x,r) |x − y| Σ\B(x,r) |x − y| 

≤ Cr (n−1−bp )/p



∞ ∫  j=0



≤ Cr (n−1−bp )/p



∞ 

Σ∩[B(x,2 j+1 r)\B(x,2 j r)]

(2 j r)−ap

j=0 

≤ Cr (n−1−bp )/p



∞  j=0

 1/p | f (y)| p dσ(y) |x − y| ap

∫ Σ∩B(x,2 j+1 r)

| f (y)| p dσ(y)

(2 j r)−ap (2 j r)λ  f  M p, λ (Σ,σ)

= Cr (λ−n−1)/p+α  f  M p, λ (Σ,σ) .

p

 1/p

 1/p

(6.2.144)

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

343

Combining (6.2.143) with (6.2.144) then yields, after solving an elementary optimization problem,      IΣ,α f (x) ≤ C · inf r α (M Σ f )(x) + r (λ−n−1)/p+α  f  M p, λ (Σ,σ) r >0

αp   1− n−1−λ αp n−1−λ = C f  M p, λ (Σ,σ) (M Σ f )(x)



1− pp∗

= C f  M p, λ (Σ,σ) (M Σ f )(x)

 pp

∗

,

∀x ∈ Σ,

(6.2.145)

where the final equality taken into account the formula

for p∗ given in (6.2.138). For each point x ∈ Σ and each scale R ∈ 0, 2 diam(Σ) we may then use (6.2.145) and Corollary 6.2.13 to estimate ⨏ ⨏  1 p  

p  p1∗  IΣ,α f  p∗ dσ p∗ ≤ C f  1−p,p∗λ M f dσ Σ M (Σ,σ) Σ∩B(x,R)

Σ∩B(x,R)

 λ−n−1  pp 1− p ∗ ≤ C f  M p,p∗λ (Σ,σ) R p  f  M p, λ (Σ,σ) ≤ C f  M p, λ (Σ,σ) R

λ−n−1 p∗

.

(6.2.146)

λ−n−1

Dividing by R p∗ and taking the supremum over all x ∈ Σ and R ∈ 0, 2 diam(Σ) then yields (6.2.141), on account of (6.2.2). 

Corollary 6.2.13 may now be thought as the limiting case (corresponding to α = 0) of the result pertaining to the action of the fractional Hardy-Littlewood maximal operator of order α on Morrey spaces, presented below. Corollary 6.2.15 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Suppose n−1−λ n−1−λ , s0

0 0 implies    f |Q  p ≤ (Q)−λ/p  f  L p (Σ,σ) −→ 0 as (Q) → ∞, (6.2.181) L (Q,(Q)−λ σ) while having λ < n − 1 forces    f |Q 

L p (Q,(Q)−λ σ)

≈ (Q)(n−1−λ)/p

⨏

| f | p dσ

 1/p

Q

≤ (Q)(n−1−λ)/p  f  L ∞ (Σ,σ) −→ 0 as (Q) → 0.

(6.2.182)

In addition, the fact that f has compact support ensures that   for ε ∈ (0, 1) fixed,  f |Q  L p (Q,(Q)−λ σ) → 0 as dist(x0, Q) → ∞ within (6.2.183) the class of cubes Q ∈ D(Σ) satisfying ε < (Q) < ε−1. In turn, from (6.2.181)-(6.2.183) and (6.2.170) (also bearing in mind (6.2.168)) we

conclude that, as wanted, Φ( f ) ∈ c0 A Σ . This finishes the proof of (6.2.177). Having established (6.2.177), the fact that Φ is continuous implies, on account of (6.2.20), that actually the quasi-isometry Φ, defined in (6.2.176),

maps the space M˚ p,λ (Σ, σ) into c0 A Σ . As a consequence,

Φ M˚ p,λ (Σ, σ) is a closed subspace of c0 A Σ and

Φ : M˚ p,λ (Σ, σ) → Φ M˚ p,λ (Σ, σ) is an isomorphism.

(6.2.184)

(6.2.185)

Henceforth we agree to denote by Φ−1 the inverse of the above isomorphism. Hence,

Φ−1 : Φ M˚ p,λ (Σ, σ) −→ M˚ p,λ (Σ, σ) (6.2.186) is itself an isomorphism with the property that

Φ−1 Φ( f ) = f for each f ∈ M˚ p,λ (Σ, σ).

(6.2.187)

∗ Step 4: Proof of the Duality Formula. Fix an arbitrary Λ ∈ M˚ p,λ (Σ, σ) . Then ˚ p,λ Λ ◦ Φ−1 is a linear continuous functional on Φ M (Σ, σ) , which is a closed subspace of c0 A Σ . The Hahn-Banach Theorem then guarantees the existence of

 on c0 A Σ satisfying some linear continuous functional Λ 

  Λ = Λ ◦ Φ−1 on Φ M˚ p,λ (Σ, σ) and ˚ p, λ (Σ,σ)) Φ( M (6.2.188) −1   Λc0 (AΣ )∗ = Λ ◦ Φ (Φ( M˚ p, λ (Σ,σ)))∗ ≈ Λ( M˚ p, λ (Σ,σ))∗ .

350

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .





 belongs to the dual of c0 A Σ , which has been identified as 1 A ∗ Hence, Λ Σ ∗

in (6.2.167). Specifically, there exists some G = {gQ }Q ∈D(Σ) ∈ 1 A Σ with the property that  ∫  Λ(F) = (F, G) = fQ gQ (Q)−λ dσ Q Q ∈D(Σ) (6.2.189)

for each F = { fQ }Q ∈D(Σ) ∈ c0 A Σ , and the second line in (6.2.188) further gives G1 (AΣ∗ ) ≈ Λ( M˚ p, λ (Σ,σ))∗ . In particular, for each f in M˚ p,λ (Σ, σ), the fact that F :=



belongs to Φ M˚ p,λ (Σ, σ) ⊆ c0 A Σ permits us to write

(6.2.190)  f Q Q ∈D(Σ) = Φ( f )



 Λ( f ) = Λ Φ−1 (Φ( f )) = (Λ ◦ Φ−1 ) Φ( f ) = Λ(F)  ∫  ∫ 

f Q gQ (Q)−λ dσ = f gQ (Q)−λ dσ, = Q ∈D(Σ)

Q

Q ∈D(Σ)

(6.2.191)

Q

thanks to (6.2.187), (6.2.188), and (6.2.189). To proceed, for each dyadic denote by  gQ the extension by zero of cube Q ∈ D(Σ)

gQ belongs the function gQ ∈ A∗Q = L q Q, (Q)−λ σ to the entire set Σ. Then each  to L q (Σ, σ) and        gQ  L q (Σ,σ) = gQ  L q (Q,σ) = (Q)λ/q gQ  L q (Q,(Q)−λ σ) . (6.2.192) In concert with (6.2.190), this implies       gQ  q gQ  L q (Σ,σ) = (Q)−λ/q  L (Q,(Q)−λ σ) Q ∈D(Σ)

Q ∈D(Σ)

= G1 (AΣ∗ ) ≈ Λ( M˚ p, λ (Σ,σ))∗ . Consequently, if r :=

q(n−1) n−1+λ(q−1)

(6.2.193)

∈ (1, q) then

       gQ (Q)−λ  L r (Σ,σ) ≤ C · gQ  L q (Σ,σ) (Q)−λ+(n−1)(1/r−1/q) 

Q ∈D(Σ)

Q ∈D(Σ)

=C·

 Q ∈D(Σ)

  (Q)−λ/q  gQ  L q (Σ,σ)

≤ CΛ( M˚ p, λ (Σ,σ))∗ < +∞,

(6.2.194)

thanks to Hölder’s inequality (together with the nature of the support of  gQ , property [133, (7.5.9)] satisfied by dyadic cubes, and the Ahlfors regularity of Σ), the fact that (n − 1)(1/r − 1/q) = λ(1 − 1/q), and (6.2.193). This proves that

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets



the series g :=

351

 gQ (Q)−λ converges absolutely in L r (Σ, σ),

(6.2.195)

Q ∈D(Σ)



 hence also pointwise σ-a.e. on Σ, as well as in Lipc (Σ) . If for each Q ∈ D(Σ) we now define   λQ := (Q)−λ/q  gQ  L q (Σ,σ) ∈ [0, ∞) (6.2.196) together with  bQ :=

  gQ (Q)−λ λQ if λQ  0,

(6.2.197)

if λQ = 0,

0

then we may express g, originally defined in (6.2.195), as 

 g= λQ bQ in L r (Σ, σ), hence also in Lipc (Σ) ,

(6.2.198)

Q ∈D(Σ)

while (6.2.193) tells that  Q ∈D(Σ)

|λQ | ≤ CΛ( M˚ p, λ (Σ,σ))∗ < +∞.

(6.2.199)

In addition, the bQ ’s are designed such that for each Q ∈ D(Σ) we have

supp bQ ⊆ Q and bQ  L q (Σ,σ) ≤ (Q)

1 λ q −1

.

(6.2.200)

From (6.2.198)-(6.2.200) and (6.2.63)-(6.2.65) we conclude (by also bearing in mind [133, (7.5.9)]) that g ∈ B q,λ (Σ, σ) and g B q, λ (Σ,σ) ≤ CΛ( M˚ p, λ (Σ,σ))∗ .

(6.2.201)

Pressing on, given any f ∈ L s (Σ, σ) with s := p(n−1) n−1−λ , we may write ∫     ∫ Λ( f ) = f gQ (Q)−λ dσ = f  gQ (Q)−λ dσ Q ∈D(Σ)

Q

∫ =

Σ

f g dσ = Λg ( f ),

Σ

Q ∈D(Σ)

(6.2.202)

where the first equality is implied by (6.2.191) and (6.2.14), the second equality is a consequence of (6.2.195) and the observation that 1/r + 1/s = 1, the third equality uses (6.2.195), while the last equality comes from (6.2.154) (again, bearing in mind (6.2.14)). Since L s (Σ, σ) is a dense subspace of M˚ p,λ (Σ, σ) (cf. (6.2.15)), and since both Λ and Λg are continuous functions on M˚ p,λ (Σ, σ), we deduce from ∗ (6.2.202) that actually Λg = Λ as functionals in M˚ p,λ (Σ, σ) . Given that Λ has

352

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .



∗ been arbitrarily selected in M˚ p,λ (Σ, σ) to begin with, this ultimately proves that the mapping defined in (6.2.154) is surjective. At this point, we may conclude that the mapping defined in (6.2.154) is actually well-defined, linear, bounded, bijective, and the estimate in (6.2.201) also shows that its inverse is also bounded. To complete the proof of Proposition 6.2.16 there remains to address the very last claim in the statement. In this regard, since Lipc (Σ) is dense in the space M˚ p,λ (Σ, σ) (cf. (6.2.20)) and since the mapping Lipc (Σ)  φ → Λ, φ ∈ C is linear and bounded with respect to the norm in M˚ p,λ (Σ, σ) (inherited from M p,λ (Σ,

∗ that σ)) it follows said mapping extends (by density) to a unique functional Θ ∈ M˚ p,λ (Σ, σ) whose norm is ≤ c C, where the constant C is as in (6.2.156) and where c ∈ (0, ∞) is a fixed number which depends only the ambient. In view of (6.2.155), this proves that q,λ there exists ∫ some function g in the space B (Σ, σ) such that g B q, λ (Σ,σ) ≤ c C and Θ( f ) = Σ f g dσ for each f ∈ M p,λ (Σ, σ). In concert with the fact that Θ(φ) = Λ, φ for each φ ∈ Lipc (Σ), this establishes (6.2.157). Finally, the uniqueness of g is a consequence of [133, Corollary 3.7.3].  One of the main reasons we have altered the standard definition of a Banach function space (as given in, e.g., [14, pp. 2-3]) is to be able to allow Morrey and block spaces to be part of this theory. We deliver on this promise in the proposition below. Proposition 6.2.17 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Pick two integrability exponents p, q ∈ (1, ∞) with 1/p+1/q = 1, along with an arbitrary parameter λ ∈ (0, n−1). Then both M p,λ (Σ, σ) and B q,λ (Σ, σ) are Generalized Banach Function Spaces (cf. Definition 5.1.4) and theirs associated spaces (in the sense of Definition 5.1.11) are, respectively, given by     (6.2.203) M p,λ (Σ, σ) = B q,λ (Σ, σ) and B q,λ (Σ, σ) = M p,λ (Σ, σ). Simply put, M p,λ (Σ, σ) and B q,λ (Σ, σ) are mutually associated with one another in the sense of Definition 5.1.11. Once the Morrey and block spaces space have been identified as Generalized Banach Function Spaces, which are actually mutually associated with one another, the abstract embedding results from Proposition 5.2.7 apply (thanks to Corollary 6.2.11 and Corollary 6.2.13) and yield embeddings like the ones in Proposition 5.2.7 for Morrey and block spaces. We shall revisit this point of view later, in Proposition 6.2.22. Proof of Proposition 6.2.17 First, the fact that the Morrey space M p,λ (Σ, σ) is a Generalized Banach Function Space may be checked from Definition 5.1.4, presently used with X := Σ, μ := σ, and with the function norm ρ : M+ (Σ, σ) → [0, ∞] given by

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

 ρ( f ) :=

sup

x ∈Σ and 0 0 with the property that ∫ | f g| dσ < ε for each σ-measurable set A ⊆ Σ with σ(A) < δ. (6.2.214) A

Also, Egoroff’s Theorem (cf., e.g., [55, Theorem 2.33, p. 62]) guarantees the existence of some σ-measurable set E ⊆ Σ with the property that σ(E) < min{ε, δ} and lim g j − g L ∞ (supp f \E,σ) = 0. j→∞

For each j ∈ N write ∫ ∫ (g − g j ) f dσ = Σ

supp f \E

∫ (g − g j ) f dσ −

E

(6.2.215)

∫ g j f dσ +

g f dσ

(6.2.216)

E

and note that, thanks to the second property in (6.2.215), ∫    lim  (g−g j ) f dσ  ≤  f  L 1 (Σ,σ) lim g j −g L ∞ (supp f \E,σ) = 0, (6.2.217) j→∞

j→∞

supp f \E

whereas (6.2.214) and the first property in (6.2.215) ensure that ∫    g f dσ  < ε. 

(6.2.218)

E

Finally, consider the middle term in the right-hand side of (6.2.216). With the exponent p ∈ (1, ∞) satisfying 1/p + 1/q = 1, we rely on (6.2.78) to estimate this term as follows: ∫  ∫      g j f dσ  ≤ |g j || f |1E dσ ≤ C sup g j  B q, λ (Σ,σ)  | f |1E  M p, λ (Σ,σ) .  j ∈N

Σ

E

(6.2.219) On the other hand, combining (6.2.5) with (6.2.9) and (6.2.215) yields    | f |1E  p, λ ≤  f  L ∞ (Σ,σ) · ε (n−1−λ)/[p(n−1)] . M (Σ,σ)

(6.2.220)

Collectively, (6.2.216), (6.2.217), (6.2.218), (6.2.219), and (6.2.220) then prove, on account of the arbitrariness of ε > 0 (and the fact that λ ∈ (0, n − 1)), that ∫ ∫ lim f g j dσ = f g dσ for each f ∈ Lipc (Σ). (6.2.221) j→∞

Σ

Σ

Next, the hypothesis of functions {g j } j ∈N is bounded in the that the sequence

∗ space B q,λ (Σ, σ) = M˚ p,λ (Σ, σ) (cf. (6.2.155) for the latter equality) together with the Sequential Banach-Alaoglu Theorem (as recalled in [133, (3.6.22)], whose

356

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

applicability in the present context is ensured by (6.2.16)), imply the existence of a sub-sequence {g jk }k ∈N of {g j } j ∈N which is weak-∗ convergent to some function h ∈ B q,λ (Σ, σ), i.e., ∫ ∫ lim f g jk dσ = f h dσ for each f ∈ M˚ p,λ (Σ, σ). (6.2.222) k→∞

Σ

Σ

In turn, from (6.2.221), (6.2.222), and (6.2.20), we see that ∫ ∫ f g dσ = f h dσ for each f ∈ Lipc (Σ). Σ

Σ

(6.2.223)

With this in hand, [133, Corollary 3.7.3] applies and gives that g = h at σ-a.e. point on Σ. In particular, g ∈ B q,λ (Σ, σ), as claimed in (6.2.211). Granted this membership, (6.2.221) self-improves to ∫ ∫ lim f g j dσ = f g dσ for each f ∈ M˚ p,λ (Σ, σ). (6.2.224) j→∞

Σ

Σ

Indeed, given any f ∈ M˚ p,λ (Σ, σ), with φ ∈ Lipc (Σ) arbitrary we may write ∫ ∫ ∫ f g j dσ = ( f − φ)g j dσ + φg j dσ for each j ∈ N. (6.2.225) Σ

Σ

Σ

Then (6.2.221) implies that ∫ ∫ ∫ ∫ lim φg j dσ = φg dσ = f g dσ − ( f − φ)g dσ,

(6.2.226)

and (6.2.78) permits us to estimate  ∫    ( f − φ)g dσ  ≤ C f − φ M p, λ (Σ,σ) g B q, λ (Σ,σ),

(6.2.227)

as well as ∫       ( f − φ)g j dσ  ≤ C f − φ M p, λ (Σ,σ) sup g j  B q, λ (Σ,σ) .

(6.2.228)

j→∞

Σ

Σ

Σ

Σ

Σ

Σ

j ∈N

The fact that φ may be selected in Lipc (Σ) so that  f − φ M p, λ (Σ,σ) becomes arbitrary small (cf. (6.2.20)) then ultimately leads to the conclusion that (6.2.224) holds. This finishes the proof of (6.2.211).  We continue by presenting a companion result to Proposition 6.2.14, detailing on the mapping properties of the fractional integral operator acting on the pre-duals of Morrey spaces. Proposition 6.2.20 Suppose Σ ⊆ Rn (where n ∈ N with n ≥ 2) is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Select q ∈ (1, ∞) along with λ ∈ (0, n − 1) and assume

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

0 n − 1 (which ultimately is implied by the first condition in (6.2.229)). This proves (6.2.230). As regards the claims pertaining to the fractional integral operator in (6.2.231), define p := (q∗ ), the Hölder conjugate exponent of q∗ , and observe that 1

−1 α p − n−1−λ

=



1 (q∗ )

α − n−1−λ

−1

−1 1 −1 α = 1− q1∗ − n−1−λ = 1− q = q , (6.2.233)

the Hölder conjugate exponent of q. The next step is to fix an arbitrary function g ∈ B q,λ (Σ, σ). Then (6.2.139) and [133, Proposition 4.1.4] ensure that

 1 IΣ,α g ∈ Lloc (Σ, σ) ⊆ Lipc (Σ) (6.2.234) and for each φ ∈ Lipc (Σ) we may write ∫  ∫      IΣ,α g, φ  =  (IΣ,α g)φ dσ  =  g(IΣ,α φ) dσ  Σ

Σ

≤ Cg B q, λ (Σ,σ) IΣ,α φ M q, λ (Σ,σ) ≤ Cg B q, λ (Σ,σ) φ M p, λ (Σ,σ) .

(6.2.235)

Above, the first equality is a consequence of [133, Proposition 4.1.4], the second equality is implied by (6.2.230) and [133, (7.8.6)], the first inequality comes from (6.2.78), and the final inequality follows from (6.2.141), bearing (6.2.233) in mind. Having established (6.2.235), the very last claim in Proposition 6.2.16 (cf. (6.2.156) (6.2.157)) then guarantees the existence of a function h ∈ B p ,λ (Σ, σ) satisfying h B q∗, λ (Σ,σ) = h B p, λ (Σ,σ) ≤ Cg B q, λ (Σ,σ),

(6.2.236)

358

6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals . . .

(where the first equality uses the fact that p = q∗ since p = (q∗ ) to begin with), and with the property that ∫ IΣ,α g, φ = h φ dσ for every φ ∈ Lipc (Σ). (6.2.237) Σ

In turn, from (6.2.237), (6.2.234), and [133, Proposition 4.1.4] we deduce that ∫ ∫ (IΣ,α g)φ dσ = h φ dσ for every φ ∈ Lipc (Σ) (6.2.238) Σ

Σ

which, in light of [133, Corollary 3.7.3], ultimately forces IΣ,α g = h at σ-a.e. point on Σ. From this and (6.2.236) we then conclude that IΣ,α g ∈ B q∗,λ (Σ, σ) and IΣ,α g B q∗, λ (Σ,σ) ≤ Cg B q, λ (Σ,σ) .

(6.2.239) 

With this in hand, all desired conclusions follow.

Proposition 6.2.16 also permits us to establish useful embeddings of weighted Lebesgue spaces into the pre-duals of Morrey spaces, of the sort discussed below. Proposition 6.2.21 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, fix q ∈ (1, ∞) along with λ ∈ (0, n − 1) and a > λ. Then one has the continuous and dense embedding   L q Σ, (1 + |x|)a(q−1) σ(x) → B q,λ (Σ, σ). (6.2.240)

1 (Σ, σ) and Proof Pick an arbitrary function g ∈ L q Σ, (1 + |x|)a(q−1) σ(x) → Lloc

 consider the distribution Λg ∈ Lipc (Σ) given by ∫   Λg, φ = gφ dσ for every φ ∈ Lipc (Σ). (6.2.241) Σ

Note that if p ∈ (1, ∞) is such that 1/p + 1/q = 1, then for each φ ∈ Lipc (Σ) we may use Hölder’s inequality and (6.2.25) to estimate ∫    g(x)|(1 + |x|)a(1−1/q) |φ(x)| dσ(x)  Λg, φ  ≤ (1 + |x|)a/p Σ ≤ Cg L q (Σ,(1+ |x |) a(q−1) σ(x)) φ L p Σ,

σ(x) 1+| x | a

≤ Cg L q (Σ,(1+ |x |) a(q−1) σ(x)) φ M p, λ (Σ,σ),

(6.2.242)

for some constant C ∈ (0, ∞) independent of g and φ. With this in hand, the last claim in Proposition 6.2.16 guarantees that there exists a (unique) function h ∈ B q,λ (Σ, σ) such that ∫   h φ dσ for every φ ∈ Lipc (Σ) (6.2.243) Λg, φ = Σ

6.2 Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets

359

and h B q, λ (Σ,σ) ≤ Cg L q (Σ,(1+ |x |) a(q−1) σ(x)) .

(6.2.244)

There remains to observe that, collectively, (6.2.241), (6.2.243), and [133, (3.7.23)] force g = h at σ-a.e. point in in Σ. With this in hand, the fact that the embedding in (6.2.240) is well-defined and continuous follows from (6.2.244). The fact that said embedding has also dense range is then seen from (6.2.70).  For example, (6.2.240) implies that, in the context of Proposition 6.2.21, if N > λ(q−1)+n−1 and f N (x) := (1 + |x|)−N for all x ∈ Σ, q then the function f N belongs to the space B q,λ (Σ, σ).

(6.2.245)

Returning to Proposition 6.2.17, we shall use it to prove the following useful embedding result of Morrey spaces into Muckenhoupt weighted Lebesgue spaces. Proposition 6.2.22 Let Σ ⊆ Rn (where n ∈ N with n ≥ 2) be a closed Ahlfors regular set, and abbreviate σ := H n−1 Σ. Then for each p0 ∈ (1, ∞) one has $ $ M p,λ (Σ, σ) ⊆ L p0 (Σ, w). (6.2.246) w ∈ A p0 (Σ,σ)

1 0. One may then check without difficulty that fR := f θ R ∈ Lipc (Σ) for each R > 0 and  f − fR  GΣ (x0,1,βo,γo ) → 0 as R → ∞. This justifies (7.1.7). Finally, observe that

Σ compact =⇒

as

⎧ 1,1 β,γ β ⎪ ⎨ G (Σ) = Lip (Σ) and G (Σ) = C (Σ); ⎪ · ⎪ ⎪ hence G β,γ (Σ) = Lip (Σ) C β (Σ) for 0 < β, γ < 1. 0 ⎩

(7.1.8)

In the same setting as above, the space of test functions with mean zero is defined ∫ 

G˚ β,γ (Σ) := f ∈ GΣ (x0, r, β, γ) : f dσ = 0 (7.1.9) Σ

which is considered equipped with the norm inherited from GΣ (x0, r, β, γ) (cf. (7.1.3)). Also, much as before, introduce

7.1 Definitions with Sharp Ranges of Indices and Basic Results

365

β,γ G˚0 (Σ) := the closure of G˚ 1,1 (Σ) in G˚ β,γ (Σ) whenever 0 < β, γ < 1.

(7.1.10)

We now proceed to introduce the scale of homogeneous Besov and Triebel-Lizorkin spaces on Ahlfors regular sets (compare with [89, Definition 5.8, p.120]). The reader is reminded that (a)+ := max{a, 0} for each a ∈ R. Definition 7.1.2 Suppose Σ ⊆ Rn is an unbounded closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Since (Σ, | · − · |, σ) is a space of homogeneous type, [133, Proposition 7.5.4] ensures the existence of a dyadic grid on Σ,  D(Σ) = Q αk k ∈Z,α∈I . (7.1.11) k

Denote by {St }t>0 the family of operators with integral kernels {St (·, ·)}t>0 as in (4.2.14) and define the conditional expectation operators {Ek }k ∈Z by setting Ek := S2−k − S2−k+1 for each k ∈ Z.

(7.1.12)

Then, if s ∈ (−1, 1),

max

 n−1 n

,

n−1 n+s



< p ≤ ∞,

0 < q ≤ ∞,



 < β < 1, max (s)+, −s + (n − 1) p1 − 1 

max s −

n−1 p , (n

− 1)

1 p

+



− 1 , −s + (n − 1)

+

1 p

−1

(7.1.13)



< γ < 1,

. p,q

the homogeneous Besov space  ∗ Bs (Σ, σ) on the set Σ is defined as the collection  β,γ of all functionals f ∈ G˚0 (Σ) for which  f B. p, q (Σ,σ) :=



s



2 Ek f  L p (Σ,σ) ks

q

 1/q < ∞,

with natural alterations when p = ∞ or q = ∞. Also, if   n−1 max n−1 s ∈ (−1, 1), max n−1 n , n+s < p ≤ ∞, n ,

n−1 n+s



< q ≤ ∞,



 < β < 1, max (s)+, −s + (n − 1) p1 − 1  max s −

n−1 p , (n

− 1)

(7.1.14)

k ∈Z

(7.1.15)

+

1 p





− 1 , −s + (n − 1) p1 − 1 < γ < 1, +

. p,q

then the homogeneous Triebel-Lizorkin  ∗ Fs (Σ, σ) on the set Σ is defined  β,γspace as the collection of all functionals f ∈ G˚0 (Σ) with the property that       ks  q 1/q   2 |Ek f |  f F. p, q (Σ,σ) :=   s k ∈Z

L p (Σ,σ)

0, which depends on Σ, p,q, and s. p,q

(ii) Given any f ∈ Fs (Σ, σ) with s, p, q, β, γ as in (7.1.33), there exist a numerical sequence λ = λQ k,ν Q k,ν ∈D∗ (Σ) , an exponent η ∈ (|s|, 1], along with η-smooth τ

τ

7.2 Atomic and Molecular Theory

379

blocks aQ κΣ ,ν of type (p, s) for τ ∈ IκΣ and ν ∈ {1, . . . , N( κΣ, τ)}, and η-smooth τ atoms aQ k,ν of type (p, s) for k ∈ Z, k ≥  κΣ + 1, τ ∈ Ik , ν ∈ {1, . . . , N(k, τ)}, τ p,q such (7.2.31) holds with convergence taking place both in Fs (Σ, σ) and ∗ ∗  that β,γ β,γ in G0 (Σ) when q < ∞, and only in G0 (Σ) when q = ∞. In addition, matters may be arranged so that λ f p, q (Σ) ≤ C f Fsp, q (Σ,σ)

(7.2.33)

for some finite constant C = C(Σ, p, q, s) > 0. Proof This follows from [92, Theorem 4, p. 75] by taking into account the renormalization we presently consider for our atoms.  In the converse direction to Theorem 7.2.7, the extent to which linear combinations of units and molecules with coefficients in a discrete Besov space belong to the corresponding continuous Besov space is studied next. Theorem 7.2.8 Assume Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, recall the family of dyadic cubes defined in (7.2.1) and recall  κΣ from (7.1.24). (i) Suppose s ∈ (−1, 1),

max

 n−1 n

,

n−1 n+s



< p ≤ ∞,

0 < q ≤ ∞,

(s)+ < η < 1, 

max (n − 1)

1 p



− 1 , −s + (n − 1)

(7.2.34)

+

1 p

−1



+

< ε < 1.

Let uQ κΣ ,ν be a (η, ε)-smooth unit of type (p, s) for each τ ∈ IκΣ and each τ ν ∈ {1, . . . , N( κΣ, τ) , and let uQ k,ν be a (η, ε)-smooth molecule of type (p, s) τ for each k ∈ Z, k ≥  κΣ + 1, τ ∈ Ik , and ν ∈ {1, . . . , N(k, τ)}. Then for every numerical sequence λ = λQ k,ν Q k,ν ∈D∗ (Σ) ∈ bp,q (Σ) it follows that the series τ

f :=

τ

(k,τ)   N k ∈Z τ ∈Ik ν=1 k ≥ κΣ

λQ k,ν uQ k,ν τ

τ

(7.2.35)

β,γ

converges in the space Bs (Σ, σ) whenever max{p, q} < ∞, and in (G0 (Σ))∗ whenever 



max (s)+, −s + (n − 1) p1 − 1 < β < 1 and 0 < γ < 1. (7.2.36) p,q

+

Furthermore, when max{p, q} < ∞ there exists C ∈ (0, ∞) such that  f Bsp, q (Σ,σ) ≤ Cλb p, q (Σ) .

(7.2.37)

380

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Moreover, when s ∈ (0, 1), the same conclusions as above continue to hold in the situation when each uQ k,ν is actually a (η, ε)-smooth unit of type (p, s) τ for every k ∈ Z, k ≥  κΣ + 1, τ ∈ Ik , and ν ∈ {1, . . . , N(k, τ)}. (7.2.38) (ii) Assume



 n−1 n−1 n−1 < p < ∞, max , , s ∈ (−1, 1), max n−1 n n+s n n+s < q ≤ ∞, (s)+ < η < 1,  

 

1 1 − 1 , −s + (n − 1) min(p,q) −1 < ε < 1. max (n − 1) min(p,q) +

+

(7.2.39) Let uQ κΣ ,ν be a (η, ε)-smooth unit of type (p, s) for every τ ∈ IκΣ and every τ ν ∈ {1, . . . , N( κΣ, τ)}, and let uQ k,ν be a (η, ε)-smooth molecule of type (p, s) τ for each k ∈ Z, k ≥  κΣ + 1, τ ∈ Ik , and ν ∈ {1, . . . , N(k, τ)}. Then for each numerical sequence λ = λQ k,ν Q k,ν ∈D∗ (Σ) ∈ f p,q (Σ) it follows that the series τ τ  β,γ  ∗ p,q (7.2.35) converges in Fs (Σ, σ) whenever q < ∞, and in G0 (Σ) whenever β, γ verify (7.2.36). Furthermore, when q < ∞ there exists C ∈ (0, ∞) with the property that (7.2.40)  f Fsp, q (Σ,σ) ≤ Cλ f p, q (Σ) . Moreover, when s ∈ (0, 1), the same conclusions as above continue to hold in the situation when each uQ k,ν is actually a (η, ε)-smooth unit of type (p, s) for τ every k ∈ Z, k ≥  κΣ + 1, τ ∈ Ik , and ν ∈ {1, . . . , N(k, τ)}. Proof This follows from [92, Theorem 5, p. 76] (cf. also [86, Theorem 2.2, p. 51] for the case when p, q ≥ 1), after readjusting notation. The last claim in the statement of the theorem is seen from an inspection of the proof of [92, Theorem 5, p. 76]. In this regard, see also the second remark in [92, § 3, p. 95].  We may further refine the decomposition (7.2.31) by retaining the atoms as they are while bundling together the blocks into a single function belonging to Lebesgue spaces, as indicated in the corollary below. Corollary 7.2.9 Assume that Σ ⊆ Rn is a closed Ahlfors regular set, and abbreviate σ := H n−1 Σ. Recall the family of dyadic cubes defined in (7.2.1) and recall the parameter  κΣ from (7.1.24). Also, suppose  n−1 s ∈ (−1, 1), max n−1 n , n+s < p ≤ 1, (7.2.41) 



 max (s)+, −s + (n − 1) p1 − 1 < β < 1, (n − 1) p1 − 1 < γ < 1.

7.2 Atomic and Molecular Theory

381

Then there exist an exponent η ∈ (|s|, 1] along with some constant C ∈ (0, ∞) p, p such that any f ∈ Bs (Σ, σ) may be decomposed as f =g+

(k,τ)   N k ∈Z τ ∈Ik ν=1 k ≥ κΣ +1

λQ k,ν aQ k,ν τ

τ

(7.2.42)

 β,γ  ∗ p,q with convergence both in Bs (Σ, σ) and in G0 (Σ) , for some function p, p

g ∈ Bs (Σ, σ) ∩

%

∗

L p (Σ, σ)

 (7.2.43)

1≤p ∗ ≤∞

satisfying gBsp, p (Σ,σ) + sup g L p ∗ (Σ,σ) ≤ C f Bsp, p (Σ,σ) 1≤p ∗ ≤∞

(7.2.44)

and for some family of η-smooth atoms aQ k,ν of type (p, s) along with some family τ of numbers λQ k,ν ∈ C, both indexed by k ∈ Z with k ≥  κΣ + 1, τ ∈ Ik , and τ ν ∈ {1, . . . , N(k, τ)}, satisfying (k,τ)    N  λ k ∈Z τ ∈Ik ν=1 k ≥ κΣ +1

Qτk,ν

 p 1/p  ≤ C f Bsp, p (Σ,σ) .

(7.2.45)

Proof From item (i) of Theorem 7.2.7, specialized to the case when s, p, β, γ are as p, p in (7.2.41) and when q := p, we know that any f ∈ Bs (Σ, σ) may be decomposed as in (7.2.42) with ( κΣ,τ)  N λQ κΣ ,ν aQ κΣ ,ν (7.2.46) g := τ ∈IκΣ

τ

ν=1

τ

for some numerical sequence λ = {λQ k,ν }Q k,ν ∈D∗ (Σ) satisfying (cf. (7.2.32)) τ

τ

λ p = λb p, p (Σ) ≤ C f Bsp, p (Σ,σ) .

(7.2.47)

To proceed, fix an arbitrary integrability exponent p∗ ∈ [1, ∞]. From (7.2.3)(7.2.3) with k :=  κΣ it follows that there exists a constant C ∈ (0, ∞) with the property that   a κΣ ,ν  p ∗ ≤ C for all τ ∈ IκΣ and ν ∈ {1, . . . , N( κΣ, τ)}. (7.2.48) L (Σ,σ) Q τ

As such, based on (7.2.46)-(7.2.48) we may estimate

382

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

   g L p ∗ (Σ,σ) = 

N ( κΣ,τ)

τ ∈IκΣ

≤

 τ ∈IκΣ

≤C

τ

ν=1

N ( κΣ,τ)

  λQ κΣ ,ν aQ κΣ ,ν 

 λ

ν=1

 τ ∈IκΣ

N ( κΣ,τ) ν=1

τ

 κ ,ν

Qτ Σ

 λ

 a

 κ ,ν

Qτ Σ

 κ ,ν

Qτ Σ

∗

L p (Σ,σ)

 

∗

L p (Σ,σ)

  ≤ Cλ p (Σ)

= Cλb p, p (Σ) ≤ C f Bsp, p (Σ,σ),

(7.2.49)

thanks to the fact that p ∈ (0, 1] (cf. (7.2.41)). In addition, from (7.2.42), (7.2.10), and item (i) of Theorem 7.2.8 we conclude that g belongs to the Besov space p, p Bs (Σ, σ) and there exists a constant C ∈ (0, ∞) independent of f with the property that gBsp, p (Σ,σ) ≤ C f Bsp, p (Σ,σ) . All together, this analysis shows that g is as in (7.2.43)-(7.2.44) and that (7.2.45) holds. 

7.3 Calderón’s Reproducing Formula and Frame Theory The result in Lemma 7.3.1 describes a general version of Calderón’s reproducing formula proved in [87, Theorem 1, p. 575], although the present formulation follows [92, Lemma 2, pp. 76-77], [199, Lemma 2.2, p. 573]. Related results may be found in [91, Theorem 4.1, p. 69], [202, Lemma 2.4, p. 100], and [89, Theorem 4.16, p. 109]. Lemma 7.3.1 Let Σ ⊆ Rn be a closed Ahlfors regular set. Abbreviate σ := H n−1 Σ and recall  κΣ from (7.1.24). Bring in the family of conditional expectation operators κΣ , denote {Ek }k ∈Z, k ≥κΣ introduced in Definition 7.1.5 and, for each k ∈ Z with k ≥  by Ek (·, ·) the integral kernel of Ek .   κΣ ,ν (·) defined on Then there exist a family of functions E τ ∈IκΣ , ν ∈ {1,..., N ( κΣ,τ)} Qτ  k (·, ·) defined on Σ × Σ, satisfying the Σ, along with a family of functions E k ∈Z, k ≥ κΣ following properties: (a) Given any ε ∈ (0, 1) there exists C ∈ (0, ∞) with the following significance. For each k ∈ Z with k ≥  κΣ one has | Ek (x, y)| ≤ for every x, x , y ∈ Σ,

(2−k

C2−kε , + |x − y|)n−1+ε

∀x, y ∈ Σ,

(7.3.1)

7.3 Calderón’s Reproducing Formula and Frame Theory

383

C2−kε |x − x | ε 2−k + |x − y|

, if |x − x | < 10 (2−k + |x − y|)n−1+2ε (7.3.2) ∫ ∫ k (x, y) dσ(y) = k (y, x) dσ(y) = 0, ∀x ∈ Σ, (7.3.3) E E

k (x , y)| ≤ | Ek (x, y) − E and

Σ

Σ

whereas for each τ ∈ IκΣ and ν ∈ {1, . . . , N( κΣ, τ)} one has     EQ κΣ ,ν (x) ≤ τ

C , (1 + |x − y|)n−1+ε

∀x ∈ Σ,

∀y ∈ QτκΣ,ν,

(7.3.4)

for every x, z ∈ Σ and y ∈ QτκΣ,ν ,     κΣ ,ν (z) ≤ EQ κΣ ,ν (x) − E Q τ

τ

C|x − z| ε 1 + |x − y| , (7.3.5) if |x − z| < 10 (1 + |x − y|)n−1+2ε ∫

and

Σ

 κΣ ,ν (x) dσ(x) = 1. E Q

(7.3.6)

τ

Moreover, the constant C appearing in (7.3.4) and (7.3.4) may be taken to be independent of jΣ , the large integer fixed earlier1 with the property that   κΣ, τ)}. diam QτκΣ,ν ≈ 2−jΣ uniformly for τ ∈ IκΣ and ν ∈ {1, . . . , N(  β,γ  ∗ (b) For each functional f ∈ G0 (Σ) with 0 < β, γ < 1 one has (with yτk,ν denoting the center of the cube Qτk,ν ) f =



N ( κΣ,τ)

τ ∈IκΣ

ν=1

+

 κΣ ,ν (·) σ(QτκΣ,ν ) mQ κΣ ,ν (EκΣ f )E Q

(k,τ)   N k ∈Z τ ∈Ik ν=1 k ≥ κΣ +1

τ

τ

(7.3.7)

k (·, yτk,ν ) σ(Qτk,ν )(Ek f )(yτk,ν )E

 β ,γ ∗ where the series converges in G0 1 1 (Σ) for any β1 ∈ (β, 1) and γ1 ∈ (γ, 1). The following two propositions provide a natural mechanism for moving back p,q and forth between discrete Besov spaces, bs (Σ), and continuous Besov spaces, p,q p,q Bs (Σ, σ), as well as between the discrete Triebel-Lizorkin spaces, fs (Σ), and p,q continuous Triebel-Lizorkin spaces, Fs (Σ, σ). Proposition 7.3.2 Let Σ ⊆ Rn be some closed Ahlfors regular set and abbreviate κΣ introduced as in (7.1.24), consider the family of functions σ := H n−1 Σ. With 

1 see the comment right before (7.1.27)

384



7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets



 κΣ ,ν (·) defined on Σ, along with the family of functions E τ ∈IκΣ , ν ∈ {1,..., N ( κΣ,τ)} Qτ k (·, ·) defined on Σ × Σ, from Lemma 7.3.1. E k ∈Z, k ≥ κΣ  n−1 Next, fix s ∈ (−1, 1) along with p ∈ (0, ∞] satisfying max n−1 n , n+s < p ≤ ∞, and suppose    

 max (s)+, −s + (n − 1) p1 − 1 + < β < 1, (n − 1) p1 − 1 + < γ < 1. (7.3.8) 

Finally, for each sequence of complex numbers of the form  λ = λτk,ν ∈ C : k ∈ Z, k ≥  κΣ, τ ∈ Ik , ν ∈ {1, . . . , N(k, τ)}

(7.3.9)

consider the (formal) series Φ(λ) :=



N ( κΣ,τ)

τ ∈IκΣ

ν=1

 κΣ ,ν (·) λτκΣ,ν E Q τ

+

(k,τ)   N k ∈Z τ ∈Ik ν=1 k ≥ κΣ +1

k (·, yτk,ν ). λτk,ν σ(Qτk,ν )E

(7.3.10)

Then the following properties hold. (1) If q ∈ (0, ∞] and λbsp, q (Σ) < ∞ then the series (7.3.10) converges to some  β,γ  ∗ p,q p,q distribution belonging to Bs (Σ, σ) both in Bs (Σ, σ) and G0 (Σ) when  β,γ  ∗ when max{p, q} < ∞, and only in G0 (Σ) in the case when max{p, q} = ∞. Moreover, in all cases there exists some constant C = C(Σ, s, p, q) ∈ (0, ∞) such that (7.3.11) Φ(λ)Bsp, q (Σ,σ) ≤ Cλbsp, q (Σ), which, in particular, implies that the application p,q

p,q

Φ : bs (Σ) −→ Bs (Σ, σ)

(7.3.12)

induced by (7.3.10) is well defined, linear, and bounded.  n−1 (2) If max n−1 < q ≤ ∞ and λ fsp, q (Σ) < ∞, then the series in (7.3.10) n , n+s p,q p,q converges s (Σ,  β,γ  ∗to some distribution belonging to Fs (Σ, σ) both in  F  ∗σ) and β,γ G0 (Σ) in the case when max{p, q} < ∞, and only in G0 (Σ) in the case when max{p, q} = ∞. Furthermore, in all cases there exists some constant C = C(Σ, s, p, q) ∈ (0, ∞) such that Φ(λ)Fsp, q (Σ,σ) ≤ Cλ fsp, q (Σ) .

(7.3.13)

Hence, the application p,q

Φ : fs

p,q

(Σ) −→ Fs (Σ, σ)

(7.3.14)

7.3 Calderón’s Reproducing Formula and Frame Theory

385

is also well-defined, linear, and bounded. (3) For each k ∈ Z with k ≥  κΣ , τ ∈ Ik , and ν ∈ {1, . . . , N(k, τ)}, define ψτk,ν :=

 κΣ,ν s − 1 + 1  ⎧ ⎪ κΣ ,  κ ,ν (·) if k =  ⎨ σ(Qτ ) n−1 p 2 E ⎪ Qτ Σ

(7.3.15)

s ⎪ − p1 + 32  ⎪ σ(Q k,ν ) n−1 κΣ + 1, Ek (·, yτk,ν ) if k ≥  ⎩ τ

(where, as before, yτk,ν is the center of Qτk,ν ). Then for each q ∈ (0, ∞] there exists a constant C = C(Σ, p, q, s) ∈ (0, ∞) with the property that   p,q each ψτk,ν belongs to Bs (Σ, σ) and ψτk,ν B p, q (Σ,σ) ≤ C. (7.3.16) s

 n−1

n−1

In addition, whenever max n , n+s < q ≤ ∞ there exists some constant C = C(Σ, p, q, s) ∈ (0, ∞) with the property that   p,q each ψτk,ν belongs to Fs (Σ, σ) and ψτk,ν F p, q (Σ,σ) ≤ C. (7.3.17) s

Proof For items (1)-(2) see [89, Proposition 7.3, p. 214] and also [199, Theorem 2.1, κΣ , along p. 575]. To deal with the claims in item (3), fix some k o ∈ Z with k o ≥  κΣ , with some τo ∈ Iko and some νo ∈ {1, . . . , N(k o, τo )}. For each k ∈ Z with k ≥  τ ∈ Ik , and ν ∈ {1, . . . , N(k, τ)}, we then define λQ k,ν := τ

s ⎧ − 1 +1  κ ,ν ⎪ ⎨ σ(QτΣo o ) n−1 p 2 if k = k o , and τ = τo , and ν = νo, ⎪

⎪ ⎪0 ⎩

(7.3.18)

if either k  k o , or τ  τo , or ν  νo .

Finally, with D∗ (Σ) as in (7.2.1), consider the numerical sequence  λ := λQ k,ν Q k,ν ∈D∗ (Σ) . τ

(7.3.19)

τ

p,q

This has only one nonzero term, hence λ ∈ bs (Σ) for each q ∈ (0, ∞] and, as seen from (7.3.18)-(7.3.19) and (7.2.28), there exists a constant C = C(Σ, s) ∈ (0, ∞) with the property that λ

p, q b s (Σ)

=

    N (k,τ)  k ∈Z k ≥ κΣ

τ ∈Ik ν=1

2

ks

1 1 p σ(Qτk,ν ) p − 2 |λQ k,ν | τ

s

= 2ko s σ(Qτkoo,νo ) p − 2 σ(Qτkoo,νo ) n−1 − p + 2 ≤ C, 1

1

1

1

 q/p  1/q

(7.3.20)

where inequality uses the fact that 2ko ≈ σ(Qτkoo,νo )− n−1 . Since, as seen from (7.3.10), (7.3.18)-(7.3.19), and (7.3.15), 1

Ψ(λ) = ψτkoo,νo ,

(7.3.21)

386

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

we may invoke the current item (1) to conclude that   p,q ψτkoo,νo ∈ Bs (Σ, σ) and ψτkoo,νo B p, q (Σ,σ) ≤ C,

(7.3.22)

s

for some constant C = C(Σ, p, q, s) ∈ (0, ∞). This establishes (7.3.16). Finally, the proof of (7.3.17) is similar, this time invoke the current item (2) and using   (k,τ)    N  q  1/q   1   λ fsp, q (Σ) =  2ks σ(Qτk,ν )− 2 λQ k,ν  1Q k,ν  τ τ   p k ∈Z τ ∈Ik ν=1 k ≥ κΣ

L (Σ,σ)

 s 1 1 1 = 2ko s σ(Qτkoo,νo )− 2 σ(Qτkoo,νo ) n−1 − p + 2 1Q k o ,νo  L p (Σ,σ) ≤ C, (7.3.23) τo



in place of (7.3.20). Here is the second proposition alluded to above.

Proposition 7.3.3 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate κΣ as in (7.1.24), consider the family of conditional expectation σ := H n−1 Σ. With  operators {Ek }k ∈Z, k ≥κΣ from Definition 7.1.5. Also, bring in the family of functions   κΣ ,ν (·) E defined on Σ, along with the family of functions τ ∈IκΣ , ν ∈ {1,..., N ( κΣ,τ)} Qτ   Ek (·, ·) k ∈Z, k ≥κΣ defined on Σ × Σ, from Lemma 7.3.1. Finally, for every functional  β,γ  ∗ f ∈ G0 (Σ) with 0 < β, γ < 1 let λτκΣ,ν := σ(QτκΣ,ν ) mQ κΣ ,ν (EκΣ f ) for τ ∈ IκΣ and ν ∈ {1, . . . , N( κΣ, τ)}, τ

κΣ + 1, τ ∈ Ik , and ν ∈ {1, . . . , N(k, τ)}, λτk,ν := (Ek f )(yτk,ν ) for k ∈ Z, k ≥  (7.3.24) where yτk,ν is the center of Qτk,ν and, with D∗ (Σ) as in (7.2.1), define  Ψ( f ) := λτk,ν Q k,ν ∈D∗ (Σ) . (7.3.25) τ

 n−1 Then for each s ∈ (−1, 1) and p ∈ (0, ∞] satisfying max n−1 n , n+s < p ≤ ∞ the following conclusions are valid.  β,γ  ∗ p,q (i) If q ∈ (0, ∞] then f ∈ Bs (Σ, σ) if and only if f ∈ G0 (Σ) for some β, γ satisfying





 < β < 1, (n −1) p1 −1 < γ < 1, (7.3.26) max (s)+, −s +(n −1) p1 −1 +



k,ν

+

the sequence λ = λτ Q k,ν ∈D∗ (Σ) := Ψ( f ) defined as in (7.3.25) belongs to τ p,q bs (Σ), and the discrete Calderón reproducing formula

7.3 Calderón’s Reproducing Formula and Frame Theory

f =



N ( κΣ,τ)

τ ∈IκΣ

ν=1

 κΣ ,ν (·) λτκΣ,ν E Q τ

(k,τ)   N

+

387

k ∈Z τ ∈Ik ν=1 k ≥ κΣ +1

k (·, yτk,ν ) λτk,ν σ(Qτk,ν )E

(7.3.27)

 β,γ  ∗ holds in G0 (Σ) . Moreover, the coefficients associated as in (7.3.25) with p,q each f ∈ Bs (Σ, σ) satisfy the following frame property: p,q

Ψ( f ) ∈ bs (Σ) and  f Bsp, q (Σ,σ) ≈ Ψ( f )bsp, q (Σ), p,q

uniformly for f ∈ Bs (Σ, σ).

(7.3.28)

  β,γ  ∗ p,q n−1 (ii) If max n−1 n , n+s < q ≤ ∞, then f ∈ Fs (Σ, σ) if and only if f ∈ G0 (Σ) for some β, γ as in (7.3.26), the discrete Calderón reproducing formula (7.3.27)  ∗ β,γ holds in G0 (Σ) , and the sequence λ = λτk,ν Q k,ν ∈D∗ (Σ) := Ψ( f ) defined as τ p,q p,q in (7.3.25) belongs to fs (Σ). In addition, for each f ∈ Fs (Σ, σ) one has p,q

Ψ( f ) ∈ fs

(Σ) and  f Fsp, q (Σ,σ) ≈ Ψ( f ) fsp, q (Σ), p,q

uniformly for f ∈ Fs (Σ, σ). Proof See [89, Theorem 7.4, p. 219] and also [199, Theorem 2.2, p. 585].

(7.3.29) 

We also have the following discrete Calderón reproducing formula in Besov spaces. Proposition 7.3.4 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Pick a smoothness index s ∈ (−1, 1) along with an integrability n−1 exponent p ∈ (0, ∞) for which max n−1 n , n+s < p, and consider β, γ satisfying



 < β < 1, max (s)+, −s + (n − 1) p1 − 1 +

(n − 1)

1 p

−1



+

< γ < 1. (7.3.30)

p,q

Then each f belonging to the space Bs (Σ, σ) with q ∈ (0, ∞), if the sequence  for λ = λτk,ν Q k,ν ∈D∗ (Σ) := Ψ( f ) is defined as in (7.3.24), one has the discrete Calderón τ reproducing formula f =



N ( κΣ,τ)

τ ∈IκΣ

ν=1

+

 κΣ ,ν (·) λτκΣ,ν E Q

(k,τ)   N k ∈Z τ ∈Ik ν=1 k ≥ κΣ +1

p,q

with convergence in Bs (Σ, σ).

τ

k (·, yτk,ν ) λτk,ν σ(Qτk,ν )E

(7.3.31)

388

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

 p,q Furthermore, if for some f ∈ Fs (Σ, σ) with max n−1 , n−1 < q < ∞, then n n+s  k,ν one considers the sequence λ = λτ Q k,ν ∈D∗ (Σ) := Ψ( f ) is defined as in (7.3.24), τ then the discrete Calderón reproducing formula (7.3.31) holds with convergence in p,q Fs (Σ, σ). p,q

Proof To begin with, since f ∈ Bs (Σ, σ), Proposition 7.3.3 gives that λ is in ∗ p,q β ,γ bs (Σ) and the series in the right-hand side of (7.3.31) converges to f in G0 1 1 (Σ) for any indices β1 ∈ (β, 1) and γ1 ∈ (γ, 1) satisfying similar properties as β, γ do in (7.3.30). On the other hand, since we have max{p, q} < ∞, Proposition 7.3.2  β ,γapplies ∗ p,q and gives that the series in (7.3.31) converges both in Bs (Σ, σ) and in G0 1 1 (Σ) for any β1 ∈ (β, 1) and γ1 ∈ (γ, 1) satisfying similar properties as β, γ do in (7.3.30). p,q As such, the series in the right-hand side of (7.3.31) to f in Bs (Σ, σ).  n−1converges p,q n−1 Finally, the case when f ∈ Fs (Σ, σ) with max n , n+s < q < ∞ is handled similarly.  When considered together, Proposition 7.3.2 and Proposition 7.3.3 yield some very useful consequences which we describe next. Proposition 7.3.5 In the context of Propositions 7.3.2-7.3.3, the bounded linear maps Φ, Ψ satisfy Φ ◦ Ψ = I, the identity operator, (7.3.32) both on the scales of Besov and Triebel-Lizorkin spaces. As a result, in the context of Propositions 7.3.2-7.3.3, Φ is onto, and Ψ is a quasi-isometric embedding (i.e., Ψ is injective and distorts quasi-norms only up to a fixed multiplicative factor) of the continuous scales of Besov and Triebel-Lizorkin spaces into their respective discrete versions.

(7.3.33)

 β,γ  ∗ Finally, when suitably interpreted, formula (7.3.32) also holds in G0 (Σ) . Proof This is a straightforward consequence of Proposition 7.3.2, Proposition 7.3.3, as well as Calderón’s reproducing formula described in Lemma 7.3.1.  The frame theory developed so far may, in turn, be used to prove the existence of some very useful approximations to the identity on Besov and Triebel-Lizorkin scales of the sort described in the proposition below. Proposition 7.3.6 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Then there exists a family of operators {PN } N ∈N satisfying the following properties.

 n−1 (1) Assume that s ∈ (−1, 1), max n−1 , n n+s < p ≤ ∞, and 0 < q ≤ ∞. Then each p,q

p,q

operator PN : Bs (Σ, σ) → Bs (Σ, σ) is linear, bounded, of finite rank, and sup PN Bsp, q (Σ,σ)→Bsp, q (Σ,σ) < +∞.

N ∈N

(7.3.34)

7.3 Calderón’s Reproducing Formula and Frame Theory

(2) Assume that s ∈ (−1, 1), max



n−1 n−1 n , n+s

some relatively compact subset O of exists N(ε) ∈ N such that



389

< p < ∞, and 0 < q < ∞. Also, fix

p,q Bs (Σ, σ).

Then for every ε > 0 there

sup  f − PN f Bsp, q (Σ,σ) < ε whenever N ≥ N(ε). f ∈O

(7.3.35)

In particular, corresponding to the case when O is a singleton, one has (with I p,q denoting the identity operator on Bs (Σ, σ)) p,q

PN → I pointwise on Bs (Σ, σ) as N → ∞.

(7.3.36)

(3) The same family of operators {PN } N ∈N enjoys similar properties as in items p,q (1)-(2) above, now formulated

on the Triebel-Lizorkin scale Fs (Σ, σ) with  n−1 s ∈ (−1, 1) and max n−1 n , n+s < p, q ≤ ∞ for item (1) and s ∈ (−1, 1) and 

n−1 max n−1 n , n+s < p, q < ∞ for item (2). Proof Recall the family of dyadic cubes defined in (7.2.1) and recall the parameter  κΣ from (7.1.24). For each k ∈ Z with k ≥  κΣ , consider a nested family {IkN } N ∈N N of finite subsets of Ik such that Ik  Ik as N → ∞. Also, fix β, γ satisfying (7.3.26).  ∗ any N ∈ N, define the operator f → PN f mapping each functional  β,γGiven f ∈ G0 (Σ) into PN f :=



N ( κΣ,τ)

τ ∈IκN

ν=1

Σ

+

 κΣ ,ν (·) σ(QτκΣ,ν ) mQ κΣ ,ν (EκΣ f )E Q τ

(k,τ)  N

 k ∈Z N ≥k ≥ κΣ

ν=1

τ ∈IkN +1

τ

k (·, yτk,ν ). σ(Qτk,ν )(Ek f )(yτk,ν )E

(7.3.37)

Note that, by design, the sums in the right-hand side run over finite sets of indices. p,q Suppose now that some arbitrary f ∈ Bs (Σ, σ) has been fixed. Then Proposik,ν tion 7.3.3, used with λ := Ψ( f ) = λτ Q k,ν ∈D∗ (Σ) as in (7.3.24)-(7.3.25), ensures τ p,q that λ ∈ bs (Σ), the equality f =



N ( κΣ,τ)

τ ∈IκΣ

ν=1

+

 κΣ ,ν (·) λτκΣ,ν E Q

(k,τ)   N k ∈Z k ≥ κΣ +1

τ ∈Ik ν=1

τ

k (·, yτk,ν ) λτk,ν σ(Qτk,ν )E

(7.3.38)

 β ,γ ∗ holds in G0 1 1 (Σ) for any β1 ∈ (β, 1) and γ1 ∈ (γ, 1) satisfying (7.3.26), and

390

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

 f Bsp, q (Σ,σ) ≈ λbsp, q (Σ) .

(7.3.39)

k,ν For each N ∈ N consider λ N := {λτ, κΣ, τ ∈Ik , 1≤ν ≤ N (k,τ) where N }k ∈Z, k ≥

 k,ν λτ, N

:=

λτk,ν if  κΣ ≤ k ≤ N, τ ∈ IkN , 1 ≤ ν ≤ N(k, τ), 0

(7.3.40)

otherwise. p,q

Then clearly λ N ∈ bs (Σ) and λ N bsp, q (Σ) ≤ λbsp, q (Σ) < ∞ for each N ∈ N. Moreover, by Proposition 7.3.2 we have PN f Bsp, q (Σ,σ) ≤ Cλ N bsp, q (Σ) for some C > 0 independent of N. In combination with (7.3.39), these properties yield PN f Bsp, q (Σ,σ) ≤ Cλ N bsp, q (Σ) ≤ Cλbsp, q (Σ) ≤ C f Bsp, q (Σ,σ) for each N ∈ N.

(7.3.41)

p,q

p,q

This proves (7.3.34). In particular, each operator PN : Bs (Σ, σ) → Bs (Σ, σ) is p,q p,q linear and bounded. The fact that each operator PN : Bs (Σ, σ) → Bs (Σ, σ) has finite rank is apparent from the definition of PN in (7.3.37). To proceed, observe that  f − PN f Bsp, q (Σ,σ)  ( κΣ,τ)   N   κΣ ,ν (·) σ(QτκΣ,ν ) mQ κΣ ,ν (EκΣ f )E = Qτ τ  N τ ∈IκΣ \Iκ

Σ

+

ν=1

(k,τ)   N

k>N τ ∈I N k

ν=1

  k (·, yτk,ν ) σ(Qτk,ν )(Ek f )(yτk,ν )E 

(7.3.42)

. p, q

Bs

(Σ,σ)

p,q

Since f ∈ Bs (Σ, σ) and we are presently assuming max{p, q} < ∞, Proposition 7.3.4 ensures that the series in the right-hand side of (7.3.38) converges to f p,q in Bs (Σ, σ). This implies that, in this case, the norm in (7.3.42) converges to 0 as N → ∞, establishing (7.3.36). We are left with proving (7.3.35). To this end, fix ε > 0, a relatively compact p,q subset O of Bs (Σ, σ), and define

 (7.3.43) C := max 1, sup PN Bsp, q (Σ,σ)→Bsp, q (Σ,σ) ∈ [1, ∞). N ∈N

To facilitate the subsequent discussion, for a generic quasi-metric space (X,  · X ), g ∈ X, and r > 0, we agree to use the notation B(g, r; X) := { f ∈ X :  f − gX < r }. (7.3.44)   p,q ε Let f0 ∈ O be arbitrary but fixed, and consider some f ∈ B f0, 3C ; Bs (Σ, σ) . By (7.3.36), there exists some N( f0 ) ∈ N such that

7.4 Interpolation of Besov and Triebel-Lizorkin Spaces via the Real Method

 f0 − PN f0 Bsp, q (Σ,σ)
max n−1 for j ∈ {0, 1}. (7.4.11) n , n+s j In this context, set s := (1 − θ)s0 + θs1 and let the number p ∈ (0, ∞] be such that 1/p = (1 − θ)/p0 + θ/p1 . Then one has  p0, p0  p ,p p, p Bs0 (Σ, σ), Bs11 1 (Σ, σ) θ, p = Bs (Σ, σ), (7.4.12) 

p , p0

p , p1

(Σ, σ), Fs11

(Σ, σ)



p, p

(Σ, σ).

(7.4.13)

Additionally, if Σ is unbounded, then .p ,p . p, p  . p0, p0  Bs0 (Σ, σ), Bs11 1 (Σ, σ) θ, p = Bs (Σ, σ),

(7.4.14)

Fs00

θ, p

= Fs

. p ,p . p, p  . p0, p0  Fs0 (Σ, σ), Fs11 1 (Σ, σ) θ, p = Fs (Σ, σ).

(ii) Suppose s0, s1 ∈ (−1, 1) with s0  s1 , max{ n−1 n ,

(7.4.15)

} < p j < ∞ for j ∈ {0, 1},   θ −1 1 ≤ q0, q1 ≤ ∞. Set s := (1 − θ)s0 + θs1 and define p := 1−θ . Then p0 + p1 

p ,q0

Fs00

p ,q1

(Σ, σ), Fs11

(Σ, σ)

 θ, p

n−1 n+s j

  p, p p, p = Bs (Σ, σ) = Fs (Σ, σ) .

 (iii) Assume that s ∈ (−1, 1), 1 ≤ p0, p1 ≤ ∞, and define p := 1−θ p0 + Then  p0, p0  p ,p p, p Fs (Σ, σ), Fs 1 1 (Σ, σ) θ, p = Fs (Σ, σ), 

p , p0

Bs 0

p , p1

(Σ, σ), Bs 1

(Σ, σ)

 θ, p

p, p

= Bs (Σ, σ).

 θ −1 p1

(7.4.16) ∈ [1, ∞]. (7.4.17) (7.4.18)

394

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Proof For formulas (7.4.12), (7.4.14) see [89, Theorem 8.7, p. 224]. Then formulas (7.4.13), (7.4.15) follow on account of these and (7.1.18), (7.1.38). For (ii)-(iii) see [199, Theorem 3.3, p. 587].  As a special case, it is useful to single out the following interpolation result. Theorem 7.4.5 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, assume n−1 n < p0 ≤ 1 < p1 < ∞. Then, if θ ∈ (0, 1) and 1 1−θ θ = + , it follows that p p0 p1 

H (Σ, σ), L (Σ, σ) p0

p1



 θ, p

=

H p (Σ, σ) if L p (Σ, σ)

n−1 n

< p ≤ 1,

(7.4.19)

if 1 < p < ∞. p ,2

Proof Use the identifications (7.1.20), ensuring that H p0 (Σ, σ) = F0 0 (Σ, σ), and p ,2  (7.1.19) according to which L p1 (Σ, σ) = F0 1 (Σ, σ), along with (7.4.6).

7.5 Complex Interpolation of Besov and Triebel-Lizorkin Spaces The complex method of interpolation, denoted by [·, ·]θ , has been originally introduced in [23] in the context of Banach spaces. Since the full scales of Besov and Triebel-Lizorkin spaces contain spaces which are not locally convex, we shall work here with the adaptation of the complex method of interpolation to analytically convex quasi-Banach spaces as described in §1.4. We begin, nonetheless, by recording the following theorem dealing with the classical complex interpolation method for the portion of the Besov and Triebel-Lizorkin scales consisting of Banach spaces. Theorem 7.5.1 Assume that Σ ⊆ Rn is an arbitrary closed Ahlfors regular set, and abbreviate σ := H n−1 Σ. Fix some θ ∈ (0, 1), along with s0, s1 ∈ (−1, 1) and p0, q0, p1, q1 ∈ (1, ∞). Finally, consider s := (1 − θ)s0 + θs1 and introduce     θ −1 θ −1 p := 1−θ together with q := 1−θ . Then p0 + p1 q0 + q1  

p ,q0

Bs00

p ,q0

Fs00

p ,q1

(Σ, σ), Bs11

p ,q1

(Σ, σ), Fs11

(Σ, σ)



(Σ, σ)



p,q

(7.5.1)

p,q

(7.5.2)

θ

= Bs (Σ, σ),

θ

= Fs (Σ, σ).

Moreover, when Σ is unbounded, similar results hold for the homogeneous scales of Besov and Triebel-Lizorkin spaces on Σ. A proof may be found in [86, Theorem 5.2, p. 64] and [90, Theorem 7.7, p. 121]. This should be compared with results in the standard Euclidean setting to the effect that if s0, s1 ∈ R, 1 ≤ p0, p1, q0, q1 ≤ ∞, and if 0 < θ < 1,

s = (1 − θ)s0 + θs1,

1 p

=

1−θ p0

+

θ p1 ,

1 q

=

1−θ q0

+

θ q1 ,

(7.5.3)

7.5 Complex Interpolation of Besov and Triebel-Lizorkin Spaces



then

p ,q0

Fs00

p ,q1

(Rn ), Fs11

(Rn )

 θ

395

p,q

= Fs (Rn ) provided

either max {p0, q0 } < ∞, or max {p1, q1 } < ∞, 

and also

p ,q0

Bs00

p ,q1

(Rn ), Bs11

(Rn )

 θ

(7.5.4)

p,q

= Bs (Rn ),

(7.5.5)

granted that min {q0, q1 } < ∞.

In addition, analogous formulas are valid for the homogeneous versions of the Besov and Triebel-Lizorkin spaces. Indeed, this particular collection of results is obtained by a careful inspection of the proofs of several well-known results in [15], [58], [166], [186]. Here we wish to mention that formulas (7.5.4), (7.5.5) have established in [127] for the full range of indices for which the scales of Besov and Triebel-Lizorkin spaces are defined (cf. also [110]). See Theorem 9.1.6 below. The main result of this section is an extension of Theorem 7.5.1 to a larger range of indices, for which the scales of Besov and Triebel-Lizorkin spaces include non locally convex spaces. More specifically, we have: Theorem 7.5.2 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, select some θ ∈ (0, 1) along with s0, s1 ∈ (−1, 1), and finally define s := (1 − θ)s0 + θs1 . Then whenever

 n−1 max n−1 , 0 < q j ≤ ∞, for j ∈ {0, 1}, n n+s j < p j ≤ ∞,     θ −1 θ −1 (7.5.6) , q := 1−θ , and p := 1−θ p0 + p1 q0 + q1 either max{p0, q0 } < ∞, or max{p1, q1 } < ∞, it follows that p ,q0

Bs00

p ,q1

(Σ, σ), Bs11

p ,q0

(Σ, σ), and Bs00

p ,q1

(Σ, σ) + Bs11

(Σ, σ)

are all analytically convex quasi-Banach spaces and



p ,q0

Bs00

p ,q1

(Σ, σ), Bs11

(Σ, σ)

 θ

p,q

= Bs (Σ, σ).

Furthermore, if 



 n−1 n−1 n−1 max n−1 < p , < ∞, max , j n n+s j n n+s < q j < ∞, for j ∈ {0, 1},     θ −1 θ −1 , q := 1−θ , and p := 1−θ p0 + p1 q0 + q1

(7.5.7)

(7.5.8)

(7.5.9)

then p ,q0

Fs00

p ,q1

(Σ, σ), Fs11

p ,q0

(Σ, σ), and Fs00

p ,q1

(Σ, σ) + Fs11

(Σ, σ)

are all analytically convex quasi-Banach spaces and

(7.5.10)

396

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets



p ,q0

p ,q1

(Σ, σ), Fs11

Fs00

(Σ, σ)

 θ

p,q

= Fs (Σ, σ).

(7.5.11)

Moreover, analogous results (for the same ranges of indices as above) are valid for the discrete scales of Besov and Triebel-Lizorkin spaces on Σ. Finally, when Σ is unbounded, similar results hold for the homogeneous scales of Besov and Triebel-Lizorkin spaces on Σ. As a preamble to the proof of Theorem 7.5.2, we discuss several useful auxiliary results. The following two results appear in [15, Theorem 5.5.3, p. 120 and Theorem 5.6.3, p. 123]. First, it is possible to identify the intermediate interpolation spaces produced by the complex method between weighted Lebesgue spaces with change of weights. Proposition 7.5.3 Let (Σ, μ) be a measure space and let ω0, ω1 : Σ → [0, ∞) be two μ-measurable functions. Then whenever 1 ≤ p0, p1 < ∞ and θ ∈ (0, 1) the following formula holds   p (7.5.12) L 0 (Σ, ω0 μ), L p1 (Σ, ω1 μ) θ = L p (Σ, ω μ), where p :=

 1−θ

+

p0

 θ −1 p1

  (1−θ )p   θ p and ω := ω0 p0 ω1 p1 ,

(7.5.13)

with equality of norms, i.e., for each f ∈ L p (Σ, ω μ) one has  f [L p0 (Σ, ω0 μ), L p1 (Σ, ω1 μ)]θ =  f  L p (Σ, ω μ) . Proof See [15, Theorem 5.5.3, p. 120].

(7.5.14) 

κΣ as in (7.1.24) and abbreviate Having fixed a background set Σ ⊆ Rn , consider   ZΣ := k ∈ Z : k ≥  (7.5.15) κΣ . Definition 7.5.4 Assume that X is a quasi-Banach space, q ∈ (0, ∞) and that s ∈ R. q In this context, define s (ZΣ, X) as the space of sequences { fk }k ∈ZΣ , with fk ∈ X for each k ∈ ZΣ , such that   { fk }k ∈Z  Σ

 q

s (ZΣ,X)

:=

∞  

2  fk X ks

q

 q1 < ∞.

(7.5.16)

k= κΣ

Remark 7.5.5 In the context of Definition 7.5.4, if μs denotes the measure defined # ks on ZΣ as ∞ k= κΣ 2 δ {k } (where δ {x } denoting the Dirac mass at a point x), it is natural to make the following identification (see (1.4.12)-(1.4.14)) q

s/q (ZΣ, X) = L q (ZΣ, μs ) ⊗ X.

(7.5.17)

Proposition 7.5.6 Let X0, X1 be two compatible Banach spaces, and suppose that 1 ≤ q0, q1 ≤ ∞, s0, s1 ∈ R, and θ ∈ (0, 1). Then the following complex interpolation result holds:

7.5 Complex Interpolation of Besov and Triebel-Lizorkin Spaces



q

q

s00 (ZΣ, X0 ), s11 (ZΣ, X1 )

where s := (1 − θ)s0 + θs1 and q :=

 1−θ q0



θ

+

397

 q = s ZΣ, [X0, X1 ]θ ,

(7.5.18)

 θ −1 . q1

Proof See [15, Theorem 5.6.3, p. 123].



To continue, assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Retaining earlier notation used in such a setting, whenever 0 < p ≤ ∞ p define the quasi-Banach space w (D∗ (Σ)) (where the purely decorative subscript “w" is meant to suggest that this is a weighted space) as  p w (D∗ (Σ)) := {λQ }Q ∈D∗ (Σ) : λQ ∈ R for every Q ∈ D∗ (Σ),

  such that {λQ }Q ∈D∗ (Σ)  p (D∗ (Σ)) < ∞ , (7.5.19) w

where  {λQ }Q ∈D

∗

  (Σ)

 p

w (D∗ (Σ))



:=



σ(Q)

1 1 p−2

|λQ |

p

 p1 .

(7.5.20)

Q ∈D∗ (Σ) p

Remark 7.5.7 If 0 < p ≤ ∞, a moment’s reflection shows that the space w (D∗ (Σ)) defined above may be naturally identified with the weighted Lebesgue space   L p D∗ (Σ), ω p m where m denotes the standard counting measure on the countable set D∗ (Σ) and ω p is the weight defined on D∗ (Σ) by p ω p (Q) := σ(Q)1− 2 for every Q ∈ D∗ (Σ).

(7.5.21)

Proposition 7.5.8 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Suppose 1 ≤ p0, p1 ≤ ∞, 1 ≤ q0, q1 ≤ ∞, and s0, s1 ∈ R. Then the following complex interpolation result holds:    q    q p p q p s00 ZΣ, w0 (D∗ (Σ)) , s11 ZΣ, w1 (D∗ (Σ)) = s ZΣ, w (D∗ (Σ)) , (7.5.22) θ

whenever θ ∈ (0, 1), p := p

 1−θ p0

+

 θ −1 , p1

q :=

 1−θ q0

+

 θ −1 , q1

and s := (1 − θ)s0 + θs1 .

p

Proof Since w0 (D∗ (Σ)) and w1 (D∗ (Σ)) are compatible Banach spaces, by Proposition 7.5.6 it suffices to show that  p0  p p w (D∗ (Σ)), w1 (D∗ (Σ)) θ = w (D∗ (Σ)). (7.5.23) This, however, is a consequence of Proposition 7.5.3 (presently employed with Σ replaced by D∗ (Σ)  equipped with the counting measure m) and the identification p w (D∗ (Σ)) = L p D∗ (Σ), ω p m valid for all p’s (cf. Remark 7.5.7 where the weight ω p is also defined), together with the observation that

398

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets



ω p0 (Q)

 (1−θ) pp  0

ω p1 (Q)

 θ pp

1

p0  (1−θ) p  p1  θ p  p0 σ(Q)1− 2 p1 = σ(Q)1− 2 p

= σ(Q)1− 2 = ω p (Q),

(7.5.24)

for each cube Q ∈ D∗ (Σ).



Proposition 7.5.9 Let Σ ⊆ Rn be a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Given q ∈ (0, ∞) along with s ∈ R, consider the weighted space L q (D∗ (Σ), ws,q m) where m is the counting measure on D∗ (Σ) and the weight ws,q is defined by   ws,q (Q) := σ(Q)−q

s 1 n−1 + 2

,

∀Q ∈ D∗ (Σ).

(7.5.25)

For each integrability exponent p ∈ (0, ∞] consider the vector-valued Lebesgue space L p (Σ, σ) ⊗ L q D∗ (Σ), ws,q m , consisting of all p-th power integrable functions on Σ with respect to the measure σ, taking values in the quasi-Banach space L q (D∗ (Σ), ws,q m) (see (1.4.12)-(1.4.14)). Then, if θ ∈ (0, 1), 1 ≤ p0, p1 ≤ ∞, 1 ≤ q0, q1 < ∞, and s0, s1 ∈ R, the following complex interpolation result (involving Banach spaces) holds:      L p0 (Σ, σ) ⊗ L q0 D∗ (Σ), ws0,q0 m , L p1 (Σ, σ) ⊗ L q1 D∗ (Σ), ws1,q1 m q

θ



= L (Σ, σ) ⊗ L D∗ (Σ), ws,q m , p

where s := (1−θ)s0 +θs1 and p, q are defined by p :=

 1−θ p0

(7.5.26) + pθ1

 −1

, q :=

 1−θ q0

+ qθ1

 −1

.

Proof The justification of (7.5.26) is analogous to the argument used in the proof of Proposition 7.5.8.  Lemma 7.5.10 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Assuming 0 < p, q ≤ ∞ and s ∈ R, consider the mapping  q p p,q R b : s ZΣ, w (D∗ (Σ)) −→ bs (Σ) (7.5.27) defined by

R b ({ fk }k ∈ZΣ ) := {λQ }Q ∈D∗ (Σ) for each  q p { fk }k ∈ZΣ ∈ s ZΣ, w (D∗ (Σ)) ,

(7.5.28)

where  for each Q  ∈ D∗ (Σ) where the integer k ∈ ZΣ is uniquely λQ := fk (Q)  ∈ Dk+jΣ (Σ). determined (thanks to (7.2.2)) by the requirement that Q Furthermore, introduce a second mapping,  p,q q p E b : bs (Σ) −→ s ZΣ, w (D∗ (Σ)) which acts according to p,q

E b ({λQ }Q ∈D∗ (Σ) ) := { fk }k ∈ZΣ for every {λQ }Q ∈D∗ (Σ) ∈ bs (Σ),

(7.5.29)

(7.5.30)

7.5 Complex Interpolation of Besov and Triebel-Lizorkin Spaces

399

p,q

where, given any {λQ }Q ∈D∗ (Σ) ∈ bs (Σ) and k ∈ ZΣ , the function fk : D∗ (Σ) → R is defined by  λQ if Q ∈ Dk+jΣ (Σ), (7.5.31) ∀Q ∈ D∗ (Σ). fk (Q) := 0 if Q ∈ D∗ (Σ) \ Dk+jΣ (Σ), Then the mappings (7.5.27), (7.5.30) are well-defined, linear, bounded and p,q

R b ◦ E b = I, the identity operator on bs (Σ).

(7.5.32)

In particular, the mapping R b in (7.5.27) is onto, i.e.,    q p p,q R b s ZΣ, w (D∗ (Σ)) = bs (Σ).

(7.5.33)

 q p Proof Given { fk }k ∈ZΣ ∈ s ZΣ, w (D∗ (Σ)) arbitrary, define the numerical sequence {λQ }Q ∈D∗ (Σ) as in (7.5.29). Based on definitions (cf. (7.2.28), (7.5.29), (7.2.2), (7.5.16), and (7.5.20)), we may then write R b ({ fk }k ∈ZΣ )bsp, q (Σ) =

=

≤

⎧ ∞  ⎪ ⎨ ⎪ ⎪ ⎪ k=κΣ ⎩ ⎧ ∞  ⎪ ⎨ ⎪ ⎪ ⎪ k=κΣ ⎩



2 σ(Q) ks

1 1 p−2

Q ∈Dk (Σ)





2ks σ(Q)



2 σ(Q) ks

1 1 p−2

Q ∈D∗ (Σ)

= { fk }k ∈ZΣ  q Z s

p Σ,w (D∗ (Σ))

,

⎫ 1/q ⎬ ⎪  p  q/p ⎪ |λQ | ⎪ ⎪ ⎭

1 1 p−2

Q ∈Dk+ jΣ (Σ)

⎧ ∞   ⎪ ⎨ ⎪ ⎪ ⎪ k=κΣ ⎩



⎫ 1/q ⎬ ⎪  p  q/p ⎪ | fk (Q)| ⎪ ⎪ ⎭

⎫ ⎬ ⎪  p  q/p ⎪ | fk (Q)| ⎪ ⎪ ⎭

1/q

(7.5.34)

which proves that the mapping R b is well defined and bounded. Clearly, the mapp,q ping R b is also linear. Assume next that {λQ }Q ∈D∗ (Σ) ∈ bs (Σ) is arbitrary and define { fk }k ∈ZΣ as in (7.5.30). Then, by unraveling definitions (cf. (7.5.16), (7.5.20), (7.5.31), (7.2.2), and (7.2.28)) we may write

400

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

E b ({λQ }Q ∈D∗ (Σ) ) q Z

p Σ,w (D∗ (Σ))

⎧ ∞ ⎪ ⎨ ⎪



s

=

=

⎪ ⎪ k=κΣ ⎩ ⎧ ∞  ⎪ ⎨ ⎪ ⎪ ⎪ k=κΣ ⎩

2

ks

 



σ(Q)

1 1 p−2

Q ∈D∗ (Σ)





2ks σ(Q)

Q ∈Dk+ jΣ (Σ)

⎫ 1/q ⎬ ⎪  p  1/p  q ⎪ | fk (Q)| ⎪ ⎪ ⎭

1 1 p−2

⎫ 1/q ⎬ ⎪  p  q/p ⎪ |λQ | ⎪ ⎪ ⎭

1/q (k,τ) ∞   N ⎧ ⎨ ⎪ ⎬ ⎪  ks  p  q/p ⎫ k,ν p1 − 12 2 σ(Qτ ) = |λQ k,ν | τ ⎪ ⎪ ⎩ k=κΣ τ ∈Ik ν=1 ⎭

= {λQ }Q ∈D∗ (Σ) bsp, q (Σ),

(7.5.35)

which shows that the mapping E b is also well-defined and isometric (hence, in particular, bounded). To justify the formula claimed in (7.5.32), let {λQ }Q ∈D∗ (Σ) ∈ D∗ (Σ) be an arbitrary fixed sequence and consider { fk }k ∈ZΣ := E b ({λQ }Q ∈D∗ (Σ) ) defined as in (7.5.30). Q }Q ∈D∗ (Σ) := R b ({ fk }k ∈ZΣ ) as defined in (7.5.29), it Then, if we further set { λ  ∈ D∗ (Σ), and for each k ∈ ZΣ such that follows that for each arbitrary, fixed cube Q    Q ∈ Dk+jΣ (Σ) we have λQ = fk (Q) = λQ . This establishes that R b ◦ E b = I on p,q bs (Σ).  Lemma 7.5.11 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Assume 0 < p, q < ∞, s ∈ R, and recall the weighted counted measure ws,q m on D∗ (Σ) introduced before (cf. (7.5.25)). In this context, define the mapping   p,q E f : fs (Σ) −→ L p (Σ, σ) ⊗ L q D∗ (Σ), ws,q m which acts according to p,q

E f ({λQ }Q ∈D∗ (Σ) ) := g for every {λQ }Q ∈D∗ (Σ) ∈ fs p,q

where, given any {λQ }Q ∈D∗ (Σ) ∈ fs

(Σ) (7.5.36)

(Σ), the function g is defined by

(g(x))(Q) := λQ 1Q for every x ∈ Σ and Q ∈ D∗ (Σ).

(7.5.37)

Then for the full range of indices p, q, s as above, the mapping (7.5.36) is well p,q defined, linear, and isometric, i.e., for each {λQ }Q ∈D∗ (Σ) ∈ fs (Σ) one has     E f ({λQ }Q ∈D (Σ) ) p = {λQ }Q ∈D∗ (Σ)  f p, q (Σ), (7.5.38) ∗ L (Σ,σ)⊗L q (D∗ (Σ),ws, q m) s

Going further, assume 1 < p, q < ∞, s ∈ R and, in this context, introduce a second mapping, namely   p,q R f : L p (Σ, σ) ⊗ L q D∗ (Σ), ws,q m −→ fs (Σ), (7.5.39)

7.5 Complex Interpolation of Besov and Triebel-Lizorkin Spaces

401

by setting   R f (g) := {λQ }Q ∈D∗ (Σ) for each g ∈ L p (Σ, σ) ⊗ L q D∗ (Σ), ws,q m ,

(7.5.40)

⨏

where λQ :=

Q

(g(x))(Q) dσ(x) for each Q ∈ D∗ (Σ).

(7.5.41)

Then, for the specified range of indices, the mapping (7.5.39) is well defined, linear, bounded and R f ◦ E f = I,

p,q

the identity operator on fs

(Σ).

(7.5.42)

In particular, whenever 1 < p, q < ∞, s ∈ R, the mapping R f in (7.5.39) is onto, i.e.,    p,q (7.5.43) R f L p (Σ, σ) ⊗ L q D∗ (Σ), ws,q m = fs (Σ). Proof The claims made about the operator E f , including (7.5.38), are straightforward from definitions (bearing in mind (1.4.14) and Lemma 1.4.11). Moving on, assume that 1 < p, q < ∞, s ∈ R. In this setting, the crux of the matter is showing With this goal in mind, pick an that R f is a well-defined and  bounded operator.  arbitrary g ∈ L p (Σ, σ) ⊗ L q D∗ (Σ), ws,q m and introduce s

gQ (x) := σ(Q)− n−1 − 2 (g(x))(Q), 1

∀Q ∈ D∗ (Σ),

∀x ∈ Σ.

(7.5.44)

Also, recall that M Σ denotes the Hardy-Littlewood maximal operator on Σ (cf. (A.0.71)). Then (with justifications to follow shortly) we may write      ⨏ q 1/q        R f (g) p, q ≈  gQ dσ  1Q    fs (Σ) Q  Q ∈D∗ (Σ)  p        q 1/q      M Σ (gQ ) ≤ C   Q ∈D∗ (Σ)         q 1/q      gQ ≤ C   Q ∈D∗ (Σ) 

L (Σ,σ)

L p (Σ,σ)

L p (Σ,σ)

     s 1

1/q    = C  σ(Q)−q n−1 + 2 |(g(·))(Q)| q    Q ∈D∗ (Σ) = Cg L p (Σ,σ)⊗L q (D∗ (Σ),ws, q m)),

L p (Σ,σ)

(7.5.45)

for some constant C ∈ (0, ∞) independent of g. Above, the first step is a matter of unraveling definitions and observing that

402

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets q

ws,q (Q) ≈ 2ksq σ(Q)− 2 ,

uniformly for Q ∈ Dk+jΣ (Σ), k ∈ ZΣ .

(7.5.46)

To justify the second step, it suffices to note that, generally speaking, there exists some constant C = C(Σ) ∈ (0, ∞) with the property that if x∗ ∈ Σ, r > 0, Δ := B(x∗, r) ∩ Σ, 1 (Σ, σ) then and h ∈ Lloc 

⨏ |h| dσ 1Δ ≤ CM Σ h pointwise on Σ. (7.5.47) Δ

The third step uses the version of a classical inequality due to Fefferman-Stein regarding the boundedness of the vector-valued Hardy-Littlewood maximal operator on spaces of homogeneous type (cf. [133, Theorem 7.6.6] for a precise formulation). This is where we make crucial use of the fact that 1 < p, q < ∞. Steps four and five are mere reinterpretations of definitions. This finishes the proof of the fact that R f is a well-defined, linear and bounded operator. Finally, formula claimed in (7.5.42) follows directly from (7.5.36)-(7.5.37) and (7.5.39)-(7.5.41).  The reader is reminded that the property being analytically convex has been defined in §1.4. Proposition 7.5.12 Assume Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. (i) If s ∈ (−1, 1),

 max

n−1 n−1 n , n+s



< p ≤ ∞,

0 < q ≤ ∞,

(7.5.48)

p,q

then the Besov space Bs (Σ, σ) is analytically convex. (ii) If



s ∈ (−1, 1),

max 

and max

n−1 n−1 n , n+s

n−1 n−1 n , n+s





< p < ∞,

< q ≤ ∞,

(7.5.49)

p,q

then the Triebel-Lizorkin space Fs (Σ, σ) is analytically convex. (iii) Similar results are valid for the associated sequence spaces. More precisely, p,q p,q bs (Σ) is analytically convex whenever (7.5.48) holds, while fs (Σ) is analytically convex if (7.5.49) holds. Finally, when Σ is unbounded, similar results hold for the homogeneous scales of Besov and Triebel-Lizorkin spaces on Σ. Proof Thanks to Proposition 7.3.5 and Proposition 1.4.6, it suffices to work with sequence spaces. In this setting, the conclusion pertaining to Besov spaces follows from the fact that the operator (7.5.30) is a linear isomorphism onto its image (see (7.5.35)), Proposition 1.4.6, and repeated applications of Proposition 1.4.15. The argument for the scale of discrete Triebel-Lizorkin spaces largely follows along these lines, making use of the first part of Lemma 7.5.11 (cf. (7.5.36)-(7.5.38)

7.5 Complex Interpolation of Besov and Triebel-Lizorkin Spaces

403

which imply that E f is a linear isomorphism onto its image), then once again invoking Proposition 1.4.6 together with Proposition 1.4.15.  Lemma 7.5.13 Let Σ ⊆ Rn be a closed Ahlfors regular set. Fix s ∈ R, 0 < p, q ≤ ∞, p,q p,q and assume that 0 < r < min{p, q}. Then the spaces fs (Σ) and bs (Σ) are lattice r-convex. Moreover, p,q

[ fs

p ,q

(Σ)]r = fs

(Σ),

p,q

p ,q

[bs (Σ)]r = bs

(Σ),

(7.5.50)

s := s + (n − 1)(r − 1)/(2r).

(7.5.51)

where the indices are related by p := p/r,

q := q/r,

Proof The first claim in the lemma is a consequence of Proposition 7.5.12 and Theorem 1.4.17. Next, it follows straight from definitions that for each r > 0 we have p,q

{|λQ | 1/r }Q ∈D∗ (Σ) ∈ fs

p ,q

(Σ) ⇐⇒ {λQ }Q ∈D∗ (Σ) ∈ fs

(Σ),

(7.5.52)

where the primed indices are as in (7.5.51) (incidentally, this can also be used to p,q show that fs (Σ) is lattice r-convex if 0 < r < min{p, q} will do). The formula in p,q (7.5.50) dealing with the scale fs (Σ) is clear from this. The treatment of the scale p,q  bs (Σ) is similar, finishing the proof of the lemma. After these preparations, we are finally ready to present the proof of Theorem 7.5.2. Proof of Theorem 7.5.2 We shall first focus on proving the claims made in Theorem 7.5.2 for the scale of Besov spaces. Fix pi, qi, p, q, for i ∈ {0, 1}, as in (7.5.6) and choose β, γ ∈ (0, 1) sufficiently close 1 (depending on the aforementioned paβ,γ rameters). Then we may take (G0 (Σ))∗ as an ambient topological vector space pi ,qi in which the Besov spaces Bsi (Σ, σ), i ∈ {0, 1}, are continuously embedded. In p ,q addition, from Proposition 7.5.12 we know that the spaces Bsii i (Σ, σ), i ∈ {0, 1}, are analytically convex (cf. also (7.5.56) below), so the version of the method of complex interpolation described in §1.4 applies. The rest of the proof is organized in a series of steps, each of which deals with a separate claim, building up to the desired conclusion. Step 1. The sum p ,q0

bs00

p ,q1

(Σ) + bs11

(Σ) is an analytically convex space.

(7.5.53)

Proof. From item (iii) in Proposition 7.5.12 we know that p ,qi

each space bsii

(Σ), i ∈ {0, 1}, is analytically convex,

(7.5.54)

whereas from definitions it is apparent that, generally speaking, p,q

bs (Σ) is separable if 0 < p < ∞, 0 < q < ∞, and s ∈ R.

(7.5.55)

404

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

In concert with Theorem 1.4.18 and the second remark following it, this proves (7.5.53). Step 2. The sum p ,q0

Bs00

p ,q1

(Σ, σ) + Bs11

(Σ, σ) is an analytically convex space.

(7.5.56)

Proof. Recall the mappings Φ and Ψ defined in Proposition 7.3.2 and Proposition 7.3.3, respectively. One may then check that the operator Ψ maps the p ,q p ,q sum space Bs00 0 (Σ, σ) + Bs11 1 (Σ, σ) isomorphically onto a closed subspace of p0,q0 p1,q1 bs0 (Σ) + bs1 (Σ). Indeed, this follows readily by taking into account that Ψ acts β,γ linearly on (G0 (Σ))∗ and that, as already pointed out, formula (7.3.32) continues to hold in this setting. Then (7.5.56) follows from this observation, (7.5.53) and Proposition 1.4.6. p ,q

Step 3. The pair of continuous Besov spaces Bsii i (Σ), i ∈ {0, 1}, is a retract of p ,q the pair of discrete Besov spaces bsii i (Σ), i ∈ {0, 1}. Proof. This follows from Proposition 7.3.5, with Φ, Ψ doing the job. Step 4. For Φ defined in Proposition 7.3.2, there holds

  p0,q0    p ,q p ,q p ,q Bs0 (Σ, σ), Bs11 1 (Σ, σ) θ = Φ bs00 0 (Σ), bs11 1 (Σ) θ .

(7.5.57)

Proof. This is a consequence of the first formula in (1.4.64). Step 5. Formula (7.5.8) holds provided  p0,q0  p ,q p,q bs0 (Σ), bs11 1 (Σ) θ = bs (Σ).

(7.5.58) p,q

p,q

Proof. This follows from (7.5.57) and the fact that Φ maps bs (Σ) onto Bs (Σ, σ) (cf. the first part of (7.3.33)). Step 6. With the mapping R b defined as in (7.5.27)-(7.5.29), there holds  p0,q0  p ,q bs0 (Σ), bs11 1 (Σ) θ

   q    q p p . (7.5.59) = R b s00 ZΣ, w0 (D∗ (Σ)) , s11 ZΣ, w1 (D∗ (Σ)) θ

Proof. This follows from Lemma 7.5.10 and (1.4.64) in Lemma 1.4.23. Step 7. One has  p ,q  p ,q p,q p0, q0, p1, q1 ≥ 1 =⇒ bs00 0 (Σ), bs11 1 (Σ) θ = bs (Σ).

(7.5.60)

Proof. This is a consequence of formula the interpolation result proved in  (7.5.59),  q p p,q Proposition 7.5.8, and the fact that R b s ZΣ, w (D∗ (Σ)) = bs (Σ) (cf. (7.5.33)).

7.6 Duality Results for Besov and Triebel-Lizorkin Spaces

405

Step 8. Under the conditions on the indices stipulated in the (first part of the) statement of Theorem 7.5.2, formula (7.5.58) holds. Proof. Suppose that 0 < r < min{p0, q0, p1, q1 }. We may then write the following string of equalities (which are individually justified shortly):    r   p ,q  1−θ  p1,q1  θ  r p ,q p ,q bs00 0 (Σ), bs11 1 (Σ) θ = bs00 0 (Σ) bs1 (Σ) =



p ,q0

bs00

(Σ)

 r  1−θ 

p ,q1

bs11

(Σ)

r θ

 p /r,q0 /r  1−θ  p1 /r,q1 /r θ bs +(n−1)(r−1)/(2r) (Σ) = bs 0+(n−1)(r−1)/(2r) (Σ) =



0

1

 p /r,q0 /r p /r,q1 /r bs 0+(n−1)(r−1)/(2r) (Σ), bs 1+(n−1)(r−1)/(2r) (Σ) θ 0 1 p/r,q/r

= bs+(n−1)(r−1)/(2r) (Σ)  p,q  r = bs (Σ) .

(7.5.61)

The first and fourth equalities in (7.5.61) are based on Theorem 1.4.18, the second and sixth equalities follow from formula (1.4.46), the third equality is a consequence of (7.5.50), while in the fifth equality we made use of the result proved in Step 7 and the fact that pi /r > 1 and qi /r> 1 for i ∈ {0, 1}.  p ,q  r  p,q  r p ,q In summary, we have that bs00 0 (Σ), bs11 1 (Σ) θ = bs (Σ) , so the claim made in Step 8 follows from this and the comment in Remark 1.4.16. Step 9. Formula (7.5.8) holds under the conditions on the indices stipulated in the (first part of the) statement of Theorem 7.5.2. Proof. This follows from the results proved in Step 5 and Step 8. This finishes the proof of the portion of Theorem 7.5.2 dealing with the scale of Besov spaces. The corresponding statements made for the scale of TriebelLizorkin spaces are proved analogously, making only natural changes in the above scheme (most notably, we use Proposition 7.5.9 in place of Proposition 7.5.8, and Lemma 7.5.11 in place of Lemma 7.5.10). The proof of Theorem 7.5.2 is therefore complete. 

7.6 Duality Results for Besov and Triebel-Lizorkin Spaces To discuss the duality theory of Besov and Triebel-Lizorkin spaces defined on Ahlfors regular sets, for any given p ∈ (0, ∞) define its Hölder conjugate exponent p as  p if 1 < p < ∞, p := p−1 (7.6.1) ∞ if 0 < p ≤ 1.

406

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Proposition 7.6.1 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, fix s ∈ (−1, 1). p ,q

p,q

(i) If 1 ≤ p < ∞, 0 < q < ∞, then the dual space of Bs (Σ, σ) is B−s (Σ, σ), where p and q are the Hölder conjugate exponents of p and q, respectively p ,q

(cf. (7.6.1)). More precisely, inone direction,  ∗ any given g ∈ B−s (Σ, σ) may be p,q identified with a functional in Bs (Σ, σ) in the following fashion. First, if  1 

, s + (n − 1) − 1 < γ < 1, (7.6.2) (−s)+ < β < 1 and max − s − n−1



p p then β,γ

p,q

G0 (Σ) ∩ Bs (Σ, σ)  f −→ Λg ( f ) :=

β,γ

G0

(Σ)

f, g

!

β,γ

(G0

(Σ))∗

(7.6.3)

is an assignment which satisfies |Λg ( f )| ≤ C f Bsp, q (Σ,σ) gB p , q (Σ,σ) −s

for each f ∈

β,γ G0 (Σ)

∩

p,q Bs (Σ, σ)

p ,q

and g ∈ B−s (Σ, σ),

(7.6.4)

for some constant C ∈ (0, ∞) which is independent of f , g. Granted this, the mapping (7.6.3) may be then extended, on account of Lemma 7.1.10, to a linear ∗ p,q functional Λ ∈ Bs (Σ, σ) satisfying Λ(Bsp, q (Σ,σ))∗ ≤ CgB p , q (Σ,σ) and −s one defines ! p, q p, q B s (Σ,σ) f , g (B s (Σ,σ))∗ := Λ( f ) (7.6.5) p,q p ,q

for each f ∈ Bs (Σ, σ) and g ∈ B−s (Σ, σ). In the opposite direction, there exists some constant C ∈ (0, ∞) with the property p,q that if Λ is any given linear functional on Bs (Σ, σ) then there exists a unique p ,q

g ∈ B−s (Σ, σ) with gB p , q (Σ,σ) ≤ CΛ and such that Λ extends Λg (defined −s above).  (ii) If 1 < p, q < ∞, or p = 1 and max n−1 < q < ∞, then the dual space of , n−1 n n+s p,q p ,q



Fs (Σ, σ) is F−s (Σ, σ), where p and q are the Hölder conjugate exponents of p and q, respectively (cf. (7.6.1)). More specifically, in one direction, any  p,q ∗ p ,q

given g ∈ F−s (Σ, σ) may be identified with a functional in Fs (Σ, σ) in the following manner. First, if β, γ are as in (7.6.2), it turns out that the assignment ! β,γ p,q G0 (Σ) ∩ Fs (Σ, σ)  f −→ Λg ( f ) := G β,γ (Σ) f , g (G β,γ (Σ))∗ (7.6.6) 0

0

satisfies |Λg ( f )| ≤ C f Fsp, q (Σ,σ) gF p , q (Σ,σ) −s

β,γ

p,q

p ,q

for each f ∈ G0 (Σ) ∩ Fs (Σ, σ) and g ∈ F−s (Σ, σ),

(7.6.7)

7.6 Duality Results for Besov and Triebel-Lizorkin Spaces

407

for some C ∈ (0, ∞) independent of f , g. Hence, this mapping  ∗ extends, p,q thanks to Lemma 7.1.10, to a linear functional Λ ∈ Fs (Σ, σ) satisfying Λ(Fsp, q (Σ,σ))∗ ≤ CgF p , q (Σ,σ) and one defines −s

p, q

Fs

f, g

(Σ,σ)

for each f ∈

!

p, q

(Fs

p,q Fs (Σ, σ)

(Σ,σ))∗

:= Λ( f ) p ,q

and g ∈ F−s (Σ, σ).

(7.6.8)

In the opposite direction, there exists C ∈ (0, ∞) with the property that if Λ is a p,q p ,q

linear functional on Fs (Σ, σ) then there exists a unique g ∈ F−s (Σ, σ) with gF p , q (Σ,σ) ≤ CΛ and such that Λ extends Λg . −s

Proof This is immediate from [89, Theorem 8.18, p. 246]. See also [91, Lemma 1.8, p. 18] or [86, Theorem 5.1, p. 64] for the case when p, q > 1.  In the context of Proposition 7.6.1, the manner in which Besov and TriebelLizorkin spaces embed into distributions (cf. Lemma 7.1.7) implies ! ! p, q = Lip (Σ,σ) f , g (Lip (Σ,σ))

p , q

B s (Σ,σ) f , g B−s (Σ,σ) (7.6.9) p ,q

whenever f ∈ Lip (Σ, σ) and g ∈ B−s (Σ, σ), and p, q

Fs

(Σ,σ)

f, g

!

p , q

F−s

(Σ,σ)

= Lip (Σ,σ) f , g

!

(Lip (Σ,σ))

p ,q

whenever f ∈ Lip (Σ, σ) and g ∈ F−s (Σ, σ).

(7.6.10)

It is significant to note that the duality pairings from Proposition 7.6.1 are actually compatible with one another. More precisely, we have the following result. Proposition 7.6.2 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. (i) Suppose 1 ≤ p0, p1 < ∞, 0 < q0, q1 < ∞, s0, s1 ∈ (−1, 1), and denote by p0 , p1 , q0 , q1 the Hölder conjugate exponents of p0, p1, q0, q1 (cf. (7.6.1)). Then one has ! ! p ,q = Bsp1, q1 (Σ,σ) f , g p1 , q1

p , q

B s 0 0 (Σ,σ) f , g 0 0 0

(Σ,σ) p ,q0

B−s0

1

p ,q1

for any f ∈ Bs00

(Σ, σ) ∩ Bs11

p0 ,q0

−s0

p1 ,q1

−s1

and g ∈ B

(Σ, σ) ∩ B

B−s1

(Σ, σ)

(Σ,σ)

(7.6.11)

(Σ, σ).

(ii) Fix two smoothness exponents s0, s1 ∈ (−1, 1) and suppose that there holds n−1 < q j < ∞ for either 1 < p0, p1, q0, q1 < ∞, or p j = 1 and max n−1 n , n+s j





j ∈ {0, 1}. Once again, denote by p0, p1, q0, q1 the Hölder conjugate exponents of p0, p1, q0, q1 (cf. (7.6.1)). Then one has

408

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets p , q0

Fs00

(Σ,σ)

f, g

!

p , q

F−s00 0 (Σ,σ) p0,q0 s0

for any f ∈ F

p0 ,q0

−s0

and g ∈ F

= Fsp1, q1 (Σ,σ) f , g

!

p ,q1

(Σ, σ) ∩ Fs11

p1 ,q1

−s1

(Σ, σ) ∩ F

p , q

1 (Σ,σ)

F−s11

1

(Σ, σ)

(7.6.12)

(Σ, σ).

Proof Fix β, γ ∈ (0, 1) sufficiently close to 1 so that the inequalities in (7.6.2) remain valid both with p, s replaced by p0, s0 and with p, s replaced by p1, s1 . Also, recall the family of operators PN , with N ∈ N, defined as in (7.3.37). Then for each p ,q

p ,q

p ,q p ,q f ∈ Bs00 0 (Σ, σ) ∩ Bs11 1 (Σ, σ) and each g ∈ B−s00 0 (Σ, σ) ∩ B−s11 1 (Σ, σ) we may write ! ! p ,q = lim Bsp0, q0 (Σ,σ) PN f , g p0 , q0

p , q

B s 0 0 (Σ,σ) f , g 0 0 B−s0

N →∞

(Σ,σ)

= lim

β,γ

= lim

p , q1 (Σ,σ)

N →∞ G0

PN f , g

(Σ)

1 N →∞ B s1

=

p , q1 (Σ,σ)

Bs 1

f, g

!

!

B−s0

β,γ

(G0

PN f , g

(Σ,σ)

(Σ))∗

!

p , q

1 (Σ,σ)

B−s11

,

p , q

1 (Σ,σ)

B−s11

(7.6.13)

thanks to Proposition 7.6.1 and (7.3.36). This establishes (7.6.11). The claim in item (ii) is dealt with analogously. 

7.7 Loose and Tight Embeddings We begin by reviewing what we refer to as loose embeddings, i.e., continuous inclusions within the Besov and Triebel-Lizorkin scales in which the smoothness and integrability exponents involved are related via inequalities. Here is the first such result. Proposition 7.7.1 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, let s ∈ (−1, 1). Then the following are true.  n−1 (i) If max n−1 n , n+s < p ≤ ∞ and 0 < q0 ≤ q1 ≤ ∞, one has p,q0

p,q

(Σ, σ) → Bs 1 (Σ, σ).   n−1 n−1 n−1 (ii) If max n−1 n , n+s < p < ∞ and max n , n+s < q0 ≤ q1 ≤ ∞, then Bs

p,q0

Fs (iii) If θ ∈ (0, 1 − s),

n−1 n

p,q1

(Σ, σ) → Fs

(Σ, σ).

(7.7.1)

(7.7.2)

< p ≤ ∞ and 0 < q0, q1 ≤ ∞, then p,q

p,q1

Bs+θ0 (Σ, σ) → Bs

(Σ, σ).

(7.7.3)

7.7 Loose and Tight Embeddings

(iv) If θ ∈ (0, 1 − s), max then

409

 n−1 n

,

n−1 n+s

< p < ∞ and max

p,q

 n−1 n

,

n−1 n+s

< q0, q1 ≤ ∞,

p,q

Fs+θ0 (Σ, σ) → Fs 1 (Σ, σ).  n−1 n−1  n−1 (v) If max n−1 n , n+s < p < ∞ and max n , n+s < q ≤ ∞, p,min{p,q }

Bs

p,max{p,q }

p,q

(Σ, σ) → Fs (Σ, σ) → Bs

(Σ, σ).

(7.7.4)

(7.7.5)

Proof See [89, Proposition 5.31, p. 140] (cf. also [91, Proposition 5.1, p. 84] in the case when 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞).  The next proposition describes loose embedding which happen to be compact, assuming the underlying closed Ahlfors regular set is bounded. Proposition 7.7.2 Assume that Σ ⊆ Rn is a compact Ahlfors regular set and abbreviate σ := H n−1 Σ. Then the following compact embeddings hold.  n−1 (i) Whenever −1 < s2 < s1 < 1, max n−1 < pi ≤ ∞ and 0 < qi ≤ ∞ for , n+s n i   i ∈ {1, 2}, and also s1 − s2 − (n − 1) p11 − p12 + > 0, it follows that the embedding p ,q1

p ,q

(Σ, σ) → Bs22 2 (Σ, σ) is compact. (7.7.6)  n−1 n−1  n−1 n−1 (ii) If −1 < s2 < s1 < 1, max n , n+si < pi < ∞, max n , n+si < qi ≤ ∞ for   i ∈ {1, 2} and s1 − s2 − (n − 1) p11 − p12 + > 0, then the embedding Bs11

p ,q1

Fs11

p ,q2

(Σ) → Fs22

(Σ, σ) is compact.

(7.7.7)

Proof See [199, Theorem 3.1, p. 586] (cf. also [91, Proposition 5.1, p.84] in the case 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞).  As a corollary of (7.7.6), (7.7.4), (7.1.55), and (7.1.38), we see that ∗

p,q

L p (Σ, σ) → B−s (Σ, σ) compactly (hence also continuously) if Σ ⊆ Rn is a compact Ahlfors regular set, σ := H n−1 Σ, and (7.7.8)   n−1 1 1 1 < p∗ < ∞, n+s < p ≤ ∞, 0 < q ≤ ∞, (n − 1) p∗ − p + < s < 1. Likewise, (7.7.6), (7.7.4), (7.1.55), and (7.1.38) imply that p,q

∗

Bs (Σ, σ) → L p (Σ, σ) compactly (hence also continuously) if Σ ⊆ Rn is a compact Ahlfors regular set, σ := H n−1 Σ, and (7.7.9)   0 < q ≤ ∞, (n − 1) p1 − p1∗ + < s < 1. 1 < p∗ < ∞, n−1 n < p ≤ ∞, We next turn to “tight” embeddings, i.e., continuous inclusions within the Besov and Triebel-Lizorkin scales in which the smoothness and integrability exponents involved are related via precise algebraic equalities (among other things). First, we record the following result of this flavor.

410

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Proposition 7.7.3 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Then the following embeddings hold.  n−1 (i) Whenever −1 < s2 < s1 < 1, max n−1 n , n+si < pi ≤ ∞ and 0 < qi ≤ ∞ for n−1 i ∈ {1, 2}, and also −1 < s1 − n−1 p1 = s2 − p2 < 1, it follows that the embedding p ,q1

Bs11

p ,q2

(Σ, σ) → Bs22

(Σ, σ) is continuous.

(7.7.10)

In addition, (7.7.10) holds whenever 1 ≤ pi = qi ≤ ∞ for i ∈ {1, 2} and n−1 0 < s0 ≤ s1 < 1 are such that s1 − n−1 p1 = s2 − p2 .   n−1 n−1 n−1 (ii) If −1 < s2 < s1 < 1, max n−1 n , n+si < pi < ∞, max n , n+si < qi ≤ ∞ for n−1 n−1 i ∈ {1, 2} and −1 < s1 − p1 = s2 − p2 < 1, then the embedding p ,q1

Fs11

p ,q2

(Σ, σ) → Fs22

(Σ, σ) is continuous.

(7.7.11)

Proof See [200, Theorem 2, p. 191]. The version of the embedding in (7.7.10) corresponding to 1 ≤ pi = qi ≤ ∞ for i ∈ {1, 2} and 0 < s0 ≤ s1 < 1 may be seen to hold thanks to [106, Proposition 5, p. 213] and Corollary 7.9.2. Parenthetically, we also wish to mention [91, Theorem 5.2, p. 89] as well as [85, Theorem 2, p. 294] for related results (valid for more restrictive ranges of indices) in the case of homogeneous Besov and Triebel-Lizorkin spaces.  The tight embedding of Besov spaces recorded in (7.7.10) may be established for other ranges of indices, as discussed in the theorem below. Ahlfors Theorem 7.7.4 Assume that Σ ⊆ Rn is a closed regular set and abbreviate  n−1 , σ := H n−1 Σ. If −1 < s0, s1 < 1 and max n−1 n n+si < pi ≤ ∞ for i ∈ {0, 1} are such that s0 s1 1 1 (7.7.12) p1 − n−1 = p0 − n−1 , then the embedding p ,q0

Bs00

p ,q1

(Σ, σ) → Bs11

(Σ, σ) is continuous

(7.7.13)

provided either 0 < q0 ≤ q1 ≤ ∞ and s0 ≥ s1,

(7.7.14)

0 < q0 < ∞, 0 < q1 ≤ ∞, and s0 > s1 .

(7.7.15)

or Proof For starters, observe that (7.2.10), (7.2.11), (7.2.27), Theorem 7.2.7, and Theorem 7.2.8 ensures that (7.7.13) holds in the case when the conditions in (7.7.14) are satisfied. To deal with the scenario in which the demands in (7.7.15) are assumed, suppose

7.7 Loose and Tight Embeddings

411

− 1 < s∗ < s < 1, 0 < q∗ < q < ∞,  n−1 max n−1 n , n+s < p ≤ ∞,  n−1 max n−1 n , n+s∗ < p∗ ≤ ∞, 1 p

−

s n−1

=

1 p∗

−

s∗ n−1 .

(7.7.16) (7.7.17) (7.7.18) (7.7.19)

Pick q0 ∈ (0, q) then define q1 := q0 . Taking q0 very close to q ensures that we may select a number θ ∈ (0, 1) satisfying

 1−s s∗ +1 n−1 n−1 . (7.7.20) , , 1 − , 1 − 1 − qq0 < θ < min 1−s 2 np (n+s)p ∗ The hypotheses in (7.7.16)-(7.7.19) guarantee that this is a meaningful demand on the parameter θ and, given that they additionally entail 0 < p < p∗ ≤ ∞ as well as p(n + s) < p∗ (n + s∗ ), we also have 

n−1 n−1 θ < min s+1 (7.7.21) 2 , 1 − np∗ , 1 − (n+s∗ )p∗ . Next, since 1/q∗ > 1/q > (1 − θ)/q0 = (1 − θ)/q1 , we may select q0, q1 > 0 small enough so that 1 1−θ θ 1 1−θ θ = + and = + . q q0 q0 q∗ q1 q1

(7.7.22)

Since (7.7.21) forces 1/p∗ < (n + s∗ )(1 − θ)/(n − 1) while (7.7.20) guarantees that ∗ −θ < s∗ < 1, we may choose some we also have −1 < s1−θ s1 ∈

 s∗ −θ 1−θ

 , s∗ ⊆ (−1, 1)

(7.7.23)

with the property that 1 (n + s1 )(1 − θ) . < p∗ n−1

(7.7.24)

Pick p1 ∈ (0, ∞) satisfying  p1 > max θp∗,

(n − 1)θ (n − 1)θ  ,  θ − s∗ + (1 − θ)s1 2 s∗ − (1 − θ)s1

(7.7.25)

 −1  then define p1 := (1 − θ) 1/p∗ − θ/ p1 ∈ (0, ∞). Hence, 1 1−θ θ = + p∗ p1 p1

(7.7.26)

and the choice of θ in (7.7.20) ensures that we have p1 > (n − 1)/n. Moreover, from n−1 hence, ultimately, (7.7.24) we see that p1 > n+s 1

412

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

max

 n−1 n

,

n−1 n+s1



< p1 < ∞.

(7.7.27)

To proceed, pick some p0 ∈ (0, ∞) which, for now, is assumed to satisfy  1

1 θ  −1 (n − 1)θ − p1 > p0 > max θ + , , θp p p∗ p1 θ − s∗ + (1 − θ)s1 + (n − 1)θ/ p1 (7.7.28) and remark that, in concert with (7.7.25), the aforementioned choice guarantees that p0 >

(n − 1)θ . θ + s∗ − (1 − θ)s1

(7.7.29)

Consider next p0 := (1 − θ)(1/p − θ/ p0 )−1 ∈ (0, ∞). The latter definition entails 1 1−θ θ = + . p p0 p0

(7.7.30)

Also, (7.7.20) forces 1/p < n(1 − θ)/(n − 1) hence 1 n(1 − θ) θ n−1 −

p n−1 p0 n

(7.7.31)

In addition, from (7.7.28) and definitions we see that 0 < p0 < p1 < ∞.

(7.7.32)

Pressing on, define s0 := s1 + (n − 1) As seen from (7.7.20), we have θ < further,

1−s 1−s∗



1 p0

−

1 p1



.

(7.7.33)

which implies s1 < s∗ < 1 −

s − s∗ 1 1 1 (1 − s1 )(1 − θ) 1 θ θ > = − > − − + n−1 n−1 p p∗ p p∗ p0 p1

1 1  (1 − θ)(s0 − s1 ) , = = (1 − θ) − p0 p1 n−1

s−s∗ 1−θ

hence,

(7.7.34)

from which we readily conclude that s0 < 1. From this, (7.7.33), (7.7.32), and (7.7.23) we see that −1 < s1 < s0 < 1 and Also, the fact that p1 >

n−1 n+s1

1 p0

−

s0 n−1

=

1 p1

−

s1 n−1 .

(7.7.35)

(cf. (7.7.27)) permits us to write

1 n s1 n s0 1 n + s1 1 = + = + + < − p1 n−1 n − 1 n − 1 n − 1 n − 1 p1 p0

(7.7.36)

7.7 Loose and Tight Embeddings

hence

1 p0


max

 n−1 n

Let us now define   s0 := θ −1 s − (1 − θ)s0 ,

,

n−1 n+s0 .

n−1 n+s0



Consequently,

< p0 < ∞.

(7.7.37)

  s1 := θ −1 s∗ − (1 − θ)s1 ,

(7.7.38)

s = (1 − θ)s0 + θ s0 and s∗ = (1 − θ)s1 + θ s1 .

(7.7.39)

which entail

Additionally, (7.7.38), (7.7.19), (7.7.33), and the definitions of p0, p1 give    s0 = θ −1 s∗ − (1 − θ)s1 + θ −1 s − s∗ − (1 − θ)(s0 − s1 ) 

1 = s1 + θ −1 (n − 1) − p 

1 = s1 + θ −1 (n − 1) − p

θ = s1 + θ −1 (n − 1) − p0

1 1 1 

− (1 − θ)(n − 1) − p∗ p0 p1

1 1 θ 1 θ 

− (n − 1) − − + p∗ p p0 p∗ p1 θ  n−1 n−1 = s1 + − . (7.7.40) p1 p0 p1

Starting with −θs∗ < θ, we may estimate s∗ < (s∗ + θ)/(1 − θ). As such, s1 < s ∗
(1 − θ)s1 − θ =⇒ s1 = > −1. (7.7.41) 1−θ θ

Also, based on (7.7.28), (7.7.19), the definitions of p0, p1 , (7.7.33), and (7.7.38) we may write s∗ − (1 − θ)s1 +(n − 1)

θ θ − (n − 1) < θ p0 p1

θ θ  max

while (7.7.44) and the first inequality in (7.7.28) give n − 1 n − 1

. , n n + s1

p1 > max

(7.7.47)

At this stage, from the version of (7.7.13) corresponding to (7.7.14) (which is presently applicable, thanks to (7.7.27), (7.7.35), (7.7.37), and bearing in mind that q0 = q1 ∈ (0, ∞)) we obtain p ,q0

Bs00

p ,q1

(Σ, σ) → Bs11

(Σ, σ) continuously,

(7.7.48)

while from part (i) in Proposition 7.7.3 (which may be invoked here with p1 := p0 , q1 := q0 , s1 := s0 , p2 := p1 , q2 := q1 , s2 := s1 , thanks to (7.7.43), (7.7.45), and (7.7.46)) we see that ) p ,) q0

Bs0 0

) p ,) q1

(Σ, σ) → Bs1 1

(Σ, σ) continuously.

(7.7.49)

In turn, from (7.7.48)-(7.7.49) and (7.5.8) in Theorem 7.5.2 (also bearing in mind (7.7.39)), we conclude that  p ,q  p,q ) p ,) q Bs (Σ, σ) = Bs00 0 (Σ, σ), Bs0 0 (Σ, σ) θ 0



p ,q1

→ Bs11 p ,q∗

= Bs∗∗ The conclusion so far is that

) p ,) q1

(Σ, σ), Bs1 1

(Σ, σ)

(Σ, σ) continuously.

 θ

(7.7.50)

7.7 Loose and Tight Embeddings

415 p ,q∗

p,q

Bs (Σ, σ) → Bs∗∗

(Σ, σ) continuously

whenever s, s∗, p, p∗, q, q∗ are as in (7.7.16)-(7.7.19).

(7.7.51)

Having established this, we may now eliminate the restriction that q∗ < q (appearing in (7.7.16)) by invoking item (i) in Proposition 7.7.1. This shows that (7.7.13) holds in the case when the conditions in (7.7.15) are satisfied.  From Proposition 7.1.9 we know that local Hardy spaces are contained in the Triebel-Lizorkin scale (cf. (7.1.56)). Using this, together with the relationship between the Triebel-Lizorkin scale and Besov scale (cf. (7.7.5)), and Theorem 7.7.4, we are then able to show that local Hardy spaces embed naturally into Besov spaces, in the precise manner described below. Corollary 7.7.5 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Then whenever  n−1 ∗ −1 < s < 1, max n−1 n , n+s < p < p ≤ 1, 1  (7.7.52) satisfy − 1 + (n − 1) p − p1∗ < s, and  n−1 n−1 max n , n+s < q < ∞, 0 < q∗ ≤ ∞, one has a well-defined continuous embedding p ∗,q ∗ (Σ, σ). s−(n−1)( p1 − p1∗ )

p,q

Fs (Σ, σ) → B

(7.7.53)

As a particular case, corresponding to s := 0 and q := 2, one has the well-defined continuous embeddings p ∗,q ∗ (Σ, σ) −(n−1)( p1 − p1∗ )

H p (Σ, σ) → h p (Σ, σ) → B n−1 n

whenever

< p < p∗ ≤ 1 and 0 < q∗ ≤ ∞.

(7.7.54)

Proof Assuming (7.7.52), the idea is to write p,q

p,max{p,q }

Fs (Σ, σ) → Bs

p ∗,q ∗ (Σ, σ), s−(n−1)( p1 − p1∗ )

(Σ, σ) → B

(7.7.55)

with the first embedding a consequence of (7.7.5), and the second embedding implied by Theorem 7.7.4. This proves (7.7.53), then (7.7.54) follows by specializing it to the case s := 0 and q := 2 (a valid choice in the context of (7.7.52)), bearing in mind the identification made in (7.1.56) and (7.1.51).  Having an adequate arsenal of embedding results at our disposal, we may now revisit the duals of Besov and Triebel-Lizorkin spaces and prove that, most of the time, these happen to be reasonably rich, even in the case when said spaces are merely quasi-Banach.

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7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Proposition 7.7.6 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate n−1 σ := H n−1 Σ. In addition, assume −1 < s < 1, max n−1 , n n+s < p ≤ ∞, and 0 < q < ∞. p,q p,q points2. Moreover, the dual of Fs (Σ, σ) Then the dual of Bs (Σ, σ) separates  n−1 its n−1 separates its points whenever max n , n+s < p, q < ∞. Proof We shall first treat the scale of Besov spaces in the scenario when  1 we  have n−1 max n−1 n , n+s < p < 1. The lower bound on p implies that s − (n − 1) p − 1 > −1, hence we may choose some exponent p∗ ∈ (1, ∞) with the property that   (7.7.56) s∗ := s − (n − 1) p1 − p1∗ > −1. Granted this, and also picking q∗ ∈ (1, ∞), we may now rely on Theorem 7.7.4 to conclude that the embedding p ,q∗

p,q

ι : Bs (Σ, σ) → Bs∗∗

(Σ, σ) is continuous.

(7.7.57)

p,q

Consider now 0  f ∈ Bs (Σ, σ). The goal is to show that there exists some linear p,q and continuous functional  : Bs (Σ, σ) → C with the property that ( f )  0. To p ,q justify this, since (7.7.57) ensures that 0  f ∈ Bs∗∗ ∗ (Σ, σ) and since the latter space is Banach (cf. item (3) in Remark 7.1.6), we may invoke ∗  p ,qthe Hahn-Banach Theorem to guarantee the existence of some functional Λ ∈ Bs∗∗ ∗ (Σ, σ) with  ∗ the p,q property that Λ( f )  0. Then  := ΛB p, q (Σ,σ) = Λ ◦ ι belongs to Bs (Σ, σ) (cf. s (7.7.57)) and ( f ) = Λ( f )  0. This finishes the proof of the fact that the dual of p,q Bs (Σ, σ) separates points in the case when p is strictly sub-unital. In the case when 1 ≤ p ≤ ∞, take q∗ := max{1, q}, and recall from item (i) of Proposition 7.7.1 that p,q

p,q∗

Bs (Σ, σ) → Bs

(Σ, σ).

(7.7.58)

p,q

Since Bs ∗ (Σ, σ) is a Banach space, the same type of reasoning as above may be p,q employed to conclude that the dual of Bs (Σ, σ) separates points in this case as well. In order to deal with the case of Triebel-Lizorkin spaces, assume next that we n−1 have max n−1 n , n+s < p, q < ∞. Then (7.7.5) gives that the embedding p,q

p,max{p,q }

Fs (Σ, σ) → Bs

(Σ, σ) is continuous.

(7.7.59)

Since we presently have max{p, q} < ∞, from what we have established in the first p,max{p,q } half of the proof we know that the dual of Bs (Σ, σ) separates points. With this in hand, the same type of reasoning then shows that the same is true for the p,q  Triebel-Lizorkin space Fs (Σ, σ).

p, q

2 in the sense that for any distinct f , g ∈ B s with the property that Λ f  Λg

 p, q ∗ (Σ, σ) there exists a functional Λ ∈ B s (Σ, σ)

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

417

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces Recall that a given topological vector space is said to be locally convex provided there exists a fundamental system of neighborhoods for the zero vector consisting of absorbing, balanced, and convex sets. We are interested in a similar yet more nuanced notion of convexity, which involves a parameter p ∈ (0, 1]. Specifically, assume X is a vector space and fix p ∈ (0, 1]. A set A ⊆ X is said to be p-convex provided  p p μ1 v1 + μ2 v2 : μ1, μ2 ≥ 0 and μ1 + μ2 = 1 ⊆ A for every v1, v2 ∈ A. (7.8.1) An alternative characterization (proved inductively on the number of vectors involved) is that A ⊆ X is p-convex if and only if A coincides with its p-convex hull, defined as M 

μ j v j : M ∈ N, {v j }1≤ j ≤M ⊆ A, {μ j }1≤ j ≤M ⊆ [0, ∞),

j=1

M 

p μ j = 1 . (7.8.2)

j=1

Given p ∈ (0, 1], call a topological vector space locally p-convex if there exists a fundamental system of neighborhoods for the zero vector consisting of absorbing, balanced, and p-convex sets. There is yet another related brand of convexity which we shall now describe. Consider a (complex) vector space X and fix p ∈ (0, 1]. The absolutely p-convex hull of a given set A ⊆ X is defined as A p :=

M  j=1

λ j v j : M ∈ N, {v j }1≤ j ≤M ⊆ A, {λ j }1≤ j ≤M ⊆ C,

M 

|λ j | p ≤ 1 .

j=1

(7.8.3) Obviously, A ⊆ A p , and A is said to be absolutely p-convex if A coincides with A p . It is clear from definitions that zA p ⊆ A p for each complex number z with |z| ≤ 1, ! A p = A p p and A p ⊆ Aq if 0 < p ≤ q ≤ 1.

(7.8.4) (7.8.5)

Also, if A is such that 0 ∈ A and za ∈ A for all a ∈ A and z ∈ C with |z| = 1, then A p is contained in the convex hull of A for any p ∈ (0, 1]. Whenever important to stress the dependence of the absolutely p-convex hull of a given set A ⊆ X on the space X,  in place of A p we shall write AX, p . space and Let X,  · X be a quasi-normed  denote by BX (0, 1) its (“open”) unit ball, that is, BX (0, 1) := x ∈ X :  xX < 1 (this is not an open set per se, as the quasi-norm is not necessarily a continuous function, but is a neighborhood of the origin, nonetheless). For each p ∈ (0, 1], define the Minkowski functional associated with the absolutely p-convex hull of the unit ball in X, i.e.,

418

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

  !

 x  := inf λ > 0 : λ−1 x ∈ BX (0, 1) , p p 

∀x ∈ X.

(7.8.6)

Whenever convenient the dependence on X of the functional in (7.8.6),     to emphasize we shall denote  ·  p by  · X, p . ! ! Remark 7.8.1 (i) Since BX (0, 1) ⊆ BX (0, 1) p , it follows that BX (0, 1) p is a neighborhood of the origin in X. In particular, for each given  x ∈ X the infimum in (7.8.6) is taken over a nonempty subset of (0, ∞). As such  x  p is a well-defined number in [0, ∞). Also,   !  x  < 1 for each x ∈ BX (0, 1) . (7.8.7) p p ! # Indeed, according to (7.8.3), any x ∈ BX (0, 1) p may be written as x = M j=1 λ j v j for # Msome pfinite family {v j }1≤ j ≤M ⊆ BX (0, 1) and with {λ j }1≤ j ≤M ⊆ C satisfying j=1 |λ j | ≤ 1. Hence, if we pick λ ∈ (0, 1) such that λ > max1≤ j ≤M v j  X , it # λ (v /λ) with {v j /λ}1≤ j ≤M still in BX (0, 1). In view of follows that λ−1 x = M   j=1 j j       (7.8.6), this forces x p ≤ λ < 1, proving (7.8.7). ! (ii) It turns out that any functional Λ ∈ X ∗ is actually bounded on BX (0, 1) p ! by ΛX ∗ . To see that this is the case, recall that any vector x ∈ BX (0, 1) p may be # written as x = M j=1 λ j v j for some finite collection of vectors {v j }1≤ j ≤M ⊆ BX (0, 1) # p and some finite family of complex numbers {λ j }1≤ j ≤M satisfying M j=1 |λ j | ≤ 1. ∗ Consequently, given any Λ ∈ X we may estimate |Λ(x)| ≤

M 

|λ j ||Λ(v j )| ≤ ΛX ∗

j=1

M 

|λ j |

j=1 M



|λ j | p

 1/p

≤ ΛX ∗ .

(7.8.8)

!  sup |Λ(x)| : x ∈ BX (0, 1) p ≤ ΛX ∗ .

(7.8.9)

≤ ΛX ∗

j=1

Hence, as claimed,

! In fact, since BX (0, 1) ⊆ BX (0, 1) p the opposite inequality in (7.8.9) is also true, so we ultimately conclude that  ! sup |Λ(x)| : x ∈ BX (0, 1) p = ΛX ∗ (7.8.10) for each functional Λ ∈ X ∗ . As a consequence, for each functional!Λ ∈ X ∗ , each vector x ∈ X, and each positive number λ such that λ−1 x ∈ BX (0, 1) p we may compute

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces



|Λ(x)| = λ|Λ(λ−1 x)| ≤ λ sup |Λ(z)| : z ∈ BX (0, 1)

419

! p

= λΛX ∗ ,

(7.8.11)

thanks to (7.8.10). Hence, taking the infimum over all λ’s with the specified properties and keeping (7.8.6) in mind, we arrive at   (7.8.12) |Λ(x)| ≤  x  p ΛX ∗ . (iii) If  · X is a p-norm on X for some p ∈ (0, 1] then the absolutely p-convex hull of unit ball in X actually coincides with the unit ball in that space, hence the quasi-norm given by (7.8.6) coincides with the quasi-norm in the original space, i.e., if  · X is p ∈ (0, 1] then  some  ! a p-norm on X for BX (0, 1) p = BX (0, 1) and  ·  p =  · X on X.

(7.8.13)

  Given a quasi-normed space X,  · X , we shall say that the dual of X separates its points (or that X is dual rich, or that X has a separating dual) provided for each x ∈ X \ {0} there exists a functional Λ ∈ X ∗ with the property that Λx  0.   Proposition 7.8.2 If X,  · X is a quasi-normed space and p ∈ (0, 1], then for each x, y ∈ X and z ∈ C one has            zx  = |z|  x  ,  x + y  p ≤  x  p +  y  p . p p p p p (7.8.14)     and  x q ≤  x  p whenever q ∈ [p, 1]. Also,

   x  ≤  xX , p

(7.8.15)

with   equality in the case when  · X is actually a p-norm (cf. (7.8.13)). In addition,  ·  satisfies the following “maximality” property: p if  ·  is some p-norm on X satisfying  · ≤  C · X for some constant C ∈ (0, ∞), then  ·  ≤ C  ·  p .

(7.8.16)

  Finally, if the dual of X separates its points, then  x  p = 0 if and only if x = 0.   Hence, in such a scenario,  ·  p is a p-norm (hence, in particular, a quasi-norm) on X. Proof If z = 0 then the first equality in (7.8.14) is clear from (7.8.6). When z  0, expressing z as ρeiθ for some ρ ∈ (0, ∞) and θ ∈ R, permits us to write

420

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

  !

 zx  = inf λ > 0 : eiθ (ρ−1 λ)−1 x ∈ BX (0, 1) p p 

 !

= inf λ > 0 : (ρ−1 λ)−1 x ∈ BX (0, 1) p

 !

= ρ inf λ > 0 : λ−1 x ∈ BX (0, 1) p     = ρ x  p = |z|  x  p,

(7.8.17)

where we have used (7.8.4) in the second equality. This establishes the first formula in (7.8.14). As regards the second formula in (7.8.14), observe that for each arbitrary ε > 0 there exist λx, λy > 0 satisfying          x  < λx <  x  + ε,  y  < λy <  y  + ε, (7.8.18) p p p p and

! λ−1 x x ∈ BX (0, 1) p,

λy−1 y ∈ BX (0, 1)

! p

.

(7.8.19)

With this in hand, one can write p (λx

λy x+y x y λx + p p 1/p = p p 1/p p 1/p λx (λx + λy ) λy + λy ) (λx + λy )  −1   −1  = μ1 λx x + μ2 λy y ,

where we have set μ1 := satisfy | μ1

|p

+ | μ2

|p

λx p p (λ x +λ y )1/p

and μ2 :=

λy . p p (λ x +λ y )1/p

(7.8.20)

Since these coefficients

= 1 we conclude from (7.8.3) and (7.8.19) that * ! + ! x+y ∈ B (0, 1) = BX (0, 1) p, X p p 1/p p p (λx + λy )

(7.8.21)

with the equality provided by the first formula in (7.8.5). As such,           1/p  x + y  ≤ (λxp + λyp )1/p <  x  + ε p +  y  + ε p . p p p

(7.8.22)

Sending ε  0 then yields the second formula in (7.8.14). Next, the third formula in (7.8.14) is  seen  directly from (7.8.6) and the second property recorded in (7.8.5). Clearly, 0 p = 0, so the claim made in (7.8.15) is obviously true when x = 0. In ! the case when x  0, for each θ ∈ (0, 1) we have θ x/ xX ∈ BX (0, 1) ⊆ BX (0, 1) p ,     hence θ  x  p = θ x  p ≤  xX by the first formula in (7.8.14) and (7.8.6). Sending θ  1 then establishes (7.8.15). Moving on, suppose  ·  is some p-norm on X satisfying  ·  ≤ C · X for some constant C ∈! (0, ∞). In light of (7.8.3), these properties imply x  ≤ C for each x ∈ BX (0, 1) p which, in concert with (7.8.6), forces  x ≤ C  x  p for each x ∈ X. This proves (7.8.16).

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

421

Finally, assume that  the  dual of X separates its points and consider some x ∈ X with the property that  x  p = 0. If x  0 then there exists Λ ∈ X ∗ such that Λ(x)  0. However, (7.8.12) written for this Λ and x then yields a contradiction. This shows that we necessarily have x = 0.    Remark 7.8.3 (i) In general, one does not expect that a finite multiple of  ·  p dominates  · X on X, as otherwise   the inequality in (7.8.15) would make the quasinorm  · X equivalent with  ·  p on X, eventually rendering X a locally p-convex space. This, of course, may not be the case for a generic quasi-normed space X (although, as a consequence of the Aoki-Rolewicz Theorem, X is locally r-convex for some r ∈ (0, 1]; cf. [138, Theorem 1.4, p. 5]).   (ii) For each p ∈ (0, 1], the quasi-norm  ·  p generates a locally p-convex topology, weaker (hence, having fewer open sets) the original topology on  than  X. Indeed, from Proposition 7.8.2 we know that  ·  p is a p-norm, while (7.8.15) implies that    BX (x, r) ⊆ x ∈ X :  x  p < r for each x ∈ X and r > 0 (7.8.23)     which, in turn, readily implies that any open set in the space X,  ·  p is also open   in the space X,  · X . We are now prepared to introduce the notion of p-envelope of a given quasinormed space X. Heuristically, this envelope may be thought of as the “smallest” locally p-convex topological space containing the original space X. Definition 7.8.4 Given a quasi-normed space X whose dual separates points and some exponent p ∈ (0, 1], the p-envelope of X,henceforth denoted by E p (X), is  defined as the completion of X in the quasi-norm  ·  p . A few comments are in order. according to the manner in which the com First,  pletion of X in the quasi-norm  ·  p is abstractly carried out, we may view E p (X) as the space of equivalence classes (which we shall indicate using brackets   [·]) of sequences of vectors in X which are Cauchy relative to the quasi-norm  ·  p . More precisely,   E p (X) = {x j } j ∈N : {x j } j ∈N ⊆ X and for each ε > 0 there exists

  N ∈ N such that  x j − xk  p < ε whenever j, k ≥ N , (7.8.24) with the convention that two sequences {x j } j ∈N and {y j } j ∈N of vectors in X belong to the same equivalence class if for each ε > 0 there exists a rank N ∈ N with the  property that  x j − y j  p < ε whenever j ≥ N.

(7.8.25)

422

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets



 Second, given any {x j } j ∈N ∈ E p (X), Proposition 7.8.2 implies that     p   p   p   ∀ j, k ∈ N.  x j p − xk p  ≤ x j − xk  p,

(7.8.26)

Hence,    x j 

p j ∈N

is a Cauchy sequence of numbers in [0, ∞).

In particular, this permits us to define        {x j } j ∈N  := lim  x j  p for all {x j } j ∈N ∈ E p (X). E p (X) j→∞

(7.8.27)

(7.8.28)

We may once again rely on Proposition 7.8.2 to see that the above definition is unambiguous, and that        p  p  {x j } j ∈N + {y j } j ∈N  p ≤  {x j } j ∈N  E p (X) +  {y j } j ∈N  E p (X) E p (X) (7.8.29)     for all {x j } j ∈N , {y j } j ∈N ∈ E p (X). Ultimately, E p (X) is a vector space and  ·  E p (X) is a p-norm on it. Third, the mapping     ιX : X,  · X −→ E p (X),  ·  E p (X) acting according to  ιX (x) := {x, x, . . . }] for each vector x ∈ X,

(7.8.30)

(7.8.31)

is linear, injective, continuous, with dense image, and satisfies

  ιX (x) E p (X) =  x  p ≤  xX for each x ∈ X,

(7.8.32)

hence, in particular, ιX X→E p (X) = sup

0x ∈X

   x 

p

 xX

≤ 1.

(7.8.33)

This is the manner in which we shall think of X as being embedded into E p (X). Remark 7.8.5 (i) When p = 1 in Definition 7.8.4, the space E p (X) corresponds to the so-called Banach envelope of X. For additional information on this matter we refer to [112], [127]. (ii) As seen from (7.8.13), if X is a p-Banach space, for some p ∈ (0, 1], whose dual separates points, then E p (X) = X and ιX is the identity operator in the context of (7.8.31).

(7.8.34)

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

423

Hence, in particular, if X is a Banach space, then for each p ∈ (0, 1] we have E p (X) = X and the mapping ιX from (7.8.31) becomes the identity operator.

(7.8.35)

A basic result is the fact that the p-envelope of a quasi-normed space with separating dual is a p-Banach space.   Proposition 7.8.6 Let X,  · X be a quasi-normed space whose duals separate points and fix some p ∈ (0, 1]. Then E p (X),  ·  E p (X) is a p-Banach space (hence, in particular, a quasi-Banach space). Proof In light of (7.8.30), there remains to show that the quasi-normed space  E p (X),  ·  E p (X) is complete. To this end, consider a Cauchy sequence {Xα }α∈N   in E p (X),  ·  E p (X) . This implies several things. First, for each  α ∈ N it follows that there exists a sequence {x αj } j ∈N ⊆ X which is Cauchy in  ·  p and such that   Xα = {x αj } j ∈N . Second, for each ε > 0 there exists Nε ∈ N with the property that  Xα − Xβ  E p (X) < ε whenever α, β ≥ Nε . According to (7.8.28), the latter property implies  β  (7.8.36) lim  x αj − x j  < ε if α, β ≥ Nε . p

j→∞

  Since for each α ∈ N fixed the sequence {x αj } j ∈N is Cauchy in  ·  p , we may select Jα ∈ N such that   α  x − x α  < α−1 if j, k ≥ Jα . (7.8.37) j k p If for each α ∈ N we now define yα := xJαα ∈ X, then (7.8.37) implies that for each α ∈ N we have    yα − x α  < α−1 if j ≥ Jα . (7.8.38) j p The claim we make at this stage is that   {yα }α∈N is a Cauchy sequence in  ·  p .

(7.8.39)

To see that this is the case, pick an arbitrary threshold ε > 0 and pick α, β ∈ N with α, β ≥ Nε . Then, if j ∈ N is such that j ≥ max{Jα, Jβ }, we may invoke (7.8.37) (twice) to estimate          yα − yβ  p ≤  yα − x α  p +  x α − x β  p +  yβ − x β  p p

j

p

j

j

p

j

p

 β  p ≤ α−p +  x αj − x j  + β−p . p

Passing to the limit j → ∞ and relying on (7.8.36) we conclude that    yα − yβ  p ≤ α−p + ε p + β−p . p

(7.8.40)

(7.8.41)

424

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

From this, the claim made in (7.8.39) follows. As a consequence of (7.8.39), it is meaningful to consider   X ∗ := {yα }α∈N ∈ E p (X). (7.8.42) To proceed, for each α ∈ N fixed consider the equivalence class of the stationary sequence whose terms are all equal to yα , i.e.,   Xα∗ := {yα, yα, . . . } ∈ E p (X). (7.8.43) Then for each α ∈ N we may compute    Xα − Xα∗  E p (X) = lim  yα − x αj  p < α−1, j→∞

(7.8.44)

on account of (7.8.28), (7.8.38), and (7.8.43). Let us also observe that for each α ∈ N fixed,   (7.8.45)  X ∗ − Xα∗  E p (X) = lim  yβ − yα  p, β→∞

thanks to (7.8.42), (7.8.43), and (7.8.28). On account of (7.8.45) and (7.8.39), we conclude that (7.8.46) Xα∗ → X ∗ as α → ∞, in E p (X). Combining this with (7.8.44) (and keeping in mind that  ·  E p (X) is a quasi-norm; cf. (7.8.30)), we finally conclude that Xα → X ∗ as α → ∞, in E p (X). (7.8.47)   This shows that the quasi-normed space E p (X),  ·  E p (X) is complete, finishing the proof of the proposition.  Remark 7.8.7 From (7.8.34) and Proposition 7.8.6 we see that if X is a quasi-normed space whose dual separates its points and, for some p ∈ (0, 1], the dual of E p (X) also separates its points then E p E p (X) = E p (X).

(7.8.48)

In fact, the following more general result holds: if X is a p-Banach space, for some p ∈ (0, 1], whose dual separates its points, then Eq (X) = X whenever 0 < q ≤ p.

(7.8.49)

Indeed, (7.8.49) follows from (7.8.34) upon noticing that if 0 < q ≤ p ≤ 1 then any p-Banach space is also a q-Banach space. As a consequence, if X is a quasi-normed space whose dual separates its points and, for some p ∈ (0,  1], the dual of E p (X) also separates its points then Eq E p (X) = E p (X) whenever 0 < q ≤ p.

(7.8.50)

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

425

Our next result identifies a dense subset of the unit ball in the envelope of a given dual-rich quasi-normed space.   Proposition 7.8.8 Let X,  · X be a quasi-normed space whose dual separates its points and fix some p ∈ (0, 1]. Also, recall the mapping ιX from (7.8.31). Then !    ιX BX (0, 1) p is a dense subset of B E p (X) (0, 1), the unit ball in E p (X),  ·  E p (X) . Proof The fact that ιX

BX (0, 1)

!  p

⊆ B E p (X) (0, 1)

(7.8.51)

is a consequence of (7.8.31), (7.8.28), and (7.8.7). To see that this inclusion is dense   with respect to the topology induced by  ·  E p (X) , pick an arbitrary {x j } j ∈N in B E p (X) (0, 1). This means that {x j } j ∈N is a sequence of vectors from  X which is Cauchy with respect to the quasi-norm  ·  p and satisfies lim  x j  p < 1. Fix an j→∞   arbitrary threshold ε > 0. Then there exist j∗ ∈ N and N ∈ N such that  x j∗  p < 1   and  x j − x j∗  p < ε whenever j ≥ N. In particular, x j∗ ∈ BX (0, 1)

! p

  and ιX (x j∗ ) = {x j∗ , x j∗ , . . . } .

(7.8.52)

Hence,       {x j } j ∈N − ιX (x j∗ )

E p (X)

    =  {x j − x j∗ } j ∈N 

E p (X)

  = lim  x j − x j∗  p < ε. j→∞

(7.8.53)

This ultimately shows that any element from B E p (X) (0, 1) may be approximated with !  in ε by an element from ιX BX (0, 1) p . Since ε > 0 has been chosen arbitrarily, it follows that the inclusion in (7.8.51) is indeed dense.  We next discuss a fundamental extrapolation procedure extending the action of a given bounded linear operator acting between two dual-rich quasi-normed spaces to their respective envelopes, with preservation of linearity and boundedness.     Proposition 7.8.9 Let X,  · X and Y,  · Y be two quasi-normed spaces whose duals separate theirs points and fix some p ∈ (0, 1]. Also, consider a linear and bounded operator     T : X,  · X −→ Y,  · Y . (7.8.54) Then T induces a linear and bounded mapping in the context         T : X,  · X, p −→ Y,  · Y, p .

(7.8.55)

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7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Moreover, the operator T from (7.8.54) extends uniquely to a linear and bounded operator     , : E p (X),  ·  E p (X) → E p (Y ),  ·  E p (Y) . (7.8.56) T Its operator norm satisfies   T (X,  | ·  | X, p )→(Y,  | ·  |Y, p ) ≤ , T  E p (X)→E p (Y) ≤ T (X,  ·  X )→(Y,  · Y )

(7.8.57)

and , ◦ ιX = ιY ◦ T on X. T

(7.8.58)     Proof Abbreviate by T X→Y the operator norm of T : X,  · X → Y,  · Y . If T X→Y = 0 the statement is obvious, so we shall assume in what follows that T X→Y > 0. For starters, we claim that     T x  ≤ T X→Y  x X, p, ∀x ∈ X. (7.8.59) Y, p ! To justify this estimate, first recall that any vector x ∈ BX (0, 1) p may be written as # x= M j=1 λ j v j for some finite collection of vectors {v j }1≤ j ≤M ⊆ BX (0, 1) and some # p finite family of complex numbers {λ j }1≤ j ≤M satisfying M j=1 |λ j | ≤ 1. Hence, since #M T x/T X→Y = j=1 λ j (T v j /T X→Y ) and each T v j /T X→Y belongs to BY (0, 1), ! we conclude from (7.8.3) that T x/T X→Y ∈ BY (0, 1) p . Thus, we ultimately have   T x  ≤ T X→Y , Y, p

∀x ∈ BX (0, 1)

! p

.

(7.8.60)

! Consider now an arbitrary x ∈ X and pick some λ > 0 such that λ−1 x ∈ BX (0, 1) p .   Then (7.8.60) permits us to estimate T(λ−1 x)Y, p ≤ T X→Y , which further implies   T x  ≤ λT X→Y for all x ∈ X and λ > 0 Y, p ! such that λ−1 x ∈ BX (0, 1) p .

(7.8.61)

With x ∈ X fixed, taking the infimum over all λ’s satisfying the aforementioned properties then yields (7.8.59) on account of (7.8.6). This proves the claim pertaining to (7.8.55). To proceed, observe that (7.8.59) implies that     T maps Cauchy sequences in X,  · X, p     (7.8.62) into Cauchy sequences in Y,  · Y, p . Next, we rely on (7.8.62) to (meaningfully) define

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces



, {x j } j ∈N T



427

, : E p (X) −→ E p (Y ) by setting T     := {T x j } j ∈N for each {x j } j ∈N ∈ E p (X).

(7.8.63)

To also show that the definition made in (7.8.63) is unambiguous start with two sequences {x j } j ∈N and {y j } j ∈N from the same equivalence class. Then (7.8.25) holds which, in concert with (7.8.62), implies that for every ε > 0 we have     T x j − T y j  ≤ T X→Y  x j − y j X, p < εT X→Y (7.8.64) Y, p whenever j ∈ N is large enough. Ultimately, this proves that {T x j } j ∈N and {T y j } j ∈N , belong to the same equivalence class, as desired. In turn, this readily implies  that T  is linear in the context of (7.8.63). Going further, for any given {x j } j ∈N ∈ E p (X) we may write         ,  =  {T x j } j ∈N  T {x j } j ∈N  E p (Y)

E p (Y)

    = lim T x j Y, p ≤ T X→Y lim  x j X, p j→∞

j→∞

  = T X→Y  {x j } j ∈N  E p (X) ,

(7.8.65)

thanks to (7.8.63), (7.8.28), and (7.8.59). , is linear and bounded To summarize, the argument so far shows the operator T the context of (7.8.63), with operator norm ≤ T X→Y (which takes care of the second estimate in (7.8.57)). Moreover, (7.8.58) is visible from (7.8.31) and (7.8.63). , : E p (X) → E p (Y ) is an extension of T : X → Y , under the This implies that T embeddings of X, Y into E p (X), E p (Y ) as in (7.8.31). The aforementioned extension is also unique since the aforementioned embeddings have dense ranges. Finally, the first estimate in (7.8.57) is justified by writing      , T {x }   j j ∈N   E p (Y) ,   sup T  E p (X)→E p (Y) =   {x } [{x j } j ∈N ]∈ E p (X)\{0} j j ∈N E p (X)

≥

sup

x ∈X\{0}

  ,  T ([{x, x, . . . }])  E p (Y)   =  {x, x, . . . }  E p (X)

= T (X,  | ·  | X, p )→(Y,  | ·  |Y, p ),

sup

x ∈X\{0}

  T x  Y, p    x  X, p (7.8.66)

thanks to definitions and (7.8.31). The proof of the proposition is therefore complete.  Remark 7.8.10 Let X be a quasi-normed space whose dual separates its points. Fix p ∈ (0, 1] and assume that the dual of E p (X) also separates its points. In particular, if Y := E p (X) then E p (Y ) = Y by (7.8.48). Recall the inclusion map ιX of X into

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7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Y from (7.8.31), and denote by , ιX : Y → Y the mapping associated with ιX as in Proposition 7.8.9. Then, based on definitions, for each x ∈ X we may write       , ιX ιX (x) = , ιX [{x, x, . . . ] = {ιX (x), ιX (x), . . . } = ιX (x). (7.8.67)         (Alternatively, , ιX ιX (x) = , ιX ◦ ιX (x) = ιY ◦ ιX (x) = IY ιX (x) = ιX (x), thanks ιX acts as the to (7.8.58) and the fact that ιY = IY .) Thus, the bounded operator , identity on the image of ιX . Since the latter is dense in Y (cf. (7.8.31)), we ultimately conclude that (7.8.68) , ιX = I E p (X) , the identity operator on E p (X). A remarkable consequence of Proposition 7.8.9 is the fact that the envelope of a dual-rich quasi-Banach space has the same dual as the original space. Proposition 7.8.11 Let X be a quasi-normed space whose dual separates its points and fix some p ∈ (0, 1]. Then one has the identification ∗  E p (X) = X ∗ isometrically via (7.8.69)   , ∈ E p (X) ∗ . X ∗  Λ −→ Λ  ∗ , for each Proof Consider the mapping Φ : X ∗ → E p (X) defined by Φ(Λ) := Λ Λ ∈ X ∗ , with the “hat” understood in the sense of (7.8.63). Since E p (C) = C (cf. (7.8.35)), Proposition 7.8.9 ensures that the mapping Φ is well defined, linear, and bounded. We claim that, in fact, Φ is an isometry. To prove this is the case, for each Λ ∈ X ∗ compute       !     , , Λ B E p (X) (0, 1)  = sup Λ ιX BX (0, 1) p  Λ(E p (X))∗ = sup ,   !   !   , ◦ ιX BX (0, 1) p  = sup |Λ(x)| : x ∈ BX (0, 1) p = sup  Λ = ΛX ∗ .

(7.8.70)

, in the space Above, first equality is simply the definition of the quasi-norm of Λ  the ∗ E p (X) (as in the past, B E p (X) (0, 1) denotes the unit ball in E p (X)). The second equality is a consequence of Proposition 7.8.8, while the third equality is plain algebra. Next, the fourth equality is justified by observing that for each x ∈ X we have   , ([{x, x, . . . }]) = [{Λx, Λx, . . . }] = Λx, , ◦ ιX (x) = Λ (7.8.71) Λ thanks to (7.8.31), (7.8.63), and the manner in which E p (C) is identified with C. Lastly, the final equality in (7.8.70) is provided by (7.8.10). This establishes (7.8.70) which, in turn, proves that Φ is indeed an isometry. ∗  ∗ Going further, let us also consider the   ∗ mapping Ψ : E p (X) → X defined by Ψ(Θ) := Θ ◦ ιX for each Θ ∈ E p (X) , where ιX is the mapping introduced in (7.8.31). Clearly, Ψ is also well-defined, linear, and bounded. Since from (7.8.71)

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

429

we know that , ◦ ιX = Λ on X, for each Λ ∈ X ∗, Λ

(7.8.72)

we conclude that Ψ(Φ(Λ)) = Λ for each Λ ∈ X ∗ . In the opposite direction, given  ∗ any Θ ∈ E p (X) , it follows that Θ ◦ ιX ∈ X ∗ satisfies 

   (7.8.73) Θ ◦ ιX ιX (x) = (Θ ◦ ιX )(x) = Θ ιX (x) for each x ∈ X, where the first equality is a consequence of (7.8.72). In turn, (7.8.73) implies that  ∗ Θ ◦ ιX and Θ are two functionals in E p (X) which agree on ιX (X), the image of the mapping ιX . Since, as noted in (7.8.31), the space ιX (X) is dense in E p (X), we ultimately  ∗ conclude that Θ ◦ ιX and Θ agree on E p (X). The fact that Θ ◦ ιX = Θ in  E p (X) translates into Φ(Ψ(Θ)) = Θ. The above reasoning shows that the mappings Φ and Ψ are inverse to one another. In particular, since Φ is an isometry it follows that Ψ is also an isometry, thus finishing the proof of (7.8.69).    On a given dual-rich quasi-Banach space, the action of the norm  · 1 may be characterized via duality, in the manner described in the proposition below.   Proposition 7.8.12 Let X,  · X be a quasi-normed space whose duals separate envelope points. Recall that E1 (X),  ·  E1 (X) is the Banach   of X (cf. item (i) of  Remark 7.8.5), and recall the embedding ιX : X,  · X → E1 (X),  ·  E1 (X) from (7.8.31) (with p := 1). Then for each x ∈ X one has 

    x  = ιX (x) E (X) = sup Λ(x) : Λ ∈ X ∗, ΛX ∗ = 1 . (7.8.74) 1 1 Proof Indeed, for each x ∈ X we may write    x  = ιX (x) E (X) 1 1  

  ∗ = sup  ιX (x)  :  ∈ E1 (X) , (E1 (X))∗ = 1  

 Λ ιX (x)  : Λ ∈ X ∗, ΛX ∗ = 1 = sup , 

 = sup Λ(x) : Λ ∈ X ∗, ΛX ∗ = 1 ,

(7.8.75)

  thanks to (7.8.32), the fact that E1 (X),  ·  E1 (X) is a Banach space (so the HahnBanach Theorem and its usual consequences are valid), the isometric identification made in Proposition 7.8.11, and (7.8.71).  For future purposes it is useful to note that taking “hats” preserves the identity.   Proposition 7.8.13 Let X,  · X be a quasi-normed space whose dual separates its points and fix some p ∈ (0, 1].

430

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

    Then the mapping ,IX : E p (X),  ·  E p (X) → E p (X),  ·  E p (X) associated as in Proposition 7.8.9 with IX , the identity operator on X, is actually I E p (X) , the identity   operator on E p (X),  ·  E p (X) . Proof With ιX denoting the mapping introduced in (7.8.31), repeated applications of (7.8.58) allow us to write ,IX ◦ ιX = ιX ◦ IX = ιX = I E p (X) ◦ ιX .

(7.8.76)

Hence, ,IX and I E p (X) coincide on ιX (X), the image of the mapping ιX . Since the latter is a dense subspace of E p (X) (cf. (7.8.31)), this ultimately forces ,IX = I E p (X) on E p (X).  It turns out that taking “hats” commutes with composition of operators. This is made precise in our next proposition.       Proposition 7.8.14 Let X,  · X , Y,  · Y , Z,  ·  Z , be three quasi-normed spaces whose duals separate their points and fix some p ∈ (0, 1]. Also, let         T : X,  · X → Y,  · Y and S : Y,  · Y → Z,  ·  Z (7.8.77) be two linear and bounded operators. Then, with “hats” considered in the sense of Proposition 7.8.9, it follows that , as operators S. ◦ T = S, ◦ T    from E p (X),  ·  E p (X) into E p (Z),  ·  E p (Z) . 

(7.8.78)

, are well-defined Proof Proposition 7.8.9 ensures that the operators S. ◦ T and S, ◦ T  linear and bounded mappings from E p (X),  ·  E p (X) into E p (Z),  ·  E p (Z) . In addition, if ιX , ιY , ι Z are the mappings associated with X, Y, Z as in (7.8.31), based on repeated applications of (7.8.58) we may write         S. ◦ T ◦ ιX = ι Z ◦ S ◦ T = ι Z ◦ S ◦ T = S, ◦ ιY ◦ T       , ◦ ιX . , ◦ ιX = S, ◦ T = S, ◦ ιY ◦ T = S, ◦ T

(7.8.79)

, agree on ιX (X), the image of the mapping ιX . Given Consequently, S. ◦ T and S, ◦ T that, as remarked in (7.8.31), the space ιX (X) is dense in E p (X), we ultimately , agree on E p (X).  conclude that S. ◦ T and S, ◦ T Other functorial properties of the “hat” operation are discussed in the proposition below.     Proposition 7.8.15 Suppose X,  · X and Y,  · Y are two quasi-normed spaces with separating duals and fix some exponent    p ∈ (0, 1]. Given some linear and bounded operator T : X,  · X → Y,  · Y , consider the linear and bounded

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

431

    , : E p (X),  ·  E p (X) → E p (Y ),  ·  E p (Y) associated with T as in operator T Proposition 7.8.9. Then the following statements are true.   −1 / , is an isomorphism and, in fact, T , = T −1 . (1) If T is an isomorphism, then T , (2) If X, Y are quasi-Banach spaces and T is surjective, then T is surjective. Proof To justify the claim made in item (1), invoke Proposition 7.8.14 and Proposition 7.8.13 to write 0 −1 = T , ◦ T/ T ◦ T −1 = ,IX = I E p (X),

(7.8.80)

−1 ◦ T −1 ◦ T = , , = T0 T/ IX = I E p (X) .

  −1 / , is an isomorphism and, in fact, T , = T −1 . This proves that T Let us now deal with the claim made in item (2). Consider an arbitrary vector y  E p (Y) ≤ 1. Then  y ∈ E p (Y ) satisfying   Proposition  7.8.8 ensures that there exists ! y∗ ∈ BY (0, 1) p with the property that  y −ιY (y∗ ) E p (Y) ≤ 1/2. The aforementioned M # membership guarantees that y∗ may be represented in the form y∗ = λ j y j for some j=1

finite family of vectors {y j }# 1≤ j ≤M ⊆ BY (0, 1) and some finite family of numbers p {λ j }1≤ j ≤M ⊆ C satisfying M j=1 |λ j | ≤ 1. Given that we are currently assuming that X, Y are quasi-Banach spaces and the operator T is surjective, we may invoke [138, Corollary 6.62, p. 423] to conclude that there exist a number C ∈ (0, ∞) and a family of vectors {x j }1≤ j ≤M ⊆ X such that T x j = y j and  x j X ≤ C for M # every j ∈ {1, . . . , M }. Define x∗ := λ j x j ∈ X and pick Co ∈ (C, ∞). Since x∗ /Co may be represented as

M # j=1

j=1

λ j (x j /Co ) with {x j /Co }1≤ j ≤M ⊆ BX (0, 1) and

    |λ j | p ≤ 1, it follows that  x∗ /Co X, p ≤ 1. Hence,  x∗ X, p ≤ Co which then j=1     implies ιX (x∗ ) E p (X) =  x∗ X, p ≤ Co . Let us also observe that by design T x∗ = y∗ which, in concert with (7.8.58), allows us to write   , ιX (x∗ ) = ιY (T x∗ ) = ιY (y∗ ). T (7.8.81) M #

     , ιX (x∗ )  Consequently,  y −T =  y − ιY (y∗ ) E p (Y) ≤ 1/2. Let us summarize E p (Y) our progress. Abbreviating  x := ιX (x∗ ) ∈ E p (X), the argument so far shows that there exists a constant Co ∈ (0, ∞) with the property that for each vector  y ∈ E p (Y ) with  y  E p (Y) ≤ 1 there exists some  x ∈ E p (X) satisfying   ,  x  E p (X) ≤ Co and such that  y −T x  E p (Y) ≤ 1/2.

(7.8.82)

With this in hand, and recalling from Proposition 7.8.6 that E p (Y ) is a quasi-Banach space, we may now rely on [138, Proposition 6.60, pp. 421–422] to conclude that , is onto, as desired. the operator T 

432

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

  Let X,  · X be a quasi-normed  space. A projection of X is a linear, bounded,  idempotent operator P on X,  · X , where the last property amounts to P2 = P on X. As a result, the image Y := P(X) of a projection P of X satisfies   Y = Ker IX − P : X → X and Y ⊕ Ker (P : X → X) = X, (7.8.83) where IX is the identity operator on X. In particular, Y is a (topologically) complemented closed subspace of X. Conversely, any (topologically) complemented closed subspace of X is the range of a projection of X. It turns out that the envelope of the a (topologically) complemented space or, equivalently, the envelope of the range of a projection, behaves in a natural fashion, as indicated in the proposition below.   Proposition 7.8.16 Let X,  · X be a quasi-normed space with separating dual and fix some p ∈ (0, 1]. Also, consider a (topologically) complemented closed subspace Y of X.   Then Y := Y,  · X is a quasi-normed space with separating dual and   (7.8.84) E p (Y ) := E p Y,  · X embeds continuously into E p (X).   In fact, if X,  · X is actually a quasi-Banach space and P : X → X is a projection , : E p (X) → E p (X) considered in the sense of Proposition 7.8.9 of X onto Y , then P is a projection of E p (X) onto E p (Y ); in particular, in such a scenario one has     , E p (X) , or E p (PX) = P , E p (X) , E p (Y ) = P

    , E p (X) , or E p Ker IX − P : X → X = P

   , : E p (X) → E p (X) . or E p Ker IX − P : X → X = Ker I E p (X) − P

(7.8.85)

  Proof Since X,  · X is a quasi-normed space with separating dual it follows that Y := Y,  · X inherits these properties. To proceed, let ι be the inclusion map of Y into X. Hence, ι : Y → X is linear and bounded. As such, Proposition 7.8.9 ensures that ι extends to a linear and bounded map   , ι : E p (Y ) := E p Y,  · X −→ E p (X). (7.8.86) We claim that , ι is also injective. To prove this claim, according to (7.8.24), (7.8.63), and (7.8.28) it suffices to show that   if {y j } j ∈N ⊆ Y is such that y j → 0 as j → ∞ in  · X, p   (7.8.87) then we necessarily have y j → 0 as j → ∞ in  · Y, p . We shall actually prove that there exists a constant C ∈ (0, ∞) with the property that      ·  ≤ C  · X, p on Y . (7.8.88) Y, p

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

433

Of course, (7.8.87) is immediate from (7.8.88). To justify   (7.8.88), pick an arbitrary vector y ∈ Y and select a positive number a >  y X, p . In view of (7.8.6), this ! guarantees the existence of some λ ∈ (0, a) such that λ−1 y ∈ BX (0, 1) X, p . Upon recalling (7.8.6), we see that we may express λ−1 y =

M 

λi vi with

i=1

M 

|λi | p ≤ 1 and {vi }1≤i ≤M ⊆ BX (0, 1).

(7.8.89)

i=1

Since Y is assumed to be a (topologically) complemented closed subspace of X, there exist a closed subspace Z of X along with a constant C ∈ (0, ∞) such that any x ∈ X may be uniquely decomposed as y + z with y ∈ Y , z ∈ Z  and, in addition, the estimate max  yX , zX ≤ C xX holds.

(7.8.90)

Use (7.8.90) to split each in (7.8.89) as vi = wi + zi with wi ∈ Y and  vi appearing zi ∈ Z satisfying max wi X , zi X ≤ Cvi X < C. Then λ−1 y =

M  i=1

λi vi =

M



M    λ i wi + λi zi .

i=1

Since Y ∩ Z = {0}, the fact that λ−1 y ∈ Y forces λ−1 y =

M 

(7.8.91)

i=1

#M

i=1

λ i wi .

λi zi = 0 and (7.8.92)

i=1

#M p Given that wi /C ∈ BY (0, 1) and also i=1 |λi | ≤ 1, from (7.8.92) and (7.8.3) we ! −1 conclude that λ y/C ∈ BY (0, 1) Y, p . On account of (7.8.6), this further implies      −1  λ y  ≤ C. Thus,  y Y, p ≤ Cλ < Ca. By letting a   y X, p we ultimately Y, p     arrive at  y Y, p ≤ C  y X, p . This establishes (7.8.88). The claim made in (7.8.87) is therefore justified and, hence, the mapping , ι in (7.8.86) is indeed injective.   Going further, strengthen the original hypotheses by assuming X,  · X is actually a quasi-Banach space, and observe that this further entails that Y,  · X is a quasi-Banach space. Also, suppose P : X → X is a projection of X onto , : E p (X) → E p (X) in the sense of Proposition 7.8.9, which is Y and consider P , known to be linear and bounded. Since Proposition 7.8.14 also guarantees that P , is idempotent, that P is a projection of the space E p (X). Next, define it follows   P# : X,  · X → Y,  · X by setting P# x := Px for each x ∈ X. Given that Y = P(X), this is a well-defined linear and bounded operator which, by design, is surjective. Granted this, from part (2) of Proposition 7.8.15 we conclude that 1 P# : E p (X) → E p (Y ) is also surjective, hence   1 P# E p (X) = E p (Y ). In addition, P = ι ◦ P# so Proposition 7.8.14 implies

(7.8.93)

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7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

,=, P ι◦1 P# .

(7.8.94)

Consequently, on account of (7.8.94) and (7.8.93) we may write

      , E p (X) = , P ι 1 P# E p (X) = , ι E p (Y ) ≡ E p (Y )

(7.8.95)

where the last step takes into consideration the specific manner in which E p (Y ) is ι, that is). Since (7.8.95) may be identified with a subspace of E p (X) in (7.8.84) (via, , interpreted as saying that the projection P of E p (X) maps onto E p (Y ), the proof of the proposition is complete.  Next we present the following general criterion for identifying the p-envelope of a given quasi-normed space with a rich dual.   space whose dual separates its Theorem 7.8.17 Let E,  · E be a quasi-normed   points. Consider a p-Banach space F,  · F , for some p ∈ (0, 1], whose dual separates its points and with the property that E ⊆ F and the inclusion ι : E → F is continuous with dense range. Then the following statements are equivalent. (1) There exists an isomorphism3 from E p (E) onto F which fixes E. More specifically, there  exists Φ : E p (E) → F isomorphism with the property that ΦE = I, the identity operator on E, where E is regarded as a subspace of E p (E) (via the embedding ιE ; cf. (7.8.31)). (7.8.96)   (2) The quasi-norms  · E, p and  · F are equivalent on E. (3) There exists a constant C ∈ (0, ∞) such that    x  ≤ C xF for each x ∈ E. (7.8.97) E, p (4) The space E has a good approximation of the identity (relative to the ambient F), in the sense that there exists a sequence of linear operators PN : E → E indexed by N ∈ N with the property that one may find a constant C ∈ (0, ∞) such that   (7.8.98) sup PN x E, p ≤ C xF for each x ∈ E, N ∈N

and PN x − xE → 0 as N → ∞ for each fixed x ∈ E.

(7.8.99)

(5) There exists a constant c ∈ (0, ∞) with the property that E ∩ BF (0, c) ⊆ BE (0, 1)

3 linear, continuous, bijective map

!

 · E E, p

(7.8.100)

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces



435



where BF (0, c) := x ∈ F :  xF < c and the vertical bar stands for closure in the topology induced by the indicated quasi-norm. (6) There exists a constant C ∈ (0, ∞) with the property that for each vector x ∈ E one may find a sequence {x N } N ∈N ⊆ E such that   sup  x N E, p ≤ C xF (7.8.101) N ∈N

and

  x = lim x N in E,  · E . N →∞

(7.8.102)

Remark 7.8.18 Whenever (7.8.96) holds we shall simply say that F is the p-envelope of E, and (via a slight abuse of notation) write F = E p (E), in place of the identification of F with E p (E) via the isomorphism Φ. Before presenting the proof of Theorem 7.8.17 we wish to make two comments. First, as evident from item (6), the operators PN : E → E with N ∈ N from item (4) need not be linear. Second, Theorem 7.8.17 specialized to the case p = 1 yields criteria for identifying the Banach envelope of a given quasi-normed space with separating dual. Proof of Theorem 7.8.17 To check the implication (1) ⇒ (2), for each x ∈ E write     xF = Φ ιE (x) F ≈ ιE (x) E p (E)     =  {x, x, . . . }  E p (E) =  x E, p . (7.8.103)  Above, the first equality is a consequence of the fact that ΦE = I, the identity operator on E, given the manner in which E is regarded as a subspace of E p (E) (via the embedding ιE ; cf. (7.8.31)). The subsequent equivalence is a consequence of the Open Mapping Theorem for quasi-Banach spaces (cf., e.g., [138, Corollary 6.62, p. 423]), bearing in mind Proposition 7.8.6. For the remaining equalities in (7.8.103) see (7.8.31), (7.8.32). Next, the implication (2) ⇒ (3) is trivial. As regards the implication (3) ⇒ (4), assume (7.8.97) holds and take PN := I, the identity operator on E, for each N ∈ N. Obviously, these are linear operators on E satisfying (7.8.98)-(7.8.99). Moving on, consider the implication (4) ⇒ (5). With C ∈ (0, ∞) as in (7.8.98), define c := C −1 . Then for each x ∈ E ∩ BF (0, c) we may rely on (7.8.98) to estimate   PN x  ≤ C xF < 1 for each N ∈ N. (7.8.104) E, p In light of this, (7.8.6) implies that for each !N ∈ N there exists some number ! λ N ∈ (0, 1) such that λ−1 N · PN x ∈ BE (0, 1) E, p . Hence, PN x ∈ BE (0, 1) E, p for each N ∈ N, thanks to (7.8.4). From this and (7.8.99) we then conclude that !  · E . In view of the arbitrariness of x, this ultimately establishes x ∈ BE (0, 1) E, p (7.8.100).

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7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Turning to the implication (5) ⇒ (6), pick an arbitrary vector x ∈ E. If x = 0, then (7.8.101)-(7.8.102) are trivially satisfied (for any C ∈ (0, ∞))by taking x N := 0 for each N ∈ N. When x  0, choose an arbitrary number λ ∈ 0, c/ xF where c ∈ (0, ∞) is as in (7.8.100), and note that this entails λ x ∈ E ∩ BF (0, c). Upon recalling that we are presently assuming (7.8.100), we then conclude that there !  exists x N in E,  · E . If we a sequence { x N } N ∈N ⊆ BE (0, 1) E, p such that λ x = lim  N →∞   x N for each N ∈ N, it follows that x = lim x N in E,  · E now define x N := λ−1  and

N →∞

    x N E, p ≤ λ−1 . sup  x N E, p = λ−1 sup 

N ∈N

N ∈N

Upon letting λ  c/ xF in (7.8.105) we therefore arrive at   sup  x N E, p ≤ c−1  xF N ∈N

(7.8.105)

(7.8.106)

which proves (7.8.101) with C := c−1 . Finally, consider the implication (6) ⇒ (1). Assume the existence of C ∈ (0, ∞) with the property that for each x ∈ E one may find a sequence {x N } N ∈N ⊆ E satisfying (7.8.101)-(7.8.102). The goal is to show that F is isomorphic to the penvelope of E, in a manner that fixes E. To get started, observe that (7.8.13) and (7.8.59) presently ensure that      xF =  x F, p ≤ ιE→F  x E, p for all x ∈ E. (7.8.107) (Alternatively, the same conclusion may be justified using (7.8.16).) Also, thanks to Proposition 7.8.9 and (7.8.34), the inclusion ι : E → F extends to a bounded, linear operator (7.8.108) , ι : E p (E) −→ F which fixes E (regarded as a subspace of E p (E) in the manner described in (7.8.31)) and acts according to the following scheme:   Start with {x j } j ∈N ∈ E p (E).  Then the sequence {x j } j ∈N ⊆ E is Cauchy with respect to  · E, p hence, thanks to the estimate in   (7.8.107), {x j } j ∈N ⊆ E is Cauchy with respect to  · F . Since F,  · F (7.8.109) is complete, the limit lim x j exists in the latter space, and we set j→∞    , ι {x j } j ∈N E = lim x j ∈ F. j→∞

The goal now becomes proving that the extension described in (7.8.108)-(7.8.109) is, in fact, an isomorphism. From (7.8.108)-(7.8.109) it is clear that this operator is one-to-one if we succeed in showing that   if {x j } j ⊆ E is Cauchy with respect to  · E, p and x j → 0 in F   (7.8.110) then necessarily x j → 0 in  · E, p as j → ∞.

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

437

  Consider a sequence as in the first line of (7.8.110). Upon recalling that  · 

is a p-norm on E (cf. Proposition 7.8.2), for each j ∈ N and N ∈ N we may estimate   p    p  p  x j  ≤  x j − (x j ) N E, p + (x j ) N E, p (7.8.111) E, p E, p

where {(x j ) N } N ∈N is the sequence associated with each vector x j as in item (6). In particular, the current hypotheses guarantee  given ε > 0 we may find  that for each some jε ∈ N such that  x jε F ≤ ε and  x jε − (x jε ) N E ≤ ε whenever N is large enough. Assuming this is the case, (7.8.15) used with X := E permits us to estimate      x j − (x j ) N  ≤  x jε − (x jε ) N E ≤ ε. (7.8.112) ε ε E, p   Also, (7.8.101) gives (x jε ) N E, p ≤ C x jε F ≤ Cε. In concert with (7.8.111),   these estimates allow us to conclude that  x jε E, p ≤ ε(1 + C p )1/p . This proves that   {x j } j ∈N contains a sub-sequence which is convergent to zero with respect to  · E, p .   Being Cauchy in the p-norm  · E, p , then the entire sequence {x j } j ∈N converges   to zero with respect to  · E, p . This finishes the proof of (7.8.110) which, in turn, shows that , ι in (7.8.108)-(7.8.109) is indeed one-to-one. Let us turn our attention to the ontoness of, ι in (7.8.108)-(7.8.109). Fix an arbitrary x ∈ E and consider the sequence {x N } N ∈N associated with the vector x as in item (6). Then, thanks to (7.8.15) and (7.8.101) (also mindful of the fact that  · E, p is a p-norm on E; cf. Proposition 7.8.2), for each N ∈ N we may write   p   p   p p p  x  ≤  x N − x E, p +  x N E, p ≤  x N − xE + C p  xF . (7.8.113) E, p Based on this and (7.8.102), sending N → ∞ yields    x  ≤ C xF for every x ∈ E, E, p

(7.8.114)

where C ∈ (0, ∞) is as in (7.8.114). Next, keeping in mind that E is viewed as a subspace of E p (E) as indicated in (7.8.31), we claim that   E ∩ BF (0, C −1 ) ⊆ , ι E ∩ B E p (E) (0, 1) .

(7.8.115)

  To see this, pick an arbitrary vector x ∈ E ∩ BF (0, C −1 ). Then {x, x, . . . } belongs to the identification of E with a subspace of E p (E) (cf. (7.8.31)) and satisfies      {x, x, . . . }  =  x E, p ≤ C xF < 1, (7.8.116) E p (E)   thanks to (7.8.114) and the fact that x ∈ BF (0, C −1 ). Thus, {x, x, . . . } belongs to

   E ∩ B E p (E) (0, 1) and (7.8.109) gives , ι {x, x, . . . } = x. The proof of the claim made in (7.8.115) is therefore finished. −1  Moving  on, we wish to remark that, in general, BF (0, C ) is not an open set in F,  · F although this is a neighborhood of the origin (cf. [138, (3.543), p. 149]).

438

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

  In particular, there exists an open set O in the space F,  · F with 0 ∈ O and such that O ⊆ BF (0, C −1 ). Since we assume that E is densely embedded into F, we may then use (7.8.115) to write O⊆O

 · F

=E∩O

 · F

⊆ E ∩ BF (0, C −1 )

 · F

   · F ⊆, ι B E p (E) (0, 1)

(7.8.117)

where, in each case, the vertical bar stands for closure in the topology induced by the indicated quasi-norm. Noting that the set on the left-most  is  side of (7.8.117) a neighborhood of the zero vector in F, and recalling that E p (E),  ·  E p (E) is a p-Banach space, hence complete (cf. Proposition 7.8.6), we may now invoke the version of the Open Mapping Theorem established in [138, Theorem 6.49, p. 408] to conclude that , ι in (7.8.108)-(7.8.109) is an open mapping. Consequently, , ι is onto (hence, ultimately, bijective) and has a continuous inverse in the context of (7.8.108).  As an example, Theorem 7.8.17 yields an elegant proof of the fact that   ∗ (7.8.118) E p∗  p (N) =  p (N) whenever 0 < p ≤ p∗ ≤ 1. ∗

Indeed, for starters, E :=  p (N) and F :=  p (N) satisfy the background hypotheses made in the statement of Theorem 7.8.17 (with p replaced by p∗ ). Next, for each N ∈ N define PN : E → E by setting    PN {λ j } j ∈N := λ1, λ2, . . . , λ N −1, λ N , 0, 0, 0, . . . (7.8.119) for each {λ j } j ∈N ∈  p (N). Hence, for each fixed {λ j } j ∈N ∈ E, we have   

1/p     |λ j | p → 0 as N → ∞. PN {λ j } j ∈N − {λ j } j ∈N  = E

(7.8.120)

j>N

Also, if for each j ∈ N we consider v j := {δ jk }k ∈N ∈ BE (0, 1), then for each N ∈ N we have N    PN {λ j } j ∈N = λ j v j for each {λ j } j ∈N ∈ E. (7.8.121) j=1

In turn, from (7.8.121), (7.8.6), and (7.8.3) we conclude that     PN {λ j } j ∈N 

E, p ∗

≤

#

N j=1

|λ j | p

∗

1/p∗

  ≤ {λ j } j ∈N F

for each N ∈ N and {λ j } j ∈N ∈ E.

(7.8.122)

Together, (7.8.120) and (7.8.122) show that the demands in item (6) of Theorem 7.8.17 are satisfied by the family of operators {PN } N ∈N . As such, Theorem 7.8.17 applies and gives (7.8.118).

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

439

In turn, Theorem 7.8.17 is the basis for other procedures capable of identifying the p-envelope of a given quasi-normed space with separating dual. In Proposition 7.8.19 below we discuss such a procedure which allows transferring the quality of being the p-envelope from one pair of quasi-normed spaces to another granted the existence of (abstract) restriction/extension operators between said pairs.   Proposition 7.8.19 Let Xi,  · Xi with i ∈ {1, 2} be two quasi-normed spaces with separating duals, satisfying X1 ⊆ X2 , and such that the inclusion ι1 : X1 → X2 is continuous. (7.8.123)   Consider also two quasi-normed spaces Yi,  · Yi with i ∈ {1, 2} satisfying Y1 ⊆ Y2 , and such that (7.8.124) the inclusion ι2 : Y1 → Y2 is continuous. Next, for i ∈ {1, 2} let         Ri : Xi,  · Xi → Yi,  · Yi and Ei : Yi,  · Yi → Xi,  · Xi

(7.8.125)

be linear and bounded operators satisfying (again, for i ∈ {1, 2}) Ri ◦ Ei = IYi , the identity operator on Yi .

(7.8.126)

In addition, assume that R2 extends R1 and E2 extends E1 , in the sense that ι2 ◦ R1 = R2 ◦ ι1 as operators from X1 into Y2,

(7.8.127)

ι1 ◦ E1 = E2 ◦ ι2 as operators from Y1 into X2 .

(7.8.128)

Then, given any p ∈ (0, 1], the following implication holds: E p (X1 ) = X2 =⇒ Y1, Y2 have separating duals and E p (Y1 ) = Y2 .

(7.8.129)

Proof Fix p ∈ (0, 1] and work under the assumption that E p (X1 ) = X2 . In particular, after eventually replacing  · X2 with an equivalent quasi-norm on X2 (something which does not affect any of the hypotheses made in the statement) we may, in light of Proposition 7.8.6 and (7.8.31), assume that   X2,  · X2 is a p-Banach space, and the inclusion (7.8.130) ι1 : X1 → X2 is continuous with dense range. We next claim that   (7.8.131) Y2,  · Y2 is a quasi-Banach space.   By assumption, Y2,  · Y2 is a quasi-normed space so we only need  to check  completeness. To this end, suppose {y j } j ∈N is a Cauchy sequence in Y2,  · Y2 .  space has Then {E2 y j } j ∈N is a Cauchy sequence in X2,  · X2 and, since the latter  been shown to be complete, it follows that {E2 y j } j ∈N converges in X2,  · X2 to

440

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

some x ∈ X2 . In concert with (7.8.126), this  allows us to conclude that the sequence {y j } j ∈N = {R2 (E2 y j )} j ∈N converges in Y2,  · Y2 to R2 x ∈ Y2 . Hence, the claim made in (7.8.131) is indeed true. To proceed, define  yY2,∗ := E2 yX1 for each y ∈ Y2 .

(7.8.132)

In relation to this, we claim that  · Y2,∗ is a p-norm on Y2 , which is equivalent (as a quasi-norm) with  · Y2 on the space Y2 . To justify this, first observe that for each y ∈ Y2 we have    yY2 =  R2 (E2 y)Y2 ≤ R2 X2 →Y2 E2 yX2 = R2 X2 →Y2  yY2,∗

(7.8.133)

(7.8.134)

and  yY2,∗ = E2 yX2 ≤ E2 Y2 →X2  yY2 .

(7.8.135)

Also, for each y ∈ Y2 and λ ∈ C we may write λyY2,∗ = E2 (λy)X2 = λE2 yX2 = |λ|E2 yX2 = |λ| yY2,∗ .

(7.8.136)

From (7.8.134)-(7.8.136) we conclude that  · Y2,∗ is a quasi-norm on Y2 , which is equivalent with  · Y2 on Y2 . Finally, for each y , y

∈ Y2 we may estimate (keeping in mind (7.8.132) and the fact that  · X2 is a p-norm)  y + y

Y2,∗ = E2 (y + y

)X2 = E2 y + E2 y

X2 p

p

p

≤ E2 y X2 + E2 y

X2 =  y Y2,∗ +  y

Y2,∗ p

p

p

p

(7.8.137)

which, ultimately, goes to show that  · Y2,∗ is a p-norm on Y2 . This finishes the proof of (7.8.133). On account of this and (7.8.131), there is therefore no loss of generality in assuming that actually   (7.8.138) Y2,  · Y2 is a p-Banach space. Next, we shall show that   Yi,  · Yi have separating duals, for i ∈ {1, 2}.

(7.8.139)

To prove this, assume first that y ∈ Y1 \ {0}. Then E1 y ∈ X1 \ {0} (since  E1 y = 0 would force y = R1 (E1 y) = 0, an impossibility). Given that X1,  · X1 is  assumed  to have a separating dual, there exists Λ1 ∈ X1∗ with the property that Λ1 E1 y  0. ∗ If we now define Λ := Λ1 ◦ E1 , it follows that  Λ ∈ Y1 and Λy = Λ1 E1 y  0.  Thus, Y1,  · Y1 has a separating dual. That Y2,  · Y2 also has a separating dual is established similarly, finishing the proof of (7.8.139). Pressing on, we claim that

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

441

the inclusion ι2 : Y1 → Y2 has dense range.

(7.8.140)

To see this is the case, pick an arbitrary y ∈ Y2 . Then E2 y ∈ X2 , so according to the last claim there exists a sequence {x j } j ∈N ⊆ X1 which converges   in (7.8.130) {R2 x j } j ∈N = {R1 x j } j ∈N ⊆ Y1 converges to to E2 y in X2,  · X2 . Consequently,  R2 (E2 y) = y in Y2,  · Y2 , proving (7.8.140). Finally, for each y ∈ Y1 we may estimate (with C ∈ (0, ∞) independent of y)        y  =  R1 (E1 y) ≤ E1 y  Y1, p

Y1, p

X1, p

≤ CE1 yX2 = CE2 yX2 ≤ C yY2

(7.8.141)

thanks to (7.8.126) (with i := 1), the first claim in Proposition 7.8.9, the estimate from (7.8.97) (with x := y, E := X1 , and F := Y2 ), thefact that E 2 extends E1 (cf.  (7.8.128)), and the boundedness of the operator E2 : Y2,  · Y2 → X2,  · X2 (cf. (7.8.125)). In turn, from (7.8.141) and the equivalence of items (3) and (1) in Theorem 7.8.17 we ultimately conclude that E p (Y1 ) = Y2 . Parenthetically, we wish to note that other equivalences from Theorem 7.8.17 work just as well as far as the present goal is concerned. For example, it may be checked without difficulty that if {PN } N ∈N is a family of linear operators on Y1 which constitute a good approximation of the identity on the space X1 (relative to the N } N ∈N ambient X2 ) in the sense of item (4) in Theorem 7.8.17, then the family { P where N := R1 ◦ PN ◦ E1 for each N ∈ N (7.8.142) P becomes a good approximation of the identity on the space Y1 (relative to the ambient Y2 ). In view of this and the equivalence of items (4) and (1) in Theorem 7.8.17 we  may then once again arrive at the conclusion that E p (Y1 ) = Y2 . We conclude the abstract considerations in this section by establishing the following criterion for identifying the Banach envelope of a given dual-rich quasi-normed space.   Proposition 7.8.20 Assume X, · X is a quasi-normed space whose dual separates  its points and suppose Y,  · Y is a Banach space with the property that X ⊆ Y and     X,  · X → Y,  · Y continuously and densely. (7.8.143) Then

E1 (X) = Y if and only if X ∗ = Y ∗

(with the last equality understood in the sense that isomorphism, with bounded inverse).

Y∗

(7.8.144)   Λ →  ΛX ∈ X ∗ is an

Proof The fact that E1 (X) = Y implies X ∗ = Y ∗ is a consequence of Proposition 7.8.11 (used with p := 1). Conversely, assume X ∗ = Y ∗ . Then   the identification E1 (X) = Y follows from Definition 7.8.4 (bearing in mind that Y,  · Y is a Banach space and that (7.8.143) holds) as soon as we show that

442

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

   x 

X,1

≈  xY uniformly for x ∈ X.

(7.8.145)

To this end, observe first that since (7.8.143) implies the existence of a constant C ∈ (0, ∞) with the property that  · Y ≤ C · X on X, we may invoke (7.8.16) to conclude that   (7.8.146)  · Y ≤ C  · X,1 on X. As far as (7.8.145) is concerned, there remains to establish a similar inequality in the opposite direction. With this goal in mind, pick an arbitrary x ∈ X. Then ιX (x) belongs to E1 (X) which, according to Proposition 7.8.6, is a Banach space when equipped with  ·  E1 (X). As such,  ∗ the Hahn-Banach Theorem guarantees the existence of a functional Λ ∈ E1 (X) with        Λ(E1 (X))∗ ≤ 1 and Λ ιX (x)  = ιX (x) E1 (X) =  x X,1 . (7.8.147)  ∗ Recall from the proof of Proposition 7.8.11 that the identification of E1 (X) with  ∗ X ∗ is via the assignment E1 (X)  Θ → Θ ◦ ιX ∈ X ∗ . Also, since we are presently ∗ ∗ assuming that X = Y (in the manner described in the statement), it follows that 2 ◦ ιX ∈ Y ∗ with Λ ◦ ιX ∈ X ∗ extends to a functional Λ   Λ 2 ◦ ιX Y ∗ ≤ CΛ ◦ ιX X ∗ ≤ CΛ(E1 (X))∗ ≤ C. (7.8.148) At this point, we may combine (7.8.147) and (7.8.148) to estimate             x  = Λ ιX (x)  =  Λ ◦ ιX (x) =  Λ 2 ◦ ιX (x) X,1   2 ◦ ιX Y ∗  xY ≤ C xY . ≤ Λ

(7.8.149)

Together with (7.8.146), this completes the proof of (7.8.145).



The considerations up to the present point in this section have been of a purely abstract nature. We shall now demonstrate the value of this body of results by identifying the envelopes of certain concrete Besov and Triebel-Lizorkin spaces. We do so in a sequence of three theorems, starting with the following result. Theorem 7.8.21 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Then whenever   ∗ n−1 n−1 −1 < s < 1, max n−1 max n−1 n , n+s < q < ∞, n , n+s < p < p ≤ 1, (7.8.150) one has  p,q  p ∗, p ∗ (7.8.151) E p∗ Fs (Σ, σ) = B 1 1 (Σ, σ). s−(n−1)( p − p ∗ )

As a particular case, corresponding to s := 0 and q := 2, one has   p ∗, p ∗ E p∗ h p (Σ, σ) = B

−(n−1)( p1 − p1∗ )

(Σ, σ) whenever

n−1 n

< p < p∗ ≤ 1,

(7.8.152)

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

and

443

  p ∗, p ∗ E p∗ H p (Σ, σ) = B

−(n−1)( p1 − p1∗ )

provided Σ is compact and

(Σ, σ)

n−1 ∗ n < p < p p,q Fs (Σ, σ) is

In particular, the Banach envelope of   p,q E1 Fs (Σ, σ) = B1,1

s−(n−1)( p1 −1)

≤ 1.

(Σ, σ),

(7.8.153)

(7.8.154)

hence, corresponding to s := 0 and q := 2, the Banach envelope of the local Hardy space  h p (Σ, σ) is E1 h p (Σ, σ) = B1,1 (Σ, σ), 1

(7.8.155)

if the set Σ is compact then   the Banach envelope of (Σ, σ). H p (Σ, σ) is E1 H p (Σ, σ) = B1,1 1

(7.8.156)

−(n−1)( p −1)

and

−(n−1)( p −1)

Proof The strategy is to invoke Theorem 7.8.17 (cf. also Remark 7.8.18) with p,q

E := Fs (Σ, σ),

p ∗, p ∗ (Σ, σ), s−(n−1)( p1 − p1∗ )

F := B

(7.8.157)

and with p∗ ∈ (0, 1] presently playing the role of the old p. As noted in (3) in Remark 7.1.6, E, F are quasi-Banach spaces. In fact, from item (4) in Remark 7.1.6 we know that F is actually a p∗ -Banach space. Also, Proposition 7.7.6 guarantees that the duals of E, F separate the points in these spaces, while Corollary 7.7.5 together with Lemma 7.1.10 ensure that E ⊆ F and the inclusion ι : E → F is continuous with dense range. As regards the existence of a good approximation of the identity for the space E, relative to the ambient F (in the sense described in item (4) of Theorem 7.8.17), consider the family {PN } N ∈N with each PN as in (7.3.37) (for some β, γ ∈ (0, 1) sufficiently close to 1). Then (7.8.99) holds for such a choice, thanks to (7.3.36). As far as the applicability of Theorem 7.8.17 is concerned, there remains to verify p,q (7.8.98). To this end, fix an arbitrary distribution f ∈ E = Fs (Σ, σ) along with some N ∈ N. We shall make use of (7.3.24) and (7.3.15) to recast (7.3.37) as PN f =



(k,τ)  N

k ∈Z τ ∈I N k N ≥k ≥ κΣ

=

 k ∈Z N ≥k ≥ κΣ

ν=1

(k,τ)  N τ ∈IkN

ν=1

s

σ(Qτk,ν )− n−1 + p − 2 λτk,ν ψτk,ν

σ(Qτk,ν )

∗

1

1

s − n−1 + p1∗ − 12

λτk,ν ψτk,ν .

(7.8.158)

where the last equality simply uses the abbreviation s∗ := s −(n−1)( p1 − p1∗ ). Observe that if we now consider the numerical sequence

444

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

 λ := λτk,ν Q k,ν ∈D∗ (Σ),

(7.8.159)

τ

∗

then the  p -norm of the coefficients intervening in the last line of (7.8.158) may be estimated as follows:  

(k,τ)  N  ν=1

k ∈Z τ ∈I N k N ≥k ≥ κΣ

≤C

σ(Qτk,ν )

∗

s − n−1 + p1∗ − 12

   N (k,τ) 

2

ks ∗

k ∈Z τ ∈Ik ν=1 k ≥ κΣ

 p∗ |λτk,ν 

 1/p∗

p − σ(Qτk,ν ) p ∗ 2 |λQ k,ν | τ 1

1

∗

 1/p∗

= Cλb p ∗, p ∗ (Σ) = CΨ( f )b p ∗, p ∗ (Σ) ≤ C f B p ∗ , p ∗ (Σ,σ) s∗

s∗

= C f B p ∗ , p ∗

1− 1 ) s−(n−1)( p p∗

s∗

,

(7.8.160)

(Σ,σ)

s∗

∗

thanks to the fact that 2s ≈ σ(Qτk,ν )− n−1 , the definitions made in (7.2.28), (7.3.25) and, finally, (7.3.28). Above, C ∈ (0, ∞) is a constant independent of f and N. In turn, from (7.8.158), (7.8.160), (7.3.17), and (7.8.3) we conclude that there exists some large constant C ∈ (0, ∞), independent of f and N, such that if f  0 then PN f C f B p ∗ , p ∗

1− 1 ) s−(n−1)( p p∗

∈ BE (0, 1) (Σ,σ)

! p∗

.

(7.8.161)

In concert, (7.8.161) and (7.8.6) ultimately prove (now also allowing f = 0) that   PN f  ∗ ≤ C f  p ∗, p ∗ (7.8.162) E, p B (Σ,σ) 1− 1 ) s−(n−1)( p p∗

for some constant C ∈ (0, ∞) independent of f and N. This establishes (7.8.98) in the current context (again, with p replaced by p∗ ). Hence, for E, F as in (7.8.157), we may now invoke the equivalence of items (1) and (4) in Theorem 7.8.17 to conclude that (7.8.151) holds. Finally, (7.8.152), (7.8.153) are particular cases of (7.8.151) (cf. (7.1.56) and (7.1.60)).  We pause to record the following intriguing observation. Remark 7.8.22 Assume that Σ ⊆ Rn is a compact Ahlfors regular set and abbreviate 1 σ := H n−1 Σ. Also, pick p ∈ n−1 n , 1 and introduce α := (n − 1)( p − 1). Then, even p α though the dual of H (Σ, σ) is C (Σ) (see Theorem 4.6.1), from Proposition 7.8.12 and (7.8.156) it follows that, for each f ∈ H p (Σ, σ), the quantity 

 sup   f , φ  : φ ∈ C α (Σ), φC α (Σ) = 1 (7.8.163) is equal to  f B1,1 (Σ,σ) , and not to  f  H p (Σ,σ) . −α

7.8 Envelopes of Besov and Triebel-Lizorkin Spaces

445

This is a striking manifestation of the failure of the Hardy space H p (Σ, σ) with  p ∈ n−1 , 1 to be a locally convex space4 (hence, in particular, a genuine normed n vector space; compare with (1.2.21)-(1.2.22)). Here is the second theorem alluded to earlier, pertaining to envelopes of Besov spaces. Theorem 7.8.23 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, assume  n−1 ∗ −1 < s < 1, 0 < q < ∞, max n−1 n , n+s < p < p ≤ 1, (7.8.164)   satisfy − 1 + (n − 1) p1 − p1∗ < s. Then

 p,q  p ∗, p ∗ E p∗ Bs (Σ, σ) = B

s−(n−1)( p1 − p1∗ )

(Σ, σ).

(7.8.165)

Proof This is established by reasoning much as in the proof of Theorem 7.8.21, p,q p ∗, p ∗ now choosing E := Bs (Σ, σ) and F := B 1 1 (Σ, σ), and relying on the s−(n−1)( p − p ∗ )

embedding (7.7.13).



We conclude with the following result in which further envelopes of Besov spaces are concretely identified. Theorem 7.8.24 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Then, whenever  n−1 −1 < s < 1, max n−1 0 < q ≤ p∗ ≤ p, (7.8.166) n , n+s < p ≤ 1, it follows that

 p,q  p, p ∗ E p∗ Bs (Σ, σ) = Bs (Σ, σ).

(7.8.167)

Proof Again, this is justified by arguing along the lines of the proof of Theop,q p, p ∗ rem 7.8.21, now taking E := Bs (Σ, σ) and F := Bs (Σ, σ), relying on the embedding (7.7.1), and keeping in mind that, as mentioned in item (4) of Remark 7.1.6, p, p ∗ the space Bs (Σ, σ) is p∗ -Banach granted the assumptions made in (7.8.166).  Lastly, we wish to remark that a special case of Theorems 7.8.21-7.8.21 worth considering is when Σ := Rn−1 × {0} (a scenario in which one may find it convenient to express the resulting envelope identifications with n replaced by n + 1 and Σ simply regarded as being Rn ). If this venue is pursued then certain redundant limitations on the indices involved are inherited for the formulation of the results in Theorems 7.8.21-7.8.21 which deal with much rougher settings. In order to avoid such an issue, in the theorem below (which refines results from [127]) we shall directly tackle the task of computing envelopes of Besov and Triebel-Lizorkin spaces in the entire Euclidean setting, albeit using the same blue-print as in the proofs of Theorems 7.8.21-7.8.21. 4 cf. [62, Theorem 6.2, p. 71] for a proof in the Euclidean setting

446

7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets

Theorem 7.8.25 For Besov and Triebel-Lizorkin spaces in Rn , one has the following envelope identifications: if s ∈ R, 0 < q < ∞, 0 < p < p∗ ≤ 1,   p,q p ∗, p ∗ n then E p∗ Fs (Rn ) = B 1 1 (R ),

(7.8.168)

if s ∈ R, 0 < q < ∞, 0 < p < p∗ ≤ 1,  p,q  p ∗, p ∗ n then E p∗ Bs (Rn ) = B 1 1 (R ),

(7.8.169)

if s ∈ R and 0 < q ≤ p∗ ≤ p ≤ 1,   p,q p, p ∗ then E p∗ Bs (Rn ) = Bs (Rn ).

(7.8.170)

s−n( p − p ∗ )

s−n( p − p ∗ )

and

Proof All formulas in the statement of the theorem may be established by reasoning along the lines of the proofs of Theorems 7.8.21-7.8.21. Once again, the idea is to implement the abstract criterion from Theorem 7.8.17, whose applicability in the present setting hinges on two ingredients. First, one requires embeddings for Besov and Triebel-Lizorkin spaces in Rn , and these are standard results (cf., e.g., [166]). Second, one needs to ensure the existence of a good approximation of the identity (in the sense described in item (4) of Theorem 7.8.17). In this regard, one may consider operators PN , with N ∈ N, mapping a distribution into its N-th partial sum associated with its series expansion with respect to a wavelet basis (which is sufficiently smooth and has sufficiently many vanishing moments). The coefficients in such an expansion are known to characterize the size of said distribution measured on Besov and Triebel-Lizorkin spaces (cf., e.g., [57], [59]) and this makes the family {PN } N ∈N an ideal replacement for the frame decomposition operators used earlier in the proofs of Theorems 7.8.21-7.8.21. 

7.9 Intrinsic Characterizations of Besov Spaces It is useful to provide intrinsic characterizations of the Besov spaces originally introduced in Definition 7.1.5, without recourse to the conditional expectation operators (7.1.30). A first result of this nature is contained in the theorem below. Theorem 7.9.1 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1 Σ. Also, assume 0 < s < 1, 1 ≤ p ≤ ∞, 0 < q ≤ ∞. Then

7.9 Intrinsic Characterizations of Besov Spaces

447

 f Bsp, q (Σ,σ) ≈  f  L p (Σ,σ)  +

∞ 

(7.9.1) 3

2

jsq

5 qp  q1

∬ 2

| f (x) − f (y)| dσ(x) dσ(y)

j(n−1)

j=0

p

(x,y)∈Σ×Σ |x−y | α − n,

(∂ Gα )(x) ≤ (8.1.6) Ce−c |x | for all |x| ≥ 1, for some fixed c > 0. Call a function u the Bessel potential of order α of f provided u = Gα ∗ f in Rn .

(8.1.7)

For each α ∈ (0, ∞), p ∈ (1, ∞), and w ∈ Ap (Rn, L n ), define the weighted Bessel potential space of order α in Rn as p Lα (Rn, wL n ) := u : Rn → C Lebesgue-measurable :

(8.1.8) u = Gα ∗ f for some f ∈ L p (Rn, wL n ) , endowed with the norm p

u Lαp (Rn,w L n ) :=  f  L p (Rn,w L n ) for each u ∈ Lα (Rn, wL n ) expressed as u = Gα ∗ f with f ∈ L p (Rn, wL n ).

(8.1.9)

In this regard, recall from [129, Theorem 3.3, p. 104] that for each given k ∈ N, p ∈ (1, ∞), and w ∈ Ap (Rn, L n ) we have p Lk (Rn, wL n ) = W k, p (Rn, wL n ) (as vector spaces) and the norms  ·  L p (Rn,w L n ) and  · W k, p (Rn,w L n ) are actually equivalent.

(8.1.10)

k

Moving on, assume next that E ⊆ Rn is a L n -measurable set which is n-thick. 1 (E, L n ), we introduce the strictly defined version Given a function f ∈ Lbdd of f at every point x ∈ E as ∫ 1 ⎧ ⎪ ⎨ lim+ n ⎪ f dL n if the limit exists, (8.1.11) [ f ]E (x) := r→0 L (E ∩ B(x, r)) E∩B(x,r) ⎪ ⎪0 otherwise. ⎩ Of course, Lebesgue’s Differentiation Theorem guarantees that [ f ]E (x) = f (x) for L n -a.e. x ∈ E, but the role of [ f ]E is to provide us with a canonical choice for a representative in the class of functions which agree with f at L n -a.e. point in E.

458

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

Our major result in this section is contained in the theorem below. This describes the traces of functions from a weighted Sobolev space in the entire Euclidean ambient on a given closed Ahlfors regular set as functions belonging to certain Besov space on that set. Theorem 8.1.1 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate σ := H n−1  Σ. Also, fix an exponent p ∈ (1, ∞) along with a number s ∈(0, 1) and ap denote a := 1 − s − p1 ∈ − p1 , 1 − p1 . Then for each u ∈ W 1, p Rn, δΣ L n the limit ⨏ (Tr

R n →Σ

u)(x) := [u] (x) := lim+ Rn

r→0

u(y) dy

(8.1.12)

B(x,r)

exists at σ-a.e. point x ∈ Σ, and the induced trace operator   ap p, p TrRn →Σ : W 1, p Rn, δΣ L n −→ Bs (Σ, σ)

(8.1.13)

is a well-defined, linear, bounded mapping. Of course, when a = 0, i.e., when s = 1 − p1 ∈ (0, 1), Theorem 8.1.1 becomes a statement about traces on Σ of functions belonging to the ordinary (i.e., unweighted) Sobolev space W 1, p (Rn ), in which case the result is well known; see (e.g., [106]). The novelty here is the consideration of the weighted case. The very format of the result in (8.1.13) is dictated by homogeneity considerations. The heuristic principle is that when measuring smoothness on the L p scale, taking the trace of a function which has a certain amount of smoothness in Rn on a “surface” of co-dimension one decreases the smoothness by 1/p units; see item (ii) in Theorem 9.4.5 in this regard. p, p So having the trace of u on Σ be in Bs (Σ, σ) would require that u has s + 1/p units n Sobolev spaces of smoothness in R . Given that we insist on working with   weighted of order one in Rn , the disagreement quantity a := 1 − s + p1 = 1 − s − p1 needs to be accounted for by the actual weight. By envisioning pointwise multiplication by γ δΣ as a fractional differentiation/integration operator of order γ, for each γ ∈ R, this provides an explanation of the nature of the power of the weight in (8.1.13). Theorem 8.1.1 is a culmination of a number of technical results, which we deal with separately in the next section (the actual proof of Theorem 8.1.1 is presented at the end of §8.2).

8.2 Technical Lemmas Before proving the above theorem we deal with several preparatory lemmas, starting with the following. Lemma 8.2.1 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate n σ := H n−1  Σ. Recall that G1 is the Bessel kernel of order  R , defined as  1 one in 1 in (8.1.5) with α := 1. Fix p ∈ (1, ∞) along with a ∈ − p , 1 − p , and consider ap f ∈ L p (Rn, δΣ L n ). Then

8.2 Technical Lemmas

459

∫ Rn

|G1 (x − y)|| f (y)| dy < +∞ for H n−1 -a.e. x ∈ Rn,

(8.2.1)

and the function ∫ u(x) := (G1 ∗ f )(x) =

Rn

G1 (x − y) f (y) dy at H n−1 -a.e. point x ∈ Rn (8.2.2)

satisfies the following two properties: u may be strictly defined at H n−1 -a.e. point in Rn,

(8.2.3)

[u]Rn = G1 ∗ f at H n−1 -a.e. point in Rn .

(8.2.4)

Proof The unweighted case (when a := 0) has been treated in [106, Proposition 1, p. 151] and our proof is an adaptation of this result. For starters observe that [133, Proposition 8.7.4] and [133, (7.7.105)] (presently used with n replaced by n + 1 and Σ := Rn ≡ Rn × {0} ⊂ Rn+1 ) ensures that  r Lloc (Rn, L n ). (8.2.5) f ∈ 1 j

1/p

∫

   ≤ C  e−c |x−y | δΣ (y)−ap dy   y ∈Rn, |x−y |>1 



1/p

∫

 ≤ CN  y ∈R n,

1/p ∫   p ap | f2, j (y)| δΣ (y) dy   n R 

 (y)−ap

|x−y |>1

 δΣ dy  N |x − y| 

 f  L p (Rn,δ a p L n ) Σ

(8.2.7)

for every N > 0. Choosing N sufficiently large then ensures (by virtue of [133, (8.7.6)]) that the last integral in (8.2.7) is finite. Thus, I I j (x) < +∞ for every ∞  E j so that H n−1 (E) = 0. Then I j (x) + I I j (x) < +∞ for x ∈ B(0, j − 1). Set E := j=2

every integer j ≥ 2 and every point x ∈ B(0, j − 1) \ E, which ultimately shows that ∫ we have Rn |G1 (x − y)|| f (y)| dy ≤ I j (x) + I I j (x) < +∞ for H n−1 -a.e. x ∈ Rn . This proves (8.2.1). Turning our attention to the claim made in (8.2.3), let us now decompose (retaining earlier notation) u(x) = u1, j (x) + u2, j (x) where, for each j ≥ 2, we have set ∫ ∫ u1, j (x) := G1 (x − y) f1, j (y) dy, u2, j (x) := G1 (x − y) f2, j (y) dy, (8.2.8) Rn

Rn

for H n−1 -a.e. x ∈ Rn . Since, as already observed, f1, j ∈ L r (Rn, L n ) for some r > 1, the unweighted version of (8.2.3) applies and gives that, for each j ≥ 2, the function u1, j can be strictly defined at each point in Rn , with the possible exception of a set E j satisfying H n−1 (E j ) = 0. On the other hand, one can show (by recycling earlier estimates and invoking Lebesgue’s Dominated Convergence Theorem) that for each j ≥ 2 the function u2, j is continuous at each point in the ball B(0, j − 1). Altogether, this shows that the function u can be strictly defined at each point in Rn not belonging ∞  E j . Observing that H n−1 (E) = 0 proves (8.2.3). Finally, (8.2.4) to the set E := j=2

can be proved in a similar fashion.



Lemma 8.2.2 Assume that Σ ⊆ Rn is a closed Ahlfors regular set and abbreviate n σ := H n−1  Σ. Recall that G1 is the Bessel kernel of order  1 one in R , defined as in (8.1.5) with α := 1. Then for each p ∈ (1, ∞) and θ ∈ 0, p there exists a constant C = C(Σ, p, θ) > 0 such that ∫ |G1 (x − y)| θ p dσ(x) ≤ C for all y ∈ Rn . (8.2.9) Σ

Proof Let p and θ be as in the statement of the lemma, and fix y ∈ Rn . Then

8.2 Technical Lemmas

∫ Σ

461

|G1 (x − y)| θ p dσ(x) ∫ =

(8.2.10) ∫

|G1 (x − y)| θ p dσ(x) +

x ∈Σ |x−y | 0 with the property that ∫     1 n |G1 (x − y)| (1−θ)p δΣ (y)−ap dy ≤ C. (8.2.13) θ > n−1 a + p − 1 + =⇒ sup x ∈Σ

Also, if 1 n−1



Rn

a+

n p

−1

 +

0 such that ∫ Rn

|G1 (x − t) − G1 (y − t)|

(1−θ)p

δΣ (t)

−ap

dt

p p

≤ C |x − y| (n−1)(θ p−1)+p−ap−1 (8.2.15)

for all x, y ∈ Σ. Proof We first prove (8.2.13). Fix x ∈ Σ arbitrary. Recall that by the second bound in (8.1.6), we have |G1 (z)| ≤ C|z| −N for all |z| ≥ 1, provided N is chosen large enough (N > np−n+1 p−1 ). Having selected such a number N, with the help of [133,

462

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

(8.7.6)] we may then write ∫



|y−x | ≥1



|G1 (x − y)| (1−θ)p δΣ (y)−ap dy ∫

≤C



|y−x | ≥1

δΣ (y)−ap dy ≤ C, |x − y| N (1−θ)p

(8.2.16)

since by assumption a < 1 − p1 . Also, making use of the first estimate in (8.1.6) (to the effect that |G1 (z)| ≤ C|z| 1−n for all z ∈ Rn \ {0}) and of [133, (8.7.3)] with N := (n − 1)(1 − θ)p, we obtain that ∫   |G1 (x − y)| (1−θ)p δΣ (y)−ap dy |y−x | ≤1

∫ ≤C



|y−x | ≤1

δΣ (y)−ap  dy ≤ C, |x − y| (n−1)(1−θ)p

(8.2.17)

provided (n − 1)(1 − θ)p < n − ap. The latter condition is equivalent with having   1 a + np − 1 . θ > n−1 (8.2.18) Now granted (8.2.18), from (8.2.16) and (8.2.17) we obtain (8.2.13) as desired. Next, we turn to the proof of (8.2.15). With p, a, θ and x as before, let y ∈ Σ, y  x, be arbitrary. Then, for some constant finite C = C(θ, p) > 0 we have ∫   |G1 (x − t) − G1 (y − t)| (1−θ)p δΣ (t)−ap dt Rn

∫





|G1 (x − t)| (1−θ)p δΣ (t)−ap dt

≤C |x−t | 0, 

 1/p

∫∫

1

1

≤ C r 1−a− p g L p (Rn,δ a p L n ) .

|u(x) − u(y)| dσ(x) dσ(y) p

r n−1

Σ

(x,y)∈Σ×Σ |x−y | n−1 a + p − 1 , + and Fubini-Tonelli’s Theorem. The fourth inequality in (8.2.27) is a consequence of (8.2.9), under the additional assumption that θ < p1 . Note that we can select θ   1 a + np − 1 < p1 . satisfying the above restrictions since a < 1 − p1 is equivalent to n−1 Next, we turn to the proof of (8.2.26). Again we start with θ ∈ (0, 1) to be specified later and fix r > 0 arbitrary. Then, based on the definition of u and Hölder’s inequality, we have

8.2 Technical Lemmas

465

∫∫ |u(x) − u(y)| p dσ(x) dσ(y)

(8.2.28)

(x,y)∈Σ×Σ |x−y | 0 with the property that given any f ∈ Wa (Ω) it is possible to find −1, p n ' f ∈ Wa (R ) such that



' f = f and  ' f W −1, p (Rn ) ≤ C f W −1, p (Ω) . (8.5.3) Ω

a

a

+ Indeed, start with a representation f = f0 + nj=1 ∂j f j which is quasi-optimal (in the + f j is the extension to Rn of f j by norm sense), then take ' f := ' f0 + nj=1 ∂j ' f j where ' setting it equal to zero outside Ω, for 0 ≤ j ≤ n.

486

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

The weighted Sobolev space of negative smoothness arise naturally as duals, in the manner described in the proposition below. The reader is reminded that the space 1, p W˚ a (Ω) has been defined in (8.3.65). Proposition 8.5.2 Let Ω ⊆ Rn be an (ε, δ)-domain with rad (Ω) > 0, whose boundary is an Ahlfors regular set, and such that Ωc := Rn \ Ω is n-thick. Also, assume p, p ∈ (1, ∞) are two exponents such that 1/p + 1/p = 1 and fix some a ∈ (−1/p, 1 − 1/p). Then  1, p ∗ −1, p W˚ a (Ω) = W−a (Ω)

(8.5.4)

and

 −1, p ∗ −1, p Wa (Ω) = W˚ −a (Ω). (8.5.5)   Note that since 1 < p < ∞ and −a ∈ − 1/p, 1 − 1/p , it makes sense to  −1, p consider the weighted Sobolev space of negative order W−a (Ω) (cf. the discussion pertaining to (8.5.1)). Proof of Proposition 8.5.2 To justify (8.5.4), we begin by defining the mapping  1, p ∗ −1, p Φ : W−a (Ω) −→ W˚ a (Ω)

(8.5.6) −1, p

by setting Φ( f ) := Λ f with the agreement that, if f ∈ W−a (Ω) admits the repre +  −ap sentation f = f0 + nj=1 ∂j f j in D (Ω) with f j ∈ L p Ω, δ∂Ω L n , 0 ≤ j ≤ n, we  1, p ∗ define Λ f ∈ W˚ a (Ω) as the functional ∫ Λ f (u) :=

Ω

u f0 dL n −

n ∫  j=1

Ω

(∂j u) f j dL n,

1, p

∀u ∈ W˚ a (Ω).

(8.5.7)

Then (8.3.65) ensures that the above definition of Λ f is independent of the particular representation of the distribution f . In particular, Φ in (8.5.6)-(8.5.7) is well defined, linear, and bounded. In fact, another application of (8.3.65) gives that Φ is one-to-one. Hence, as far as (8.5.4) is concerned, there remains to show that Φ is onto (once this is established, we conclude that Φ is an isomorphism, which finishes the proof of ∗  1, p (8.5.4)). With this goal in mind, fix an arbitrary functional Λ ∈ W˚ a (Ω) . Our aim  −1, p is to find f ∈ W−a (Ω) such that Λ f = Λ. To get started, consider the application     n+1 1, p ap ι : W˚ a (Ω) −→ L p Ω, δ∂Ω L n 1, p defined by ι(u) := (u, ∇u) for each u ∈ W˚ a (Ω).

(8.5.8)

Clearly, ι maps its domain in a one-to-one fashion onto its range, Range ι. Thanks to Proposition 8.3.10, the latter happens to be a closed subspace of the Banach     n+1 ap . In particular, ι is an isomorphism onto the Banach space space L p Ω, δ∂Ω L n

8.5 Weighted Sobolev Spaces of Negative Order, Duality, and Conormal Derivatives

487

1, p Range ι. Note that if ι−1 denotes the inverse of ι : W˚ a (Ω) → Range ι, the Open Mapping Theorem ensures that ι−1 is continuous, hence

    n+1 ap Λ ◦ ι−1 : Range ι −→ L p Ω, δ∂Ω L n

(8.5.9)

is a bounded linear map. By the Extension Theorem, this map may  * Hahn-Banach ∗  ap n " n+1 p . We may then invoke Riesz’s be extended to a functional in L Ω, δ∂Ω L   −ap Representation Theorem to conclude that there exist f j ∈ L p Ω, δ∂Ω L n with 1, p 0 ≤ j ≤ n such that for all u ∈ W˚ a (Ω) we have    Λ(u) = Λ ◦ ι−1 ι(u) =

∫ Ω

u f0 dL n −

n ∫  j=1

Ω

(∂j u) f j dL n .

(8.5.10)

+ −1, p As a consequence, we have Λ = Λ f with f := f0 + nj=1 ∂j f j ∈ W−a (Ω), as wanted. Having proved (8.5.4), formula (8.5.5) follows from this and (8.3.66) (after readjusting notation).  Moving on, the goal is to associate a natural notion of conormal derivative operator with any given second-order system. To set the stage, fix an open set Ω ⊆ Rn with an Ahlfors regular boundary. Also, suppose 1 < p, p < ∞ satisfy 1/p + 1/p = 1 and select some a ∈ (−1/p, 1 − 1/p). Then, as seen from (8.5.1)-(8.5.2), any given ' × M system L in Rn second-order homogeneous constant (complex) coefficient M becomes a linear and bounded operator in the context * 1, p "M * −1, p "M , L : Wa (Ω) −→ Wa (Ω) .

(8.5.11)

* 1, p "M We wish to associate a notion of conormal derivative for functions in Wa (Ω) . * 1, p "M Our definition of the conormal derivative of a function u ∈ Wa (Ω) strongly depends on the choice of an extension f of Lu, which is a vector-distribution in "M * 1, p * −1, p ∗ , , Wa (Ω) , to a vector functional belonging to the space W−a (Ω) ] M . By * 1, p ∗" M , satisfies this, it is meant that f ∈ W−a (Ω)  f , ψ = Lu, ψ,

* "M , ∀ψ ∈ Cc∞ (Ω) ,

(8.5.12)

where the left side is understood in the sense of the duality paring between the *  1, p * 1, p "M ∗" M , , functional f ∈ W−a (Ω) and ψ, viewed as an element in W−a (Ω) , while the right side is considered in the sense of distributional pairing in Ω. To facilitate the subsequent discussion we make the following convention.

488

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

*  1, p ∗" M , given an arbitrary vector functional f ∈ W−a (Ω) , we henceforth agree to denote by f Ω the vector distribution induced by f on the open set Ω, that is, the linear assignment . * ∞ "M ,  ψ → [(W 1, p (Ω))∗ ] M Cc (Ω) 1, p  , f, ψ , ∈ C. [W (Ω)] M −a

(8.5.13)

−a

In this notation, the condition demanded in (8.5.12) then simply reads Lu = f Ω in *  "M , D (Ω) . Proposition 8.5.3 Let Ω ⊆ Rn be an (ε, δ)-domain with a compact Ahlfors regular boundary, and such that Rn \ Ω is n-thick. Abbreviate σ := H n−1 ∂Ω, and fix  p ∈ (1, ∞) along with s ∈ (0, 1), then define a := 1 − s − p1 ∈ − p1 , 1 − p1 .  −1  Also, let p := 1 − p1 be the Hölder conjugate exponent of p. Next, for some   ' M ∈ N, select a coefficient tensor A = aαβ M, , with constant (complex) jk

1≤α ≤ M 1≤β ≤M 1≤ j,k ≤n

entries, and associate with this coefficient tensor the homogeneous   second-order n ' × M system L A := aαβ ∂j ∂k constant (complex) coefficient M , 1≤α ≤ M in R . In this jk 1≤β ≤M

setting, consider the conormal derivative map   * 1, p " M *  1, p * "M ∗" M , , ∂νA : (u, f ) ∈ Wa (Ω) ⊕ W−a (Ω) : L Au = f Ω in D (Ω) "M * p, p , −→ Bs−1 (∂Ω, σ)

(8.5.14)

* 1, p "M assigning to each pair (u, f ), where u = (uβ )1≤β ≤M ∈ Wa (Ω) and where *  "M *  1, p ∗" M , , has the property that L Au = f Ω in D (Ω) , the functional f ∈ W−a (Ω) *

∂νA(u, f ) ∈

p, p

B1−s (∂Ω, σ)

"M ,

∗

* p, p "M , = Bs−1 (∂Ω, σ)

(8.5.15)

acting according to (with the summation convention over repeated indices in effect) - A . . αβ p , p  p , p  , ∗ ∂ν (u, f ), ϕ , := a jk ∂k uβ , ∂ j Φα +  f , Φ ([B (∂Ω,σ)] M ) [B (∂Ω,σ)] M 1−s

1−s

*

p, p

for all ϕ ∈ B1−s (∂Ω, σ)

"M ,

* 1, p "M , and Φ = (Φα )α ∈ W−a (Ω)

with the property that TrΩ→∂Ω Φ = ϕ, (8.5.16) where the first set of brackets in the right side is understood as the integral pairing     ap −ap between functions from L p Ω, δ∂Ω L n and functions from L p Ω, δ∂Ω L n (Ω), and the second set of brackets in the right side is the canonical duality pairing between *  1, p * 1, p "M ∗" M , , W−a (Ω) and W−a (Ω) . Then the operator (8.5.14)-(8.5.16) is well defined, linear, and bounded in the sense that there exists a constant C ∈ (0, ∞) with the property that

8.5 Weighted Sobolev Spaces of Negative Order, Duality, and Conormal Derivatives



! A ! !∂ (u, f )! p, p +  f [(W 1, p (Ω))∗ ] M 1, p , ≤ C u , ν [B (∂Ω,σ)] M [W (Ω)] M a

s−1

489

 (8.5.17)

−a

for each pair (u, f ) belonging to the domain of ∂νA (cf. (8.5.14)). * 1, p "M and Furthermore, for each pair (u, f ) with u = (uβ )1≤β ≤M ∈ Wa (Ω) *  "M *  1, p ∗" M , , such that L Au = f Ω in D (Ω) one has the following f ∈ W−a (Ω) generalized “half” Green’s formula: - A . (8.5.18) p , p  p , p  , ∗ ∂ν (u, f ), TrΩ→∂Ω w , ([B (∂Ω,σ)] M ) [B (∂Ω,σ)] M 1−s

αβ

= a jk

∫ Ω

1−s

. ∂k uβ ∂j wα dL n + [(W 1, p (Ω))∗ ] M 1, p  , f, w , [W (Ω)] M −a

−a

* 1, p "M , for each w = (wα )α ∈ W−a (Ω) .

" M * 1, p * 1, p ∗" M , ⊕ W−a (Ω) such Finally, for any two given pairs, (u, f ) ∈ Wa (Ω) *  "M * 1, p "M ∗" M , , * 1, p such that that L Au = f Ω in D (Ω) and (w, g) ∈ W−a (Ω) ⊕ Wa (Ω) *  "M   L A w = gΩ in D (Ω) (where A is the transpose of A), one has the following generalized “full” (or “symmetric”) Green’s formula: - A . (8.5.19) p , p  p , p  , ∗ ∂ν (u, f ), TrΩ→∂Ω w , ([B (∂Ω,σ)] M ) [B (∂Ω,σ)] M 1−s

1−s

-

. − ([Bsp, p (∂Ω,σ)] M )∗ ∂ν (w, g), TrΩ→∂Ω u [Bsp, p (∂Ω,σ)] M . - . g, u [W 1, p (Ω)] M . = [(W 1, p (Ω))∗ ] M 1, p 1, p  , f, w , − [(W (Ω))∗ ] M [W (Ω)] M −a

A

−a

a

a

Proof Let us check that the definition of the action of the functional (8.5.15) on any * p, p "M , given function ϕ ∈ B1−s (∂Ω, σ) (as described in (8.5.16)) does not depend on "M * 1, p , as long as TrΩ→∂Ω Φ = ϕ. With this goal in mind, the choice of Φ ∈ W−a (Ω) * 1, p "M , and TrΩ→∂Ω Φ j = ϕ for j ∈ {1, 2}, then Φ := Φ1 − Φ2 assuming Φ j ∈ W−a (Ω) * 1, p "M , belongs to W−a (Ω) and has TrΩ→∂Ω Φ = 0. Granted these properties, we may invoke Proposition 8.3.10 to conclude that there exists a sequence of (vector* "M , indexed by i ∈ N, for which valued) test functions, say Ψ(i) = (Ψα(i) )α ∈ Cc∞ (Ω) * 1, p "M , (i) lim Ψ = Φ in W−a (Ω) . Then, if (Φα )α denote the scalar components of Φ, i→∞

for each pair (u, f ) belonging to the domain of the conormal derivative operator (8.5.14) we may write

490

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces αβ -

.

. a jk ∂k uβ, ∂j Φα + f , Φ . . αβ = a jk lim ∂k uβ, ∂j Ψα(i) + lim f , Ψ(i) i→∞

i→∞



αβ a jk

= lim

i→∞

∫ Ω

 = lim

i→∞

−

αβ a jk

∂k uβ ∂j Ψα(i) dL n

∫ Ω

-

+ [D  (Ω)] M L Au, Ψ

uβ ∂k ∂j Ψα(i) dL n

-

(i)

.

+ [D  (Ω)] M L Au, Ψ

/ [D(Ω)] M

(i)

.

/ [D(Ω)] M

= lim 0 = 0,

(8.5.20)

i→∞

thanks to the convention made in (8.5.13). This proves that the definition made in (8.5.16) is indeed coherent. Once this has been established, we may then conclude that the conormal operator is well defined and linear in the context of (8.5.14). To prove * 1, p "M *  1, p ∗" M , its boundedness, fix a pair (u, f ) ∈ Wa (Ω) ⊕ ∈ W−a (Ω) satisfying *  "M * p, p "M , , L Au = f Ω in D (Ω) , and pick an arbitrary ϕ ∈ B1−s (∂Ω, σ) . Then, if Ex∂Ω→Ω denotes the vector version of the extension operator from Theorem 8.4.1, the function Φ := Ex∂Ω→Ω ϕ satisfies "M * 1, p , and Φ[W 1, p (Ω)] M Φ ∈ W−a (Ω) p , p  , ≤ Cϕ , [B (∂Ω,σ)] M −a

(8.5.21)

1−s

for some constant C ∈ (0, ∞) independent of ϕ. Based on (8.5.21) and (8.5.16) we may then estimate



- A .

 

([B p , p (∂Ω,σ)] M p , p  , ∗ ∂ν (u, f ), ϕ ,

) [B (∂Ω,σ)] M 1−s

1−s

αβ

≤ a jk ∂k uβ  L p (Ω,δ a p L n ) ∂j Φα  L p (Ω,δ −a p L n ) ∂Ω

∂Ω

+  f [(W 1, p (Ω))∗ ] M 1, p  , Φ , [W (Ω)] M −a

−a

 ≤ C u[W 1, p (Ω)] M +  f [(W 1, p (Ω))∗ ] M p , p  , ϕ ,. [(B (∂Ω,σ)] M 

a

−a

(8.5.22)

1−s

"M * p, p , On account of this, the arbitrariness of ϕ ∈ B1−s (∂Ω, σ) , and Proposition 7.6.1 we conclude that estimate (8.5.17) holds. This finishes the proof of the fact that the conormal operator is also bounded in the context of (8.5.14). Finally, the generalized “half” Green’s formula (8.5.18) is a consequence of (8.5.16) used with ϕ := TrΩ→∂Ω w and Φ := w, while the “full” Green’s formula (8.5.19) is obtained by subtracting two versions of (8.5.18) (one written for A, and one written for the  transpose coefficient tensor A ).

8.5 Weighted Sobolev Spaces of Negative Order, Duality, and Conormal Derivatives

491

Occasionally, we need the notion of conormal derivative in the more general setting described below. Remark 8.5.4 Retain the assumptions on Ω, A, p, s, a, from Proposition 8.5.3. In addition, fix an arbitrary cutoff function ψ ∈ Cc∞ (Rn ) with ψ ≡ 1 near ∂Ω. "M * 1, p (cf. Convention 8.3.7) and each For each u = (uβ )1≤β ≤M ∈ Wa (Ω)bdd * "M *  1, p ∗" M , , with the property that L Au = f Ω in D (Ω) define f ∈ W−a (Ω) "M   * p, p , u, ' f ∈ Bs−1 (∂Ω, σ) ∂νA(u, f ) := ∂νA '

(8.5.23)

with "M *  1, p * 1, p ∗" M , and ' f ∈ W−a (Ω) given by ' u := ψu ∈ Wa (Ω)

 αβ αβ αβ ' f := a jk (∂j ∂k ψ)uβ + a jk (∂j ψ)(∂k uβ ) + a jk (∂k ψ)(∂j uβ )

+ψf. (8.5.24) Then the above definition is meaningful, and does not depend on the particular * 1, p "M , cutoff function ψ. Furthermore, for each function w = (wα )α ∈ W−a (Ω) which vanishes outside a bounded subset of Ω the following generalized “half” Green’s formula holds: - A . (8.5.25) p , p  p , p  , ∗ ∂ν (u, f ), TrΩ→∂Ω w , ([B (∂Ω,σ)] M ) [B (∂Ω,σ)] M 1−s

αβ

= a jk

∫ Ω

, 1≤α ≤ M

1−s

. ∂k uβ ∂j wα dL n + [(W 1, p (Ω))∗ ] M 1, p  , f, w ,. [W (Ω)] M −a

−a

* 1, p "M *  1, p ∗" M , Finally, for each pair (w, g) ∈ W−a (Ω) ⊕ Wa (Ω) with L A w = gΩ *  "M in D (Ω) (where A is the transpose of A) and with w vanishing outside a bounded subset of Ω, the generalized “full” Green’s formula - A . (8.5.26) p , p  p , p  , ∗ ∂ν (u, f ), TrΩ→∂Ω w , ([B (∂Ω,σ)] M ) [B (∂Ω,σ)] M 1−s

1−s

-

. − ([Bsp, p (∂Ω,σ)] M )∗ ∂ν (w, g), TrΩ→∂Ω u [Bsp, p (∂Ω,σ)] M . . g, ψu [W 1, p (Ω)] M = [(W 1, p (Ω))∗ ] M 1, p 1, p  , f, w , − [(W (Ω))∗ ] M [W (Ω)] M A

−a

holds for each cutoff function ψ ∈ of w.

−a

Cc∞ (Rn ) with ψ

a

a

≡ 1 near both ∂Ω and the support

A couple of clarifications regarding Remark 8.5.4 are in order. First, the fact that * 1, p "M * 1, p "M u belongs to Wa (Ω)bdd puts ' u := ψu in Wa (Ω) (cf. Convention 8.3.7).  1, p ∗ 1, p Second, since W−a (Ω) is a module over Cc∞ (Rn ), so is W−a (Ω) , via duality. As *  1, p ∗" M , such, ψ f ∈ W−a (Ω) . In addition,

492

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces αβ

αβ

αβ

a jk (∂j ∂k ψ)uβ + a jk (∂j ψ)(∂k uβ ) + a jk (∂k ψ)(∂j uβ ) belongs to ∗    1, p ap ' }. L p Ω, δ∂Ω L n → W−a (Ω) for each α ∈ {1, . . . , M

(8.5.27)

*  1, p ∗" M , . It is also clear from Collectively, these properties show that ' f ∈ W−a (Ω) *  "M , u = ' f Ω in D (Ω) , in the sense of (8.5.13). Together with definitions that L A' Proposition 8.5.3, these observations show that the definition made in (8.5.23) is indeed meaningful. Moreover, the fact that said definition is independent of the cutoff function follows from the fact that, in the context of Proposition 8.5.3, the conormal derivative ∂νA(u, f ) is the zero distribution whenever u vanishes near ∂Ω. Finally, the generalized “half” Green’s formula (8.5.25) is seen from (8.5.23) and (8.5.18), while the generalized “full” Green’s formula (8.5.26) is implied by (8.5.19) and (8.5.23). While the value p = ∞ is excluded from Proposition 8.5.3, we nonetheless have the following more restrictive result corresponding to this end-point: Proposition 8.5.5 Let Ω ⊆ Rn be a bounded (ε, δ)-domain with an Ahlfors regular boundary. Abbreviate σ := H n−1 ∂Ω and denote by δ∂Ω the distance  function to ∂Ω.  ' M ∈ N, consider a coefficient tensor A = aαβ Also, for some M, , with conjk 1≤α ≤ M 1≤β ≤M 1≤ j,k ≤n

stant (complex) entries, and associate with this coefficient tensor the second-order  ' × M system L A := aαβ ∂j ∂k homogeneous constant (complex) coefficient M , 1≤α ≤ M jk 1≤β ≤M

in Rn (with the summation convention over repeated indices understood throughout). In this context, for each fixed s∗ ∈ (0, 1) define the conormal derivative operator  * "M 1−s∗ 1,1 (Ω) : δ∂Ω |∇u| ∈ L ∞ (Ω, L n ) ∂νA : u ∈ L ∞ (Ω, L n ) ∩ Wloc / (8.5.28) *  "M "M * ∞,∞ , , and L Au = 0 in D (Ω) −→ Bs∗ −1 (∂Ω, σ) associating to each u = (uβ )1≤β ≤M belonging to the domain of ∂νA in (8.5.28) the functional * "M * "M , ∗ , 1,1 (∂Ω, σ) = Bs∞,∞ (∂Ω, σ) (8.5.29) ∂νAu ∈ B1−s ∗ ∗ −1 defined according to 1,1 , ∗ ([B1−s (∂Ω,σ)] M ) ∗

for all ϕ ∈

*

-

.

∫

∂νAu, ϕ [B1,1 (∂Ω,σ)] M , 1−s∗

"M , 1,1 B1−s (∂Ω, σ) ∗

:=

Ω

αβ

a jk (∂k uβ )(∂j Φα ) dL n

(8.5.30)

where (Φα )1≤α ≤ M , := Ex∂Ω→Ω ϕ.

Above, the summation convention over repeated indices is in effect, and Ex∂Ω→Ω denotes the extension operator from Theorem 8.4.1 (acting componentwise).

8.5 Weighted Sobolev Spaces of Negative Order, Duality, and Conormal Derivatives

493

Then the conormal derivative operator ∂νA considered in (8.5.28)-(8.5.30) is well defined, linear, and bounded in the sense that there exists a constant C ∈ (0, ∞) with the property that ! ! ! A ! ! 1−s∗



! !∂ u! ∞,∞ ∇u (8.5.31) ! ∞ , ≤ C !δ M ν ∂Ω [B (∂Ω,σ)] n L (Ω, L )

s∗ −1

for each function u belonging to the domain of ∂νA (cf. (8.5.28)). Moreover, the conormal derivative operator defined in (8.5.28)-(8.5.30) is compatible with the conormal derivative operator introduced in Proposition 8.5.3 in the following sense: Given any s ∈ (0, s∗ ) and p ∈ (1, ∞) it follows that any func* 1, p "M with tion u belonging to the domain of ∂νA in (8.5.28) satisfies u ∈ Wa (Ω) a := 1 − s − 1/p and * "M , (∂Ω, σ) defined as in the conormal derivative ∂νAu ∈ Bs∞,∞ ∗ −1 (8.5.30)-(8.5.30) coincides with the earlier conormal derivative * p, p "M , ∂νA(u, 0) ∈ Bs−1 (∂Ω, σ) defined as in (8.5.16) (with f = 0).

(8.5.32)

' = M and A is weakly elliptic, the domain of ∂νA in It is worth noting that when M "M * s (8.5.28) simply becomes C ∗ (Ω) ∩ Ker L A, i.e., the space of C M -valued Hölder functions of order s∗ which are null-solutions of L A in Ω. That the domain of ∂νA in (8.5.28) is, in the aforementioned circumstances, con"M * ∩ Ker L A may be seen by combining two observations. First, tained in C s∗ (Ω) by elliptic regularity, any distribution which is a null-solution of L A in Ω is smooth. Second, in view of item (1) in [133, Proposition 5.11.14], the set Ω is a locally uniform domain. Granted these, from [133, (5.11.75)] we conclude that ! ! ! ! + Cu[L ∞ (Ω, L n )] M , (8.5.33) u[C s∗ (Ω)] M ≤ C !|∇u| · dist(·, ∂Ω)1−s∗ ! ∞ n L (Ω, L )

*

hence ultimately u ∈ C s∗ (Ω)

"M

.

* "M Conversely, given any null-solution u of L A belonging to C s∗ (Ω) , elliptic regularity implies that u is smooth in Ω. Also, by interior estimates (cf. [133, Theorem 6.5.7]), for each x ∈ Ω we have ⨏ −1 |(∇u)(z)| ≤ Cδ∂Ω (x) |u(x) − u(y)| dy sup B(x,δ ∂Ω (x))

z ∈B(x,δ ∂Ω (x)/2)

≤ Cδ∂Ω (x)s∗ −1,

(8.5.34)

for some constant C = C(A, n) ∈ (0, ∞). In turn, this estimate readily implies that 1−s∗ δ∂Ω |∇u| ∈ L ∞ (Ω, L n ), and the desired conclusion follows. We now turn to the task of presenting the proof of Proposition 8.5.5. Proof of Proposition 8.5.5 From estimate (8.4.10) (presently used with p := 1, * 1,1 "M , k := 0, and s := 1 − s∗ ) we know that given any ϕ ∈ B1−s (∂Ω, σ) , the function ∗

494

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

Φ := Ex∂Ω→Ω ϕ satisfies ! ! ! s∗ −1



! !δ∂Ω ∇Φ !

L 1 (Ω, L n )

≤ Cϕ[B1,1

1−s∗

, (∂Ω,σ)] M

,

(8.5.35)

for some constant C ∈ (0, ∞) independent of ϕ. From (8.5.35) we then conclude that whenever the function u = (uβ )1≤β ≤M belongs to the domain of ∂νA in (8.5.28) and Φ = (Φα )1≤α ≤ M , is as above we may estimate ∫ ! ! ! ! ! 1−s∗



! ! s∗ −1



! αβ ∇Φ ! 1 ∇u a jk |∂k uβ ||∂j Φα | dL n ≤ C !δ∂Ω ! ∞ !δ ∂Ω n n L (Ω, L )

Ω

≤ Cϕ[B1,1

1−s∗

! ! ! 1−s∗



! ∇u ! , !δ M (∂Ω,σ)] ∂Ω

L (Ω, L )

L ∞ (Ω, L n )

(8.5.36)

for some constant C = C(Ω, A, s∗ ) ∈ (0, ∞). This proves that the conormal derivative operator introduced in (8.5.28)-(8.5.30) is indeed well defined, linear, and bounded. To deal with the last claim in the statement, fix s ∈ (0, s∗ ), p ∈ (1, ∞), and suppose the function u belongs to the domain of ∂νA in (8.5.28). Then [133, (8.7.3)] (presently * 1, p "M if used with Σ := ∂Ω, N := 0, α := 1 − (s∗ − s)p) implies that u ∈ Wa (Ω) a := 1 − s − 1/p. With this in hand, the claim in (8.5.32) is now seen from (8.5.16), (8.5.30), and (8.4.3) (used with p in place of p and 1 − s in place of s, a scenario in  which 1 − (1 − s) − 1/p becomes −a). It turns out that for smooth function we may actually compute the conormal derivative, originally considered as in Proposition 8.5.3, in a pointwise sense, as indicated below. Proposition 8.5.6 Let Ω ⊆ Rn be a bounded (ε, δ)-domain with an Ahlfors regular boundary, and such that Rn \ Ω is n-thick. Abbreviate σ := H n−1 ∂Ω and denote by ν = (ν1, . . . , νn ) the geometric measure theoretic outward unit normal to Ω. Also, fix p ∈ (1, ∞), denote by p its Hölder conjugate exponent, pick some s ∈ (0, 1), ' M ∈ N, and consider a and define a := 1 − s − p1 . Next, fix two integers M,  αβ  coefficient tensor A = a jk 1≤α ≤ M , with constant (complex) entries. Associate with 1≤β ≤M 1≤ j,k ≤n

this coefficient tensor the homogeneous constant (complex) coefficient   second-order n ' × M system L A := aαβ ∂j ∂k M , in R (as usual, the summation convention 1≤α ≤ M jk 1≤β ≤M

over repeated indices is understood throughout). Finally, consider an arbitrary * "M * 1, p "M vector valued function u = (uβ )1≤β ≤M ∈ C ∞ (Ω) ⊆ Wa (Ω) and define  , ∗ M 1, p f ∈ W−a (Ω) by setting (bearing (8.3.35) in mind) 1, p  , [(W−a (Ω))∗ ] M

-

f, F

.

∫ 1, p  , [W−a (Ω)] M

for each F = (Fα )1≤α ≤ M ,

:=

Ω

(L Au)α Fα dL n

 M , 1, p ∈ W−a (Ω) .

(8.5.37)

8.5 Weighted Sobolev Spaces of Negative Order, Duality, and Conormal Derivatives

495

* "M * "M , , Then the function L Au ∈ C ∞ (Ω) satisfies L Au = f Ω in D (Ω) and, * p, p "M , A considered in the sense of with the conormal derivative ∂ν (u, f ) ∈ Bs−1 (∂Ω, σ) Proposition 8.5.3, one has 

αβ ∂νA(u, f ) = ν j a jk (∂k uβ ) ∂Ω . (8.5.38) , 1≤α ≤ M

* "M , Proof From (8.5.37) and definitions it is clear that L Au = f Ω in D (Ω) . To * "M , and consider check (8.5.38), pick some arbitrary ϕ = (ϕα )1≤α ≤ M , ∈ Lip (∂Ω)

* " , M n Φ = (Φα ) such that Φ = ϕ. Then , ∈ Lip (R ) 1≤α ≤ M p , p 

([B1−s

∂Ω

c

, ∗ (∂Ω,σ)] M )

∂νA(u, f ), ϕ ∫ =

Ω

p , p 

[B1−s

, (∂Ω,σ)] M

∫

αβ

a jk (∂k uβ )(∂j Φα ) dL n +

∫ =

.

∂Ω

Ω

(L Au)α Φα dL n

αβ ν j a jk (∂k uβ ) ∂Ω ϕα dσ,

(8.5.39)

thanks to (8.5.16), the definition of f in (8.5.37), and the De Giorgi-Federer version of Divergence Formula from [133, Theorem 1.1.1] (bearing in mind that the present geometric assumptions entail ∂∗ Ω = ∂Ω; cf. (A.0.89), [133, (5.11.35)], and [133, Lemma 5.11.9)]). With this in hand, (8.5.38) now follows with the help of Lemma 7.1.10.  Proposition 8.5.7, stated below, deals with the definition and properties of what we shall refer to as the principal symbol map. To state the stage, recall the convention made in (8.5.13). Proposition 8.5.7 Suppose Ω ⊆ Rn is an (ε, δ)-domain with a compact Ahlfors regular boundary, and such that Rn \ Ω is n-thick. Abbreviate σ := H n−1 ∂Ω, and fix two exponents p, p ∈ (1, ∞) satisfying 1/p + 1p = 1 along with some s ∈ (0, 1), then set a := 1 − s − p1 ∈ − p1 , 1 − p1 . Also, let D be an N × M homogeneous first-order system with constant (complex) coefficients, say D=

n  j=k

γβ

bk ∂k

1≤γ ≤ N . 1≤β ≤M

(8.5.40)

In this setting, consider the map  *  " M * 1, p ∗" N ap (−i)Sym(D; ν) : (u, f ) ∈ L p Ω, δ∂Ω L n ⊕ W−a (Ω) : * p, p * "N "N −→ Bs−1 (∂Ω, σ) Du = f Ω in D (Ω)

(8.5.41)

496

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

*  *  1, p ∗" N " M ap assigning to each pair (u, f ) with u ∈ L p Ω, δ∂Ω L n and f ∈ W−a (Ω) *  "N with the property that Du = f Ω in D (Ω) the functional (−i)Sym(D; ν)(u, f ) ∈

*

p, p

B1−s (∂Ω, σ)

"N

∗

* p, p "N = Bs−1 (∂Ω, σ)

(8.5.42)

acting according to (with the summation convention over repeated indices in effect) . (−i)Sym(D; ν)(u, f ), ϕ p , p  p , p  N ∗ ([B (∂Ω,σ)] ) [B (∂Ω,σ)] N 1−s

:=  f , Φ − for all ϕ ∈

*

∫ Ω

"N p, p B1−s (∂Ω, σ)

.

1−s

u, D Φ dL n

(8.5.43)

* 1, p "N and Φ ∈ W−a (Ω)

with the property that TrΩ→∂Ω Φ = ϕ, where the first set of brackets in the right-hand side of the first line of (8.5.43) is the * 1, p "N * 1, p ∗" N and W−a (Ω) . canonical duality pairing between W−a (Ω) Then, in relation to this “symbol” map, the following statements are true. (1) The operator (8.5.41)-(8.5.43) is well defined, linear, and bounded in the sense that there exists a constant C ∈ (0, ∞) with the property that ! ! !(−i)Sym(D; ν)(u, f )! p, p (8.5.44) [B (∂Ω,σ)] N s−1



≤ C u[L p (Ω,δ a p L n )] M +  f [(W 1, p (Ω))∗ ] N ∂Ω



−a

for each pair (u, f ) belonging to the domain of (−i)Sym(D; ν) (cf. (8.5.41)). *  * 1, p " M ∗" N ap (2) For each pair (u, f ) with u ∈ L p Ω, δ∂Ω L n and f ∈ W−a (Ω) such *  "N that Du = f Ω in D (Ω) one has the following generalized integration by parts formula: . (−i)Sym(D; ν)(u, f ), TrΩ→∂Ω w [B p, p (∂Ω,σ)] N p , p  ([B (∂Ω,σ)] N )∗ 1−s

1−s

∫ =  f , w −

-

Ω

. * 1, p "N u, D w dL n for each w ∈ W−a (Ω) . (8.5.45)

(3) In addition to the N × M system D from (8.5.40), consider a homogeneous, ' × N system D ' in Rn , say constant (complex) coefficient, first-order M '= D

n  j=1

αγ ' b j ∂j

1≤α ≤M , 1≤γ ≤ N

(8.5.46)

8.5 Weighted Sobolev Spaces of Negative Order, Duality, and Conormal Derivatives

497

' and define the homogeneous, constant (complex) coefficient, second-order M×M system ' L := DD. (8.5.47) Also, define the coefficient tensor (with the summation convention over repeated indices in effect)  αβ  αβ 'αγ γβ AD,D := a jk 1≤α ≤ M (8.5.48) , where each a jk := b j bk . ' 1≤β ≤M 1≤ j,k ≤n

* 1, p "M * 1, p ∗" M , Then for each pair (u, f ) with u ∈ Wa (Ω) and f ∈ W−a (Ω) such *  "M , one has that Lu = f Ω in D (Ω) A D, ,D

∂ν

' ν)(Du, f ), (u, f ) = (−i)Sym( D;

(8.5.49)

where the conormal derivative in the left-hand side above is defined in the sense of Proposition 8.5.3, and the principal symbol map in the right-hand side is ' defined as in (8.5.41) (with u replaced by Du and D replaced by D). Proof The claim in item (1) may be justified in a very similar fashion to the proof of Proposition 8.5.3. Also, the claim in item (2) is implied by item (1), while the claim in item (3) is clear from (8.5.43) and (8.5.16).  We conclude this section by recording the following Green-type integration by parts formula involving a null-solution of a system L and a null-solution of the transpose system L  , belonging to certain suitably weighted Sobolev spaces with dual integrability exponents. Corollary 8.5.8 Assume Ω ⊆ Rn is an (ε, δ)-domain with a compact Ahlfors regular boundary, and such that Rn \ Ω is n-thick. Abbreviate σ := H n−1 ∂Ω, and fix two  satisfying exponents p, p ∈ (1,  1/p + 1p = 1 along with some s ∈ (0, 1), then set  ∞) 1 1 1 a := 1 − s − p ∈ − p , 1 − p . Next, consider a homogeneous, constant (complex) ' × N system D ' in Rn , along with a homogeneous, constant coefficient, first-order M (complex) coefficient, first-order N × M system D in Rn , and define the homogeneous, ' × M system constant (complex) coefficient, second-order M ' L := DD.

(8.5.50)

Then for any two vector-valued functions, * 1, p "M u ∈ Wa (Ω) with Lu = 0 in Ω, and "M * 1, p , with L  w = 0 in Ω, w ∈ W−a (Ω) one has

(8.5.51)

498

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

0 p, p , [B s−1 (∂Ω,σ)] M

' ν)(Du, 0), TrΩ→∂Ω w (−i)Sym( D;

1 p , p 

[B1−s

, (∂Ω,σ)] M

(8.5.52)

0 1 ' w, 0) p, p = [Bsp, p (∂Ω,σ)] M TrΩ→∂Ω u, (−i)Sym(D ; ν)( D . [B (∂Ω,σ)] M −s

Proof This follows from (8.5.19) and (8.5.49).



8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n We begin by introducing a special brand of Sobolev spaces, requiring the membership of distributional derivatives to solid maximal Lebesgue spaces defined as in (A.0.63) with μ a power weighted Lebesgue measure. Specifically, we make the following definition. Definition 8.6.1 Let Ω be an open, nonempty, proper subset of Rn , and pick an integrability exponent p ∈ (0, ∞) along with an integer k ∈ N0 and some power k, p a ∈ R. In this context, define the weighted maximal Sobolev space Wa, (Ω) as    k, p p ap k,1 (Ω) : ∂ α u ∈ L Ω, δ∂Ω L n for all α ∈ N0n with |α| ≤ k Wa, (Ω) := u ∈ Wloc (8.6.1) k, p and equip it with the quasi-norm given for each u ∈ Wa, (Ω) by  uW k, p (Ω) := ∂ α u Lp (Ω,δ a p L n ) a,

∂Ω

|α | ≤k

≈

 ∫



(∂ α u),θ p δ ap dL n |α | ≤k

∂Ω

Ω

1 p

for θ ∈ (0, 1).

(8.6.2)

In the same setting as above, let us also consider the homogeneous weighted . k, p maximal Sobolev space homogeneous weighted maximal Sobolev Wa, (Ω) defined as   . k, p  p ap k,1 Wa, (Ω) := u ∈ Wloc (Ω) : ∂ α u ∈ L Ω, δ∂Ω L n for all α ∈ N0n with |α| = k (8.6.3) . k, p and for each u ∈ Wa, (Ω) set  uW. k, p (Ω) := ∂ α u Lp (Ω,δ a p L n ) . (8.6.4) a,

|α |=k

∂Ω

In contrast to the standard generic L p -based Sobolev spaces in Ω which turn out to have good functional analytic properties only when p ≥ 1, our weighted maximal Sobolev spaces (of the sort introduced in Definition 8.6.1) turn out to

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

499

behave remarkably well even when p < 1. This point is underscored by the trace result proved in Theorem 8.6.8, the regularity results established in Corollary 9.2.31 and Corollary 9.2.38, and the fact that basic singular integral operators, like the double and single layer potential operators associated with an elliptic system, behave naturally on this scale of spaces (a topic addressed at length in [136, Chapter 4]). For now, we wish to make several remarks in relation to the brand of weighted Sobolev spaces from Definition 8.3.4. First, given any open nonempty proper subset Ω of Rn , from [133, (6.6.44)] we conclude that we have the continuous embedding k, p

k, p

Wa, (Ω) → Wa (Ω) for all p ∈ (0, ∞), k ∈ N0, and a ∈ R.

(8.6.5)

The second remark concerning the weighted Sobolev spaces from Definition 8.6.1 and Definition 8.3.4 is contained in the lemma below. To facilitate its statement, we make the following convention. Given a homogeneous constant (complex) coefficient second-order weakly elliptic M × M system L in Rn along with an arbitrary open set Ω ⊆ Rn , we agree to abbreviate   "M * : Lu = 0 in Ω . (8.6.6) Ker L := u ∈ C ∞ (Ω) The reader is also reminded that the characteristic matrix of L is defined in (A.0.64). Lemma 8.6.2 With M, m ∈ N, let L be a constant (complex) coefficient homogeneous M × M system of order 2m in Rn , with the property that det [L(ξ)]  0 for each ξ ∈ Rn \ {0}. Also, assume Ω ⊆ Rn is an open, nonempty, proper subset of Rn , and suppose (8.6.7) 0 < p < ∞, k ∈ N0, a ∈ R. Then * k, p "M * k, p "M Wa, (Ω) ∩ Ker L = Wa (Ω) ∩ Ker L with equivalent norms.

(8.6.8)

Proof This is a consequence of (8.6.5), [133, (6.5.40) in Theorem 6.5.7], and [133, (6.6.91)].  Our third remark pertaining to the relation between the weighted Sobolev spaces from Definition 8.6.1 and Definition 8.3.4 is discussed in the next proposition. Proposition 8.6.3 Let Ω be an open, nonempty, proper subset of Rn . Suppose   n−1 (n − 1) p1 − 1 < s < 1, k ∈ N0, n < p ≤ 1,    then set a := 1 − s − p1 ∈ − p1 , −n p1 − 1 (8.6.9) and q :=

np n−1+p−sp

∈ (1, n).

Then one has the continuous embedding k, p

Wa, (Ω) → W k,q (Ω).

(8.6.10)

500

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

Proof From the embedding in [133, (6.6.74)] used with α := (1 − p + sp)/(np) (which, thanks to (8.6.9), is a number belonging to the interval (0, 1/p)) we see that we have the continuous embedding p p(1−s)−1 n  L Ω, δ∂Ω L → L q (Ω, L n ).

In turn, this readily implies (8.6.10), upon noting that ap = p(1 − s) − 1.

(8.6.11) 

The fourth (and last) remark we make is contained in the following lemma, where we compare smooth truncations with restrictions to bounded sets (something which is going to be relevant later on). Lemma 8.6.4 Let Ω be an open (nonempty, proper) subset of Rn with compact boundary, and for each R > 0 define ΩR := Ω ∩ B(0, R). Also, fix k ∈ N0 . Suppose u ∈ C k (Ω) is a function whose derivatives of order ≤ k are all subaveraging in Ω (in the sense of [133, Definition 6.5.1]). Finally, select some p ∈ (0, ∞) along with a > −1/p. Then the following equivalence holds: ψu ∈ Wa (Ω) for each ψ ∈ Cc∞ (Rn ) k, p

(8.6.12)

if and only if

k, p u Ω R ∈ Wa, (ΩR ) for each sufficiently large R > 0.

(8.6.13)

Proof The property formulated in (8.6.12) is readily seen to be equivalent to demanding that   a L n for each sufficiently large R > 0 ∂ α u ∈ L p ΩR, δ∂Ω (8.6.14) and each multi-index α ∈ N0n with |α| ≤ k. Note that for each sufficiently large R > 0 we have  ∫ ap δ∂Ω R |∂ α u| p dL n < +∞ |α | ≤k

Ω R \Ω R/2

∫   since u ∈ C k ΩR \ ΩR/2 and Ω

R \Ω R/2

(8.6.15)

ap

δ∂Ω R dL n < +∞ given that ap > −1. Also,

δ∂Ω R ≈ δ∂Ω on ΩR/2 .

(8.6.16)

From (8.6.15)-(8.6.16) we then see that (8.6.14) is further equivalent to having

k, p u Ω R ∈ Wa (ΩR ) for each sufficiently large R > 0. (8.6.17) Finally, that this is equivalent to (8.6.13) is clear from [133, Lemma 6.6.7] (bearing in mind [133, (6.5.2)]).  Our principal result in this section is Theorem 8.6.8 dealing with the nature of the trace operator acting from power weighted maximal Sobolev spaces, of the

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

501

sort introduced in Definition 8.6.1. The aforementioned trace theorem is stated and proved later. To facilitated dealing with this, for the time being we discuss several key technical results, starting with the following lemma.   Lemma 8.6.5 Suppose Φ : Rn × Rn \ diag → C is a function with the property that there exist a constant A ∈ (0, ∞) along with two exponents, M ∈ (0, ∞) and α ∈ (0, 1], such that |Φ(x, z)| ≤

A if x, z ∈ Rn with x  z, |x − z| M

(8.6.18)

α



Φ(x, z) − Φ(y, z) ≤ A|x − y| if x, y, z ∈ Rn with |x − y| < 1 |x − z|. (8.6.19) 2 M+α |x − z|

Also, assume θ ∈ (0, 1), and

max n − 1, 2(n − 1) − M p < μ < n − 1 + αp.

p ∈ (0, ∞),

(8.6.20)

Finally, having considered a closed upper Ahlfors regular set Σ ⊆ Rn , denote by δΣ the distance function to Σ, and abbreviate σ := H n−1 Σ. Then there exists a constant C ∈ (0, ∞), depending only on A, M, α, p, θ, μ, n, with the property that p ∫ ∫ ⨏



Φ(x, z) − Φ(y, z) dz dσ(x) dσ(y) ≤ CδΣ (z0 )2(n−1)−μ−M p |x − y| μ Σ Σ B(z0,θ δΣ (z0 )) (8.6.21) for each point z0 ∈ Rn \ Σ.   Proof Fix ε ∈ 0, (1 − θ)/4 and consider an arbitrary point z0 ∈ Rn \ Σ. Then for   each x, y ∈ Σ and z ∈ B z0, θδΣ (z0 ) we have  θ   θ  δΣ (z) ≤ 1−θ |z − x|, (8.6.22) |z − z0 | < θδΣ (z0 ) ≤ 1−θ so |x − z0 | ≤ |x − z| + |z − z0 | < |x − z| +



θ 1−θ



|z − x| = (1 − θ)−1 |x − z|. (8.6.23)

In particular, |x − y| < ε|x − z0 | =⇒ |x − y| < 12 |x − z|.

(8.6.24)

Work now under the assumption that |x − y| < 12 |y − z0 | (which, given that ε < 1/2, is the case if |x − y| < ε|y − z0 |). Then |y − z0 | ≤ |y − x| + |x − z0 | < 12 |y − z0 | + |x − z0 |, hence |y − z0 | < 2|x − z0 | which, in combination with (8.6.22), gives  2  |x − z|. (8.6.25) |y − z0 | < 2|x − z0 | ≤ 2|x − z| + 2|z − z0 | ≤ 1−θ

502

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

Thus, |x − y| < ε|y − z0 | =⇒ |x − y| < 12 |x − z|. Moving on, consider 

 S := (x, y) ∈ Σ × Σ : |x − y| < ε max |x − z0 |, |y − z0 | .

(8.6.26)

(8.6.27)

Then S = S1 ∪ S2, where

(8.6.28)

   S1 := (x, y) ∈ Σ × Σ : y ∈ B x, ε|x − z0 | ,    S2 := (x, y) ∈ Σ × Σ : x ∈ B y, ε|y − z0 | ,

(8.6.29)

and   |x − y| < 12 |x − z| whenever (x, y) ∈ S and z ∈ B z0, θδΣ (z0 ) . Rely on (8.6.28) to write ⨏ ∬ (x,y)∈S

B(z0,θ δΣ (z0 ))



Φ(x, z) − Φ(y, z) dz

p

(8.6.30)

dσ(x) dσ(y) ≤ I1 + I2, (8.6.31) |x − y| μ

where, for j ∈ {1, 2}, ⨏

∬ I j :=

(x,y)∈S j

Note that

B(z0,θ δΣ (z0 ))

A|x − y| α dz |x − z| M+α

p

dσ(x) dσ(y) . |x − y| μ

(8.6.32)

∬

|x − y| αp dσ(x) dσ(y) (M+α)p |x − y| μ (x,y)∈S1 |x − z0 | ∫  ∫ 1 dσ(y) =C dσ(x) μ−αp (M+α)p x ∈Σ |x − z0 | y ∈Σ∩B(x,ε |x−z0 |) |x − y|

I1 ≤ C

∫ ≤C

x ∈Σ

dσ(x) ≤ CδΣ (z0 )2(n−1)−μ−M p . |x − z0 | M p+μ−(n−1)

(8.6.33)

Above, the first inequality is based on (8.6.30) and (8.6.19), plus the observation  that |x − z| ≈ |x − z0 | uniformly for x ∈ Σ, z0 ∈ Rn \ Σ, and z ∈ B z0, θδΣ (z0 ) . The subsequent equality uses the description of S1 given in (8.6.29). The penultimate

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

503

inequality is a consequence of the first estimate in [133, (7.2.5)] (since μ−αp < n−1), while the last inequality in (8.6.33) is implied by [133, (8.7.92)] (presently used with Ω := Rn \ Σ), bearing in mind that M p + μ − (n − 1) > n − 1. Going further, we write ∬ |x − y| αp dσ(x) dσ(y) I2 ≤ C (M+α)p |x − y| μ (x,y)∈S2 |y − z0 | ∫  ∫ 1 dσ(x) =C dσ(y) μ−αp (M+α)p y ∈Σ |y − z0 | x ∈Σ∩B(y,ε |y−z0 |) |x − y| ∫ ≤C

y ∈Σ

dσ(y) ≤ CδΣ (z0 )2(n−1)−μ−M p . |y − z0 | M p+μ−(n−1)

(8.6.34)

The first estimate in (8.6.34) is seen from (8.6.30), (8.6.19), and (8.6.25). The following equality in (8.6.34) is based on the description of S2 from (8.6.29). The last two inequalities in (8.6.34) are justified as the case of (8.6.33), based on the first estimate in [133, (7.2.5)] and [133, (8.7.92)]. Going further, make use of (8.6.28) to write, for some constant C ∈ (0, ∞) depending only on the exponent p, p ⨏ ∬



Φ(x, z) − Φ(y, z) dz dσ(x) dσ(y) |x − y| μ (x,y)∈(Σ×Σ)\S B(z0,θ δΣ (z0 ))   ≤ C II1 + II2 ,

(8.6.35)

where ∬

p

⨏

II1 :=

B(z0,θ δΣ (z0 ))

(x,y)∈Σ×Σ |x−y | ≥ε |x−z0 |

|Φ(x, z)| dz

dσ(x) dσ(y) |x − y| μ

(8.6.36)

dσ(x) dσ(y) . |x − y| μ

(8.6.37)

and ∬ II2 := (x,y)∈Σ×Σ |x−y | ≥ε |y−z0 |

p

⨏ B(z0,θ δΣ (z0 ))

|Φ(y, z)| dz

  In turn, since |x−z| ≈ |x−z0 | uniformly for x ∈ Σ, z0 ∈ Rn \Σ, and z ∈ B z0, θδΣ (z0 ) , we may invoke (8.6.18), the second inequality in [133, (7.2.5)] (keeping in mind that μ > n − 1), and [133, (8.7.92)] to estimate

504

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

∫ II1 ≤ C

x ∈Σ

∫ ≤C

x ∈Σ

1 |x − z0 | M p

∫ y ∈Σ\B(x,ε |x−z0 |)

 dσ(y) dσ(x) |x − y| μ

dσ(x) ≤ CδΣ (z0 )2(n−1)−μ−M p . |x − z0 | M p+μ−(n−1)

(8.6.38)

n Likewise, since we  have |y − z| ≈ |y − z0 | uniformly for y ∈ Σ, z0 ∈ R \ Σ, and  z ∈ B z0, θδΣ (z0 ) , we may rely on (8.6.18), the second inequality in [133, (7.2.5)], and [133, (8.7.92)] to write ∫  ∫ 1 dσ(x) dσ(y) II2 ≤ μ Mp y ∈Σ |y − z0 | x ∈Σ\B(y,ε |y−z0 |) |x − y|

∫ ≤C

y ∈Σ

dσ(y) ≤ CδΣ (z0 )2(n−1)−μ−M p . |y − z0 | M p+μ−(n−1)

(8.6.39)

At this stage, the estimate claimed in (8.6.21) follows by combining (8.6.31), (8.6.33), (8.6.34), (8.6.35), (8.6.38), and (8.6.39).  The proposition below prefigures the format of the main trace result, discuss later in Theorem 8.6.8.   Proposition 8.6.6 Assume Φ : Rn × Rn \ diag → C (where n ∈ N, with n ≥ 2) is a function satisfying, for some constant C ∈ (0, ∞), |Φ(x, z)| ≤

C if x, z ∈ Rn with x  z, |x − z| n−1



Φ(x, z) − Φ(y, z) ≤ C|x − y| if x, y, z ∈ Rn with |x − y| < 1 |x − z|. 2 |x − z| n

(8.6.40) (8.6.41)

Also, consider an open, nonempty, proper subset Ω of Rn with an upper Ahlfors regular boundary. Abbreviate σ := H n−1 ∂Ω, and recall that δ∂Ω denotes the distance function to ∂Ω. Finally, suppose   n−1 (n − 1) p1 − 1 < s < 1, (8.6.42) n < p ≤ 1, and fix a function Then

p p(1−s)−1 n  L . ϕ ∈ L Ω, δ∂Ω

(8.6.43)

∫ Ω

|Φ(x, z)||ϕ(z)| dz < +∞ for H n−1 -a.e. point x ∈ Rn,

(8.6.44)

and the function (defined by means of an absolutely convergent integral) as ∫ u(x) := Φ(x, z)ϕ(z) dz for H n−1 -a.e. x ∈ Rn (8.6.45) Ω

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

505

satisfies, for some constant C ∈ (0, ∞) independent of ϕ, ∫

∫

∂Ω

∂Ω

|u(x) − u(y)| p dσ(x) dσ(y) |x − y| n−1+sp

 p1 ≤ Cϕ L p (Ω,δ p(1−s)−1 L n ) . 

∂Ω

(8.6.46)

Moreover, the limit ⨏ (TrRn →∂Ω u)(x) := [u]Rn (x) := lim+ r→0

u(y) dy B(x,r)

(8.6.47)

exists and equals u(x) at σ-a.e. point x ∈ ∂Ω. To offer an example, work in the two-dimensional setting. Specifically, consider an open, nonempty, proper subset Ω of R2 ≡ C with an upper Ahlfors regular boundary, and pick a function p p(1−s)−1 2  ϕ ∈ L Ω, δ∂Ω L where

Then

∫ Ω

1 2

< p ≤ 1 and

1 p

− 1 < s < 1.

|ϕ(ζ)| dL 2 (ζ) < +∞ for H 1 -a.e. point z ∈ C, |ζ − z|

and the function

∫ u(z) :=

Ω

ϕ(ζ) dL 2 (ζ) for H 1 -a.e. z ∈ C ζ−z

(8.6.48)

(8.6.49)

(8.6.50)

(which is well defined, since it is given by an absolutely convergence integral) satisfies, for some constant C ∈ (0, ∞) independent of ϕ, ∫ ∂Ω

∫ ∂Ω

|u(z) − u(w)| p dH 1 (z) dH 1 (w) |z − w| 1+sp

 p1 ≤ Cϕ L p (Ω,δ p(1−s)−1 L 2 ), 

∂Ω

(8.6.51)

and has the property that ⨏ the limit (TrC→∂Ω u)(z) := lim+ exists and equals u(z) at

r→0 1 H -a.e.

u dL 2 B(z,r)

(8.6.52)

point z ∈ ∂Ω.

Here is the proof of Proposition 8.6.6. Proof of Proposition 8.6.6 Introduce q :=

np n−1+p−sp

∈ (1, n),

(8.6.53)

with the membership implied by (8.6.42). Then, if ϕ ' denotes the extension of ϕ to Rn by zero outside Ω, based on (8.6.11) we see that

506

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

ϕ ' ∈ L q (Rn, L n ) and  ϕ ' L q (Rn, L n ) ≤ Cϕ L p (Ω,δ p(1−s)−1 L n ) 

∂Ω

for some constant C ∈ (0, ∞) independent of ϕ. We shall show next that ∫ |ϕ(z)| dz < +∞ for H n−1 -a.e. point x ∈ Rn . n−1 |x − z| Ω

(8.6.54)

(8.6.55)

To this end, [61, Theorem 4.1, p. 294] (used with α := 1 and p := 1) plus [49, Remark on p. 156, Theorem 3 on p. 193] give ∫ |ϕ '(z)| dz < +∞ for H n−1 -a.e. point x ∈ Rn . (8.6.56) n−1 B(x,1) |x − z| See also [5, §4] and the comment in relation to [105, (7.8), p. 178] in this regard. Let us next observe that q , the Hölder conjugate exponent of q ∈ (1, n), satisfies q  > n/(n − 1). Consequently, for each x ∈ Rn we may write ∫ ∫ ∫ |ϕ(z)| |ϕ '(z)| |ϕ '(z)| dz = dz + dz n−1 n−1 n |x − z| |x − z| |x − z| n−1 Ω B(x,1) R \B(x,1) ∫ 1 ∫ |ϕ '(z)| dz q q n n ≤C dz + C ϕ '  L (R , L )  n−1 (n−1)q B(x,1) |x − z| R n \B(0,1) |z| < +∞ for H n−1 -a.e. point x ∈ Rn,

(8.6.57)

thanks to (8.6.56), Hölder’s inequality, (8.6.54), and the fact that (n − 1)q  > n. This proves (8.6.44). In turn, (8.6.55) is a consequence of (8.6.44) and (8.6.40). Turning our attention to (8.6.46), consider a Whitney decomposition of Ω into Euclidean cubes Q j of side-length (Q j ) (cf. [133, Proposition 7.5.3]). Concretely, suppose 

j ∈N Q j

=Ω=



j ∈N 2Q j ,

Q˚ j ∩ Q˚ k =  if j  k,

δ∂Ω (x) ≈ (Q j ) uniformly for j ∈ N and x ∈ 2Q j , + and j ∈N 12Q j ≤ C for a constant C = Cn ∈ (0, ∞). Having fixed some θ ∈ (0, 1) we may then estimate

(8.6.58)

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

∫

∂Ω

507

∫

|u(x) − u(y)| p dσ(x) dσ(y) n−1+sp ∂Ω |x − y| /p ∫ ∫ ∫



dσ(x) dσ(y) ≤ |ϕ(z)| Φ(x, z) − Φ(y, z) dz |x − y| n−1+sp ∂Ω ∂Ω Ω ∫ =

∂Ω

∫ ≤



∫ ∂Ω



j

∫



∂Ω

∫



∂Ω

∫



|ϕ(z)| Φ(x, z) − Φ(y, z) dz

Qj

j

∫

∂Ω

≤C

∫

2Q j

j

∫



ϕ,θ p dL n

1 p

/p

Qj

≤C

 ⨏ 2Q j

j

× ≤C

2Q j

j

∂Ω

2Q j

j

≤C

∫ 2Q j

j

= Cϕ

p

p

∂Ω



Φ(x, z) − Φ(y, z) dz

/p

Qj

dσ(x) dσ(y) |x − y| n−1+sp



ϕ,θ p δ n(p−1) dL n × ∂Ω

×  ∫

∫

∫

∫

≤C

dσ(x) dσ(y) |x − y| n−1+sp



ϕ,θ p dL n × ∫

 ∫

dσ(x) dσ(y) |x − y| n−1+sp

×



Φ(x, z) − Φ(y, z) dz

×

/p

Qj

 ⨏

∂Ω

dσ(x) dσ(y) |x − y| n−1+sp



Φ(x, z) − Φ(y, z) dz

sup |ϕ| Qj

/p

⨏

∫

∂Ω

∂Ω



Φ(x, z) − Φ(y, z) dz Qj

/p

dσ(x) dσ(y) |x − y| n−1+sp



ϕ,θ p δ n(p−1) dL n (Q j )(n−1)(1−p)−sp ∂Ω



ϕ,θ p δ p(1−s)−1 dL n ≤ C ∂Ω p(1−s)−1

L (Ω,δ ∂Ω

Ln)

.

∫ Ω



ϕ,θ p δ p(1−s)−1 dL n ∂Ω

(8.6.59)

The first inequality above uses (8.6.44)-(8.6.45), and the subsequent equality is based on the first equality in the first line of (8.6.58). Next, the second inequality in (8.6.59) is obvious, while the third inequality in (8.6.59) is a consequence of [133, Proposition 6.6.3], [133, Lemma 6.5.2], [133, (6.5.13)], and [133, (6.6.46)]. Going

508

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

further, the fourth inequality in (8.6.59) is implied by the fact that p ∈ (0, 1], while the fifth inequality in (8.6.59) (where the integral average has been relocated) is justified by the second line in (8.6.58). The sixth inequality in (8.6.59) is consequence of Lemma 8.6.5 used here with Σ := ∂Ω and μ := n − 1 + sp, keeping in mind that the current function Φ satisfies (8.6.18)-(8.6.19) with M := n − 1 and α := 1 (so the inequalities in (8.6.20) are satisfied by these choices, thanks to (8.6.42)). Going further, the seventh inequality in (8.6.59) originates from the second line in (8.6.58), whereas the eighth inequality in (8.6.59) comes from the second equality in the first line of (8.6.58) and the third line in (8.6.58). The final equality in (8.6.59) is a simple consequence of the definition of the quasi-norm in the maximal Lebesgue space p p(1−s)−1 n  L . L Ω, δ∂Ω To deal with the very last property in the statement of the proposition, we first make the claim that if α ∈ (0, n) and if Iα (x) := |x| α−n for each x ∈ Rn , then there exists a finite constant C = C(n, α) > 0 with the property that 1B(0,r) ∗ Iα ≤ C · Iα everywhere in Rn for each radius r ∈ (0, 1). To prove (8.6.60), fix x ∈ Rn \ {0} along with r ∈ (0, 1) and observe that ∫   dy . 1B(0,r) ∗ Iα (x) = n−α B(x,r) |y|

(8.6.60)

(8.6.61)

In the case when |x| ≥ 2 any point y ∈ B(x,  r) satisfies |y| ≥ |x|/2 which, in  light of (8.6.61), further implies 1B(0,r) ∗ Iα (x) ≤ 2n−α Iα (x). Finally, in the case when |x| ≤ ∫2 we have 2α−n ≤ Iα (x) as well as B(x, r) ⊆ B(0, 3). In view of the formula B(0,3) |y| α−n dy = 3α n−1 ωn−1 , we ultimately conclude that we have   1B(0,r) ∗Iα (x) ≤ 2n−α 3α n−1 ωn−1 Iα (x) in this case. The claim in (8.6.60) is therefore justified. Going further, consider a point x ∈ Rn with the property that ∫ |ϕ(z)| dz < +∞. (8.6.62) |x − z| n−1 Ω With I1 (w) := |w| 1−n for each w ∈ Rn , it follows from (8.6.40) and (8.6.60) (presently employed with α := 1) that there exists some C ∈ (0, ∞) with the property that for each z ∈ Rn and r ∈ (0, 1) we have ∫ ∫   dy |Φ(y, z)| dy ≤ C = C 1B(0,r) ∗ I1 (x − z) n−1 B(x,r) B(x,r) |y − z| ≤ C · I1 (x − z) =

C . |x − z| n−1

Granted (8.6.62)-(8.6.63), Fubini’s Theorem allows us to write

(8.6.63)

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

∫

∫

∫

u(y) dy = B(x,r)

B(x,r)

=

∫ ∫ Ω

Ω

509

Φ(y, z)ϕ(z) dz dy Φ(y, z) dy ϕ(z) dz for each r ∈ (0, 1).

(8.6.64)

B(x,r)

Since (8.6.41) implies that for each z ∈ Rn the function Φ(·, z) is continuous in Rn \ {z}, we have ⨏ lim+ Φ(y, z) dy = Φ(x, z) for each z ∈ Rn \ {x}. (8.6.65) r→0

B(x,r)

Based on (8.6.64)-(8.6.65) and Lebesgue’s Dominated Convergence Theorem we then conclude that ⨏ ∫ ⨏ lim+ u(y) dy = lim+ Φ(y, z) dy ϕ(z) dz r→0

B(x,r)

r→0

Ω

∫ = lim+ r→0

Ω

B(x,r)

Φ(x, z)ϕ(z) dz = u(x).

(8.6.66)

The above argument shows that ⨏ the limit [u]Rn (x) := lim+ r→0

equals u(x) at each point x ∈

u(y) dy exists and B(x,r) Rn where

(8.6.67)

(8.6.62) holds.

Then the claim in (8.6.47) becomes a consequence of this and (8.6.55).



In order to wrap up the preparations for dealing with the main trace result in this section, subsequently presented in Theorem 8.6.8, we need to revisit the arguments from [103] leading up to Jones’ extension theorem (recalled in Theorem 8.3.2) and explore the prospect of incorporating weights (in the form of powers of the distance to the boundary function) in the estimates satisfied by Jones’ extension operator (8.3.23). This task is accomplished in the following lemma. Lemma 8.6.7 Let Ω be an (ε, δ)-domain in Rn with rad (Ω) > 0, and recall that δ∂Ω denotes the distance function to ∂Ω. Fix some k ∈ N and bring in Jones’ extension operator Λk from Theorem 8.3.2. Given a function u ∈ W k,q (Ω) with q ∈ [1, ∞], define





+ + G := |α | ≤k ∂ α Λk u ∈ L q (Rn, L n ), f := |α | ≤k ∂ α u ∈ L q (Ω, L n ),

and set g := G Rn \Ω ∈ L q (Rn \ Ω, L n ). (8.6.68) Finally, pick θ ∈ (0, 1), p ∈ (0, ∞), and γ ∈ R arbitrary. Then there exists a constant C ∈ (0, ∞) independent of u with the property that

510

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

∫ p

γ

∫ p

g,θ δ∂Ω dL n ≤ C R n \Ω

γ

f,θ δ∂Ω dL n

(8.6.69)

Ω

where the solid maximal functions f,θ and g,θ are considered (as in (A.0.111)) relative to the open sets Ω and Rn \ Ω where the functions f , g are defined. Proof For starters, observe that simple connectivity arguments give that       dist x, ∂(Rn \ Ω) = dist x, ∂(Ω) = dist x, Ω = δ∂Ω (x) for each point x ∈ Rn \ Ω.

(8.6.70)

To proceed, pick some Whitney decomposition W(Ω) of Ω, along with a Whitney  decomposition W Rn \ Ω of the open set (Rn \ Ω)◦ = Rn \ Ω (with the convention     that W Rn \ Ω =  if Ω = Rn ), and define Ws Rn \ Ω as in (8.3.19). Next, as in [103, p. 77], abbreviate     W1 := W(Ω), W2 := W Rn \ Ω , W3 := Ws Rn \ Ω . (8.6.71) Fix a large constant Co ∈ (1, ∞) and a large integer m ∈ N (depending only geometry). For each Q ∈ W1 then define the cluster ' [Q]m,θ to be the union of all dilated cubes of the form (1 + Co θ) · Q ' where Q ∈ W1 is such that there exists a family {Q j }1≤ j ≤m ⊆ W1 ' and 4−1 ≤ (Q j )/ (Q j+1 ) ≤ 4 for each with Q1 = Q, Q m = Q, j ∈ {1, . . . , m − 1}.

(8.6.72)

In addition, define the cluster [Q]m as [Q]m,θ in (8.6.72) with θ := 0. In this notation, given any Q j , Q k ∈ W3 with Q j ∩ Q k   it follows that each chain Fj,k connecting Q∗j with Q∗k as in [103, Lemma 2.8, p. 78] is contained in the cluster [Q∗j ]m . With F(Q j ) defined for each Q j ∈ W3 as in [103, p. 80] we then have (recall that Q∗ denotes the reflexion of Q across the boundary of Ω; cf. Theorem 8.3.2) F(Q) ⊆ [Q∗ ]m for each Q ∈ W3 .

(8.6.73)

A geometric argument (based on (8.3.12)-(8.3.16); cf. [103, (3.1)-(3.2), p. 80]) also gives that  1[Q∗ ]m, θ ≤ M for some M ∈ (0, ∞). (8.6.74) Q ∈W3

Granted (8.6.73), from [103, Lemma 3.2, p. 80] used with p := ∞ we conclude (keeping in mind that (Q) ≤ εδ/(16n) for each Q ∈ W3 ) that there exists some C ∈ (0, ∞) such that g L ∞ (Q, L n ) ≤ C f  L ∞ ([Q∗ ]m, L n ) for each Q ∈ W3 .

(8.6.75)

In turn, (8.6.75) implies that for each fixed λ ∈ (1, 3) (cf. (8.3.17)-(8.3.18)) we have

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

g L ∞ (λQ, L n ) ≤ C ·

max

' ' ∈W3, Q∩Q Q ' (Q)≈(Q)

 f  L ∞ ([Q'∗ ]m, L n ) for each Q ∈ W3 .

511

(8.6.76)

In particular, if θ ∈ (0, 1) is small enough from (8.6.76) and (A.0.111) we conclude that, on the one hand, g,θ (x) ≤ C ·

max

' ' ∈W3, Q∩Q Q ' (Q)≈(Q)

 f  L ∞ ([Q'∗ ]m, L n ) whenever x ∈ Q ∈ W3 .

On the other hand, given any Q ∈ W1 we may write ⨏ p f,θ dL n  f  L ∞ (Q, L n ) ≤ sup x ∈Q

≤C

(8.6.77)

1 p

B(x,θ δ ∂Ω (x))

⨏

p

(1+Co θ)·Q

f,θ dL n

1 p

,

(8.6.78)

thanks to [133, (6.6.6)], [133, (6.6.36)] (presently  used with u := f and s := p), and the observation that for each x ∈ Q we have B x, θδ∂Ω (x) ⊆ (1 + Co θ) · Q. As a consequence of (8.6.78) and (8.6.72) we therefore have 1 ⨏ p p f,θ dL n for each Q ∈ W1 . (8.6.79)  f  L ∞ ([Q]m, L n ) ≤ C [Q] m, θ

By combining (8.6.77) with (8.6.79) we arrive at the conclusion that ⨏ p g,θ (x) p ≤ C · max f,θ dL n whenever x ∈ Q ∈ W3 .

(8.6.80)

Integrating over x ∈ Q then yields ∫ ∫ p g,θ dL n ≤ C · max

(8.6.81)

' ' ∈W3, Q∩Q Q ' (Q)≈(Q)

'∗ ] m, θ [Q

' ' ∈W3, Q∩Q Q ' (Q)≈(Q)

Q

p

'∗ ] m, θ [Q

f,θ dL n for each Q ∈ W3,

and since δ∂Ω ≈ (Q) on each Q ∈ W3 (cf. (8.6.70) and (8.3.14)), we also obtain from (8.6.81) that ∫ ∫ p γ p γ n g,θ δ∂Ω dL ≤ C · max f,θ δ∂Ω dL n for each Q ∈ W3 . ' ' ∈W3, Q∩Q Q ' (Q)≈(Q)

Q

'∗ ] m, θ [Q

(8.6.82) Summing over Q ∈ W3 and keeping (8.6.74) in mind finally produces ∫ ∫  p γ p γ g,θ δ∂Ω dL n ≤ C f,θ δ∂Ω dL n where D3 := Q. D3

Ω

Q ∈W3

(8.6.83)

512

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

Moving on, [103, Lemma 3.3, p. 81] used with p := ∞ gives that  g L ∞ (Q, L n ) ≤ C  f  L ∞ (Q'∗, L n ) for each Q ∈ W2 \ W3 .

(8.6.84)

' ' ∈W3, Q∩Q Q

Much as before, this implies (assuming θ ∈ (0, 1) is small) that 1 ⨏ p p g,θ (x) ≤ C · max f,θ dL n whenever x ∈ Q ∈ W2 \ W3, (8.6.85) ' Q

' (1+Co θ)·Q

' ∈ W3 with the property that there where the maximum is taken over all cubes Q ' ∩ Q o  . Ultimately, exists Q o ∈ W2 such that (Q o ) ≈ (Q), Q o ∩ Q  , and Q this permits us to deduce that ∫ ∫  p γ p γ g,θ δ∂Ω dL n ≤ C f,θ δ∂Ω dL n where D2 := Q. (8.6.86) Ω

D2

Q ∈W2 \W3

At this stage, given that D2 ∪ D3 = Rn \ Ω, the estimate claimed in (8.6.69) follows from (8.6.83) and (8.6.86) assuming θ ∈ (0, 1) is small enough. Finally, the restriction that θ is small may then be lifted by invoking [133, (6.6.28)] (bearing in mind (8.6.70)).  We are now in a position to present our main result concerning traces of functions belonging to the power weighted maximal Sobolev spaces introduced in Definition 8.6.1, complementing the trace result from Theorem 8.3.6. Theorem 8.6.8 Let Ω ⊆ Rn be an (ε, δ)-domain with rad (Ω) > 0 whose boundary is an Ahlfors regular set, and abbreviate σ := H n−1 ∂Ω. Also, fix   n−1 (n − 1) p1 − 1 < s < 1, n < p ≤ 1, (8.6.87)    and set a := 1 − s − p1 ∈ − p1 , −n p1 − 1 . 1, p

Then for each function u ∈ Wa, (Ω) the boundary trace ⨏ (TrΩ→∂Ω u)(x) := [u]Ω (x) := lim+ u(y) dy r→0

B(x,r)∩Ω

(8.6.88)

exists at σ-a.e. point x ∈ ∂Ω,

and the following Gagliardo semi-norm estimate holds ∫ ∂Ω

∫ ∂Ω



(TrΩ→∂Ω u)(x) − (TrΩ→∂Ω u)(y) p |x − y| n−1+sp

 p1 dσ(x) dσ(y)

≤ CuW 1, p (Ω) a,

(8.6.89) for some constant C ∈ (0, ∞) independent of u.

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

513

  Moreover, if for each z ∈ ∂Ω and r ∈ 0, diam Ω some interior corkscrew point Ar (z) for z at scale r has been selected, and if C > large constant  0 is a sufficiently  so that B(Ar (z), r/C) ⊆ Ω for all z ∈ ∂Ω and r ∈ 0, diam Ω , then for each function 1, p u ∈ Wa, (Ω) one has ⨏ (TrΩ→∂Ω u)(z) = lim+ u(y) dy for σ-a.e. z ∈ ∂Ω. (8.6.90) r→0

B(Ar (z),r/C)

In addition, there exists some aperture parameter κ > 0 with the property that for 1, p each function u ∈ Wa, (Ω) which has a κ-nontangential pointwise limit at σ-a.e. point on ∂Ω it follows

κ−n.t. TrΩ→∂Ω u = u ∂Ω at σ-a.e. point on ∂Ω.

(8.6.91)

A few comments are in order here. First, thanks to (8.6.5), Theorem 8.3.6, and (7.9.10)-(7.9.11), the above trace result is in fact also true in the range 1 < p < ∞. Second, the Gagliardo semi-norm estimate (8.6.89) actually becomes a genuine p, p membership of the trace to the boundary Besov space Bs (∂Ω, σ) at least if Ω is 1, p a Lipschitz domain; cf. (7.9.22). Hence, in such a setting, TrΩ→∂Ω maps Wa, (Ω) p, p boundedly into Bs (∂Ω, σ) for the range of indices specified in (8.6.87). < p ≤ 1 and Third, when p, s are as in the first line of (8.6.87), i.e., n−1 n   (n − 1) p1 − 1 < s < 1, the disagreement parameter a := 1 − s − p1 is necessarily negative (which is in  sharp  contrast to the case when p was allowed to be in (1, ∞)). In  fact, a ∈ − p1 , −n p1 − 1 and this interval contains only strictly negative numbers. Thus, in order to have a trace result when p is sub-unital we are naturally led to considering genuinely weighted (maximal) Sobolev space. We now turn to the task of providing the proof of Theorem 8.6.8. 1, p

Proof of Theorem 8.6.8 Given u ∈ Wa, (Ω), from Proposition 8.6.3 (used with k := 1) we conclude that u ∈ W 1,q (Ω) with q :=

np n−1+p−sp

∈ (1, n)

and uW 1, q (Ω) ≤ CuW 1, p (Ω)

(8.6.92)

a,

for some C ∈ (0, ∞) independent of u. Bring in Jones’ extension operator Λ1 defined as in Theorem 8.3.2 with k := 1. Then Λ1 : W 1, p (Ω) −→ W 1, p (Rn ) linearly and boundedly,

(8.6.93)

hence there exists a constant C ∈ (0, ∞), independent of u, such that w := Λ1 u ∈ W 1,q (Rn ), w W 1, q (Rn ) ≤ CuW 1, q (Ω),

and w Ω = u at L n -a.e. point in Ω.

(8.6.94)

514

8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

Moreover, from (8.3.40) in Theorem 8.3.6 (applied here with p := q and a := 0) we see that the boundary trace TrΩ→∂Ω u, originally defined as in (8.6.88), satisfies ⨏ (TrΩ→∂Ω u)(x) = lim+ w(y) dy at σ-a.e. point x ∈ ∂Ω. (8.6.95) r→0

B(x,r)

Going further, since Cc∞ (Rn ) embeds densely into the Sobolev space W 1,q (Rn ), we may find a sequence {w j } j ∈N ⊆ Cc∞ (Rn ) such that w j → w in W 1,q (Rn ) as j → ∞. By eventually passing to a sub-sequence, there is no loss of generality in assuming that we also have w j → w at L n -a.e. point in Rn as j → ∞. Let En be the standard radial fundamental solution for the Laplacian in Rn . Fix some arbitrary point x ∈ Rn and denote by δx the Dirac distribution in Rn with mass at x. Then for each j ∈ N we may write . w j (x) = δx, w j  = Δ[En (x − ·)], w j . . = − ∇[En (x − ·)], ∇w j = (∇En )(x − ·), ∇w j ∫ (∇En )(x − ·), ∇w j  dL n . (8.6.96) = Rn

* "n Given that ∇w j → ∇w in L q (Rn, L n ) as j → ∞, the Fractional Integration  −1  Theorem (cf. [133, (7.8.7)]) guarantees that if q∗ := q1 − n1 then ∫

∫ Rn

(∇En )(· − y), (∇w j )(y) dy −→ in

∗ L q (Rn, L n )

Rn

(∇En )(· − y), (∇w)(y) dy

(8.6.97)

as j → ∞.

In particular, by passing to a sub-sequence we may also ensure that the above convergence takes place at L n -a.e. point in Rn . From this and (8.6.96) we then conclude that ∫ w(x) = (∇En )(x − y), (∇w)(y) dy at L n -a.e. point x ∈ Rn . (8.6.98) Rn

To proceed, consider the open set D := Rn \ ∂Ω. Since L n (∂Ω) = 0 given that ∂Ω is upper Ahlfors regular, it follows that ∂Ω has an empty interior. In turn, this implies (8.6.99) ∂D = ∂Ω hence also δ∂D = δ∂Ω in Rn .  n  Keeping this in mind and observing that D = Ω ∪ R \ Ω (disjoint union), Lemma 8.6.7 implies that * p p(1−s)−1 n " n ∇w ∈ L D, δ∂D L and ∇w [L p (D,δ p(1−s)−1 L n )]n ≤ CuW 1, p (Ω) 

∂D

a,

(8.6.100)

8.6 Traces from Weighted Maximal Sobolev Spaces in Open Subsets of R n

515

for some C ∈ (0, ∞) independent of u. Since Rn \ D = ∂Ω and L n (∂Ω) = 0, we may refashion (8.6.98) as ∫ (∇En )(x − z), (∇w)(z) dz at L n -a.e. point x ∈ Rn . (8.6.101) w(x) = D

Upon noting that the function defined as Φ(x, z) := (∇En )(x − z) for each x, z ∈ Rn with x  z satisfies (8.6.40)-(8.6.41), we may now invoke Proposition 8.6.6 (with Ω := D and ϕ := ∇w which, thanks to (8.6.100), satisfies (8.6.43)) and conclude, based on the integral representation formula in (8.6.101), that (cf. (8.6.47)) ⨏ w(y) dy = w(x) at H n−1 -a.e. point x ∈ ∂D = ∂Ω, (8.6.102) lim+ r→0

B(x,r)

and (cf. (8.6.46)) ∫ ∂D

∫ ∂D

|w(x) − w(y)| p dH n−1 (x) dH n−1 (y) |x − y| n−1+sp

 p1 ≤ C∇w [L p (D,δ p(1−s)−1 L n )]n . 

∂D

(8.6.103) At this stage, the Gagliardo semi-norm estimate claimed in (8.6.89) follows by combining (8.6.103), (8.6.102), (8.6.95), (8.6.100), and (8.6.99). Finally, the claims made in (8.6.90) and (8.6.91) are consequences of Proposition 8.3.8, Corollary 8.3.9 (both used with a := 0), and Proposition 8.6.3 (cf. (8.6.92)). 

Chapter 9

Besov and Triebel-Lizorkin Spaces in Open Sets

We begin by reviewing Besov and Triebel-Lizorkin spaces in the entire Euclidean setting in §9.1. Besov and Triebel-Lizorkin in arbitrary open subsets of Rn are then defined in §9.2 via restriction from the corresponding scales in the Euclidean setting (in the sense of distributions). Certain quasi-Banach envelopes of Besov and Triebel-Lizorkin spaces are concretely identified in §9.3. The topic of traces of functions belonging to Besov and Triebel-Lizorkin spaces is treated in §9.4. In this regard, we first consider said spaces in the entire Euclidean setting and take traces on arbitrary Ahlfors regular sets in Rn , then we use these results to establish mapping properties for trace operators from spaces defined in open subsets of Rn onto theirs boundaries. This body of results is further augmented in §9.5 by proving mapping properties for conormal derivative operators acting on functions belonging to Besov and Triebel-Lizorkin spaces in certain open subsets of Rn . Finally, in §9.6 we construct extension operators from boundary Besov spaces into Besov and Triebel-Lizorkin spaces, which turn out to be inverses from the right for the the trace operators acting from Besov and Triebel-Lizorkin spaces defined in the open set in question considered earlier, in §9.4.

9.1 Besov and Triebel-Lizorkin Spaces in R n Informative accounts pertaining to the topic of Besov and Triebel-Lizorkin spaces in the Euclidean setting may be found in many references, including [189], [188], [187], [59], [60], [15], [166]. One convenient point of view is offered by the classical Littlewood-Paley theory (cf., e.g., [166], [187], [188]). More specifically, let Ξ be the collection of all families {ζ j }∞ j=0 of Schwartz functions with the following properties: (i) There exist finite positive constants A, B, C such that  supp (ζ0 ) ⊂ {x : |x| ≤ A} and (9.1.1) supp (ζ j ) ⊂ {x : B2 j−1 ≤ |x| ≤ C2 j+1 } if j ∈ N. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1_9

517

518

9 Besov and Triebel-Lizorkin Spaces in Open Sets

(ii) For every multi-index α ∈ N0n there exists a positive finite constant Cα such that (9.1.2) sup sup 2 j |α | |∂ α ζ j (x)| ≤ Cα . x ∈R n j ∈N

(iii) One has

∞ 

ζ j (x) = 1 for every x ∈ Rn .

(9.1.3)

j=0  n Fix some family {ζ j }∞ j=0 ∈ Ξ. Also, let F and S (R ) denote, respectively, the Fourier transform and the class of tempered distributions in Rn . Then the Triebelp,q Lizorkin space Fs (Rn ) is defined for 0 < p < ∞, 0 < q ≤ ∞, and s ∈ R as

  p,q Fs (Rn ) := f ∈ S (Rn ) :  f Fsp, q (Rn ) < +∞

(9.1.4)

where for each f ∈ S (Rn ) we have set ∞    1/q    |2s j F −1 (ζ j F f )| q  f Fsp, q (Rn ) :=   j=0

(9.1.5)

L p (R n, L n )

with the inner  q quasi-norm replaced by sup over j ∈ N0 when q = ∞. The case p = ∞ is somewhat special, in that a suitable version of (9.1.5) needs to be used (see, e.g., [166, p. 9]). First, for each k ∈ Z and l = (l1, . . . , ln ) ∈ Zn , we introduce the notation   Q k,l := x = (x1, . . . , xn ) ∈ Rn : 2−k li ≤ xi ≤ 2−k (li + 1), i ∈ {1, . . . , n} . (9.1.6) Then, corresponding to p = ∞, 0 < q ≤ ∞, and s ∈ R, for each f ∈ S (Rn ) we set f

∫

 ∞, q Fs (R n )

:= sup sup 2

jn

j ∈N0 l ∈Z n

∞ 

Q j, l k=j

|2sk F −1 (ζ j F f )(x)| q dx

 1/q

.

(9.1.7)

p,q

The scale of Besov space Bs (Rn ), indexed by 0 < p, q ≤ ∞ and s ∈ R, is defined as   p,q (9.1.8) Bs (Rn ) := f ∈ S (Rn ) :  f Bsp, q (Rn ) < +∞ where, for each f ∈ S (Rn ),  f Bsp, q (Rn ) :=

∞ 

2s j F −1 (ζ j F f ) L p (Rn, L n ) q

 1/q

,

(9.1.9)

j=0

with the outer  q quasi-norm replaced by sup over j ∈ N0 when q = ∞. Different choices of the family {ζ j }∞ j=0 in the class Ξ yield the same spaces (9.1.4)-(9.1.8), albeit equipped with equivalent quasi-norms. As is well known (see, e.g., [187, p,q § 2.3.3]), Bs (Rn ) is a quasi-Banach space for s ∈ R, 0 < p, q ≤ ∞ (Banach space if 1 ≤ p, q ≤ ∞) and

9.1 Besov and Triebel-Lizorkin Spaces in R n

519

S (Rn ) → Bs (Rn ) → S (Rn ). p,q

(9.1.10) p,q

in the sense of continuous (topological) embeddings. Similarly, Fs (Rn ) is a quasiBanach space for s ∈ R, 0 < p < ∞, 0 < q ≤ ∞ (Banach space if 1 ≤ p < ∞, 1 ≤ q ≤ ∞) and p,q (9.1.11) S (Rn ) → Fs (Rn ) → S (Rn ). Moreover, given s ∈ R, p,q

S (Rn ) → Fs (Rn ) densely, if and only if q < ∞,

(9.1.12)

p,q

S (Rn ) → Bs (Rn ) densely, if and only if max {p, q} < ∞. Furthermore, (cf., e.g., [166, p. 30] and [187, p. 47]) p,min{p,q }

Bs

p,max{p,q }

p,q

(Rn ) → Fs (Rn ) → Bs

(Rn )

(9.1.13)

for 0 < p, q ≤ ∞ and s ∈ R

For indices p, q, s such that n/(n + 1) < p, q ≤ ∞ and n(1/p − 1)+ < s < 1 one has

∫  1/q q  f (· + t) − f (·) L p (Rn, L n ) dt . (9.1.14)  f Bsp, q (Rn ) ≈  f  L p (Rn, L n ) + |t| n+sq Rn This result is proved, for instance, in [187, § 2.5.12], where one can also find an analogous characterization of Triebel-Lizorkin spaces as well as intrinsic descriptions of both of Besov and Triebel-Lizorkin spaces with larger amounts of smoothness, involving higher order differences. Note that by specializing (9.1.14) to the case of the diagonal Besov scale we obtain

n if n+1 < p, q ≤ ∞ and n p1 − 1 + < s < 1 then ∫ f

p, p B s (R n )

≈  f  L p (Rn, L n ) +

∫

Rn

Rn

| f (x) − f (y)| p dx dy |x − y| n+sp

 1/p

(9.1.15) .

In the next several theorems we collect other basic properties of Besov and TriebelLizorkin spaces, starting with a theorem describing the monotonicity of the spaces p,q As (Rn ), with A := B or A := F, with respect to s and q (cf. [187, Proposition 2, p. 47], [166, Proposition, p. 29]). Theorem 9.1.1 Let s ∈ R, ε > 0, 0 < q0 ≤ q1 ≤ ∞, and 0 < p ≤ ∞. Then p,q

p,q0

As+ε1 (Rn ) → As

p,q1

(Rn ) → As

(Rn ),

A ∈ {B, F},

(9.1.16)

with the convention that p < ∞ if A = F. The next theorem describes lifting results for Besov and Triebel-Lizorkin spaces.

520

9 Besov and Triebel-Lizorkin Spaces in Open Sets

Theorem 9.1.2 Assume that 0 < p, q ≤ ∞, s ∈ R, and A ∈ {B, F}. Then As (Rn ) = (I − Δ)μ As+μ (Rn ) for each μ ∈ R. p,q

p,q

(9.1.17)

Also, for each m ∈ N one has   p,q p,q As (Rn ) = f ∈ S (Rn ) : ∂ α f ∈ As−m (Rn ) for all α ∈ N0n with |α| ≤ m   p,q p,q = f ∈ As−m (Rn ) : ∂ α f ∈ As−m (Rn ) for all α ∈ N0n with |α| = m (9.1.18) and q  f  Ap, n ≈ s (R )

 |α | ≤m

q ∂ α f  Ap, n s−m (R )

q ≈  f  Ap, n + s−m (R )

In particular,

 |α |=m

q ∂ α f  Ap, n . s−m (R )

(9.1.19)

∂ α : As (Rn ) −→ As− |α | (Rn ) p,q

p,q

(9.1.20)

is well defined, linear, and bounded for each multi-index α ∈ N0n . We now collect the well-known embedding properties for Besov and TriebelLizorkin spaces on Rn in the following theorem. Theorem 9.1.3 (i) Let 0 < p, q, a, b ≤ ∞ and s ∈ R. Then the embeddings p,a

p,q

p,b

Bs (Rn ) → Fs (Rn ) → Bs (Rn )

(9.1.21)

are continuous if and only if 0 < a ≤ min{p, q} and max{p, q} ≤ b ≤ ∞.

(9.1.22)

(ii) Assume 0 < p0 < p < p1 ≤ ∞ and s0, s1, s ∈ R satisfy 1 p0

−

s0 n

=

1 p

−

s n

=

1 p1

−

s1 n.

(9.1.23)

Then, whenever 0 < a, b ≤ ∞, p ,a

p,q

p ,b

Bs00 (Rn ) → Fs (Rn ) → Bs11 (Rn )

(9.1.24)

are well-defined continuous embeddings if and only if 0 < a ≤ p ≤ b ≤ ∞. (iii) Let 0 < p < p1 ≤ ∞, s, s1 ∈ R, 0 < r, q ≤ ∞. Assume that Then p,r p ,q Fs (Rn ) → Bs11 (Rn ).

(9.1.25) 1 p

−

s n

=

1 p1

−

s1 n.

(9.1.26)

9.1 Besov and Triebel-Lizorkin Spaces in R n

521

Proof Part (i) of the theorem can be found in [174, Theorem 3.1.1]. See also [166, Theorem, p. 30]. Parts (ii) and (iii) were first proved in [101, Theorem 2.1, pp. 95-96] for homogeneous spaces, then in [174, Theorem 3.2.1] for inhomogeneous spaces. See also [166, Theorem, p. 31].  We continue discussing embedding results, of the following tight nature. Theorem 9.1.4 Given 0 < p0 ≤ p1 ≤ ∞, s0, s1 ∈ R, and 0 < q0 ≤ q1 ≤ ∞ with the property that p10 − sn0 = p11 − sn1 , one has the continuous embedding p ,q0

Bs00

p ,q1

(Rn )

(9.1.27)

p ,q1

(Rn ),

(9.1.28)

1 1 s0 s1 = − − , p0 n p1 n

(9.1.29)

(Rn ) → Bs11

Moreover, one has the continuous embedding p ,q0

Fs00

(Rn ) → Fs11

provided either 0 < p0 < p1 < ∞,

0 < q0, q1 ≤ ∞, and

or 0 < p0 = p1 < ∞,

0 < q0 ≤ q1 ≤ ∞, and s0 = s1 .

(9.1.30)

Proof The embedding in (9.1.27) follows from item (ii) in Theorem 9.1.3 and Theorem 9.1.1, while the embedding claimed in (9.1.28) may be justified by combining Theorem 9.1.1 with [166, Theorem 2.2.3(5), p. 31] (cf. also [166, Remark 2, p. 31]). Turning to the issue of duality, we have the following result. Theorem 9.1.5 For s ∈ R and 0 < p, q < ∞, one has

p,q

∗

p,q

∗

p,q

∗

Bs (Rn ) Fs (Rn ) Fs (Rn )

p,q

= B−s+n(1/p−1)+ (Rn ) p,q

(9.1.31)

= F−s (Rn ) if p > 1,

(9.1.32)

∞,∞ = B−s+n(1/p−1) (Rn ) if p < 1,

(9.1.33)

where p and q  are, respectively, the Hölder conjugate exponents of p and q, considered in the sense of (7.6.1). Proofs and other related results may be found in, e.g., [187], [166] (cf. also the references therein). Here we only want to point out that (9.1.32) with 0 < q < 1 is not usually covered in the literature. An argument may be found in [127]. The reader is reminded that the version of the complex interpolation method employed in the theorem below has been reviewed in §1.4. Theorem 9.1.6 Fix n ∈ N and assume

522

9 Besov and Triebel-Lizorkin Spaces in Open Sets

θ ∈ (0, 1),

α0, α1 ∈ R,

α := (1 − θ)α0 + θα1,

0 < p j ≤ ∞, 0 < q j ≤ ∞, for j ∈ {0, 1},

θ −1 θ −1 , q := 1−θ . p := 1−θ p0 + p1 q0 + q1

(9.1.34)

Then, for A ∈ {B, F}, p ,q0

Aα00

p ,q1

(Rn ), Aα11

p ,q0

(Rn ), and Aα00

p ,q1

(Rn ) + Aα11

(Rn )

are all analytically convex quasi-Banach spaces.

(9.1.35)

In addition, 

p ,q0

Fα00

p ,q1

(Rn ), Fα11

(Rn )

 θ

p,q

= Fα (Rn ),

if either max{p0, q0 } < ∞, or max{p1, q1 } < ∞, and



p ,q0

Bα00

p ,q1

(Rn ), Bα11

(Rn )

 θ

(9.1.36)

p,q

= Bα (Rn ),

(9.1.37)

provided min {q0, q1 } < ∞.

Finally, similar results are valid for the homogeneous Besov and Triebel-Lizorkin spaces in Rn , and also for the discrete versions of these scales of spaces. Proof See [110, Theorem 9.1] and its proof.



Regarding the real method of interpolation, we have the following classical result (see, e.g., [187]). Theorem 9.1.7 Assume that 0 < q0, q1, q ≤ ∞, 0 < θ < 1, α0, α1 ∈ R with α0  α1 , and define α := (1 − θ)α0 + θα1 . Then

p,q0 n p,q p,q (9.1.38) Fα0 (R ), Fα1 1 (Rn ) θ,q = Bα (Rn ), 0 < p < ∞, p,q0 n p,q1 n p,q Bα0 (R ), Bα1 (R ) θ,q = Bα (Rn ), 0 < p ≤ ∞. (9.1.39) p,q

p,q

A powerful tool in the study of the spaces Bs (Rn ) and Fs (Rn ) is the atomic decomposition of these spaces. Here we review some of the main results from [57] and [59] which, in turn, build on the work of many other people. Turning to specifics, denote by J (Rn ) the collection of dyadic cubes in Rn and fix some s ∈ R along with p ∈ (0, ∞]. Also, consider a parameter J ∈ R and pick two integers K, L satisfying   K ≥ ([s] + 1)+ and L ≥ max [J − n − s], −1 . (9.1.40) Given an arbitrary cube Q ∈ J (Rn ) with side-length (Q) ≤ 1, call a function aQ ∈ W K,∞ (Rn ) a (s, p)-atom of type (K, L, J) if the following properties hold:

9.1 Besov and Triebel-Lizorkin Spaces in R n

523

(1) supp (aQ ) ⊆ 3Q,

(9.1.41)

(2) ∂γ aQ  L ∞ (Rn, L n ) ≤ L n (Q)s/n−1/p− |γ |/n whenever |γ| ≤ K, ∫ (3) xγ aQ (x) dx = 0 whenever |γ| ≤ L and (Q) < 1,

(9.1.42)

Rn

(9.1.43)

with the convention that property (3) is omitted if L < 0. For a proof of the following theorem see [57], [59]. n Theorem 9.1.8 If s ∈ R, 0 < p, q ≤ ∞, J := min{1, p } , and K, L are two integers as in (9.1.40), then there exists some finite constant C = C(s, p, n, K, L) > 0 with the property that for any numerical sequence {λQ }Q ∈ J(Rn ),l(Q)≤1 and any family {aQ }Q ∈ J(Rn ), (Q)≤1 of (s, p)-atoms of type (K, L, J) there holds

    

 Q ∈ J(R n ) (Q)≤1

   λQ aQ  

1/q

p, q

Bs

(R n )

∞    q/p     ≤ C |λQ | p    j=0 Q ∈ J(R n ) − j   (Q)=2

.

(9.1.44)

n Conversely, if s ∈ R, 0 < p, q ≤ ∞, and J := min{1, p } , then having fixed two integers K, L satisfying (9.1.40), it follows that for each tempered distribution p,q f ∈ Bs (Rn ) one may find some family {aQ }Q ∈ J(Rn ), (Q)≤1 of (s, p)-atoms of type (K, L, J), along with some sequence of complex numbers {λQ }Q ∈ J(Rn ), (Q)≤1 with the property that 1/q

 f Bsp, q (Rn )

∞   q/p    ≈ |λQ | p   j=0 Q ∈ J(Rn ): (Q)=2− j 

,

(9.1.45)

such that f may be represented in the form  f = λQ aQ with convergence in S (Rn ).

(9.1.46)

In particular, (9.1.44) implies that  p,q λQ aQ convergence to f in Bs (Rn ) if max{p, q} < ∞.

(9.1.47)

Q ∈ J(R n ) (Q)≤1

Q ∈ J(R n ) (Q)≤1

Results similar in spirit to those presented above are also valid for TriebelLizorkin spaces. Concretely, if s ∈ R, 0 < p < ∞, 0 < q ≤ ∞ J := min{1,n p,q } , and K, L are two integers satisfying (9.1.40), then there exists some finite constant C = C(s, p, n, K, L) > 0 with the property that

524

9 Besov and Triebel-Lizorkin Spaces in Open Sets

 1/q              q        L n (Q)−1/p |λQ |1Q   λQ aQ  ≤ C        p, q n n n  Q ∈ J(R ) ∈ J(R ) Fs (R )   Q(Q)≤1 (Q)≤1   L p (R n, L n ) (9.1.48) for any numerical sequence {λQ }Q ∈ J(Rn ), (Q)≤1 and family {aQ }Q ∈ J(Rn ), (Q)≤1 of (s, p)-atoms of type (K, L, J). In the converse direction, if s ∈ R, 0 < p < ∞, 0 < q ≤ ∞, J := min{1,n p,q } , and K, L are two integers satisfying (9.1.40), it follows that each tempered distribution p,q f ∈ Fs (Rn ) may be written as  f = λQ aQ with convergence in S (Rn ), (9.1.49) Q ∈ J(R n ) (Q)≤1

for some family {aQ }Q ∈ J(Rn ),l(Q)≤1 of (s, p)-atoms of type (K, L, J) and some numerical sequence {λQ }Q ∈ J(Rn ), (Q)≤1 with the property that

 f Fsp, q (Rn )

 1/q         q    L n (Q)−1/p |λQ |1Q   ≈  .   Q∈J(Rn )  p n n  (Q)≤1   L (R , L )

In particular, (9.1.48) implies that  p,q λQ aQ convergence to f in Fs (Rn ) if q < ∞.

(9.1.50)

(9.1.51)

Q ∈ J(R n ) (Q)≤1

It has long been known that many classical smoothness spaces are encompassed by the Besov and Triebel-Lizorkin scales. For example, Bs∞,∞ (Rn ) = C s (Rn ) if 0 < s  N, p,2

F0 (Rn ) = L p (Rn, L n ) if 1 < p < ∞, p,2

p

Fs (Rn ) = Ls (Rn ) if 1 < p < ∞ and s ∈ R,

(9.1.52) (9.1.53) (9.1.54)

p,2

(9.1.55)

p,2

(9.1.56)

Fk (Rn ) = W k, p (Rn ) if 1 < p < ∞ and k ∈ Z, F0 (Rn ) = h p (Rn ) if 0 < p ≤ 1,

∗ F0∞,2 (Rn ) = bmo(Rn ) = h1 (Rn ) .

(9.1.57)

Let us comment on the nature of the spaces appearing above and, in the process, discuss other identifications. First, there is a suitable notion of homogeneous TriebelLizorkin space (see, e.g., [57]), and the standard John-Nirenberg of functions of bounded mean oscillations fits on this scale, namely

9.1 Besov and Triebel-Lizorkin Spaces in R n

525

.

F0∞,2 (Rn ) = BMO(Rn ).

(9.1.58)

Second, recall the classical Bessel potential spaces Ls (Rn, L n ) := (I − Δ)−s/2 L p (Rn, L n ) p

for each s ∈ R and p ∈ (1, ∞)

(9.1.59)

(compare with the weighted version in (8.1.8)). Often, these are referred to as L p based fractional Sobolev spaces of order s. The case p = 2 is particularly ubiquitous. In this scenario, the L 2 -based Sobolev space of fractional order s ∈ R in Rn are sometimes simply denoted by H s (Rn ). These may be defined directly as    ∈ L 2 (Rn, L n ) (9.1.60) H s (Rn ) := U ∈ S (Rn ) : (1 + |ξ | 2 )s/2U (where “hat” denotes the Fourier transform in Rn ), with norm    2 n n . U  H s (Rn ) := (1 + |ξ | 2 )s/2U L (R , L )

(9.1.61)

Hence, Fs2,2 (Rn ) = H s (Rn ) for each s ∈ R.

(9.1.62)

p,q

Third, we briefly discuss the scale Cα (Rn ) introduced in [172]. Concretely, for each given m ∈ N0 set Pm := polynomials in Rn of degree ≤ m.

(9.1.63)

Also, given an integer m ∈ N0 , a point x ∈ Rn , a scale t > 0, and a Lebesguemeasurable function u in B(x, t) ⊂ Rn , define ⨏ oscm (u, x, t) := inf |u − P| dL n . (9.1.64) P ∈ Pm

B(x,t)

Whenever 0 < p < ∞, 0 < q ≤ ∞, α > 0, and m ∈ N with m > α, set   p,q 1 Cα (Rn ) := u ∈ Lloc (Rn, L n ) : uCαp, q (Rn ) < +∞ ,

(9.1.65)

where the quasi-norm  · Cαp, q (Rn ) is defined as  ∫  +

dt  1/q   p n n L (R , L ) t 0 (9.1.66) for each Lebesgue measurable function u in Rn . Whenever necessary to indicate the p,q,m n p,q (R ) in place of Cα (Rn ). Then, dependence on the integer m we shall write Cα with this piece of notation, [172, Theorem 1, p. 393] implies that u

p, q Cα (R n )

p,q

:= u

L p (R n, L n )

∞

[t −α oscm−1 (u, ·, t)]q

p,q,m

Fα (Rn ) = Cα

(Rn ) whenever 0 < p < ∞, 0 < q ≤ ∞,

1 and m ∈ N with m > α > n min {p,q } − 1 +,

(9.1.67)

526

9 Besov and Triebel-Lizorkin Spaces in Open Sets

with equivalence of quasi-norms. In particular, for the above range of indices, the p,q,m n space Cα (R ) is actually independent of m. n Fourth, recall the classical scale of ∫ Hardy spaces in R . Fix a test function ψ ∈ Cc∞ (Rn ) with the property that Rn ψ dL n = 1, and set ψt := t −n ψ(·/t) for each t > 0. Given a tempered distribution u ∈ S (Rn ) we define its radial maximal function and its truncated version, respectively, by setting u++ := sup |ψt ∗ u|, 0 α > n min {p,q } − 1 + , then there exists a p,q,m p,q bounded extension operator E : Cα (Ω) −→ Cα (Rn ).

(9.2.27)

9.2 Besov and Triebel-Lizorkin Spaces in Open Subsets of R n

531

Granted this, elementary considerations then imply that, in the same context as p,q,m n p,q above, the restriction operator R Rn →Ω : Cα (R ) → Cα (Ω) is well defined, p,q linear, bounded and onto. On the other hand, so is R Rn →Ω acting from Fα (Rn ) p,q into Fα (Ω). In concert with (9.1.67), this analysis proves the following (which is essentially [172, Corollary 1, p. 398]): Proposition 9.2.3 Let Ω ⊆ Rn be an (ε, δ)-domain. Then, with equivalence of quasinorms, p,q,m

Cα

p,q

(Ω) = Fα (Ω) whenever 0 < p < ∞, 0 < q ≤ ∞,

1 and m ∈ N with m > α > n min {p,q } − 1 +.

(9.2.28)

The family of spaces (9.2.25)-(9.2.26) interfaces tightly with another scale introduced by R. DeVore and R. Sharpley in [45]. Specifically, given an open set Ω ⊆ Rn along with p ∈ (0, ∞) and α ≥ 0, they have considered     p 1 (Ω, L n ) : uCαp (Ω) := u L p (Ω, L n ) + uα#  L p (Ω, L n ) < ∞ , Cα (Ω) := u ∈ Lloc (9.2.29) where   ∫ 1 # n uα (x) := sup |u − P| dL , inf n+α ∀x ∈ Ω. (9.2.30) B(x,r) 0 n

(9.2.32) 1 p

− 1 +.

# , a variant of the sharp function (9.2.30), defined Proof It is useful to introduce uα,λ for each fixed parameter λ ∈ (0, 1] much as before, except that the supremum is now taken over 0 < t < λ · δ∂Ω  (x).Then from (9.2.26) with m = [α] + 1, we have uCαp,∞ (Ω) ≈ u L p (Ω, L n ) + u# 1  L p (Ω, L n ) . As such, everything comes down to

checking that

α, 2

   #  u 1  p ≈ uα#  L p (Ω, L n ) . α, L (Ω, L n )

(9.2.33)

2

In turn, for the current range of indices, this is a direct consequence of the estimate established in [149, Lemma 2.3, p. 65].  Combining Proposition 9.2.3 with Proposition 9.2.4 then yields the following identification (for related results in smooth domains, see also [190, p. 50 and p. 248]).

532

9 Besov and Triebel-Lizorkin Spaces in Open Sets

Corollary 9.2.5 If Ω ⊆ Rn is an (ε, δ)-domain then p

p,∞

Cα (Ω) = Fα

(Ω) whenever

0 < p < ∞ and α > n p1 − 1 + .

(9.2.34)

Moving on, we shall now consider local Hardy spaces and local Hardy-based Sobolev spaces in open subsets of the Euclidean ambient. To get started, in same spirit as (9.2.1), given an open set Ω ⊆ Rn and p ∈ (0, ∞) define    h p (Ω) := u ∈ D (Ω) : there exists U ∈ h p (Rn ) such that U Ω = u ,   (9.2.35)  uh p (Ω) := inf U h p (Rn ) : U ∈ h p (Rn ), U Ω = u , ∀u ∈ h p (Ω). Then from (9.2.35), (9.2.1), and (9.1.56) we conclude that for each open set Ω ⊆ Rn we have p,2

F0 (Ω) = h p (Ω) for 0 < p < ∞

(9.2.36)

and p,2

F0 (Ω) = h p (Ω) = L p (Ω, L n ) provided 1 < p < ∞.

(9.2.37)

In order to be able to discuss a useful characterization of the local Hardy

p (Ω), we need some notation. Concretely, fix ψ ∈ C ∞ B(0, 1) such that space h c ∫ ψ dL n = 1 and set ψt := t −n ψ(·/t) for each t > 0. The radial maximal B(0,1) function of a distribution u ∈ D (Ω) is defined as (see [145, p. 205]) + uΩ (x) :=

sup 0 n 1/p − 1 . Then p

p,2

hk (Ω) = Fk (Ω).

(9.2.45) p

p,2

Proof Assume first that k < 0 and note that the inclusion hk (Ω) → Fk (Ω) is immediate from definitions, (9.2.36), (9.2.8), and (9.2.9). To see the opposite p,2 p,2 inclusion fix u ∈ Fk (Ω), say u = w|Ω for some w ∈ Fk (Rn ). Since, by (9.1.74), w γ may be represented in the form w = |γ | ≤−k ∂ wγ with each wγ ∈ h p (Rn ), it follows  p γ that u = |γ | ≤−k ∂ (wγ |Ω ) and wγ |Ω ∈ h p (Ω). Consequently, u ∈ hk (Ω), proving the right-to-left inclusion in (9.2.45). At this stage, we may conclude that (9.2.45) holds in the case when k < 0.

The case when k ∈ Z satisfies k > n 1/p − 1 makes essential use of Miyachi’s work in [147], [146]. Specifically, as in [146, Remark (h), p. 80], let us introduce   (9.2.46) Wpk (Ω) := u ∈ h p (Ω) : ∂γ u ∈ h p (Ω) for all γ ∈ N0n with |γ| = k and observe that, by virtue of the last remark of §4 on p. 80 in [146], p

Wpk (Ω) = Ck (Ω), p

(9.2.47)

where Ck (Ω) is the DeVore-Sharpley space defined as in (9.2.29)-(9.2.30) (with α := k). On account of the extension result recalled in (9.2.31) it follows that

534

9 Besov and Triebel-Lizorkin Spaces in Open Sets

   Wpk (Ω) = uΩ : u ∈ Wpk (Rn ) . However, thanks to (9.1.18) and (9.1.56), we have p,2 Wpk (Rn ) = Fk (Rn ) so, all together, p,2

Wpk (Ω) = Fk (Ω). p

(9.2.48)

p,2

Consequently, hk (Ω) → Wpk (Ω) = Fk (Ω), proving the left-to-right inclusion in (9.2.45). The opposite inclusion is a consequence of (9.2.36), (9.2.8), (9.2.9), and definitions.  Suppose Ω ⊆ Rn is an arbitrary open set and fix p ∈ (1, ∞) along with k ∈ N0 . Then the intrinsically defined L p -based Sobolev space of order k in Ω is defined as   W k, p (Ω) := u ∈ L p (Ω, L n ) : ∂γ u ∈ L p (Ω, L n ) for all γ ∈ N0n with |γ| ≤ k . (9.2.49) As is well known, this is a Banach space when equipped with the natural norm  uW k, p (Ω) := |γ | ≤k ∂γ u L p (Ω, L n ) for each u ∈ W k, p (Ω). We may also consider L p -based Sobolev spaces of negative order. With Ω, p, k as above, these are intrinsically defined via    ∂γ uγ with each uγ ∈ L p (Ω, L n ) , (9.2.50) W −k, p (Ω) := u ∈ D (Ω) : u = |γ | ≤k

 and are equipped with the norm uW −k, p (Ω) := inf |γ | ≤k uγ  L p (Ω, L n ) where the infimum is taken over all representations of the distribution u ∈ W −k, p (Ω) as in (9.2.50). Based on these definitions and (9.2.37) it follows that for each open set Ω ⊆ Rn we have p

W k, p (Ω) = hk (Ω) if 1 < p < ∞ and k ∈ Z.

(9.2.51)

Also, reasoning much as in Remark 8.5.1 we see that for each open set Ω ⊆ Rn we have    W −k, p (Ω) = uΩ : u ∈ W −k, p (Rn ) (9.2.52) whenever p ∈ (1, ∞) and k ∈ N0, in a quantitative fashion. The above definitions and remarks set the stage for the following natural identification result. Corollary 9.2.8 Let Ω be a bounded (ε, δ)-domain in Rn . Then for each k ∈ Z and p ∈ (0, ∞) one has  p

hk (Ω) if 0 < p ≤ 1 and either k ≤ 0, or k > n p1 − 1 , p,2 Fk (Ω) = (9.2.53) W k, p (Ω) if 1 < p < ∞ and k ∈ Z. Proof This is a consequence of Proposition 9.1.9, (9.2.52), (9.2.1), Proposition 9.2.7, and the fact that, in the current geometric setting, the space W k, p (Ω) may be described

9.2 Besov and Triebel-Lizorkin Spaces in Open Subsets of R n

535

   as uΩ : u ∈ W k, p (Rn ) in a quantitative manner whenever p ∈ (1, ∞) and k ∈ N0 (which is established in [103]).  There are geometric and analytic assumptions guaranteeing the coincidence of the spaces defined in (9.2.1)-(9.2.2). A result of this favor is contained in the proposition below. Proposition 9.2.9 Let Ω ⊆ Rn be an open set with a compact Ahlfors regular boundary. Then for each A ∈ {B, F} one has p,q p,q As (Ω) = A˚ s (Ω) provided p, q ∈ (0, ∞) and s
p1∗ − 1 such p,q p ,q that As (Rn ) ⊆ As∗∗ ∗ (Rn ) and the desired conclusion follows from the first part of the proof.  A useful consequence of Lemma 9.2.10, identifying a context in which the restriction operator to an open set is actually an isomorphism, is given next. Lemma 9.2.11 Let Ω ⊆ Rn be an open set with an Ahlfors regular boundary. Then p,q

p,q

R Rn →Ω : As,0 (Ω) −→ As,z (Ω), provided 0 < p < ∞,

A ∈ {B, F}, isomorphically 

 0 < q ≤ ∞, max p1 − 1, n p1 − 1 < s.

(9.2.57)

As a consequence, for the above range of indices one may canonically identify p,q p,q As,0 (Ω) with As,z (Ω). p,q

Proof From (9.2.3)-(9.2.4) we see that R Rn →Ω maps the space As,0 (Ω) onto p,q As,z (Ω), in a linear and bounded fashion whenever 0 < p, q ≤ ∞ and s ∈ R.

536

9 Besov and Triebel-Lizorkin Spaces in Open Sets

There remains to observe that assumptions on p, q, s made in (9.2.57) this operap,q tor is also injective. To this end, assume u ∈ As,0 (Ω) with p, q, s as in (9.2.57) is

such that R Rn →Ω u = 0. Since we know that supp u ⊆ Ω to begin with, this forces supp u ⊆ Ω ∩ (Rn \ Ω) = ∂Ω. Having established this, Lemma 9.2.10 applies (with Σ := ∂Ω) and gives u = 0. This proves the injectivity of the restriction operator in the context of (9.2.57), so the desired conclusion follows.  We next turn our attention to the issue of extension operators with preservation of class. For starters, we collect some remarks of a general nature in the following lemma. Lemma 9.2.12 Let Ω ⊆ Rn be an open set and assume that 0 < p, q ≤ ∞, s ∈ R, and A ∈ {B, F} are such that there exists a linear and bounded extension operator, i.e., a linear and bounded mapping p,q

p,q

EΩ→Rn : As (Ω) −→ As (Rn ),

(9.2.58)

p,q R Rn →Ω EΩ→Rn u = u for all u ∈ As (Ω).

(9.2.59)

with the property that

Then   EΩ→Rn u

p, q

As

(R n )

p,q

q ≈ u Ap, uniformly for u ∈ As (Ω). s (Ω)

(9.2.60)

As a consequence, EΩ→Rn has closed range and maps isomorphically onto its image, p,q p,q hence As (Ω) is isomorphic to a closed subspace of As (Rn ). In particular, p,q

As (Ω) is a reflexive Banach space if 1 < p, q < ∞.

(9.2.61)

p,q

Moreover, with I denoting the identity operator on As (Rn ), it follows that p,q

p,q

P := I − EΩ→Rn ◦ R Rn →Ω : As (Rn ) −→ As (Rn ) is a projection (i.e., P is linear, bounded, and satisfies P2 = P). Finally, p,q

p,q Im P := P As (Rn ) = As,0 Rn \ Ω provided ∂Ω = ∂(Ω).

(9.2.62)

(9.2.63)

As a consequence,

p,q Im P = As,0 Rn \ Ω whenever Ω satisfies an exterior corkscrew condition.

(9.2.64)

Proof Indeed, one inequality in (9.2.60) follows from the boundedness of the opp,q p,q erator EΩ→Rn from As (Ω) into As (Rn ), while the opposite inequality is a consequence of the definition in (9.2.1) and of the identity (9.2.59). Next, that P given in (9.2.62) is a projection satisfying (9.2.63) is readily seen from definitions (the

9.2 Besov and Triebel-Lizorkin Spaces in Open Subsets of R n

537

demand that ∂Ω = ∂(Ω) ensures that Rn \ Ω = Rn \ Ω). Finally, (9.2.64) is a consequence of (9.2.63) and [133, (5.1.10)].  Our next two lemmas establish that the Besov and Triebel-Lizorkin scales in open sets (and related variants) consists of analytically convex quasi-Banach spaces (a functional analytic concept introduced and studied in §1.4). Lemma 9.2.13 Assume that Ω ⊆ Rn is an open set, and suppose 0 < p, q ≤ ∞, s ∈ R, and A ∈ {B, F} are such that a linear and bounded extension operator as in p,q p,q (9.2.58) exists. Then the spaces As (Ω) and As,0 (Ω) are analytically convex. p,q

Proof The claim regarding As (Ω) follows from the last part of Lemma 9.2.12, p,q Proposition 1.4.6, and the corresponding result for As (Rn ) (cf. Theorem 9.1.6). p,q The claim about As,0 (Ω) is a consequence of Proposition 1.4.6, definition (9.2.3) and Theorem 9.1.6.  Lemma 9.2.14 Let Ω ⊆ Rn be an open set. Fix pi, qi ∈ (0, ∞] along with si ∈ R, for i ∈ {0, 1}, and let A ∈ {B, F}. (1) If there exists a common linear and bounded extension operator p ,qi

EΩ→Rn : Asii

p ,qi

(Ω) −→ Asii

p ,q

(Rn ),

i ∈ {0, 1},

(9.2.65)

p ,q

then the sum space As00 0 (Ω) + As11 1 (Ω) is analytically convex. (2) If ∂Ω = ∂(Ω) and there exists a common linear and bounded extension operator p ,qi

ERn \Ω→Rn : Asii

p ,qi

(Rn \ Ω) −→ Asii

p ,q

(Rn ),

i ∈ {0, 1},

(9.2.66)

p ,q

then the sum space As00,0 0 (Ω) + As11,0 1 (Ω) is analytically convex. p ,q

p ,q

Proof Let us abbreviate Xi := Aαii i (Ω) and Yi := Aαii ,0 i (Ω) for i ∈ {0, 1}. Granted (9.2.65), the space X0 +X1 is analytically convex by the first part of Lemma 1.4.23 and Theorem 9.1.6. This establishes the claim in part (1) of the lemma. The hypotheses in part (2) together with Lemma 9.2.12 ensure that the operator p ,qi

P := I − ERn \Ω→Rn ◦ R Rn →Rn \Ω : Aαii

p ,q

(Rn ) −→ Aαii ,0 i (Ω),

i ∈ {0, 1}, (9.2.67)

p ,q

is a common projection for the spaces Aαii i (Rn ), i ∈ {0, 1}. Hence, in such a scenario, the second part of Lemma 1.4.23 applies and, in concert with Theorem 9.1.6,  proves that Y0 + Y1 is analytically convex. The next theorem is a particular case of a more general extension result proved by H. Triebel in [191]. Theorem 9.2.15 (i) Let Ω be a nonempty open subset of Rn satisfying an interior corkscrew condition and with the property that Ω  Rn . Then for any given smoothness threshold M > 0 there exists a common extension operator

538

9 Besov and Triebel-Lizorkin Spaces in Open Sets p,q

p,q

M EΩ→R (Ω) −→ As (Rn ) n : As

(9.2.68)

linearly and boundedly, for A ∈ {B, F}, whenever

0 < q ≤ ∞, n p1 − 1 + < s < M if A = B,

1 0 < q ≤ ∞, n min{p,q } − 1 + < s < M if A = F.

0 < p ≤ ∞, 0 < p < ∞,

(9.2.69) (9.2.70)

(ii) Suppose Ω ⊆ Rn is a nonempty open set with an Ahlfors regular boundary and such that Ω = Rn . Then there exists a common extension operator p,q

p,q

EΩ→Rn : As (Ω) −→ As (Rn )

(9.2.71)

linearly and boundedly, for A ∈ {B, F}, whenever 0 < p < ∞,

0 < q ≤ ∞, and max

1 p

− 1, n

1 p

−1



< s.

(9.2.72)

Proof The claim in part (i) follows from [191, Theorem 4.4, p. 103] where such an extension result is proved for nonempty open sets Ω ⊆ Rn satisfying Ω  Rn , L n (∂Ω) = 0, and which are I-thick. To elaborate on the nature of the latter condition, recall the reflection technique Q → Q∗ used in the construction of Jones’ extension n operator from [103]. The key geometrical requirement c ◦ for the open set Ω−j⊆ R is the ability to associate to each cube Q ⊆ Ω such that (Q) ≈ 2 and dist(Q, ∂Ω) ≈ 2−j where j ∈ N is larger than or equal to a suitably chosen threshold jo ∈ N, a so-called reflected cube Q∗ ⊆ Ω with the property that (Q∗ ) ≈ 2−j , dist(Q∗, ∂Ω) ≈ 2−j , and dist(Q∗, Q) ≈ 2−j . Nonempty, open, proper subsets Ω of Rn satisfying this condition are called I-thick (interior thick) by H. Triebel in [191, Definition 3.1(iii), p. 70]. A routine argument shows that any open, nonempty, proper subset of Rn satisfying the interior corkscrew condition is automatically I-thick.

(9.2.73)

Granted this, and keeping in mind [133, (5.1.6)] as well as [133, Lemma 5.1.2], we may invoke [191, Theorem 4.4, p. 103] which provides the conclusion in part (i). Turning to part (ii), consider now the scenario when Ω ⊆ Rn is a nonempty open set satisfying Ω = Rn and such that ∂Ω is Ahlfors regular. Then Σ := ∂Ω is a closed Ahlfors regular set satisfying Σ = Rn \ Ω. Recall from (9.2.6) that the restriction to Ω (in the sense of distributions) is a linear, bounded, surjective operator p,q

p,q

R Rn →Ω : As (Rn ) −→ As (Ω), A ∈ {B, F}, provided 

 0 < p < ∞, 0 < q ≤ ∞, max p1 − 1, n p1 − 1 < s.

(9.2.74)

We claim that under the present assumptions this operator is also injective. Indeed, p,q if u ∈ As (Rn ), with the indices p, q, s as above, is such that R Rn →Ω u = 0 it follows

9.2 Besov and Triebel-Lizorkin Spaces in Open Subsets of R n

539

that supp u ⊆ Rn \ Ω = Σ. Granted this, Lemma 9.2.10 forces u = 0, proving that R Rn →Ω in (9.2.74) is injective. Hence, ultimately, R Rn →Ω is a linear isomorphism in the context of (9.2.74). Taking EΩ→Rn to be the inverse of this operator, yields a mapping p,q

p,q

EΩ→Rn : As (Ω) −→ As (Rn ) linear and bounded, for A ∈ {B, F}, p,q

with the property that R Rn →Ω ◦ EΩ→Rn = I, the identity on As (Ω), 

 provided 0 < p < ∞, 0 < q ≤ ∞, and max p1 − 1, n p1 − 1 < s. Thus, EΩ→Rn is an extension operator with preservation of class.

(9.2.75)



The next two theorems below describe extension operators in the negative (or close to negative) regime of smoothness. Theorem 9.2.16 Let Ω  Rn be a nonempty open set satisfying an exterior corkscrew condition. Then for any ε ∈ (0, 1) there exists a common extension operator ε EΩ→R (Ω) −→ As (Rn ) linearly and boundedly, n : As p,q

p,q

(9.2.76)

whenever ε < p ≤ ∞, 0 < q ≤ ∞, − ε1 < s < 0 if A = B

(9.2.77)

ε < p < ∞, ε < q ≤ ∞, − ε1 < s < 0 if A = F.

(9.2.78)

Proof This is implied by [191, Theorem 4.7, p. 105].



Theorem 9.2.17 Let Ω  Rn be an open set satisfying an exterior corkscrew condition and whose boundary is Ahlfors regular. Then for any ε ∈ (0, 1) there exists a common extension operator ε EΩ→R (Ω) −→ As (Rn ) linearly and boundedly, n : As p,q

p,q

(9.2.79)

whenever

and

⎧ ε < p ≤ ∞, 0 < q ≤ ∞, − ε1 < s < 0, ⎪ ⎪ ⎨ ⎪ or ⎪ ⎪ ⎪ 1 < p < ∞, 0 < q ≤ ∞, 0 ≤ s < 1 , ⎩ p

if A = B,

(9.2.80)

⎧ ε < p < ∞, ε < q ≤ ∞, − ε1 < s < 0, ⎪ ⎪ ⎨ ⎪ or ⎪ ⎪ ⎪ 1 < p < ∞, 1 ≤ q < ∞, 0 ≤ s < 1 , ⎩ p

if A = F.

(9.2.81)

Proof This is implied by [191, Theorem 4.10 p. 106].



540

9 Besov and Triebel-Lizorkin Spaces in Open Sets

We now discuss the interpolation of the Besov and Triebel-Lizorkin spaces and establish the following analogue of Theorem 9.1.6 in certain classes of open subsets of the Euclidean ambient. We begin by considering the complex method for analytically convex quasi-Banach spaces developed in §1.4 (cf. Lemmas 9.2.13-9.2.14 in this regard). Theorem 9.2.18 Assume Ω ⊂ Rn is a nonempty open set with the property that Ω  Rn . Also, suppose 0 < p0, p1 ≤ ∞, 0 < q0, q1 ≤ ∞ with min {q0, q1 } < ∞, α0, α1 ∈ R, θ ∈ (0, 1), and define p :=

1−θ p0

+

θ −1 , p1

q :=

1−θ q0

+

θ −1 , q1

α := (1 − θ)α0 + θα1 .

Then, with A ∈ {B, F}, one has  p0,q0  p ,q p,q Aα0 (Ω), Aα11 1 (Ω) θ = Aα (Ω)

(9.2.82)

(9.2.83)

in each of the following scenarios: (i) The set Ω satisfies an interior corkscrew condition and

n p1i − 1 + < αi for i ∈ {0, 1}, if A = B,

0 < pi < ∞, n min{p1i ,qi } − 1 + < αi for i ∈ {0, 1}, if A = F.

(9.2.84)

(ii) The set Ω satisfies an exterior corkscrew condition and αi < 0 for i ∈ {0, 1}, if A = B,

(9.2.85)

0 < pi < ∞ and αi < 0 for i ∈ {0, 1}, if A = F.

(iii) The set Ω satisfies an exterior corkscrew condition, has an Ahlfors regular boundary, and 1 < pi < ∞ and 0 ≤ αi
0 : λ X p X, p p functional associated with the absolutely p-convex hull of the unit ball in X (7.8.6)  Xbdd (Ω) = X (Ω)bdd the collection of distributions u in Ω satisfying ψ Ω u ∈ X (Ω) for each cutoff function ψ ∈ Cc∞ (Rn ) (8.3.42)

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174. W. Sickel and H. Triebel, Hölder inequalities and sharp embeddings in function spaces of s s B p, q and Fp, q type, Z. Anal. Anwendungen, 14 (1995), no. 1, 105–140. 175. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. 176. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, Vol. 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. 177. E.M. Stein and G. Weiss, On the theory of harmonic functions of several variables, I. The theory of H p spaces, Acta Math., 103 (1960), 25–62. 178. E.M. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math., 90 (1968), no. 1., 163–196. 179. E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971. 180. E. Straube, Interpolation between Sobolev and between Lipschitz spaces of analytic functions on star-shaped domains, Trans. Amer. Math. Soc., 316 (1989), no. 2, 653–671. 181. M.H. Taibleson and G. Weiss, The molecular characterization of Hardy spaces, pp. 281– 287 in Harmonic Analysis in Euclidean Spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Proc. Sympos. Pure Math., Vol. XXXV, Amer. Math. Soc., Providence, R.I., 1979. 182. M.H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, pp. 67–149 in Representation Theorems for Hardy Spaces, Astérisque, Vol. 77, Soc. Math. France, Paris, 1980. 183. M. Taylor, Partial Differential Equations, Vol. I, Springer-Verlag, 1996 (2nd ed., 2011). 184. M.E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Vol. 81, Math. Surveys and Monographs, AMS, Providence, R.I.,2000. 185. A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Dover Publications, 2004. 186. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978 (2nd ed., Barth, Heidelberg, 1995). 187. H. Triebel, Theory of Function Spaces, Birkhäuser, Berlin, 1983. 188. H. Triebel, Theory of Function Spaces, II, Monographs in Mathematics, Vol. 84, Birkhäuser Verlag, Basel, 1992. 189. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2-nd edition, Heidelberg-Leipzig, Barth, 1995. 190. H. Triebel, Theory of function spaces, III, Monographs in Mathematics, Vol. 100, Birkhäuser Verlag, Basel, 2006. 191. H. Triebel, Function Spaces and Wavelets on Domains, Tracts in Mathematics, Vol. 7, European Mathematical Society, Zürich, Switzerland, 2008. 192. B.O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, Vol. 1736, Springer-Verlag, 2000. 193. P. Turpin, Convexités dans les espaces vectoriels topologiques generaux, Dissertationes Math., Vol. 131, 1974. 194. S.V. Uspenski˘ı, Imbedding theorems for classes with weights, Trudy Mat. Inst. Steklov., 60 (1961), 282–303. 195. A. Uchiyama, A maximal function characterization of H p on the space of homogeneous type, Trans. Amer. Math. Soc., 262 (1980), no. 2, 579–592. 196. N. Varopoulos, A remark on BMO and bounded harmonic functions, Pacific J. Math., 73 (1977), 257–259. 197. X. Wu and X. Wu, Weak Hardy spaces H p,∞ on spaces of homogeneous type and their applications, Taiwanese J. Math., 16 (2012), no. 6, 2239–2258. 198. T. Xiangxing, Noncontinuous data boundary value problems for Schrödinger equation in Lipschitz domains, Front. Math. China, 4 (2006), 589–603. 199. D. Yang, Frame characterizations of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications, Georgian Math. J., 9 (2002), no. 3, 567–590. 200. D. Yang, Embedding theorems of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, Sci. China Ser. A, 46 (2003), 187–199.

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Subject Index

(ε, δ)-domain, 878 p-convex set, 417 p-envelope, 421 p-norm, 5 r-convexification, 47 s-norm, 59 P-maximal function, 607 Pg -maximal function, 679 H q,λ,θ -dome, 300 absolutely p-convex hull (of a set), 417 absolutely p-convex set, 417 absolutely continuous norm, 59 admissible family of functions, 37 Ahlfors regular domain, 869 analytic function one variable, 36 approximation to the identity (ATTI), 84, 115 associated space, 225 atom (p, q), 143 η-smooth of type (p, s), 373 p, 371 H q,λ , 297 of type (K, L, J), 522

Banach function space (classical), 219 Besov space homogeneous, 365 in Rn , 518 inhomogeneous, 368 block η-smooth of type (p, s), 374 p, 371 B q,λ , 326 block-based homogeneous Sobolev space, 861 block-based negative Sobolev space, 801 boundary measure theoretic, 896 boundary-to-boundary Cauchy-Clifford integral operator, 874 Boyd indices, 277 bullet product, 894 Calderón product, 48 Calderón’s maximal operator, 780 Cauchy sequence (in a topological group), 56 Cauchy-Clifford integral operator boundary-to-boundary, left-handed, 629

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1

915

916

boundary-to-boundary, right-handed, 630 boundary-to-domain, left-handed, 629 boundary-to-domain, right-handed, 630 truncated, 874 charge, 166 compatible quasi-normed spaces, 25 complementary function, 261 complete topological group, 56 complex interpolation intermediate space, 48 quasi-norm, 48 conditional expectation operators, 382 conormal derivative from weighted Sobolev spaces, 488 countably simple (function), 39 dilation indices of a Young function, 255 distribution on an arbitrary set, 887 distributional tangential derivatives, 703 domain (ε, δ), 878 Ahlfors regular, 869 doubling Young function, 254 essential norm, 14 extension operator, 536 exterior derivative operator, 877 extrapolation theorem for Generalized Banach Function Spaces, 233 Fefferman-Stein grand maximal function, 108 filtering operator, 212 fractional Hajłasz-Sobolev space, 824 Fredholm radius, 77 function analytic (one variable), 36

SUBJECT INDEX

complementary, 261 strictly defined, 457 Young, 253 function norm, 218 Generalized Banach Function Space, 219 grand maximal function, 532 Hardy inequalities, 132 Hardy space atomic, 143 ionic characterization of, 166 Lebesgue-based, 114 local, 371 Lorentz-based, 117 Hardy-based homogeneous Sobolev space, 816 inhomogeneous Sobolev space, 818 harmonic double layer (boundary-to-domain), 730 Hodge star operator, 900 homogeneous Besov space, 365 Hardy space, 115 Triebel-Lizorkin space, 365 homogeneous Sobolev space block-based, 861 Hardy-based, 816 Morrey-based, 848 index function, 64, 74 indices Boyd, 277 dilation, 255 inhomogeneous Besov space, 368 inhomogeneous Sobolev space Hardy-based, 818 inhomogeneous Triebel-Lizorkin space, 368 integration by parts on the boundary both functions are smooth, 683 both functions belong to Sobolev spaces, 779

SUBJECT INDEX

both functions in Besov spaces, 846 one Hardy-based Sobolev function, one smooth function, 812, 814, 816 one Lebesgue function, one Sobolev function, 799 one Sobolev function, one smooth function, 685 interpolation space inner complex, 48 outer complex, 48 ion, 166

917

(q, λ)-dull, 359 H p , 174 H q,λ,θ , 301 Morrey space, 314 Morrey-based homogeneous Sobolev space, 848 negative Sobolev space, 801 Sobolev space, 793 Morrey-Campanato space homogeneous, 295 inhomogeneous, 297 Muckenhoupt weighted negative Sobolev space, 801 Sobolev space, 793

Jones extension operator, 470 lattice of functions r-convex, 46 local (boundary) Sobolev space, 684 local Hardy space, 371 local John condition, 886 locally p-convex (topological vector space), 417 Luxemburg norm on an Orlicz space, 256 maximal “altered” Cauchy integral operator, 874 maximal Cauchy-Clifford integral operator, 874 maximal Riesz transform, 898 measure feeble, 56 push-forward, 785 support of, 900 measure theoretic exterior, 879 measure theoretic interior, 885 modified (boundary-to-boundary) harmonic double layer, 733 modified (boundary-to-domain) harmonic double layer, 732 modulus of concavity, 5 molecule (η, ε)-smooth of type (p, s), 374 (p, q, ε), 174 (p, qo, ε, s)-rough, 375

nontangential maximal operator, 893 pull-back, 884 nontangential maximal function truncated, 893 norm Luxemburg, on an Orlicz space, 256 operator L p -filtering, 212 bounded, 3, 61 finite-rank, 2 Fredholm on Banach spaces, 63 Fredholm on TVS, 73 quasi-subadditive, 33 semi-Fredholm, 64 Orlicz space, 256 pointwise variation, 902 Pompeiu-Hausdorff distance, 877 principal symbol, 622 principal symbol of a first-order system, 900 pseudo-quasi-Banach space, 59 pseudo-quasi-norm, 59 pseudo-quasi-normed space, 59 push-forward of a measure, 785 quasi-Banach analytically convex, 38 quasi-isometry, 347

918

quasi-normed lattice of functions, 33 quotient topology, 9 radial maximal function, 532 real interpolation intermediate space, 29 quasi-norm, 29 rearrangement invariant Banach function space, 257 retract, 52 Riemannian metric tensor, 679 Riesz transform boundary-to-boundary, 899 in Rn , 633 truncated, 899 Sarason space VMO, 82 simple functions with support of finite measure, 900 Sobolev space L p -based homogeneous, 749 block-based, 795 block-based homogeneous, 861 homogeneous, 749 vanishing Morrey-based, 865 local, 684 local off-diagonal (boundary), 684 Lorentz-based, 792, 793 Morrey-based homogeneous, 848 negative block-based, 801 definition, 796 Morrey-based, 801 Muckenhoupt weighted, 801 off-diagonal, 800 vanishing Morrey-based, 803 off-diagonal (boundary), 684 off-diagonal Lorentz-based, 792 off-diagonal Morrey-based, 794 vanishing Morrey-based, 794 weighted, 792 space (q, λ)-midway, 360 L p -based Sobolev, 684 L p -based homogeneous Sobolev, 749

SUBJECT INDEX

p-Banach, 5 associated, 225 atomic Hardy, 143 block, 326 Hardy (Lebesgue-based), 114 Hardy (Lorentz-based), 117 homogeneous Besov, 365 homogeneous Hardy, 115 homogeneous Morrey-Campanato, 295 homogeneous Triebel-Lizorkin, 365 inhomogeneous Besov, 368 inhomogeneous Morrey-Campanato, 297 inhomogeneous Triebel-Lizorkin, 368 local Hardy, 371 local off-diagonal (boundary) Sobolev, 684 local BMO, 372 locally bounded, 3 Lorentz-based Sobolev, 792 Morrey, 314 Morrey-based Sobolev, 793 Muckenhoupt weighted Sobolev, 793 negative Sobolev, 796 of functions of bounded variation, 871 of functions of locally bounded variation, 871 off-diagonal Morrey-based Sobolev, 794 off-diagonal negative Sobolev, 800 Orlicz, 256 pseudo-quasi-Banach, 59 pseudo-quasi-normed, 59 quasi-Banach, 5 rearrangement invariant, 257 Sarason, VMO, 82 weighted Bessel potential, 457 weighted maximal Sobolev, 498 weighted Sobolev, 792 Zygmund, 274

SUBJECT INDEX

spectral radius definition, 76 standard fundamental solution for the Laplacian, 879 strictly defined version of a function, 457 strongly μ-measurable (function), 39 support of a measurable function, 901 surface-to-surface change of variable formula, 785 system of auxiliary function, 879 tangential derivative distributional, 703 weak, 896 tangential derivative operator, 896 in a pointwise sense, 682 tangential gradient, 728 test functions of type (x0, r, β, γ), 363 topologically bounded set, 2 Triebel-Lizorkin space homogeneous, 365 in Rn , 518 inhomogeneous, 368 truncated

919

Cauchy-Clifford integral operator, 874 Riesz transform, 899 unit, (η, ε)-smooth of type (p, s), 374 vanishing Hölder space, 98 vanishing mean oscillations, 82 weak conormal derivative definition, 658 weak normal derivative, 667 weak tangential derivative, 896 weakly compatible (TVS), 61 weighted L p Lebesgue space over Ω, 467 maximal Sobolev space, 498 Sobolev space, 792 Sobolev space in Ω, for the weight ap w := δ∂Ω , 472 Sobolev spaces in Rn , 456 Whitney decomposition lemma, 469 Young function, 253 doubling, 254 Zygmund’s space, 274

Symbol Index

∗ Hodge star operator, 900 ∧ exterior product of differential forms, 895 ∨ interior product of differential forms, 895 Δ Laplace operator, 877 ∇u gradient (Jacobian matrix) of u, 881 ∇ gradient operator in Rn−1 , 881 ∇tan tangential gradient, 728 Δ = Δ(x, r) surface ball, 878 UV symmetric difference of U and V, 878 U # V the union of two disjoint sets U, V, 902 ·, · D(Ω) distributional pairing D (Ω) in Ω, 877  X ∗ ·, · X , 904 (Lip c (Σ)) ·, · Lip c (Σ) (or simply ·, · ) & ' distributional pairing, 887 ·, · E pointwise (real) pairing in the fibers of Hermitian vector bundle E, 895 ·, · Λ T M (real) pointwise pairing on Λ T M, 895 Lip c (∂Ω) ·, · Lip c (∂Ω) pairing between Lipc (∂Ω) and its dual Lipc (∂Ω) , 899

u · w = u, w dot product of two vectors u, w ∈ Cn , 884 u · w = u, w dot product of two vectors u, w ∈ Rn , 877  ∞ contribution of F at infinity, [F] 879

· X→Y operator norm, 6 ess

·

X→Y essential norm, 14

·

Minkowski functional of p& ' BX (0, 1) p , 417

· (X0,X1 )θ, q real interpolation quasi-norm, 29

· [X0,X1 ]θ complex interpolation quasi-norm, 48 (a)+ := max{a, 0}, 897 X/Y quotient space, 1 [x]X/Y equivalence class of x ∈ X in X/Y , 1 1E characteristic function of E, 895 ⨏fΔ integral⨏ average of f in Δ, 758 f dμ, E f dμ integral average of E f on E, 885 U closure of the set U, 873 V ⊥ annihilator of a subspace V of a Banach space X, 66 ⊥ W annihilator of a subspace W of X ∗ , where X is Banach, 66

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Mitrea et al., Geometric Harmonic Analysis II, Developments in Mathematics 73, https://doi.org/10.1007/978-3-031-13718-1

921

922

√ i = −1 ∈ C complex imaginary unit, 884, 900 [A; B] := [A, B] := AB − BA the commutator of A and B, 871 { A; B} := AB + BA the anti-commutator of A and B, 871 d exterior derivative operator, 877 δ formal adjoint of the exterior derivative operator d, 877 δ jk Kronecker symbol, 877 δx Dirac distribution with mass at x, 877 δF distance function to the set F, 877 δ∂Ω (·) distance function to the boundary, 878 εBA generalized Kronecker symbol, 879 Φ(X → Y ) Fredholm operators from X into Y , 63, 74 Φ+ (X → Y ) finite-dim kernel semi-Fredholm operators from X into Y , 64 Φ− (X → Y ) finite-dim cokernel semi-Fredholm operators from X into Y , 64 Γκ (x) nontangential approach region, 881 λQ concentric dilate of the cube Q by the factor λ, 469 Λk Jones extension operator, 470 Λ∗,q Calderón’s maximal operator, 780 Λ T M the -th exterior power of the vector bundle on M, 886 % μ Cauchy-Clifford transform of the measure μ, 882 νg GMT unit normal induced by the metric tensor g, 894 ν E GMT unit normal induced by the standard Euclidean metric, 894  ν • F the bullet product of ν with F, 894 ν • u the Clifford bullet product of ν with u, 652

SYMBOL INDEX

ωn−1 surface area of S n−1 , 895 ρinv (T; X) spectral radius of T ∈ Bd (X), 76 ρFred (T; X) Fredholm (or essential spectral) radius of T ∈ Bd (X), 77 ρsym the symmetrized version of ρ, 901 ρ# the regularized version of ρ, 901 σg surface measure induced by the metric tensor g, 900 σ E surface measure induced by the standard Euclidean metric, 900 σ∗ = H n−1 ∂∗ Ω surface measure, 900 σ = H n−1 ∂Ω, the surface measure on ∂Ω, 900 ∂nta Ω nontangentially accessible boundary of Ω, 897 ∂lfp Ω, 897 ∂∗ E measure theoretic boundary of E, 896 ∂ ∗ E reduced boundary of E, 896 ∂T E, 896 ∂ N E, 896 ∂νA conormal derivative operator with respect to the coefficient tensor A acting from Besov and Triebel-Lizorkin spaces, 594 ∂νA(·, ·) conormal derivative operator with respect to the coefficient tensor A acting from weighted . A Sobolev spaces, 488 ∂ν weak conormal derivative operator with respect to the coefficient tensor A, 658 ∂τ j k pointwise tangential derivative operator, 682 ∂τ j k tangential derivative operator, . 685, 703 ∂τ j k weak tangential derivative, 896 ∂τXY tangential derivative operator on manifolds, 896  projection map onto Λ , 897 Πm m

SYMBOL INDEX

πκ (E), πΩ,κ (E) “shadow” (or projection) of E ⊆ Ω onto ∂Ω, 897 τt dilation by a factor of t, 902 τρ topology induced by the quasi-distance ρ, 902 Aκ (∂Ω) accessibility set, 869 Ap (X, ρ, μ) Muckenhoupt class, 869 [w] A p characteristic of the Muckenhoupt weight w, 869 A∞ (X, ρ, μ) Muckenhoupt class, 869 A p absolutely p-convex hull of the set A, 417 p,q As,z (Ω) restrictions to Ω of p,q distributions from As,0 (Ω), 528 q

· Ap, quasi-norm in s, z (Ω) Besov/Triebel-Lizorkin space of restrictions to Ω of p,q distributions from As,0 (Ω), 528 p,q As (Ω) Besov/Triebel-Lizorkin space in Ω, 527 q

· Ap, quasi-norm in s (Ω) Besov/Triebel-Lizorkin space in Ω, 527 p,q A˚ s (Ω) closure of Cc∞ (Ω) in p,q As (Ω), 527 p,q p,q As,0 (Ω) distributions from As (Rn )

supported in Ω, 527

· Ap, q (Ω) quasi-norm in s,0 Besov/Triebel-Lizorkin space p,q of distributions from As (Rn ) supported in Ω, 527 A q,κ L q -based area-function, 565 B(X → Y ) linear and (topologically) bounded operators from X to Y, 3 Bd(X) linear and bounded operators on X, 7  Bd X → Y linear and (norm) bounded operators from X to Y, 7 BMO−1 (∂Ω, σ), 820

923

BMO(X, μ) space of functions of bounded mean oscillations, 871 . , homogeneous BMO

· BMO(X,μ) semi-norm, 871

· BMO(X,μ) inhomogeneous BMO “norm”, 871 bmo (Σ, σ) local BMO space on the set Σ, 372

· bmo (Σ,σ) local BMO semi-norm, 372

f ∗ (Δ) local BMO norm of f on Δ, 870  BMO(X, μ) the space BMO modulo constants, 871 BV(O) space of functions of bounded variation in O, 871 BVloc (O) space of functions of locally bounded variation in O, 871 Bn−1 (x , r) open ball with center x and radius r in Rn−1 , 870 Bρ (x, r) ρ-ball with center at x and . p,q radius r, 870 Bs (Σ, σ) homogeneous Besov space on the set Σ, 365

· B. p, q (Σ,σ) homogeneous Besov s quasi-seminorm, 365 p,q Bs (Σ, σ) (inhomogeneous) Besov space on the set Σ, 368

· Bsp, q (Σ,σ) (inhomogeneous) Besov quasi-norm, 368 p,q bs (Σ) discrete Besov space on Σ, 377

· bsp, q (Σ) quasi-norm on discrete Besov space, 377 bp,q (Σ), 376

· b p, q (Σ) , 376 p,q Bs (Rn ) Besov space in Rn , 518

· Bsp, q (Rn ) quasi-norm in the Besov space in Rn , 518 q,λ B1 (∂Ω, σ) block-based Sobolev . q,λ space, 795 B1 (∂Ω, σ) block-based homogeneous Sobolev space, 861

924

B q,λ (Σ, σ) block space, 326

· B q, λ (Σ,σ) norm on block space, 327

· M q, λ (Σ,σ) norm on midway space, 360 q,λ B−1 (∂Ω, σ) block-based negative Sobolev space, 801 Borelτ (X) Borelians of the topological space (X, τ), 871 BL(Σ) bounded Lipschitz functions on Σ, 871 Bψ ( f , r), 56 CL left-handed boundary-to-domain Cauchy-Clifford integral operator, 629 CR right-handed boundary-to-domain Cauchy-Clifford integral operator, 630 Cmax maximal Cauchy-Clifford integral operator, 874 alt Cmax maximal “altered” Cauchy integral operator, 874 Cε truncated Cauchy-Clifford integral operator, 874 C boundary-to-boundary Cauchy-Clifford integral operator, 874 C L left-handed boundary-to-boundary Cauchy-Clifford integral operator, 629 C R right-handed boundary-to-boundary Cauchy-Clifford integral operator, 630 C k (Ω) functions of class C k in an open neighborhood of Ω, 873 Cck (Ω) functions of class C k with compact support in the open set Ω, 873 Cbk (Ω) bounded functions of class C k in Ω, 873 k C -singsup u singular support of u, 900

SYMBOL INDEX

Cc∞ (Ω) restrictions to Ω of functions in C ∞ (Rn ), 528  ∞ ∗c Cb (Ω) the algebraic dual of Cb∞ (Ω), 874

.

C α (U, ρ) homogeneous Hölder space, 875

· C.α (U,ρ) homogeneous Hölder space semi-norm, 875 .α C (U, ρ)/∼ homogeneous Hölder space modulo constants, 875 .α Cloc (U, ρ) local homogeneous Hölder space, 875 C α (U, ρ) inhomogeneous Hölder space, 876

· C α (U,ρ) inhomogeneous Hölder space norm, 875 α Cc (U, ρ) Hölder functions with ρ-bounded support, 876 .γ Cvan (Σ) homogeneous vanishing Hölder space, 98 γ Cvan (Σ) inhomogeneous vanishing Hölder space, 99 CBM(Ω) complex Borel measures in Ω, 874 CBM(X, τ) complex Borel measures in the topological space (X, τ), 874 CMO(Σ, σ), 184 p,q Cα (Ω), 530

· Cαp, q (Ω) , 530 p,q,m Cα (Ω), 530 p Cα (Ω), 531 C n Clifford algebra generated by n imaginary units, 875 Cp(X → Y ) space of compact linear operators from X into Y , 3 Cp(X) space of compact linear operators from X into itself, 3 Cθ,b (x, h), 874 Cρ , 875 ρ , 875 C D(X) dyadic grid on X, 878

SYMBOL INDEX

D harmonic double layer potential operator (boundary-to-domain), 730 Kmod boundary-to-boundary modified harmonic double layer potential operator, 733 Dmod boundary-to-domain modified harmonic double layer potential operator, 732 Dk (X), 878 D (Ω) space of distributions in Ω, 877  D = nj=1 e j  ∂j Dirac operator in Rn , 877 D first-order system, 877 D (real) transpose of the first-order system D, 877 D complex conjugate of the first-order system D, 877 D∗ Hermitian adjoint of the first-order system D, 877 D∗ (Σ), 373 D L Dirac operator acting from the left, 878 DR Dirac operator acting from the right, 878 Dist [E, F] Pompeiu-Hausdorff distance between E and F, 877 diamρ (A) ρ-diameter of the set A, 878 dg (x, y) geodesic distance between x and y, 877 divF the divergence of the vector  877 field F, divg differential geometric divergence (associated with the metric tensor g), 877 dim X dimension of X, 1 dVg volume element on M induced by the metric tensor g, 902 {Ek }k ∈Z, k ≥κΣ conditional expectation operators, 382 EΔ standard fundamental solution for the Laplacian, 879

925

E (Ω) distributions compactly supported in Ω, 879 EK (Ω) distributions in Ω supported in K, 879 E p (X) p-envelope of X, 421 Ex∂Ω→Ω extension operator from ∂Ω to Ω, 481 EΩ→Rn extension operator from Ω to Rn , 536 ext∗ (E) measure theoretic exterior of E, 879 e j standard j-th unit vector in Rn , 879 {e j }1≤ j ≤n standard orthonormal basis in Rn , 879 p,q n Fs (R ) Triebel-Lizorkin space in Rn , 518

· Fsp, q (Rn ) quasi-norm in the Triebel-Lizorkin space in Rn , 518 f p,q (Σ), 377

· f p, q (Σ) , 377 . p,q Fs (Σ, σ) homogeneous Triebel-Lizorkin space on the set Σ, 365

· F. p, q (Σ,σ) homogeneous s Triebel-Lizorkin quasi-seminorm, 365 p,q Fs (Σ, σ) (inhomogeneous) Triebel-Lizorkin space on the set Σ, 368 p,

· Fs q (Σ,σ) (inhomogeneous) Triebel-Lizorkin quasi-norm, 368 p,q fs (Σ) discrete Triebel-Lizorkin space on Σ, 378

· fsp, q (Σ) quasi-norm on discrete Triebel-Lizorkin space, 378 fBρ (x,r) integral average of f over Bρ (x, r), 885 fE∗ non-increasing rearrangement of f : E → R, 880 [ f ]E strictly defined version of the function f on the set E, 457

926

fγ Fefferman-Stein grand maximal function of f , 108 fp# L p -based Fefferman-Stein sharp maximal function, 901 Gα Bessel kernel of order α, 457 GΣ (x0, r, β, γ), 364 β,γ G˚0 (Σ), 364 β,γ

G0 (Σ) test functions on Σ, 364 β,γ

Gb (Σ), 373  g = 1≤ j,k ≤n g jk dx j ⊗ dxk Riemannian metric tensor, 881  g = 1≤ j,k ≤n g jk dx j ⊗ dxk Riemannian metric tensor, 679 H n−1 , the (n − 1)-dimensional Hausdorff measure in Rn , 882 n−1 Hg (n − 1)-dimensional Hausdorff measure associated with the metric g, 882 H s s-dimensional Hausdorff measure in Rn , 882 s H∗ s-dimensional Hausdorff outer-measure in Rn , 882 p H L -filtering operator, 212 H s (Ω) L 2 -based fractional Sobolev space in Ω, 530

· H s (Ω) norm in the L 2 -based fractional Sobolev space in Ω, . p 530 H1 (∂Ω, σ) Hardy-based homogeneous Sobolev space, . p 816  H1 (∂Ω, σ) ∼ classes of equivalence, modulo . p constants, of functions in H1 (∂Ω, σ), 817 p,q H1 (∂Ω, σ) Hardy-based inhomogeneous Sobolev space, 818 H. p (Σ, σ) Hardy space, 114 H p (Σ, σ) homogeneous Hardy space, 115

· H p (Σ,σ) quasi-norm on Hardy space, 114 H p,q (Σ, σ) Lorentz-based Hardy space, 117

SYMBOL INDEX

· H p, q (Σ,σ) quasi-norm on Lorentz-based Hardy space, 117 p,q Hat (Σ, σ) atomic Hardy space, 143 p,q Hfin (Σ, σ) finite linear combinations of atoms, 164

· H p, q (Σ,σ) quasi-norm on fin p,q Hfin (Σ, σ), 164 q,λ H (Σ, σ), 297

· H q, λ (Σ,σ) , 297 h p (Σ, σ) local Hardy space, 371

· h p (Σ,σ) quasi-norm on local Hardy space, 371 h p (Ω) local Hardy space in Ω, 532

· h p (Ω) quasi-norm in the local Hardy space in Ω, 532 p hk (Ω) local Hardy-based Sobolev space in Ω, 533 IE,α fractional integral operator of  order α on E, 885 Im T : X → Y image of T : X → Y , 2 index index function, 64, 74 int∗ (E) measure theoretic interior of E, 885 i(Φ) lower dilation index of Φ, 255 I(Φ) upper dilation index of Φ, 255 ι∗ pull-back map induced by the canonical inclusion ι, 884 ι∗# sharp pull-back, 884 ι∗n.t. nontangential pull-back, 884 KΔ boundary-to-boundary harmonic double layer potential, 886 KΔ# transpose harmonic double layer  potential, 886 Ker T : X → Y kernel of T : X → Y, 2 Ker L null-space of the system L, 499 L ρ Köthe function space associated with ρ, 55 L(ξ) characteristic matrix of L, 888 L(X → Y ) linear and continuous operators from X to Y , 3 L β (Σ), 143 L n Lebesgue measure in Rn , 886

SYMBOL INDEX

Lgn measure associated with the n-form dVg , 886 Lip(X) space of Lipschitz functions on X, 886 Lipc (X) space of Lipschitz functions with bounded support in X, 887  Lipc (Σ) distributions on Σ, 887 L 0 (X, μ) measurable functions which are a.e. pointwise finite, 886 r (X, μ) L r -integrable functions on Lfin sets of finite μ-measure, 887 ∞ essentially bounded functions Lcomp with compact support, 886 p Lbdd (Ω, L n ) p-th power integrable functions over bounded subsets of Ω, 886 L p (Ω, wL n ) weighted L p Lebesgue space over Ω, 467 p L (Ω, μ) maximal Lebesgue space, 887 p L1 (∂∗ Ω, σ∗ ) L p -based (boundary) Sobolev space, 684

· L p (∂∗ Ω,σ∗ ) norm on Sobolev 1 space, 693 p L1,loc (∂∗ Ω, σ∗ ) local (boundary) Sobolev space, 684 p L1 (∂∗ Ω, σg∗ ) ⊗ E global (boundary) Sobolev space on manifolds, 791 p L1,loc (∂∗ Ω, σg∗ ) ⊗ E local (boundary) Sobolev space on manifolds, 789 p L1 (∂Ω, w) Muckenhoupt weighted (boundary) Sobolev space, 793 p L1 (∂∗ Ω, wσ∗ ) weighted Sobolev . p space, 792 L1 (∂Ω, σ) homogeneous Sobolev space, 749 L p,q (X, μ) Lorentz space on X with respect to the measure μ, 887

· L p, q (X,μ) Lorentz space quasi-norm, 887 p,q L (Ω, μ) maximal Lorentz space, 887

927 p,q

L1 (∂Ω, σ) Lorentz-based Sobolev space, 793 p Ls (Ω) Bessel potential space in Ω, 529

· Lsp (Ω) norm in the Bessel potential space in Ω, 529 p Lα (Rn, wL n ) weighted Bessel potential space in Rn , 457

· Lαp (Rn,w L n ) norm on p Lα (Rn, wL n ), 457 p L exp (logθ L) Orlicz space, 276 L Φ (X, μ) Orlicz space, 256

· L Φ (X,μ) Luxemburg norm on the Orlicz space L Φ (X, μ), 256 p L (log L)α Zygmund’s space, 274 p,q L1 (∂∗ Ω, σ∗ ) off-diagonal (boundary) Sobolev space, 684 p,q L1,loc (∂∗ Ω, σ∗ ) local off-diagonal (boundary) Sobolev space, 684 p L−1 (∂∗ Ω, σ∗ ) negative Sobolev space, 796 p L−1 (∂Ω, w) Muckenhoupt weighted negative Sobolev space, 801 p,q L−1 (∂∗ Ω, σ∗ ) off-diagonal negative Sobolev space, 800 log+ positive ln, 888 . L p,λ (Σ, σ) homogeneous Morrey-Campanato space, 295

· L. p, λ (Σ,σ) Morrey-Campanato semi-norm, 295 p,λ L (Σ, σ) inhomogeneous Morrey-Campanato space, 297

· L p, λ (Σ,σ) Morrey-Campanato norm, 297 (Q) side-length of the cube Q, 469 p w (D∗ (Σ)), 397

· wp (D∗ (Σ)) , 397 Mγ∗ (F) upper γ-dimensional Minkowski content of F, 890 M p,λ (Σ, σ) Morrey space, 314

· M p, λ (Σ,σ) norm on Morrey space, 314 M˚ p,λ (Σ, σ) vanishing Morrey space, 316

928 p,λ

M1 (∂Ω, σ) Morrey-based Sobolev space, 793 p,q,λ M1 (∂Ω, σ) off-diagonal Morrey-based Sobolev space, 794 p,q,λ (∂Ω, σ) off-diagonal M˚ 1 vanishing Morrey-based . p,λ Sobolev space, 794 M1 (∂Ω, σ) homogeneous Morrey-based Sobolev space, 848 p,λ M˚ 1 (∂Ω, σ) vanishing Morrey-based Sobolev space, . p,λ 794 M1 (∂Ω, σ) homogeneous vanishing Morrey-based Sobolev space, 865 p,λ M−1 (∂Ω, σ) Morrey-based negative Sobolev space, 801 M+ (X, μ) non-negative μ-measurable functions on X, 218 M (X, μ) μ-measurable functions on X, 218 M X Hardy-Littlewood maximal operator on X, 891 M A,s,α fractional Hardy-Littlewood maximal operator, 891 M X,s L s -based Hardy-Littlewood maximal operator, 891 R local L s -based M X,s Hardy-Littlewood maximal operator, 891 M X,s,α fractional Hardy-Littlewood maximal operator, 891 mE (λ, f ), 891 M q,λ (Σ, σ) (q, λ)-mid space, 360 N0 = N ∪ {0}, 893 p Nκ (Ω; μ), 894 N Φ ( f ) modular size of f , 258 Nκ nontangential maximal operator, 893 NκE the nontangential maximal operator restricted to E, 893

SYMBOL INDEX

Nκε the nontangential maximal function truncated at height ε, 893 κ,θ,r averaged nontangential N maximal function, 893 m osc (u, x, t) oscillation of order m of the function u at location x and scale t, 525 Oε one-sided collar neighborhood of ∂Ω, 895 osc p ( f ; R) L p -based mean oscillation of f at scales up to R, 895 pX lower Boyd index, 277 qX upper Boyd index, 277 Φ ∈ Δ2 doubling Young function, 254 P maximal function of Carleson type, 607 Pg maximal function of Carleson 679  type on manifolds, P.V. b k(x − ·)|Σ principal-value distribution on the set Σ, 804 R+n upper half-space in Rn , 898 R−n lower half-space in Rn , 898 R j boundary-to-boundary Riesz transform, 899 R j,ε truncated Riesz transform, 899 R j,max maximal Riesz transform, 898 R Rn →Ω restriction operator from Rn to Ω, 528 RRn →∂Ω restriction operator from Rn to ∂Ω, 898 RΩ→∂Ω higher-order restriction operator from Ω to ∂Ω, 573 RHq (X, ρ, μ) reverse Hölder class, 898 [w]RHq reverse Hölder constant of a weight in RHq (X, ρ, μ), 898 rad(Ω), 899 regsupp u regular support of a distribution u ∈ D (Ω), 898 n−1 unit sphere in Rn , 900 S n−1 S± upper/lower hemispheres of S n−1 , 900

SYMBOL INDEX

S(X, μ) simple functions on (X, μ), 900 Sfin (X, μ) simple functions on (X, μ) with support of finite measure, 900 Sym(D; ξ) principal symbol of the first-order system D, 900 Sym(D; ν) • F bullet product of F with the principal symbol of the first-order system D, 651 S (Rn ) Schwartz functions, 901 S (Rn ) tempered distributions, 901 supp μ support of the measure μ, 900 supp f support of the measurable function f , 900 T ∗ adjoint of T, 4 TrΩ→∂Ω trace operator from Ω to ∂Ω, 586 TrRn →Σ trace operator from Rn to Σ, 458 TrΩ→∂Ω trace operator from Ω to ∂Ω, 473 Tγ (x) bump functions centered at x,  κ−n.t.107 u|∂Ω (x) nontangential trace of u at x ∈ ∂Ω, 894 uα# , 531 + radial maximal function of u in uΩ Ω, 532 ∗ grand maximal function of u in uk,Ω Ω, 532 u,θ solid maximal function of u, 901 E local solid maximal function of u,θ u, 901 tangential maximal function of umax M u, 891 uscal scalar part of u, 901 uvect vector part of u, 903 VarF pointwise variation of F, 902 V( f ; O) variation of f in O, 903 VMO(X, μ) space of functions of vanishing mean oscillations, 82 VMO−1 (∂Ω, σ), 820 W k, p (Ω) L p -based Sobolev space of order k in Ω, 903

929 k, p

Wbdd (Ω), 903 k, p

Wloc (Ω) local L p -based Sobolev space of order k in Ω, 903 k, p W (Ω, wL n ) weighted Sobolev space in Ω, 468

· W k, p (Ω,w L n ) norm in weighted Sobolev space in Ω, 468 W k, p (Rn, wL n ) weighted Sobolev spaces in Rn , 456

· W k, p (Rn,w L n ) norm in the weighted Sobolev spaces in Rn , 456 k, p Wa (Ω) weighted Sobolev space in ap Ω, for the weight w := δ∂Ω , 472

· W k, p (Ω) quasi-norm in the a weighted Sobolev space k, p Wa (Ω), 472 1, p Wa (Ω)bdd , 475 1, p W˚ a (Ω) closure of Cc∞ (Ω) in 1, p Wa (Ω), 479 k, p Wa, (Ω) weighted maximal Sobolev space, 498

· W k, p (Ω) quasi-norm in weighted a, maximal Sobolev space, 498 . k, p Wa, (Ω) homogeneous weighted maximal Sobolev space, 498

· W. k, p (Ω) homogeneous weighted a, maximal Sobolev space, 498 −1, p Wa (Ω) weighted Sobolev space of order −1 in Ω, 485

· W −1, p (Ω) norm on weighted a Sobolev space of order −1 in Ω, 485 W s, p (Σ, σ) fractional Hajłasz-Sobolev space, 824 X01−θ X1θ Calderón product, 48 (X0, X1 )θ,q real interpolation intermediate space, 29 [X0, X1 ]θ complex interpolation intermediate space, 48 [X]r r-convexification of X, 47 Xbdd (Ω), X (Ω)bdd , 474

930

SYMBOL INDEX

X Generalized Banach Function Space on the measure space (X, M, μ), 219

· X norm on the Generalized Banach Function Space X, 219 X

associated space of X, 225

· X norm on the associated space of X , 225 ∞ ˚ closure of Lcomp in the Generalized X Banach Function Space X, 247 X  , 900 ξ  , 900