Functional analysis, harmonic analysis, and image processing: in honor of Bjorn Jawerth

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693

Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth

Michael Cwikel Mario Milman Editors

American Mathematical Society

Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth

Michael Cwikel Mario Milman Editors

693

Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth

Michael Cwikel Mario Milman Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

2010 Mathematics Subject Classification. Primary 42B20, 42B25, 42B35, 42B37, 42C15, 42C40, 46E30, 46E35, 46B70.

Library of Congress Cataloging-in-Publication Data Names: Jawerth, Bj¨ orn, 1952–2013 | Cwikel, M. (Michael), 1948– editor. | Milman, Mario, editor. Title: Functional analysis, harmonic analysis, and image processing: A collection of papers in honor of Bj¨ orn Jawerth / Michael Cwikel, Mario Milman, editors. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Contemporary mathematics; volume 693 | Includes bibliographical references. Identifiers: LCCN 2016055558 |ISBN 9781470428365 (alk. paper) Subjects: LCSH: Harmonic analysis. | Fourier analysis. | Image processing. | AMS: Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Singular and oscillatory integrals (Calder´ on-Zygmund, etc.). msc | Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Maximal functions, Littlewood-Paley theory. msc | Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Function spaces arising in harmonic analysis. msc | Harmonic analysis on Euclidean spaces – Nontrigonometric harmonic analysis – General harmonic expansions, frames. msc | Harmonic analysis on Euclidean spaces – Nontrigonometric harmonic analysis – Wavelets and other special systems. msc | Functional analysis – Linear function spaces and their duals – Spaces of measurable functions (Lp -spaces, Orlicz spaces, K¨ othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.). msc | Functional analysis – Linear function spaces and their duals – Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems. msc | Functional analysis – Normed linear spaces and Banach spaces; Banach lattices – Interpolation between normed linear spaces. msc Classification: LCC QA403 .F86 2017 | DDC 515/.2433–dc23 LC record available at https://lccn.loc.gov/2016055558 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/693

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Contents

Preface

vii

Bj¨ orn David Jawerth (1952–2013) Michael Cwikel, Michael Frazier, Louise M. Jawerth, and Mario Milman

1

Jawerth–Milman extrapolation theory: Some recent developments with applications Sergey V. Astashkin and Konstantin V. Lykov

7

Uncertainty principles and weighted norm inequalities John J. Benedetto and Matthew Dellatorre

55

The discrete Calder´ on reproducing formula of Frazier and Jawerth ´ a ´ d B´ Arp enyi and Rodolfo H. Torres

79

A characterisation of the Besov-Lipschitz and Triebel-Lizorkin spaces using Poisson like kernels Huy-Qui Bui and Timothy Candy

109

An approximation problem in multiplicatively invariant spaces C. Cabrelli, C. A. Mosquera, and V. Paternostro

143

Discrete decomposition of homogeneous mixed-norm Besov spaces G. Cleanthous, A. G. Georgiadis, and M. Nielsen

167

From Frazier-Jawerth characterizations of Besov spaces to wavelets and decomposition spaces H. G. Feichtinger and F. Voigtlaender

185

Traces and extensions of weighted Sobolev and potential spaces Michael Frazier and Svetlana Roudenko

217

Compact embeddings of weighted smoothness spaces of Morrey type: An example Dorothee D. Haroske and Leszek Skrzypczak

235

Tracking the structural deformation of a sheared biopolymer network Louise M. Jawerth and David A. Weitz

255

Extrapolation, a technique to estimate ´ szlo ´ Lempert La

271

v

vi

CONTENTS

On a dual property of the maximal operator on weighted variable Lp spaces Andrei K. Lerner

283

Is the Dirichlet space a quotient of DAn ? Richard Rochberg

301

1

Characterizations of the Hardy space H (R) and BMO(R) Wael Abu-Shammala, Ji-Liang Shiu, and Alberto Torchinsky

309

Four proofs of cocompactness for Sobolev embeddings Cyril Tintarev

321

Tempered homogeneous function spaces, II Hans Triebel

331

Isotropic and dominating mixed Besov spaces: A comparison Van Kien Nguyen and Winfried Sickel

363

An iteratively reweighted least squares algorithm for sparse regularization Sergey Voronin and Ingrid Daubechies

391

Preface Bj¨ orn Jawerth’s passing was a painful personal and professional loss for us. As time passed, we came to feel that the best way to render a tribute to him would be to “let the mathematics do the talking”. Thus, with the invaluable help of former collaborators, students, and colleagues of Bj¨ orn, we have put together this collection of surveys and research articles on some of the areas in mathematics where his contributions were decisive. We have also included a brief biographical article about Bj¨orn, which we co-authored with Michael Frazier and Louise Jawerth. That article includes a list of Bj¨ orn’s papers, all those that were reviewed in the Mathematical Reviews, as well as a list of his doctoral students, which we compiled using the database of the Mathematics Genealogy Project. As we believe this volume attests, Bj¨orn made very important mathematical contributions of lasting value in a wide range of topics. Our hope is that the collection offered here can serve as a source of inspiration for a new generation of mathematicians. We are very grateful to all the contributing authors and the anonymous referees. We are also grateful to John Benedetto, Simeon Reich, and Alexander Zaslavski for their help with editorial matters and their encouragement at crucial stages of this project. As on previous occasions, we were fortunate to have the expert editorial help of Christine Thivierge of the AMS, and we also thank her very warmly. Michael Cwikel (Haifa)

Mario Milman (Buenos Aires and Delray Beach)

vii

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13953

Bj¨ orn David Jawerth (1952–2013) Michael Cwikel, Michael Frazier, Louise M. Jawerth, and Mario Milman Bj¨ orn was born in Stockholm, Sweden, on November 25, 1952. His aptitude for mathematics was already apparent at an early age. As a teenager he combined school with helping the Jawerths’ family business, the engineering firm Jawerth Systems. In its day Jawerth Systems engaged in the construction of suspension roofs on stadiums and at airports all over Europe, and Bj¨ orn contributed to these activities by doing calculations and programming work. By the time he was 24 years old he had a Ph.D. in mathematics and dual masters degrees in physics and electrical engineering from Lund University. The combination of mathematics with concrete applications and business would remain a hallmark throughout his subsequent career. At Lund his Ph.D. advisor was Jaak Peetre and his dissertation on function spaces and embeddings (cf. [1977], [1977b], [1978]) was very well received. Indeed, it is still cited nowadays in the literature on these topics. After his Ph.D. he went to Indiana University, Bloomington on a postdoctoral fellowship. Bloomington proved to be an important stepping stone for Bj¨orn’s future in the US. He developed a collaboration with Alberto Torchinsky. Their joint work on local maximal functions (cf. [1985b]), weighted norm inequalities (cf. [1984]) and interpolation (cf. [1986d]), continues to be relevant today. On the personal side his daughter Louise (a.k.a. Lolo) was born in Bloomington. In 1982 Bj¨orn took up a visiting position at Washington University, Missouri, where he would be promoted to a tenured position in 1984. At WashU his research and teaching flourished. He continued his work on weighted norm inequalities (cf. [1986c]), and he collaborated with Richard Rochberg and Guido Weiss on interpolation theory (cf. [1986]). At the same time he also started a collaboration with Michael Frazier (a.k.a. Mike) in which they developed the theory of the “phitransform”, an early and close precursor of the influential theory of wavelets due to Ingrid Daubechies, Yves Meyer, and many others. Frazier and Jawerth used the phi-transform to obtain a unified decomposition theory of function spaces (cf. [1990b], [1988], [1985]). Several articles in this volume give accounts of some of 2010 Mathematics Subject Classification. Primary 01A70. M. Cwikel’s work was supported by the Technion V.P.R. Fund and by the Fund for Promotion of Research at the Technion, L. M. Jawerth was supported by the National Science Foundation (DMR-1310266) and the Harvard Materials Research Science and Engineering Center (DMR1420570), M. Milman was partially supported by a grant from the Simons Foundation (#207929 to Mario Milman). c 2017 American Mathematical Society

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M. CWIKEL, M. FRAZIER, L. M. JAWERTH, AND M. MILMAN

the contributions of Frazier-Jawerth and their influence in harmonic analysis, the theory of function spaces, and their applications to image processing (cf. [BT], [FV], [HS], [NS]). An introduction to this work, as it stood at the time of its development, appeared in the influential set of lecture notes that Bj¨ orn wrote jointly with Mike and Guido Weiss (cf. [1991b]). At WashU, and later at the University of South Carolina, Bj¨ orn attracted and trained many bright graduate students, some of whom would go on to become prominent research mathematicians in their own right. Among them we should mention Marius Mitrea (SC), Carlos Perez (WashU) and Rodolfo Torres (WashU). Below we have compiled a complete listing of Bj¨ orn’s Ph.D. students. Also during his time at Washington U, Bj¨ orn started a long term collaboration with Mario Milman which, in particular, resulted in the development of a theory of extrapolation spaces (cf. [1991c], and the article by Astashkin-Lykov [AL] in this volume). Bj¨ orn and Mario also collaborated with Michael Cwikel and Richard Rochberg on other topics connected with interpolation theory (cf. [1990d], [1989b]). Bj¨ orn’s family was also growing, and his second daughter (Nicole a.k.a. Niki) was born in St. Louis. In 1988 Bj¨orn moved to the University of South Carolina where he became the David W. Robinson Palmetto Professor of Mathematics and Adjunct Professor of Computer Science. At USC he helped launch the Interdisciplinary Mathematics Institute (IMI) which continues to operate to this day. By this time Bj¨ orn was working in earnest to develop applications of wavelet theory to image processing (cf. [1993]). In South Carolina he collaborated with Ron DeVore, Brad Lucier, Vasil Popov, and others, discovering influential ways for applying harmonic analysis to image processing (see e.g., [1992e], [1992f], [1992g]). His interest in developing applications would also lead Bj¨ orn to form, in parallel, his own company, which he named Summus. Eventually he found himself focusing all his energy on Summus, and departed academia at an early age. At Summus he did what founders of start-ups usually do: everything! He was involved in all aspects of the business. He continued with his research, which by now had become very applied. He continued to train young people. Among his postdoctoral mentees at that stage we can mention Vilhelm Adolfsson and Wim Sweldens1 . He would eventually move Summus to the newly created technological complex being developed in Raleigh, North Carolina, where in 2004 he created yet another company to which he gave the enigmatic name 5 examples. At 5 examples he would continue to develop new technologies and, in the process of doing so, he was awarded several new patents2 .

1 Although Wim Sweldens does not appear in the formal list of Bj¨ orn’s Ph.D. students below, Bj¨ orn is listed as an advisor in Wim Sweldens’ Ph.D. thesis (cf. W. Sweldens, The Construction and Application of Wavelets in Numerical Analysis, Ph.D. thesis, Leuven, 1995; https://lirias.kuleuven.be/cv?u=U0015479 and http://cm-belllabs.github.io/who/wim/thesis/thesis.pdf) 2 We refer to the Jawerth Memorial website https://bjornjawerthmemorial.wordpress.com /about-bjorn-jawerth/ where the reader can find more information about Bj¨ orn, including his technological patents and publications in other fields.

¨ BJORN DAVID JAWERTH (1952–2013)

3

During the last years of his life he also entered into a new research collaboration with his old friend and collaborator Mario. But the results of their work still await publication3 . Bj¨ orn is survived by his two daughters, Louise and Nicole; his sister, AnneSophie Sj¨ oberg; his half-brother, Kjell Jawerth; and his partner of 13 years, Mona Rozovich. He did not live to witness the birth of his granddaughter Katharina. We fondly salute the memory of a remarkable person and exceptional scientist, who was very special for each of us. Publications (as per Math. Reviews) Research Papers [2007] B. Jawerth and M. Milman, Weakly rearrangement invariant spaces and approximation by largest elements, in “Interpolation theory and applications”, Contemp. Math. 445 (2007), 103-110, MR 2381889. [1999] B. Jawerth, P. Lin and E. Sinzinger, Lattice Boltzmann models for anisotropic diffusion of images, Journal of Mathematical Imaging and Vision 11 (1999), 231-237. MR 1731973. [1996] B. Jawerth and M. Mitrea, Acoustic scattering, Galerkin estimates and Clifford algebras, in “Clifford algebras in analysis and related topics (Fayetteville, AR, 1993)”, Stud. Adv. Math. 1996, pp 199-216. MR 1383106. [1995] E. Cornea, B. Jawerth and W. Zheng, Wavelets and interactive surface modeling, in “Wavelets and interactive surface modeling”, Approximation theory VIII, Vol. 2, Ser. Approx. Decompos. 6, World Sci. Publ., River Edge, NJ, pp 39–46. MR 1471774. [1995b] B. Jawerth and W. Sweldens, Biorthogonal smooth local trigonometric bases, J. Fourier Anal. Appl. 2 (1995), 109-133. MR 1365201. [1995c] B. Jawerth and M. Mitrea, On the spectra of the higher-dimensional Maxwell operators on nonsmooth domains, in “Harmonic analysis and operator theory”, Contemp. Math. 189, 1995, pp. 309-325. MR 1347020. [1995d] B. Jawerth and M. Mitrea, Higher-dimensional electromagnetic scattering theory on C 1 and Lipschitz domains, Amer. J. Math. 117 (1995), 929–963. MR 1342836. [1994] B. Jawerth and M. Mitrea, Clifford wavelets, Hardy spaces, and elliptic boundary value problems, in “Wavelets and their applications”, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 442 (1994) 261–290. MR 1339778. [1994b] B. Jawerth and W. Sweldens, An overview of wavelet based multiresolution analyses, SIAM Rev. 36 (1994), 377–412. MR 1292642. [1994c] V. Adolphsson, B. Jawerth and R. Torres, A boundary integral method for parabolic equations in non-smooth domains, Comm. Pure Appl. Math. 47 (1994), 861–892. MR 1280992. [1994d] L. Andersson, B. Jawerth and M. Mitrea, The Cauchy singular integral operator and Clifford wavelets, in “Wavelets: mathematics and applications”, Stud. Adv. Math., 1994, pp 525-546, CRC, Boca Raton, FL. MR 1247527. [1994e] C.-c. Hsiao, B. Jawerth, B. J. Lucier and X. M. Yu, Near optimal compression of orthonormal wavelet expansions, in “Wavelets: mathematics and 3 The collection of articles that resulted from this collaboration is currently being curated by Mario.

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applications”, Stud. Adv. Math. 1994, 425-446, CRC, Boca Raton, FL. MR 1247523. [1994f] L. Andersson, N. Hall, B. Jawerth and G. Peters, Wavelets on closed subsets of the real line, in “Recent advances in wavelet analysis”, Wavelet Anal. Appl. 3, Academic Press, Boston, MA, 1994, pp 1-61. MR 1244602. [1993] B. Jawerth, M. L. Hilton and T. L. Huntsberger, Local enhancement of compressed images, in “Wavelet theory and application”, Kluwer Acad. Publ., Boston, MA 1993, pp 39-49. MR 1266655. [1993b] A. Cohen, I. Daubechies, B. Jawerth and P. Vial, Multiresolution analysis, wavelets and fast algorithms on an interval, C. R. Acad. Sci. Paris S´er. I Math. 316 (1993), 417-421. MR 1209259. [1992] B. Jawerth and M. Milman, Wavelets and best approximation in Besov spaces, in “Interpolation spaces and related topics”, Haifa, 1990, Israel Math. Conf. Proc. 5, 1992, pp 107–112. MR 1206494. [1992b] B. Jawerth and M. Milman, New results and applications of extrapolation theory, in “Interpolation spaces and related topics”, Haifa, 1990, Israel Math. Conf. Proc. 5, 1992, pp 81–105. MR 1206493. [1992c] M. Frazier and B. Jawerth, Applications of the φ and wavelet transforms to the theory of function spaces, in “Wavelets and their applications”, Jones and Bartlett, Boston, MA, 1992, pp 377-417. MR 1187350. [1992d] V. Adolfsson, M. Goldberg, B. Jawerth and H. Lennerstad, Localized Galerkin estimates for boundary integral equations on Lipschitz domains, SIAM J. Math. Anal. 23 (1992), 1356–1374. MR 1177796. [1992e] R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), 737–785. MR 1175690. [1992f] R. DeVore, B. Jawerth and B. Lucier, Surface compression, Comput. Aided Geom. Design 9 (1992), 219–239. MR 1175287. [1992g] R. DeVore, B. Jawerth and B. Lucier, Image compression through wavelet transform coding, IEEE Trans. Inform. Theory 38 (1992), 719-746. MR 1162221. [1992h] A. Deliu and B. Jawerth, Geometrical dimension versus smoothness, Constr. Approx. 8 (1992), 211–222. MR 1152878. [1991] V. Adolfsson, B. Jawerth and R. Torres, Singular integral equations, spaces of homogeneous type and boundary elements in nonsmooth domains, Rev. Un. Mat. Argentina 37 (1991), 163–183. MR 1266681. [1990] Y. S. Han, B. Jawerth, M. Taibleson and G. Weiss, Littlewood-Paley theory and ε families of operators, Colloq. Math. 60/61 (1990), 321-359. MR 1096383. [1990b] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. MR 1070037. [1990c] B. Jawerth, C. Perez and G. Welland, The positive cone in TriebelLizorkin spaces and the relation among potential and maximal operators, Contemp. Math. 107 (1990), 71-91. MR 1066471. [1990d] M. Cwikel, B. Jawerth and M. Milman, On the fundamental lemma of interpolation theory, J. Approx. Theory 60 (1990), 70-82. MR 1028895. [1990e] M. Cwikel, B. Jawerth and M. Milman, The domain spaces of quasilogarithmic operators, Trans. Amer. Math. Soc. 317 (1990), 599-609. MR 974512.

¨ BJORN DAVID JAWERTH (1952–2013)

5

[1989] M. Frazier, Y.-S. Han, B. Jawerth and G. Weiss, The T 1 theorem for Triebel-Lizorkin spaces, Lecture Notes in Math. 1384, 1989, pp 168–181, Springer, Berlin. MR 1013823. [1989b] M. Cwikel, B. Jawerth, M. Milman and R. Rochberg, Differential estimates and commutators in interpolation theory, in “Analysis at Urbana, Vol. 2”, London Math. Soc. Lecture Note Ser. 138, pp 170-220, 1989, Cambridge Univ. Press, Cambridge. MR 1009191. [1989c] B. Jawerth and M. Milman, A theory of extrapolation spaces. First applications, C. R. Acad. Sci. Paris S´er. I Math. 308 (1989), 175-179. MR 984917. [1989d] B. Jawerth and M. Milman, A theory of extrapolation spaces, further applications, C. R. Acad. Sci. Paris S´er. I Math. 309 (1989), 225-229. MR 1006735. [1989e] B. Jawerth and M. Milman, Interpolation of weak type spaces, Math. Z. 201 (1989), 509-519. MR 1004171. [1988] M. Frazier and B. Jawerth, The φ-transform and applications to distribution spaces, Lecture Notes in Math. 1302, 1988 pp 223–246, Springer, Berlin. MR 942271. [1986] B. Jawerth, R. Rochberg and G. Weiss, Commutator and other second order estimates in real interpolation theory, Ark. Mat. 24 (1986), 191–219. MR 884187. [1986b] B. Jawerth, The K−functional for H p and BM O in the poly-disk, Proc. Amer. Math. Soc. 98 (1986), 232–238. MR 854025. [1986c] B. Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108 (1986), 361-414. MR 833361. [1986d] B. Jawerth and A. Torchinsky, A note on real interpolation of Hardy spaces in the polydisk, Proc. Amer. Math. Soc. 96 (1986), 227–232. MR 818449. [1985] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799. MR 808825. [1985b] B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), 231–270. MR 779906. [1984] B. Jawerth and A. Torchinsky, The strong maximal function with respect to measures, Studia Math. 80 (1984), 261-285. MR 783994. [1984b] B. Jawerth and A. Torchinsky, On a Hardy and Littlewood imbedding theorem, Michigan Math. J. 31 (1984), 131-137. MR 752250. [1984c] B. Jawerth, The K−functional for H 1 and BM O, Proc. Amer. Math. Soc. 92 (1984), 67-71. MR 749893. [1978] B. Jawerth, The trace of Sobolev and Besov spaces if 0 < p < 1, Studia Math. 62 (1978), 65-71. MR 0482141. [1977] B. Jawerth, Weighted norm inequalities for functions of exponential type, Ark. Mat. 15 (1977), 223-228. MR 0463783 [1977b] B. Jawerth, Some observations on Besov and Lizorkin-Triebel spaces, Math. Scand. 40 (1977), 94-104. MR 0454618.

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Books and Research Monographs [1991b] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics 79, 1991, American Mathematical Society, Providence, RI, pp viii+132. MR 1107300. [1991c] B. Jawerth and M. Milman, Extrapolation theory with applications, Mem. Amer. Math. Soc. 89 (1991), pp iv+82. MR 1046185. References to articles in this volume [AL] S. Astashkin and S. Lykov, Jawerth–Milman extrapolation theory: some recent developments with applications. [BT] A. B´enyi and R. H. Torres, The Discrete Calder´ on Reproducing Formula of Frazier and Jawerth. [FV] H. G. Feichtinger and F. Voigtlaender, From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces. [HS] D. D. Haroske and L. Skrzypczak, Compact embeddings of weighted smoothness spaces of Morrey type: an example. [NS] V. K. Nguyen and W. Sickel, Isotropic and Dominating Mixed Besov Spaces – a Comparison.

• • • • • • • • • • • • •

Doctoral Students (as per Math. Genealogy Project) 1988 Anca Deliu, Washington University in St. Louis 1989 Carlos Perez, Washington University in St. Louis 1989 Rodolfo Torres, Washington University in St. Louis 1992 Chia-chang Hsiao, University of South Carolina 1992 Kenneth Yarnall, University of South Carolina 1993 Baiqaio Deng, University of South Carolina 1993 Mong-shu Lee, University of South Carolina 1994 Marius Mitrea, University of South Carolina 1994 Qun Wu, University of South Carolina 1996 Anping Chen, University of South Carolina 1996 Weimin Zheng, University of South Carolina. 1997 Emil-Adrian Cornea, University of South Carolina 1999 Eric Sinzinger, University of South Carolina

Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail address: [email protected] Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996 E-mail address: [email protected] Department of Physics, Harvard University, Cambridge, Massachusetts 02138 Current address: Max Planck Institute for the Physics of Complex Systems, 01187 Dresden Germany E-mail address: [email protected] Instituto Argentino de Matematica, Buenos Aires, Argentina E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13945

Jawerth–Milman extrapolation theory: Some recent developments with applications Sergey V. Astashkin and Konstantin V. Lykov Abstract. The main aim of this paper is to survey a part of remarkable work of Bj¨ orn Jawerth and Mario Milman in extrapolation theory. We focus on a few of their fundamental ideas and results and then trace some more recent developments of this theory. The paper contains also some new results. By using a family of extrapolation functors (F-methods), which widely generalize the Σ-and Δ-functors, we discuss connections between the interpolation and extrapolation processes. In the case of the scale of Lp -spaces we give a characterization of Marcinkiewicz spaces which can be obtained by F-methods and extend some stability relations from Jawerth-Milman’s theory to a rather broad family of non-tempered weights. Basing on a generalization of JawerthMilman’s idea of a tempered weight, we single out a subclass of rearrangement invariant spaces X such that the corresponding extrapolation parameter related to appropriate F-functor is explicitly determined by X itself and find a new description of such spaces by using the real K-method of interpolation. The paper contains also applications of the above constructions to the classical power moment problem and to extrapolation of sublinear operators with a quasi-Banach target space.

1. Introduction and preliminaries The starting point for extrapolation of operators was the following remarkable theorem proved by Japanese mathematician Sh.Yano in 1951 (special cases of this result had been considered earlier by Titchmarsh and Zygmund [73, p. 332]). Theorem 1.1 ([71]). (a) Let an operator T be defined on the space L1 [0, 1] with values in the set of measurable functions on [0, 1] and let T satisfy the sublinearity ∞ condition: for some B > 0 and all xj ∈ L1 [0, 1] such that the series j=1 xj converges in L1 [0, 1] we have ∞ ∞        xj (t)  B |T xj (t)| a.e. on [0, 1]. T j=1

j=1

2010 Mathematics Subject Classification. Primary 46B70, 46E30; Secondary 60E05, 44A60. Key words and phrases. Extrapolation spaces, Yano’s theorem, interpolation of operators, K-functional, real method of interpolation, rearrangement invariant space, Marcinkiewicz space, Lorentz space, Zygmund space, moment problem, Carleman condition, sublinear operator. This work was supported by the Ministry of Education and Science of the Russian Federation and by the RFBR grant 17-01-00138. The second author was also supported by the RFBR grants 14-01-31452-mol-a and 16-41630676. 7

c 2017 American Mathematical Society

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SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

Suppose also, that T is bounded in Lp [0, 1] for every p ∈ (1, p0 ), p0 > 1, and (1.1)

T Lp →Lp  C(p − 1)−β , p ∈ (1, p0 ),

for some β > 0, with a constant C > 0 independent of p. Then, T : L(log L)β → L1 , where the Zygmund space L(log L)β consists of all measurable on [0, 1] functions x(t) such that 1 xL(log L)β = x∗ (t) logβ (e/t) dt < ∞ 0 ∗

(x (t) stands for the left-continuous, non-increasing rearrangement of |x(t)|). (b) If T is an operator bounded in Lp [0, 1] for all p > p0 and (1.2)

T Lp →Lp  Cp1/β , p ∈ (p0 , ∞),

for some β > 0, with a constant C > 0 independent of p, then T : L∞ → Exp Lβ . The Zygmund space Exp Lβ consists of all measurable functions x(t) such that xExp Lβ = sup (x∗ (t) log−1/β (e/t)) < ∞. 0 0. I(R, R) = t ρ(t) If I is exact, then its characteristic function is quasi-concave (i.e., ρ(t) is increasing and ρ(t)/t is decreasing). Conversely, if ρ(t) is quasi-concave, then there are many exact interpolation methods with the characteristic function ρ and among them K J the functors of the real method of interpolation A ρ,∞ and Aρ,1 are maximal and minimal, respectively. More precisely, if I is an exact interpolation functor with the characteristic function ρ, then we have 1 1  ⊂ K J ⊂ I(A) A A ρ,∞ ρ,1

(1.3)

 (see [31] or [59]). for every Banach couple A Recall that, for every quasi-concave function ρ and 1  q  ∞, the space of K the real K–method of interpolation A ρ,q consists of all a ∈ A0 + A1 such that ⎞1/q ⎛∞ q 

K(t, a; A , A ) dt 0 1 ⎠ < ∞ if q < ∞ aA K := ⎝ ρ,q ρ(t) t 0

and

K(t, a; A0 , A1 ) < ∞, ρ(t) where the Peetre K–functional is defined as follows: aA K

ρ,∞

:= sup

0 0, we get the second part of (1.3).

JAWERTH–MILMAN EXTRAPOLATION THEORY

11

 and an intermediate space A with respect to it, by A◦ Given a Banach couple A  in A. Moreover, let A˜ be the Gagliardo completion we denote the closure of Δ(A)   representable as Σ(A)-limits  of A in Σ(A), i.e., the set of all elements a ∈ Σ(A) of ˜ bounded sequences in A. Then, for every i = 0, 1, a ∈ Ai if and only if (1.6)

aA˜i := sup t>0

 K(t, a, A) 0, min(1, )J(s, u(s); A) s s 0 for some universal constant γ. Note that the converse inequality  ∞ t    ds , t > 0, K(t, a; A) min(1, )J(s, u(s); A) s s 0 ∞ ds for any representation a = 0 u(s) s , is an immediate consequence of the triangle inequality. Let us pass to the extrapolation theory. The main object here are families {Aθ }θ∈Θ of Banach spaces indexed by some set Θ. These spaces are required to be compatible, i.e., there is a Hausdorff topological vector space TA such that the 1 continuous inclusions Aθ ⊂ TA , θ ∈ Θ, hold. We write T : {Aθ }θ∈Θ → {Bθ }θ∈Θ if T is a linear continuous operator from TA into TB and its restriction to Aθ maps Aθ into Bθ with norm  1 for each θ ∈ Θ. We say that the Banach spaces A and B are extrapolation spaces with respect to the families {Aθ }θ∈Θ and {Bθ }θ∈Θ 1 of compatible spaces if the condition T : {Aθ }θ∈Θ → {Bθ }θ∈Θ implies that T : A → B. By an extrapolation method E we mean a functor defined on a collection Dom(E) of families of compatible spaces such that E({Aθ }θ∈Θ ) and E({Bθ }θ∈Θ ) are extrapolation spaces for all {Aθ }θ∈Θ and {Bθ }θ∈Θ from Dom(E) 1 . The simplest and simultaneously the most important extrapolation methods are the functors of the sum and the intersection of families of Banach spaces. Let {Aθ }θ∈Θ be a family of compatible spaces such that for some Banach space Σ the continuous inclusions Aθ ⊂ Σ, θ ∈ Θ, hold with uniformly bounded norms. Then we set     aθ in Σ : aθ ∈ Aθ and aθ Aθ < ∞ , Σ(Aθ ) := a = θ

with the norm aΣ(Aθ ) := inf



θ

aθ Aθ , where the infimum is taken over all ad-

θ

missible representations of a ∈ Σ(Aθ ). Note that the space Σ(Aθ ) does not depend on the ambient space Σ. Analogously, if the inclusions Δ ⊂ Aθ , θ ∈ Θ, are uniformly bounded for some Banach space Δ, we can define the intersection Δ(Aθ ) of 1 These definitions are rather flexible and so they seem to be more suitable than those given in [35, 36, 55], based on the more restrictive concept of a family of strongly compatible spaces.

12

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

the family {Aθ }θ∈Θ , which consists of all a ∈

 θ∈Θ

Aθ such that

aΔ(Aθ ) = sup aAθ < ∞. θ∈Θ

Jawerth–Milman’s approach to the problem of identification of extrapolation spaces is based largely on two key facts. At first, it turns out that the above Σand Δ-functors commute with appropriate interpolation functors.  be a Banach couple and {ρθ }01 1/ϕp,∞

28

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

(v) There exists a constant C > 0 such that 1/ϕp  C 1/ϕp,∞ ,

1 < p < ∞.

Proof. We first observe that from our hypothesis on ϕ and from Theorem 3.2 it follows that M (ϕ) = LL∞ (w2 ) . Therefore, by [10, Proposition 2.3] and [43, Lemma 2.1.4], we conclude that there exists a constant C > 0 such that  t t ds t  C , 0 < t  1. ϕ(t) ϕ(s) ϕ(t) 0 Thus, again by [43, Theorem 2.4.3], (3.6)

ϕ  p,∞ 1/ϕp,∞ and ϕ  p 1/ϕp , 1 < p < ∞.

Moreover, from Theorem 3.2, equivalence (3.6) and the inequality 1/ϕp,∞ 

t1/p , 0 < t  1, ϕ(t)

it follows that ϕ(t) sup

(3.7)

p>1

t1/p , 0 < t  1. 1/ϕp,∞

Implication (i) ⇒ (ii) is clear. (ii) ⇒ (iii). If M (ϕ) ∈ EF , then, by Theorem 3.1, we have M (ϕ) = LL∞ (w1 ) . Combining this together with the equality M (ϕ) = LL∞ (w2 ) , we infer that LL∞ (w1 ) = LL∞ (w2 ) . From (3.6) it follows that condition (iv) means equivalence of the fundamental functions of the spaces LL∞ (w1 ) and LL∞ (w2 ) . Hence, the implication (iii) ⇒ (iv) is obvious. (iv) ⇒ (v). We observe that, by H¨ older inequality, the function M1 given by M1 (s) = log 1/ϕ1/s , 0 < s < 1, is convex on (0, 1). Moreover, clearly,   1 t1/p   = exp sup log t − log ϕ(t) , 1 < p < ∞, ϕ  p,∞ = sup 0 0, and let the function ϕ be defined by relation (3.2). Then, one can easily check that ω is tempered at ∞ if and only if ψ ∈ Δ2 , which means that ψ(t)  C  ψ(t2 ), 0 < t  1. Furthermore, in this case the functions ψ and ϕ are equivalent on [0, 1] and M (ϕ) ∈ EF [7, Theorem 2]. The next theorem implies that formula (3.1) holds for every weight of the form p → ψ(e−p ) defined on (1, ∞), where ψ is an increasing function on [0, 1] satisfying (3.8). Moreover, we prove that the same assertion holds even for much faster ”falling p down” at infinity weights ω of the form ω(p) = ψ(e−e ), 1 < p < ∞. Theorem 3.5 ([10, Theorem 3.1]). Let ω(p) = ψ(e−p ) or ω(p) = ψ(e−e ) for some increasing function ψ on [0, 1] satisfying condition (3.8) and let the function ϕ be defined by (3.2). Then extrapolation formula (3.1) holds. Moreover, the Marcinkiewicz space M (ϕ) coincides with the Orlicz space LM generated by the function M (u) := supp>1 ω(p)p up , u  0. p

30

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

Proof. We consider only the case when ω(p) = ψ(e−e ), because for a weight of the form ω(p) = ψ(e−p ) the proof can be carried on by the same lines. At first, for p ∈ [log k, log(k + 1)], k  2, we have p

ψ(e−e ) xp  ψ(e−k ) xlog(k+1)  Cψ(e−(k+1) ) xlog(k+1) , p

which yields (3.9)

sup ψ(e−e ) xp p

p>1

sup

ψ(e−k ) xlog k .

k3, k∈N

Further, it is clear that LL∞ (ω) is an r.i. space on [0, 1] with the fundamental function equivalent to ϕ. Since M (ϕ) is the largest space among of all r.i. spaces with the fundamental function ϕ [43, Theorem 2.5.7], we obtain the continuous embedding LL∞ (ω) ⊂ M (ϕ). So, to complete the proof of (3.1) it is enough to show that the reverse embedding holds. Since 1 xM (ϕ) , t ∈ (0, 1], x∗ (t)  ϕ(t) for every x ∈ M (ϕ), the latter embedding is an immediate consequence of the fact that 1/ϕ ∈ LL∞ (ω) , or, equivalently, by (3.9), that sup ψ(e−k ) 1/ϕlog k < ∞.

(3.10)

k3

Let us prove (3.10). First, from the definition of ϕ it follows that 1 1 1 ϕ(t)  ψ(e−(k+1) )t log(k+1)  ψ(e−k )t log(k+1) , C whence 1 ψ(e−k )  Ct− log(k+1) ϕ(t) for all k  3 and 0 < t  1. Hence,

 1  1  log1 k log k log(k + 1) log k ψ(e−k ) 1/ϕlog k  C t− log(k+1) dt  C sup < ∞, k3,N log(1 + 1/k) 0

and (3.1) (together with (3.10)) is proved. We now prove that the space LL∞ (ω) coincides with the Orlicz space LM . 1 Observe, at first, that M (|x(t)|) dt  1, provided xLM  1. Then, using the 0

definition of M , we deduce that 1 ω(p)

|x(t)|p dt  1,

p

1 < p < ∞.

0

Hence, it follows that x ∈ LL∞ (ω) and xLL∞ (ω)  1. Conversely, let xLL∞ (ω)  1. Without loss of generality, we may assume that ψ(1) = 1. By this assumption, C −k  ψ(e−k )  1 and hence for every k  3 ψ(e−k )log(k−1)

= ψ(e−k )− log k−1 ψ(e−k )log k k

 C k log k−1 ψ(e−k )log k  C 3/2 ψ(e−k )log k . k

JAWERTH–MILMAN EXTRAPOLATION THEORY

31

Therefore, we have M (u) = sup ψ(e−e )p up  ψ(e−e )u + sup C 3/2+log(k−1) ψ(e−k )log k ulog k p

p>1

k3

 ψ(e−e )u +

∞ 

C 4 log k ψ(e−k )log k ulog k ,

k=3

and so  1

|x(t)| M 4 4 dt e C

−e

 ψ(e

0

1 )

∞

 |x(t)| dt + C 4 log k ψ(e−k )log k 4 4 e C k=3

0 ∞



1

|x(t)| e4 C 4

log k dt

0

 1 1 + < 1. 4 e k4 k=3

This shows that xLM  e C 4 .



4

Now, we identify the class of quasi-concave functions ϕ such that M (ϕ) ∈ EF \EF . Proposition 3.6. Let γ > 0 and let N : (0, ∞) → (0, ∞) be an Orlicz function such that N (t) = t(log t)γ for all t  tγ , where tγ is large enough. Suppose that ϕ is defined by ϕ(t) = exp(−N −1 (log(1/t))), 0 < t  1, where N −1 is the inverse function of N . (i) If γ  1, then M (ϕ) ∈ EF and in addition xM (ϕ) xLM sup e−

N ∗ (p) p

p>1

xp ,

where LM is the Orlicz space, generated by the function M (u) = exp(N (log u)) for large u > 0, and N ∗ (p) := sups>0 {ps − N (s)}. (ii) If 0 < γ < 1, then M (ϕ) ∈ EF \EF . Proof. For the proof of part (i) see [9, Example 4.14]. To prove part (ii) we first observe that t1/p , 01 ); so, for X ∈ SEF the corresponding extrapolation parameter F is explicitly determined by X. More precisely, the norm of any function in every strong extrapolation r.i. space X is equivalent to the norm in X of the function t ∈ [0, 1] → xlog(e/t) , i.e., xX  xlog(e/t) X . Let us emphasize that the class SEF is rather wide. In particular, according to (3.3), all Zygmund spaces Exp Lβ , with β > 0, belong to SEF . Moreover, applying Corollary 2.6, one can verify that the same result holds also for an exponential Orlicz space Exp LN provided that N is an Orlicz function. Finally, a Marcinkiewicz space M(ϕ) (a Lorentz space Λ(ϕ)) belongs to the class SEF if and only if ϕ(t) ϕ(t2 ) [8, Theorem 2.10]. In [9] (see also [8]), by using the operator S2 x(t) = x(t2 ) and the following generalization of Jawerth-Milman’s idea of a tempered weight, some characterizations of strong extrapolation spaces have been found. Definition 4.2. A parameter F of an extrapolation F-method is tempered if the operator Df (p) := f (2p) is bounded in F . Here, we complement the above results from [9] in the following way. Theorem 4.3. For any r.i. space X on [0, 1] the following conditions are equivalent: (1) X ∈ SE; (2) the operator S2 x(t) = x(t2 ) is bounded in X;

34

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

(3) X = LF with some tempered extrapolation parameter F ; (4) with constants independent of x ∈ X and t > 0 we have K(t, x; L∞ , X) K(t, xlog(e/·) ; L∞ , X),

(4.1)

where xlog(e/s) denotes the Lp -norm of a function x with p = log(e/s); (5) there are 1  p = q < ∞ and an intermediate between L∞ and L∞ (1/t) Banach lattice G on [0, ∞) such that K X = (Lp , L∞ )K G = (Lq , L∞ )G ;

(6) there is an intermediate between L∞ and L∞ (1/t) Banach lattice G on [0, ∞) such that for every p  1 X = (Lp , L∞ )K G; (7) there is an intermediate between L∞ and L∞ (1/t) Banach lattice G on [0, ∞) such that the operator T f (t) := f (t2 )/t is bounded on G and X = (L1 , L∞ )K G. Proof. Equivalence of conditions (1), (2), (3) and (4) is proved in [9, Theorems 2.2 and 3.1(i)]. Since the implication (6)⇒(5) is obvious, it suffices only to prove the implications (2)⇒(6), (5)⇒(2), (2)⇒(7) and (7)⇒(5). (2)⇒(6). Let p > 1 be arbitrary. Since Lr is an exact interpolation space with respect to the couple (L1 , L∞ ) for each 1  r  ∞, from Calderon-Mityagin theorem (see, for instance, [43, Theorem 2.4.3]) it follows easily that every X from the class SEF is an interpolation space between L1 and L∞ . So, recalling that conditions (1) and (2) are equivalent, by Brudnyi-Krugljak theorem [16, Theorem 4.4.5], we have X = (L1 , L∞ )K G (with equivalent norms), for some intermediate Banach lattice G between L∞ and L∞ (1/t). Moreover, one can easily check that the operator Sp x(t) := x(tp ) is bounded in X if and only if S2 is. So, the estimate t K(t, Sp x; L1 , L∞ ) = 0

x∗ (sp ) ds =

tp

x∗ (s) d(s1/p )

= K(t, x; Lp,1 , L∞ )

0

 K(t, x; Lp , L∞ ) implies xX  Cx(L1 ,L∞ )K  Cx(Lp ,L∞ )K  CSp x(L1 ,L∞ )K  CSp X→X xX , G

G

G

and, as a result, we obtain (6). (5)⇒(2). Let, say, 1  q < p < ∞. Then, it can be easily verified that the above operator Sr is bounded from Lp into Lq provided that 1 < r < p/q. Hence, by interpolation argument and the hypothesis, Sr acts boundedly in X, and so does S2 . (2)⇒(7). Denote by G the space of all measurable functions f on (0, ∞) such that f G := f (t)/t · χ[0,1] X + f · χ[1,∞) L < ∞. ∞

Now, all required conditions for G follow from the fact that X is an interpolation space between Lp and L∞ for each 1  p < ∞ and from the boundedness of t Hardy-Littlewood operator x → 1t 0 x(s) ds on Lp if 1 < p  ∞.

JAWERTH–MILMAN EXTRAPOLATION THEORY

35

(7)⇒(5). Let us show that condition (5) holds for q = 1 and p = 4/3. Clearly, we need only to prove that X ⊂ (L4/3 , L∞ )K G. n 2n 2n Noting that T f (t) = t · f (t )/t , for each 0 < t  1/2, we have t 

4/3

K(t, x; L4/3 , L∞ ) 

K(t, x; L4/3,1 , L∞ ) =

x∗ (s) ds3/4

0

t2 

∗

x (s) ds + s1/4

0



∞ 

t−2

n=1



∞ 

t2

n−1



t2 ∞  

t2

n=1 2n+1 t

x∗ (s) ds3/4 

n

1 x∗ (s) ds + t

t2

x∗ (s) ds + tx∗ (t2 ) s1/4

x∗ (s) ds

0

t2n+1 2n

t

2n−1 −1

n=1 ∞ 

n

t4/3 

·

t

t

t2n

x∗ (s) ds + T (K(t, x; L1 , L∞ ))

0 n−1

21−2

T n (K(t, x; L1 , L∞ )) + T (K(t, x; L1 , L∞ )).

n=1

Consequently,

∞    n−1   K(t, x; L4/3 , L∞ )χ[0,1/2] G   22−2 T n (K(t, x; L1 , L∞ ))   n=1



∞ 

n−1

22−2

G

n

T G→G K(t, x; L1 , L∞ )G  CxX .

n=1

On the other hand, it is clear that K(t, x; L4/3 , L∞ )G  C  K(t, x; L4/3 , L∞ ) · χ[0,1/2] G . with some constant C  independent of x ∈ L4/3 , and the result follows.



More detailed information on the class SEF may be found in the papers [8] and [9]. We mention here only that, by using Theorem 4.3, one can extend Corollaries 2.10 and 2.11 to the case of any increasing function Φ satisfying the inequality 2Φ(u)  Φ(Cu) for some constant C and all large enough u > 0. Indeed, the latter condition ensures that Exp LΦ ∈ SEF , and so applying equivalence (4.1) we come to desired result. In particular, the above corollaries hold for Zygmund spaces Exp Lβ with every β > 0 (see [5]). We omit details. 5. Probabilistic moment problem As we have seen in the preceding sections, the extrapolation theory, applied to the scale of Lp -spaces, allows to determine whether a random variable with the given Lp -norms belongs to some limiting space with respect to this scale. Due to that we obtain some information concerning the distribution of the given random variable, which can be also useful from probabilistic point of view (see, for instance, the book [57], where extrapolation estimates are applied to studying random fields and some problems of mathematical statistics). The aim of this section is to highlight

36

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

some recent results, which are related to the classical power moment problem and were obtained by using extrapolation techniques. We shall restrict ourselves to consideration of measurable functions on [0, 1] equipped with the Lebesgue measure μ; however, all results of this section can be easily extended also to random variables defined on an arbitrary non-atomic  probability space. Denote by L the linear space p τ } = μ{t ∈ [0, 1] : y(t) > τ }

for all τ ∈ R.

Otherwise, we say that the Hamburger moment problem is indeterminate. It is well known that the sets D and L\D are non-empty. For instance, if g is the standard Gaussian random variable, i.e., ∞ u2 1 e− 2 du, μ{t ∈ [0, 1] : g(t) > τ } = √ 2π τ

then g ∈ D but g3 ∈ L\D [13]. Furthermore, the Hamburger moment problem is determinate for every function x satisfying the following Cramer condition: 1 eε|x(t)| dt < ∞ for some ε > 0 0

or, equivalently, if x ∈ ExpL. One can easily check also that the latter condition may be restated by using Lp −norms (moments) of x: sup p∈N

xp p

0, can be decomposed into a sum of two disjointly supported functions, which correspond, after renorming, to densities of random variables satisfying the Carleman condition. So, the random variable with the density p, having indeterminate moment problem, may be represented as a sum of two disjointly supported random variables, either of which has determinate moment problem. An immediate consequence of this result is the fact that the class D is not a linear space. Below we give a different proof of the latter assertion, which additionally provides such an explicit representation of some (other than in [63]) random variable, having indeterminate moment problem. Theorem 5.1. There exist functions x, y ∈ C such that the sum x + y does not belong to the class D.

38

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

Proof. At first, we observe that the Stieltjes moment problem and so the Hamburger one are indeterminate for the nonnegative decreasing function z(t) = log3 (1/t). In fact, μ{t ∈ [0, 1] : z(t) > τ } = e−τ

1/3

,

τ  0.

Therefore, the density of z is the function p(u) = 13 u− 3 e−u , which satisfies condition (5.3). ∞ Next, setting t0 = 1, define two sequences of reals {tk }∞ k=0 and {pk }k=1 : 1/3

2

2 ek

tk = e−e

k2

pk = z(tk )2 = log6 (1/tk ) = e6e ,

and

k ∈ N.

Let us show that for any k ∈ N we have tk+1

z(t)pk dt < 1. 0

In fact, it is easy to see that



log(1/t) = inf

β>0



β − β1 t e

, t > 0,

whence

β − β1 t e for all positive t and β. Applying the latter inequality for β > 3pk , we obtain the estimate tk+1 tk+1

3pk tk+1 3pk β z(t)pk dt = log3pk (1/t) dt  t− β dt e log(1/t) 

0

0

0

= Letting here β = log(1/tk+1 ) = ee Hence, tk+1

z(t)pk dt  e(e 0

(k+1)2

−1)·3pk

(k+1)2

−1

3pk

3p − k +1 β 3pk +1 · − · tk+1β . e β −1/β

, we get tk+1 = e and − 3pβk + 1 > 1/e.

  2 (k+1)2 · e · e3pk · tk+1 = exp 3pk e(k+1) + 1 − ee =

  k2 2 (k+1)2  1, = exp 3e6e +(k+1) + 1 − ee

and the desired inequality is proved. Now, we set ∞  x(t) = z(t) · χ(t2k+1 ,t2k ] (t) and

y(t) = z(t) ·

k=0

∞ 

χ(t2k+2 ,t2k+1 ] (t).

k=0

Then z = x + y and either of these functions satisfies the Carleman condition. We check the latter fact only for x; the case of y may be considered in the same way. If p  p2n+1 , we have xp  xp2n+1

 

z · χ(0,t2n+2 ] p

+ z · χ(t2n+1 ,1] p 2n+1 √ 1 + z(t2n+1 ) = 1 + p2n+1 . 2n+1

JAWERTH–MILMAN EXTRAPOLATION THEORY

39

Consequently, ∞ 1

dp  xp

p 2n+1

1

dp xp

p 2n+1



1+

dp p2n+1 − 1 = √ √ p2n+1 1 + p2n+1

1

(2n+1)2 √ = p2n+1 − 1 = e3e − 1.

Since this inequality holds for every n ∈ N, we have (5.1) and the proof is complete.  Remark 5.2. Among other results, in the paper [28], it is proved (see Proposition 3.10) the existence of positive logarithmic convex sequences of reals {an }∞ n=1 and {bn }∞ n=1 such that ∞  n=1

1 = ∞, √ n a n

∞  n=1

∞ 

1 √ = ∞, n bn

n=1

1 " < ∞. n max{an , bn }

This implies that the class of quasianalytic weights fails to be a linear space with respect to the usual sum of its elements. The latter result is also an immediate n n consequence of Theorem 5.1. Indeed, it suffices to put an = xn and bn = yn , n ∈ N, where the functions x and y are taken from Theorem 5.1. Remark 5.3. Similarly as in the proof of Theorem 5.1, by using the property of absolute continuity of integral, we can obtain the following more general result: every function x ∈ L is representable as a sum of two disjointly supported functions from the class C (and hence from D); see details in [52]. From probabilistic point of view, this means that an arbitrary distribution with all finite moments is a mixture of two ones with determinate Hamburger moment problem. Remark 5.4. Let us denote by Cα , with a fixed α > 0, the class of all functions x satisfying the condition  1 = ∞. xα p p∈N

In particular, the Stieltjes moment problem is determinate for each nonnegative function from the class C1/2 . At the same time, it makes sense to study the class Cα also for other values of α. For instance, if x, y ∈ C2 are arbitrary nonnegative functions, then the coincidence of their distributions follows from that of their moments of order 4k for all k ∈ N. Arguing as in the proof of Theorem 5.1, one can check that Cα + Cα = L for each α > 0. Though the class C fails to be a linear space (by Theorem 5.1), it can be represented as a certain union of (Banach) Orlicz spaces. Indeed, if y ∈ C, then every function x satisfying the condition sup p∈N

xp yp

0 and all t ∈ [0, 1]. Theorem 5.5. Let ϕ ∈ Δ2 . Either of the inclusions 1) M(ϕ) ⊂ C, 2) M(ϕ) ⊂ D, 3) M0 (ϕ) ⊂ C, 4) M0 (ϕ) ⊂ D, is equivalent to the condition 5) 1 dt = ∞. ϕ(t) t 0

Proof. It is easy to see that we need to prove only the implications 5) ⇒ 1) and 4) ⇒ 5). 5) ⇒ 1). Since ϕ ∈ Δ2 , then we have x

x

x

xM(ϕ) sup ϕ(t)x∗ (t) sup ϕ(e−p )xp sup ϕ(e−p )xp , p1

t∈(0,1]

p∈N

p

whence 1/ϕ ∈ M(ϕ) and 1/ϕp 1/ϕ(e−p ) (see [7, Theorem 2]). Therefore, the inclusion M(ϕ) ⊂ C holds if ∞ ϕ(e−p ) dp = ∞, 1

which is equivalent to the condition 5). 4) ⇒ 5). On the contrary, assume that 5) fails. Then there is an increasing continuous positive function ψ(t) on [0, 1], limt→0+ ϕ(t)/ψ(t) = 0, satisfying (similarly as ϕ) the condition 1 dt < ∞. ψ(t) t 0

Thus, there is a symmetrically distributed decreasing function, which both coincides with 1/ψ on some interval (0, δ), where δ > 0, and satisfies the Krein condition (5.2) [50, see the proof of Theorem 1]. Since additionally 1/ψ ∈ M0 (ϕ), then M0 (ϕ) ⊂ D, which contradicts 4). So, the implication 4) ⇒ 5) is also proved.  Let M (u) be an Orlicz function. Suppose that there exists C > 0 such that (5.4)

M (u)2  M (Cu)

for all sufficiently large u. Then one can easily check that the Orlicz space LM coincides with the Marcinkiewicz space M(ϕ), where ϕ(t) := 1/M −1 (1/t) ∈ Δ2 . Then, Theorem 5.5 implies the following result (see [50, Theorem 2]; to be compared with [27, Theorem 1.1]).

JAWERTH–MILMAN EXTRAPOLATION THEORY

41

Theorem 5.6. If an Orlicz function M (u) satisfies condition (5.4), then the following conditions are equivalent: ∞  M (u) du = ∞. (1)LM ⊂ C, (2)LM ⊂ D and (3) M (u) u 1

In particular, any function M (u) such that u

M (u) ∼ e (log u)·(log log u)·...·(log... log u) , with arbitrary multiplicity of logarithm, satisfies inequality (5.4). Thus, applying Theorem 5.6, we obtain the following corollary (see also the paper [27], where a similar result related to the multidimensional moment problem was proved in a different way). Corollary 5.7. Let a random variable ξ be such that, for sufficiently large C > 0, the random variable ξ η= log(|ξ| + C) · log log(|ξ| + C) · . . . · log log . . . log(|ξ| + C) satisfies the Cramer condition. Then the Hamburger moment problem is determinate for ξ. The next corollary refines the results of the paper [13] related to the moment problem for powers of the standard Gaussian random variable g. Corollary 5.8. Let a > 0, bi ∈ R, i = 1, 2, . . . , n, and let C ∈ R be such that n−tuple logarithm log log . . . log C is well-defined and positive. Then the Hamburger moment problem is determinate for the random variable η = sign(g) · |g|a · (log(|g| + C))b1 · (log log(|g| + C))b2 . . . (log log . . . log(|g| + C))bn if and only if one of the following n + 1 conditions holds: 1) a ∈ (0, 2); 2) a = 2, b1 < 1; ... n) a = 2, b1 = . . . = bn−2 = 1, bn−1 < 1; n+1) a = 2, b1 = . . . = bn−1 = 1, bn  1. Remark 5.9. It is worthwhile to note that the conditions of determination of the (Hamburger!) moment problem for the random variable ζ = |g|a · (log(|g| + C))b1 · (log log(|g| + C))b2 . . . (log log . . . log(|g| + C))bn , which is equal to modulus of the random variable η from Corollary 5.8, are completely different, namely, 1) a ∈ (0, 4); 2) a = 4, b1 < 2; ... n) a = 4, b1 = . . . = bn−2 = 2, bn−1 < 2; n+1) a = 4, b1 = . . . = bn−1 = 2, bn  2. The latter result is a consequence of two following facts: a) the condition ∞  M (u) du √ =∞ M (u) u 1

42

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

ensures that the Stieltjes moment problem is determinate for a random variable from the Orlicz space LM [50, Theorem 4]; b) if the Stieltjes moment problem is determinate for a positive random variable, then so is the Hamburger moment problem [30, Theorem A]. Let ϕ be an increasing concave function on [0, 1] and r  1. Denote by Λr (ϕ) the Lorentz space with the norm ⎛ 1 ⎞ r1  xΛr (ϕ) = ⎝ x∗ (t)r dϕ(t)⎠ . 0

Theorem 5.10. Let r  1 and let a concave function ϕ ∈ Δ2 be representable in the form ∞ ϕ(t) = g(s)r+1 ds, log 1/t

where the nonnegative function g(s) satisfies the condition ∞ g(s) ds = +∞. 1

Then Λr (ϕ) ⊂ C. Proof. Since ϕ ∈ Δ2 , then by [7, Theorem 3], we have ⎛∞ ⎞ r1  x xΛr (ϕ) ⎝ xrp e−p ϕ (e−p ) dp⎠ . 1

Moreover, from the hypothesis of the theorem it follows that tϕ (t) = g(log 1/t)r+1 and so for every x ∈ Λr (ϕ) ∞ ∞ r r+1 xp g(p) dp = xrp e−p ϕ (e−p ) dp < ∞. 1

1

On the other hand, applying H¨ older inequality, we obtain ∞ ∞ r r r+1 g(p) dp = xpr+1 g(p) · x− dp p 1

1

1 r ⎛∞ ⎞ r+1 ⎞ r+1 ⎛∞   dp ⎠  ⎝ xrp g(p)r+1 dp⎠ ·⎝ , xp

1

1

and from divergence of the integral from the left-hand side of this inequality it follows that the Carleman condition (5.1) holds for all x ∈ Λr (ϕ).  Corollary 5.11. Let ϕα (t) log−1 (e/t) logα (log ee /t). The following conditions are equivalent: 1) Λ(ϕα ) ⊂ C; 2) Λ(ϕα ) ⊂ D; 3) α  −2.

JAWERTH–MILMAN EXTRAPOLATION THEORY

43

Proof. Noting that the function ϕ(t) = log−1 (ee /t) log−2 (log ee /t) satisfies the conditions of Theorem 5.10 with the function # 1 + 2 log−1 (p + e) g(s) = , (p + e) log(p + e) we infer that for α  −2

Λ(ϕα ) ⊂ Λ(ϕ) ⊂ C ⊂ D. Next, assuming that α < −2, let us represent α in the form α = −2 − β, β > 0. Then, one can check that the function x(t) = x∗ (t) = log(ee /t) log1+β/2 log(ee /t) belongs to the Lorentz space Λ(ϕα ). Indeed, 1 xΛ(ϕa )

  log(ee /t) log1+β/2 (log ee /t) d log−1 (ee /t) logα (log ee /t) =

0

∞ =−

u 1+β/2

e u

  d e−u u−2−β =

1

∞   u−1−β/2 + (2 + β)u−2−β/2 du < ∞. 1

Therefore, the embedding M(1/x) ⊂ Λ(ϕ) holds (M(1/x) is the Marcinkiewicz space generated by the function 1/x). Moreover, since 1

1 dt = x(t) t

1

0

0

dt t log(ee /t) log1+β/2

log(ee /t)

< ∞,

the fundamental function 1/x of the Marcinkiewicz space M(1/x) does not satisfy the condition 5) of Theorem 5.5. As a result we see that in the case when α < −2 the space Λ(ϕα ) contains a function, for which the moment problem is indeterminate.  Remark 5.12. From the proof of Corollary 5.11 it follows that Λ(ϕ) ⊂ C, where ϕ(t) = log−1 (ee /t) log−2 (log ee /t). It is perhaps noteworthy to emphasize that this result is not covered by Theorem 5.5, because there is no Marcinkiewicz space M(ψ) such that Λ(ϕ) ⊂ M(ψ) ⊂ D. Indeed, if Λ(ϕ) ⊂ M(ψ), then ψ(t)  Cϕ(t) with some constant C > 0 independent of t ∈ [0, 1]. Hence, 1 ψ(t) 0

dt C t

1

log−1 (ee /t) log−2 (log ee /t)

dt < ∞, t

0

and combining this together with Theorem 5.5 we infer that there is x ∈ M(ψ) \ D, which is a contradiction. Thus, Theorem 5.10 gives us new examples of r.i. spaces from the classes C and D. This is a classical result (see [1, the proof of the Theorem 5.5.4, p. 212] or [44, Theorem 1]) that the coincidence of Lp -norms of two nonnegative functions for all p ∈ [p0 , p1 ], 0 < p0 < p1 , implies the coincidence of their distributions. At the same time, the fact of existence of random variables with indeterminate moment problem indicates that the knowledge of all integer moments of a random variable, generally, is not sufficient to restore its distribution. Moreover, it turns out that the

44

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

above knowledge does not allow even to control half-integer Lp −norms. Namely, in [49, Proposition 1], the following result is proved. Theorem 5.13. For arbitrary sequence {Cj }∞ j=1 of positive numbers there exist nonnegative functions x and y such that both xj = yj < ∞ and xj+1/2  Cj yj+1/2 for all j ∈ N. Furthermore, the estimates for Lp −norms of functions generally do not imply these not only for their distributions but even for their K−functionals in the couple (L1 , L∞ ). Indeed, applying Proposition 3.6 from Section 3 on the existence of quasi-concave functions ϕ such that M (ϕ) ∈ EF \EF , we get the following result (see [49, Lemma 6]). Theorem 5.14. There are two non-increasing nonnegative functions x and y on (0, 1] such that q

q

q

yq  xq  2yq < ∞ for all q > 0 and

t lim sup t→0+

0 t

x(s) ds = +∞. y(s) ds

0

Since

⎛ tp ⎞ p1  t,x K(t, x; Lp , L∞ ) ⎝ (x∗ (s))p ds⎠ , 0

(see, for instance, [14, Theorem 5.2.1]), from Theorem 5.14 it follows Corollary 5.15. For every p > 0 there exist functions x and y such that q q q yq  xq  2yq < ∞ for all q > 0 and lim sup t→0+

K(t, x; Lp , L∞ ) = +∞. K(t, y; Lp , L∞ )

It is well known a crucial role in interpolation theory played by the property of K-divisibility of the K-functional of a Banach couple [16, Ch. 3]. The following result shows, however, that Lp -norms of a measurable function generally fail to have analogous property. Corollary 5.16. For every p  1 there are non-increasing nonnegative functions x and yk , k ∈ N, satisfying the conditions: for all k ∈ N;  1) yk ∈ L∞  ∞ ∞ 2) xq   k=1 yk q  k=1 yk q < ∞ for all q ∈ [1, ∞); ∞ 3) for arbitrary representation x = k=1 xk (with convergence in measure) we have sup k∈N,q∈[p,∞)

xk q yk q

= ∞.

Proof. According to Corollary 5.15, we can find functions x = x∗ and y = y ∗ such that xq  yq if q  1 and (5.5)

lim sup t→0+

K(t, x; Lp , L∞ ) = +∞. K(t, y; Lp , L∞ )

JAWERTH–MILMAN EXTRAPOLATION THEORY

45

Let {tj }∞ j=1 be a decreasing sequence of reals from the interval (0, 1) tending to zero. Denoting y˜0 := y, we introduce two sequences of functions on [0, 1]: y˜j (t) := (y(t) − y(tpj )) · χ(0,tpj ) (t) and

yj (t) := y˜j−1 (t) − y˜j (t), j ∈ N.

Clearly, we have ∞ 

y˜j =

yk , j = 0, 1, 2, . . .

k=j+1

and (5.6) ˜ yj p + tj · y − y˜j ∞

⎛ p ⎞ p1 ⎛ p ⎞ p1 tj tj ⎜ ⎟ ⎜ ⎟  ⎝ (y(s))p ds⎠ + tj · y(tpj )  2 ⎝ (y(s))p ds⎠ . 0

0

Using (5.5), we can choose a sequence {tj }∞ j=1 , satisfying the above conditions, in such a way that ⎛ p ⎞ p1 tj ⎜ ⎟ (5.7) K(tj , x; Lp , L∞ )  j · ⎝ (y(s))p ds⎠ 0

and 2˜ yk q  yk q

for q = p, 2p, . . . , kp, k ∈ N.

From the latter estimate it follows that, for q = np, n ∈ N and all k  n, we have yk+1 q  yk q /2, which yields for all j  n − 1 (5.8)

∞ 

yk q 

k=j+1

∞ 

2j+1−k · yk q = 2yj+1 q  2˜ yj q  2yq < ∞.

k=j+1

Therefore, the sequence {yk }∞ k=1 satisfies the condition 2). Since 1) also holds (in view of the definition of {yk }∞ k=1 ), it remains only to prove  3). Assuming the contrary, find a representation x = ∞ k=1 xk and a constant C such that for all q  p and k ∈ N we have xk q  Cyk q . Then, applying inequalities (5.8), with q = p, and (5.6), we obtain     j    j ∞      ∞     + tj ·  K(tj , x; Lp , L∞ )   x x  x  + t · xk ∞  k k k j p     k=j+1  k=1 k=j+1 k=1 ∞ p ⎞ ⎛ j ∞    C⎝ yk p + tj yk ∞ ⎠ k=j+1



k=1

⎞ p1 ⎛ p tj   ⎟ ⎜ 2C ˜ yj p + tj y − y˜j ∞  4C ⎝ (y(s))p ds⎠ . 0

Since the latter inequality contradicts (5.7), the result follows.



46

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

6. Extrapolation of operators with a quasi-Banach target space Here, we consider operators acting from a scale of Banach spaces {Xθ }θ∈Θ into a fixed quasi-Banach space Y . In the case when Y is a Banach space and the norms of an operator T : Xθ → Y are uniformly bounded from the triangle inequality in Y  it follows easily, under some natural conditions, that T is bounded from the sum θ Xθ into Y . But if the triangle inequality fails in Y (say, it holds with some constant larger than 1), this argument generally does not work. However, as we shall see, a little bit more accurate application of the Σ-method of extrapolation still allows us to extend the boundedness of an operator T : Xθ → Y , with a quasiBanach Y , beyond the scale {Xθ }θ∈Θ (a more detailed exposition of results of this section see in [51]). A different approach to extrapolation of operators with values in quasi-normed Marcinkiewicz spaces, is presented in [19]. A quasi-norm on a real vector space Y is a mapping y → yY from Y into R such that for some K < ∞ we have (1) yY > 0 if 0 = y ∈ Y ; (2) λyY = |λ|yY , λ ∈ R, y ∈ Y ; (3) y1 + y2 Y  K(y1 Y +y2 Y ), y1 , y2 ∈ Y . A space Y equipped with a quasi-norm (which induces a locally bounded topology on Y ) is a quasinormed space. Throughout this section the letter K  1 will mean the modulus of concavity of the quasi-norm in Y given by the generalized triangle inequality in the above definition. A complete quasi-normed space is called a quasi-Banach space. An example of a quasi-Banach (and not Banach) space is Lp [0, 1], 0 < p < 1 with  1/p 1 (in this case for K we can take 2(1−p)/p ). the quasi-norm xp := 0 |x(t)|p dt Clearly, for every quasi-normed space Y and arbitrary n ∈ N we have     n−1 n     n j n−1    y K y  + K y   K j yj Y . j j n Y Y    j=1  j=1 j=1 Y

Moreover, it is not difficult to prove the following result on convergence of series in quasi-Banach spaces (see [51, Lemma 2] or [54, Theorem 1.1]).  Lemma 6.1. Let Y be a quasi-Banach space. If y = ∞ j=1 yj (convergence in Y ), then we have ∞  K j yj Y . yY  j=1

Conversely, from the condition ∞ 

K j yj Y < ∞

j=1

it follows the existence of an element y ∈ Y such that y =

∞

j=1 yj .

We say that an operator T acting from X into Y , where X and Y are quasinormed spaces, is bounded if its quasi-norm T xY T X→Y := sup x=0 xX is finite, and, in addition, T 0 = 0. Let S be the set of all measurable functions f : [0, 1] → R. By S we denote the set of all measurable functions f : [0, 1] → R ∪ {±∞} and by S + the subset of S consisting of all nonnegative functions. Observe that in S + the pointwise

JAWERTH–MILMAN EXTRAPOLATION THEORY

47

addition of elements is well-defined. The following definition is largely inspired by the content of Yano’s theorem. Definition 6.2. An operator T defined on a Banach space X and taking its values in S is said to be sublinear if for some B > 0 and an arbitrary expansion ∞ x = j=1 xj , where the series converges in X, we have |T x(t)|  B

∞ 

|T xj (t)| a.e. on [0, 1].

j=1

In this section, we focus on results related to sublinear operators acting into S. However, similar results hold also for sublinear operators with a range contained into an arbitrary lattice and, under some natural conditions, for linear operators acting into an arbitrary quasi-Banach space (see [51]). Theorem 6.3. Let Y ⊂ S be a quasi-Banach lattice and let Xj , j ∈ N, be Banach spaces continuously embedded into a Banach space A with xA  K j xXj

for all x ∈ Xj and j ∈ N.

Moreover, suppose that T is a sublinear operator defined on the space X :=

∞ 

K j Xj

j=1

and T xY  xXj for all x ∈ Xj and j ∈ N. Then T is bounded from X into Y and T X→Y  B, where B is the constant from definition 6.2. Proof. At first, we observe that the space X is well-defined. Moreover, by definition of the Σ−functor, for every x ∈ X and arbitrary ε > 0 there exist elements xj ∈ Xj , j ∈ N, such that x=

∞ 

xj (convergence in A) and

j=1

∞

∞ 

K j xj Xj < xX + ε,

j=1

which implies that the series j=1 xj converges not only in A but in X also. Therefore, by the sublinearity of T , the lattice property of Y and Lemma 6.1, we have      ∞ ∞    ∞  j  |T xj |  B K T x   B K j xj Xj  B(xX + ε). T xY  B j Y   j=1  j=1 j=1 Y

Since ε > 0 is arbitrary, the result follows.



Theorem 6.3, applied to the scale of Lp,1 -spaces, and the identification of its limiting spaces give the following extension of Yano’s theorem to the case of operators with a quasi-Banach target space. Theorem 6.4. Let α > 0 and let Y ⊂ S be a quasi-Banach lattice with the modulus of concavity of the quasi-norm K. Suppose that T is a sublinear operator defined on the Lorentz space Λ(ψ) on [0, 1] with " ψ(t) t logα (e/t) exp(2 (α + ε) log K · log log(e/t)), where ε > 0.

48

SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

Moreover, let

T Lp,1 →Y  C

(6.1)

p p−1

α ,

for some C > 0 and all p > 1. Then T acts boundedly from the space Λ(ψ) into Y . Sketch of the proof. (see details in [51, Theorem 5]). Assume that {Λ(ψθ )}θ∈Θ is an arbitrary family of Lorentz spaces uniformly embedded into L1 [0, 1]. Then, since a Lorentz space is the smallest one in the class of r.i. spaces with the same fundamental function, we get the following isometric equation  Λ(ψθ ) = Λ(ψ), θ∈Θ

where ψ(t) = inf θ∈Θ ψθ (t) (see also [51, Lemma 4]). Theorem 6.3, we deduce that the Now, denoting Cp := C(p/(p − 1))α , by ∞ operator T is boundedly acts from the space j=1 K j Cpj Lpj ,1 into Y for arbitrary j j sequence {pj }∞ j=1 . Setting pj = a /(a − 1), a > K, after some calculation we obtain ∞  K j Cpj Lpj ,1 = Λ(ϕa ), where ϕa (t) = aα t logα+loga K (eα+loga K /t). j=1

Observe that the embeddings Λ(ϕa ) ⊂

∞ 

K j Cpj Lpj ,1

j=1

hold with some constant independent of a > K. Hence, T xY  CxΛ(ϕa ) . Therefore, applying Theorem 6.3 once more, we conclude that for every sequence ∞ j {aj }∞ j=1 the operator T is bounded from the sum j=1 K Λ(ϕaj ) into Y . Finally, taking aj = K uj , u > 1, we infer that T is bounded from the Lorentz space Λ(ψ) into Y , where 1

1

inf K j+αuj t logα+ uj (eα+ uj /t) j∈N " t logα (e/t) exp(2 (α + 1/u) log K log log(e/t)).

ψ(t) =

 A short comment seems to be in order. There is the essential difference in extrapolation of operators with Banach and quasi-Banach target spaces. Indeed, assume that an operator T is bounded from every space Lp,1 , with p > 1, into Y with the norm estimate as in Theorem 6.4. Then, passing to the infimum, it is easy to deduce that T (λχA )Y  C|λ|χA Λ(ϕ) ,

where ϕ(t) = t logα (eα /t),

for each characteristic function χA , A ⊂ [0, 1], and all λ ∈ R. Now, if Y is a Banach space and T satisfies some mild conditions, from the latter estimate it follows that T is bounded from the whole space Λ(ϕ) into Y [43, Lemma II.5.2]. In contrast to that we cannot draw such a conclusion, in general, in the quasi-Banach case.

JAWERTH–MILMAN EXTRAPOLATION THEORY

49

Theorem 6.5 ([51, Theorem 11]). Let ϕ be an increasing continuous concave function on [0, 1], ϕ(0) = 0, such that Λ(ϕ) = L1 , and let Y ⊂ S be a quasi-Banach lattice which fails to be a normed space. Then there is an operator T defined on L1 with values in S + satisfying the following conditions: 1) from |x|  |y| ∈ L1 it follows T x  T y; 2) T (λx) = |λ|T x for all x ∈ L1 and λ ∈ R; 3) ∞ ∞   if x = xi (convergence in L1 ), then T x  T xi ; i=1

4)

i=1

T χA Y < ∞; A⊂[0,1] ϕ(μA) sup

5) sup x∈P

T xY = ∞, xΛ(ϕ)

where P is the set of all measurable finite-valued functions on [0, 1]. By some additional assumptions with respect to a quasi-Banach space Y , the extrapolation domain of the operator T from Theorem 6.4 can be essentially extended. The following important concept was introduced by Kalton in the paper [37] (see Definition 3.2 and Theorem 3.6). Definition 6.6. A quasi-Banach space Y is called logconvex if there is a constant C > 0 such that for all yj ∈ Y , j ∈ N, we have   ∞  ∞      y  C (1 + log j)yj Y j  j=1  j=1 Y

An example of such a space is L1,∞ (i.e., weak L1 ) with the quasi-norm x1,∞ := sup tx∗ (t) = sup τ μ{t : |x(t)| > τ } t∈(0,1]

τ >0

(for the proof see [37, Theorem 3.4] or [68, Lemma 2.3]). Theorem 6.7 ([51, Theorem 9]). Suppose that Y ⊂ S is a logconvex quasiBanach lattice and T is a sublinear operator defined on the Lorentz space Λ(ψ) with t ψ(t) t logα (b/t) log log log(b/t), where b > ee . Then, if T acts boundedly from Lp,1 into Y for all p > 1 with norm estimate (6.1), then T is bounded from Λ(ψ) into Y . Finally, we survey briefly some other results related to the latter theorem. According to Theorem 5.7.1 from [35], the assumption

α 1 p p xp,1 , p > 1 (6.2) sup τ · μ {t : |T x(t)| > τ }  C p−1 τ >0 (which is stronger than (6.1) in the case when Y = L1,∞ ) implies the boundedness of T from the smaller space L logα log log L into L1,∞ . Later on, in [19], Carro and Martin strengthened this result, replacing “double logarithm” with “triple logarithm” as in Theorem 6.7. There is also a series of theorems, in which the

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SERGEY V. ASTASHKIN AND KONSTANTIN V. LYKOV

latter result was proved by using (6.1) or (6.2) only for characteristic functions [2–4, 17, 18, 21, 23]. However, in all above cited papers either the stronger estimate (6.2) was used or only operators with some specific properties were considered (for instance, see [18, Corollary 3.2], where this result is proved for the so-called (ε, δ)-atomic approximable operators). In contrast to that, in Theorem 6.7, we do not impose any extra conditions on T and exploit only estimate (6.1). A somewhat close result is stated at the end of the paper [22]; however, instead of (6.1) a similar estimate for the operator norm from Lp,∞ into Y is there employed. References [1] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR0184042 [2] N. Yu. Antonov, Convergence of Fourier series, Proceedings of the XX Workshop on Function Theory (Moscow, 1995), East J. Approx. 2 (1996), no. 2, 187–196. MR1407066 [3] J. Arias-de-Reyna, Pointwise convergence of Fourier series, J. London Math. Soc. (2) 65 (2002), no. 1, 139–153, DOI 10.1112/S0024610701002824. MR1875141 [4] Juan Arias de Reyna, Pointwise convergence of Fourier series, Lecture Notes in Mathematics, vol. 1785, Springer-Verlag, Berlin, 2002. MR1906800 [5] S. V. Astashkin, Extrapolation properties of the scale of Lp -spaces (Russian, with Russian summary), Mat. Sb. 194 (2003), no. 6, 23–42, DOI 10.1070/SM2003v194n06ABEH000740; English transl., Sb. Math. 194 (2003), no. 5-6, 813–832. MR1992175 [6] S. V. Astashkin, Extrapolation functors on a family of scales generated by the real interpolation method (Russian, with Russian summary), Sibirsk. Mat. Zh. 46 (2005), no. 2, 264–289, DOI 10.1007/s11202-005-0021-2; English transl., Siberian Math. J. 46 (2005), no. 2, 205–225. MR2141194 [7] S. V. Astashkin and K. V. Lykov, Extrapolation description of Lorentz and Marcinkiewicz spaces “close” to L∞ (Russian, with Russian summary), Sibirsk. Mat. Zh. 47 (2006), no. 5, 974–992, DOI 10.1007/s11202-006-0090-x; English transl., Siberian Math. J. 47 (2006), no. 5, 797–812. MR2266510 [8] S. V. Astashkin and K. V. Lykov, Strong extrapolation spaces and interpolation (Russian, with Russian summary), Sibirsk. Mat. Zh. 50 (2009), no. 2, 250–266, DOI 10.1007/s11202009-0023-6; English transl., Sib. Math. J. 50 (2009), no. 2, 199–213. MR2531751 [9] Sergey Astashkin and Konstantin Lykov, Extrapolation description of rearrangement invariant spaces and related problems, Banach and function spaces III (ISBFS 2009), Yokohama Publ., Yokohama, 2011, pp. 1–52. MR3013119 [10] S. V. Astashkin, K. V. Lykov, and M. Mastylo, On extrapolation of rearrangement invariant spaces, Nonlinear Anal. 75 (2012), no. 5, 2735–2749, DOI 10.1016/j.na.2011.11.016. MR2878470 [11] Colin Bennett, Banach function spaces and interpolation methods. I. The abstract theory, J. Functional Analysis 17 (1974), 409–440. MR0361826 [12] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR928802 [13] Christian Berg, The cube of a normal distribution is indeterminate, Ann. Probab. 16 (1988), no. 2, 910–913. MR929086 [14] J¨ oran Bergh and J¨ orgen L¨ ofstr¨ om, Interpolation spaces. An introduction, SpringerVerlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR0482275 [15] A. Yu. Brudnyi, N. Ya. Krugljak, Real Interpolation Functors, Yaroslavl, Yaroslavl University, 1981 (Deposited in VINITI 13.05.1981, Dep. N 2620-81) 211 pp. (in Russian). [16] Yu. A. Brudny˘ı and N. Ya. Krugljak, Interpolation functors and interpolation spaces. Vol. I, North-Holland Mathematical Library, vol. 47, North-Holland Publishing Co., Amsterdam, 1991. Translated from the Russian by Natalie Wadhwa; With a preface by Jaak Peetre. MR1107298 [17] Mar´ıa J. Carro, From restricted weak type to strong type estimates, J. London Math. Soc. (2) 70 (2004), no. 3, 750–762, DOI 10.1112/S0024610704005812. MR2096875

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[66] Stephen Simons, Minimax and monotonicity, Lecture Notes in Mathematics, vol. 1693, Springer-Verlag, Berlin, 1998. MR1723737 [67] Eric V. Slud, The moment problem for polynomial forms in normal random variables, Ann. Probab. 21 (1993), no. 4, 2200–2214. MR1245307 [68] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35–54. MR0241685 [69] Jordan Stoyanov, Krein condition in probabilistic moment problems, Bernoulli 6 (2000), no. 5, 939–949, DOI 10.2307/3318763. MR1791909 [70] J. M. Stoyanov, Counterexamples in Probability, Third edition, Dover Publications, New York, 2013. [71] Shigeki Yano, Notes on Fourier analysis. XXIX. An extrapolation theorem, J. Math. Soc. Japan 3 (1951), 296–305. MR0048619 [72] V. I. Judoviˇ c, Some estimates connected with integral operators and with solutions of elliptic equations (Russian), Dokl. Akad. Nauk SSSR 138 (1961), 805–808. MR0140822 [73] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR0107776 Department of Mathematics, Samara National Research University, Moskovskoye shosse 34, 443086, Samara, Russia E-mail address: [email protected] Image Processing Systems Institute, Molodogvardejskaya st. 151, 443001 Samara, Russia, and Samara National Research University, Moskovskoye shosse 34, 443086, Samara, Russia E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13944

Uncertainty principles and weighted norm inequalities John J. Benedetto and Matthew Dellatorre This paper is dedicated to the memory of Bj¨ orn Jawerth Abstract. The focus of this paper is weighted uncertainty principle inequalities in harmonic analysis. We start by reviewing the classical uncertainty principle inequality, and then proceed to extensions and refinements by modifying two major results necessary to prove the classical case. These are integration by parts and the Plancherel theorem. The modifications are made by means of generalizations of Hardy’s inequality and weighted Fourier transform norm inequalities, respectively. Finally, the traditional Hilbert space formulation is given in order to construct new examples.

1. Introduction 1.1. Background and theme. Uncertainty principle inequalities abound in harmonic analysis, e.g., see [62], [25], [28], [30], [29], [18], [27], [66], [8], [26], [9], [42], [20], [32], [38], [56]. Having been developed in the context of quantum mechanics, the classical Heisenberg uncertainty principle is deeply rooted in physics, see [45], [72], [71], [34]. The classical mathematical uncertainty principle inequality was first stated and proved in the setting of L2 pRq, the space of Lebesgue measurable square-integrable functions on the real line R, in 1924 by Norbert Wiener at a G¨ottingen seminar [3], also see [49]. This is Theorem 1.1. The proof of the basic inequality, (1.1) below, invokes integration by parts, H¨older’s inequality, and the Plancherel theorem, see (1.3). For more complete proofs, see, for example, [72], [9], [32], [38]. Theorem 1.1. (The classical uncertainty principle inequality) If f P L2 pRq and x0 , γ0 P R, then (1.1) ||f ||2 ď 4π||px ´ x0 qf pxq||2 ||pγ ´ γ0 qfppγq||2 , 2

2010 Mathematics Subject Classification. Primary 42Bxx; Secondary 42-06, 42-02. Key words and phrases. Harmonic analysis, uncertainty principle inequalities, weighted norm inequalities, Hardy inequalities. The first author gratefully acknowledges the support of ARO Grant W911NF-15-1-0112, DTRA Grant 1-13-1-0015, and ARO Grant W911NF-16-1-0008. He is also thankful for the wonderful collaboration on the subject matter, beginning with Hans Heinig around 1980, and continuing with Heinig and Ray Johnson for 20 years. The second author gratefully acknowledges the support of the Norbert Wiener Center. The authors are also indebted to the blog of T. Tao, Math Stack Exchange, and Math Overflow for numerous insights, explanations, and discussions. Finally, the authors appreciate the referral by the anonymous referee to the work of Meyers and Serrin [55] (1964). c 2017 American Mathematical Society

55

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JOHN J. BENEDETTO AND MATTHEW DELLATORRE

and there is equality if and only if 2

f pxq “ Ce2πixγ0 e´spx´x0 q ,

(1.2)

for C P C and s ą 0. p|| ¨ ||2 designates the L2 norm, and the Fourier transform fp of f is formally defined as ż fppγq “ f pxqe´2πixγ dx.q R

The uncertainty principle inequality (1.1) is a consequence of the following calculation for the case px0 , γ0 q “ p0, 0q and for f P S pRq, the Schwartz class of infinitely differentiable rapidly decreasing functions defined on R. ˆż ˙2 ˆż ˙2 ||f ||42 “ x|f pxq2 |1 dx ď |x||f pxq2 |1 dx R

(1.3)

R

ˆż ď4 R

|xfĘ pxqf 1 pxq|dx

˙2

ď 4||xf pxq||22 ||f 1 pxq||22 “ 16π 2 ||xf pxq||22 ||γ fppγq||22 . Integration by parts gives the first equality and the Plancherel theorem gives the second equality. The third inequality is a consequence of H¨ older’s inequality. There is a result analogous to Theorem 1.1 for the case d ą 1. This is Theorem 2.3, that is given in Section 2. The main difficulty in the d ą 1 case is that the square integrability of the distributional derivatives of f in the inequality, arising p of γ fppγq, does not afford easy technical manipulation, e.g., from the analogue on R being able to deduce absolute continuity, see [33], [48]. One way to remedy this is to introduce the notion of a bi-Sobolev space. In this context, the argument is reduced to proving the uncertainty principle for smooth compactly supported functions on Rd , and extending to L2 pRd q by means of a density argument. This was originally done in [8]. Using more abstract ideas, Folland and Sitarum also gave a proof of the result for L2 pRd q as a special case, see [32], pages 210-213. The approach in Section 2, following [8], has the advantage of using the same method of integration by parts, H¨older’s inequality, and the Plancherel theorem, as in the one-dimensional case, in order to obtain versions of the classical uncertainty principle inequality on L2 pRd q. It thereby serves as a stepping stone to proving more difficult classical cases involving weighted spaces as well as extending its theoretical tentacles far beyond Theorems 1.1 and 2.3. Remark 1.2 (The additive uncertainty principle). Cowling and Price [23] proved the following strong additive version of the classical uncertainty principle inequality on R for arbitrary p, q P r1, 8s and a, b ą 0, and for the class of tempered functions f , i.e., essentially polynomial growth, for which fp is a function. There is C ą 0 such that ´ ¯ (1.4) @f, ||f ||22 ď C |||x|a f pxq||p ` |||γ|b fppγq||q if and only if 1 1 1 1 ´ and b ą ´ . 2 p 2 q (|| ¨ ||p and || ¨ ||q designate the Lp and Lq -norms.) aą

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

57

Remark 1.3. The relevance of Theorem 1.1 for quantum mechanics can be illustrated by considering a freely moving mass point with varying location l P R. The term ||xf pxq||22 represents the average distance of l from its expected value x0 “ 0. In fact, the position l is interpreted as a random variable depending on the state function f ; more precisely, the probability that x is in a given region A Ď R is defined as ż |f pxq|2 dx,

A

and ||xf pxq||22 is the variance of x. Our theme is as follows. We shall extend and refine Theorems 1.1 and 2.3 in several ways. The main ingredients of our proofs, however, will remain the same: integration by parts will give way to conceptually similar ideas such as generalizations of Hardy’s inequality, and the Plancherel theorem will be generalized to weighted Fourier transform norm inequalities. 1.2. Outline. In Section 2, we give a detailed proof of the classical uncertainty principle on Rd . Because of our theme for generalizing the classical uncertainty principle inequality, Sections 3 and 4 are devoted to Hardy’s inequality and weighted Fourier transform norm inequalities, respectively. Then, in Section 5, the results in Sections 3 and 4 are used to obtain a variety of uncertainty principle inequalities. In Section 6 we provide a proof of the traditional uncertainty principle inequality for general Hilbert spaces in order to exhibit several elementary and some new examples. We conclude with a brief Epilogue. Remark 1.4. Most of these topics have a long history with contributions by some of the most profound harmonic analysts. Our presentation has to be viewed in that context, notwithstanding the considerable number of references to the first named author. It was his intention to put together various uncertainty principle inequalities in which he was involved and that had a common point of view. 1.3. Notation. Generally, our notation is standard from modern analysis texts, e.g., [68], [63], [31], [24], [10], [11]. The Fourier transform fp of a complex-valued Lebesgue measurable function f : Rd Ñ C on Euclidean space Rd is formally defined as ż p f pxqe´2πix¨γ dx, f pγq “ Rd

p d “ Rd and R p denotes a frequency or spectral domain. where γ P R In particular, we use the d-dimensional multi-index notation, where if α is a d-tuple of natural numbers, α “ pα1 , α2 , . . . , αd q, then α ď β means αi ď βi for each i P t1, . . . , du. Also, we write αd 1 α2 xα “ xα 1 x2 . . . xd

and B α “ B1α1 B2α2 . . . Bdαd ,

i where Biαi “ B αi {Bxα i , and |α| “ α1 ` ¨ ¨ ¨ ` αd . Further, we use the following notation. If p ě 1, then p1 is defined by p1 ` p11 “ 1. Let R` “ r0, 8q. If x P Rd and r ą 0, then Bpx, rq Ď Rd is the open ball of radius r centered at x. Let v ě 0 on Rd and let p ě 1. Lpv pRd q is the weighted

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JOHN J. BENEDETTO AND MATTHEW DELLATORRE

space of Borel measurable functions f : Rd Ñ C for which ˆż ˙ p1 p |f pxq| vpxqdx ă 8. ||f ||p,v “ Rd

We shall usually omit the domain of integration in integrals when the setting is clear. Finally, if X and Y are topological vector spaces, then L pX, Y q is the space of continuous linear mappings X Ñ Y . 2. The classical uncertainty principle inequality for L2 pRd ) In this section we describe the classical uncertainty principle inequality on Rd . Definition 2.1. Given integers m, n ě 0, and let 1 ď p ď 8. The Sobolev space W m,p pRd q is the Banach space of functions f P Lp pRd q with norm, ÿ ||f ||m,p “ ||B α f ||p ă 8. |α|ďm

The weighted space norm,

Lp0,n pRd q

is the Banach space of functions f P Lp pRd q with

||f ||p,n “

ÿ

||tβ f ptq||p ă 8.

|β|ďn

The bi-Sobolev space Lpm,n pRd q is the Banach space of functions f P W m,p pRd q X Lp0,n pRd q with norm, ||f ||m,n,p “ ||f ||m,p ` ||f ||p,n ă 8. The following result is a variant of a theorem of Meyers-Serrin [55] (1964). It is to be expected by a natural approximate identity strategy combined with truncations on larger and larger domains. We provide full details to show that the strategy works and because of the expository nature of a chapter such as this. 2.2. Given integers m, n ě 0. Cc8 pRd q is dense in the Hilbert space ` 2 Theorem ˘ d Lm,n pR q, ||...||m,2,n , with inner product, ÿ ÿ xB α f, B α gy ` x tβ f, tβ gy, xxf, gyy “ |α|ďm

|β|ďn 2

d

where x¨ , ¨y is the usual inner product on L pR q. Proof. i. Although it is well-known, we first show that Cc8 pRd q is dense in W pRd q, 1 ď p ă 8. Let f P W m,p pRd q, and let thj u Ď Cc8 pRd q be an L1 approximate identity [10], Section 1.6. Assume without loss of generality that each suppphj q Ď Bp0, 1q. Choose u P Cc8 pRd q such that 0 ď u ď 1 and u “ 1 on Bp0, 1q, and define uj ptq “ upt{jq. Now fix α with the property |α| ď m. Not only does B α pf ˚ hj q “ f ˚ B α hj , but, by integration by parts, m,p

B α pf ˚ hj q “ B α f ˚ hj . Thus, by Young’s inequality, ||B α pf ˚ hj q||p ď ||B α f ||p ||hj ||1 , and so each f ˚ hj is an element of W m,p pRd q, as is each uj pf ˚ hj q.

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

59

The desired density will follow from the triangle inequality once we prove that (2.1)

@α satisfying |α| ď m,

lim ||B α rpf ˚ hj qpuj ´ 1qs ||p “ 0.

jÑ8

To this end, note that Leibniz’ formula gives ||B α rpf ˚ hj qpuj ´ 1qs ||p ď ||puj ´ 1qB α pf ˚ hj q||p ÿ (2.2) ` |Cαβ |j ´|β| ||B α´β pf ˚ hj qptqB β upt{jq||p . βďα,|β|ě1

The dominated convergence theorem and Young’s inequality allow us to show that the first term on the right side of (2.2) tends to 0 as j Ñ 8. Again, Young’s inequality and the fact that lim j ´|β| ||B β u||8 “ 0

jÑ8

for |β| ě 1 show that the remaining terms on the right side of (2.2) tend to 0 as j Ñ 8. Thus, (2.1) is proved. This density in W m,p pRd q can also be proved by an equicontinuity argument, much like the one we now give in part ii. ii. It is sufficient to prove that @f P L20,m pRd q,

(2.3)

lim ||Fβj pf q||2 “ 0

jÑ8

for each β for which |β| ď n, where Fβj pf q “ Fj pf q “ tβ puj pf ˚ hj q ´ f q. To this end we first show that sup ||Fj pf q||2 “ Cpf q ă 8.

(2.4)

j

This is accomplished by the estimate, ||Fj pf q||2 ´ ||tβ f ptq||2 ď Cpβq||B β pfpp hj q||2 ÿ ÿ ď |Cβγ |||B β´γ fpB γ p |Cβγ |||B β´γ fp||2 sup ||B γ p hj ||2 ď hj ||8 , γďβ

γďβ

γďβ

and the fact, in the case hj is the dilation j d hpjtq, that ˇ ˇ ż ˇ ˇ γp γ ´|γ| ´2πipu{jq¨λ ˇ e duˇˇ ď Kpγqj ´|γ| |B hj pλq| “ ˇCpγq hpuqu j since suppphq is compact. This last estimate used the Plancherel theorem so we note the fact that the distribution B β pfpp hj q is an element of L2 pRd q. It is straightforward to check that the elements of L20,n pRd q having compact support are dense in L20,n pRd q and that @β satisfying |β| ď n,

tFβj u Ď LpL20,n pRd q, L2 pRd qq.

Because of p2.4q we can invoke the uniform boundedness principle and obtain sup ||Fj ||2 “ C ă 8. Thus, tFj u is equicontinuous. On the other hand, it is routine to check that limjÑ8 ||Fj pf q||2 “ 0 for compactly supported functions f P L20,n pRd q. This convergence on a dense subset of L20,n pRd q combined with the equicontinuity yield convergence on L20,n pRd q, and the resulting limit F pf q for f P L20,n pRd q deter mines an element F P LpL20,n pRd q, L2 pRd qq. Thus, p2.3q is obtained.

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Because of the caveat mentioned after Remark 1.2, Theorem 2.2 or a similar result is needed to prove the following Rd uncertainty principle inequalities in Theorem 2.3. The remainder of the proof of Theorem 2.3 is an adaptation of the basic calculation (1.3) on R given after the statement of Theorem 1.1. Theorem 2.3 (The classical uncertainty principle inequality). If f P L2 pRd q p d , then and px0 , γ0 q P Rd ˆ R (2.5)

||f ||22 ď 4π||pxj ´ x0,j qf pxq||2 ||pγj ´ γ0,j qfppγq||2

@j “ 1, . . . , d,

and 4π |||x ´ x0 |f pxq||2 |||γ ´ γ0 |fppγq||2 , d where, for example, x0 “ px0,1 , . . . , x0,d q and ¸1{2 ˜ d ÿ 2 |x ´ x0 | “ pxj ´ x0,j q .

(2.6)

||f ||22 ď

j“1

The constant 4π{d is optimal since equality is obtained in (2.6) for f pxq “ expp´π|x|2 q, x0 “ γ0 “ 0. 3. Hardy type inequalities 3.1. Hardy’s classical inequality. In this subsection we state Hardy’s inequality, Theorem 3.2. This is background for Section 3.2, where we shall discuss a Hardy type inequality on R`d due to Hernandez [46]. These inequalities can be viewed in a certain sense as generalizations of integration by parts. as

Definition 3.1. The Hardy operator is the positive linear operator Pd defined ż xd ż x1 ż ... f pt1 , . . . , td qdt1 . . . dtd “ f ptqdt Pd pf qpxq “ 0

0

x0,xy

for Borel measurable functions f on R`d . The region x0, xy Ď Rd is tt “ pt1 , . . . , td q : xj ą 0 and 0 ă tj ă xj for each j “ 1, . . . , du. The dual Hardy operator Pd1 is defined as ż8 ż8 ż Pd1 pf qpxq “ ... f pt1 , . . . td qdt1 . . . dtd “ f ptqdt. xd

x1

xx,8y

The unbounded region xx, 8y Ď R is defined analagously to x0, xy. d

Theorem 3.2 (Hardy’s inequality (1920) [40]). Let f ě 0 (f ‰ 0) be Borel measurable and p ą 1. Then, ˙p ż 8 ˆ ż8 p P1 pf qptqp t´p dt ă f ptqp dt. (3.1) p ´ 1 0 0 G.H. Hardy, along with E. Landau, G. P´ olya, I. Schur, M. Riesz, proved this inequality as well as the following discrete version between 1920 and 1925 [39]. Theorem 3.3 (Hardy’s discrete inequality). Let p ą 1 and let tak u8 k“1 be a sequence of non-negative real numbers. Then, ˜ ¸p ˆ ˙p ÿ 8 n 8 ÿ 1 ÿ p (3.2) ak ď apn . n p ´ 1 n“1 n“1 k“1

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

61

´ ¯p p Since the constant p´1 is sharp, Theorems 3.2 and 3.3 not only express the fact that the Hardy operators are bounded mappings from Lp into Lp and lp into p . lp , respectively, for p ą 1, but that each has norm p1 “ p´1 Remark 3.4. a. It is not difficult to see that restricting to step functions in the integral inequality (3.1) gives the discrete version. However, historically, a weaker form of the integral version was proved first, followed by the discrete version (as stated), and then finally the integral version (as stated) was proved. b. Hardy’s original motivation in studying these types of inequalities was to find a simpler proof for Hilbert’s inequality [47] for double series. In fact, it can be shown that Hilbert’s inequality follows from the discrete version. See [50] for a history of Hardy’s inequality. c. Hardy’s inequality is striking in that it is an Lp inequality with an explicit optimal constant and that the only function for which equality is satisfied is the zero function. Remark 3.5. a. For the sake of context, we mention here that Hardy’s inequality is a fundamental inequality in analysis that demonstrates two very useful principles. Using notation from the fractional calculus, the first principle is that an inverse power weight such as 1{|x|α may be dominated in an Lp sense, by the corresponding derivative |∇|α . Certain higher dimensional generalizations of Hardy’s inequality on Rd take the form, ||f pxq{|x|α ||p ď Cα,p,d |||∇|α f ||p , where α, p, d, and f satisfy certain conditions. These inequalities are fundamental in the study of partial differential equations that involve singular potentials or weights such as 1{|x|α , e.g., [43], [61], [54], also see Section 4.4. b. The second principle exemplified by Hardy’s inequality is that a maximal average of a function is in many cases dominated in an Lp sense by the function itself. This can be seen by a different type of generalization, namely, the HardyLittlewood maximal function inequality, and its variants, in terms of the HardyLittlewood maximal function M defined in the following way. Given a locally integrable function f P L1loc pRd q, M f : Rd Ñ R is a function that at each point x P Rd gives the maximum average value that f can take on balls centered at that point. More precisely, letting Bpx, rq Ď Rd denote the open ball of radius r centered at x and letting |Bpx, rq| be its d-dimensional Lebesgue measure, M f pxq is defined as ż 1 |f pyq|dy. M f pxq “ sup rą0 |Bpx, rq| Bpx,rq Theorem 3.6. (Hardy-Littlewood maximal function inequality) Let d ě 1. There is a constant Cd ą 0 such that Cd ||f ||1 , @f P L1 pRd q and @λ ą 0 |tM f ą λu| ă λ where |tM f ą λu| is the Lebesgue measure of the set tx P Rd : M f pxq ą λu. This inequality is fundamental in harmonic analysis, ranging from the study of singular integral operators, for example the Hilbert transform, to the convergence of Fourier series, e.g., see [58], [57], [68], [67]. Both the discrete and continuous Hardy inequalities have been generalized and applied to problems in analysis and differential equations, e.g., see [41], [59], [51], [50].

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JOHN J. BENEDETTO AND MATTHEW DELLATORRE

3.2. Hernandez’ weighted Hardy inequality. Hernandez [46] proved a far-reaching classical extension of Hardy’s inequality that we now state in Theorem 3.8. The following result is needed for its proof. The proof of Lemma 3.7 is based on H¨ older’s inequality for positive linear operators. It should be compared with the Schur test for positive integral operators, see [37], Appendix A. Lemma 3.7 ([46], Theorem 3.1). Given 1 ă p ď q ă 8 and non-negative Borel measurable functions u and v on X Ď Rd . Suppose P : Lpv pXq Ñ Lqu pXq is a 1 1 positive linear operator with canonical dual operator P 1 : Lqu´q1 {q pXq Ñ Lqv´p1 {p pXq, defined by the duality ż ż P pf qpxqgpxqdx “ f pxqP 1 pgqpxqdx. X

X

Assume there exist K1 , K2 ą 0 such that 1

@g P Lpq{pq pXq, for which g ě 0 and ||g||pq{pq1 ď 1, there are non-negative functions, f1 P Lpv pXq,

1

h1 P Lpup{q g pXq,

1

f2 P Lpu´p1 {q g pXq,

h2 P Lpv´p1 {p pXq,

with the properties, P pf1 q ď K1 h1 and P 1 pf2 gq ď K2 h2

(3.3) and

´p{p1

´q{p1 q{p

v “ f1

h2 and u “ h1 f2 . ´ 1 ¯ 1 Then P P L pLpv pXq, Lqu pXqq, P 1 P L Lqu´q1 {q pXq, Lpv´p1 {p pXq , and 1{p1

||P ||, ||P 1 || ď K1

1{p

K2 .

Notice that if we set 1

1

1

1

f1 “ v ´p {p Pd pv ´p {p q´1{p and h1 “ Pd pv ´p {p q´1{p , 1

1

1

f2 “ up{q Pd1 puq´p{pqp q and h2 “ Pd pv ´p {p q´1{p , 1

then (3.3) is valid for any non-negative g P Lpq{pq pR`d q, for which ||g||pq{pq1 ď 1, as long as (3.4), (3.5), and (3.6) are assumed. As a result, Hernandez obtained the following version of Hardy’s inequality on R`d . Theorem 3.8 ([46], Section 4.2). Given 1 ă p ď q ă 8 and non-negative Borel measurable functions u and v on R`d . Assume there exist positive K, C1 ppq, and C2 ppq such that ˜ż ¸1{q ˜ż ¸1{p1 (3.4)

sup są0

(3.5)

1

vpxq´p {p dx

upxqdx xs,8y

@x P R`d ,

“ K,

x0,sy

´ ¯ 1 1 1 1 Pd v ´p {p pPd pv ´p {p q´1{p pxq ď C1 ppqPd pv ´p {p q´1{p ,

and ´ ¯ 1 Pd1 upPd1 puqq´1{p pxq ď C2 ppqq{p pPd1 uq1{p . ´ 1 ¯ ` ˘ 1 Then, Pd P L Lpv pR`d q, Lqu pR`d q , Pd1 P L Lqu´q1 {q pR`d q, Lpv´p1 {p pR`d q , and ||Pd ||,

(3.6)

@x P R`d ,

1

||Pd1 || ď KC1 ppq1{p C2 ppq1{p .

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

63

Remark 3.9. Condition (3.4) is necessary and sufficient for weighted Hardy inequalities on R and necessary on Rd , d ą 1. Conditions (3.5) and (3.6) are automatically satisfied on R. Conditions (3.4), (3.5), and (3.6), are sufficient but not necessary on Rd , d ą 1. A different but important direction for establishing Hardy inequalities is found in [13]. 3.3. The regrouping lemma. The following lemma will allow us to use the results from Subsection 3.2 to derive an uncertainty principle inequality in Subsection 5.3. Let Ω be the subgroup of the orthogonal group whose corresponding matrices with respect to the standard basis are diagonal with ˘1 entries. Each element ω P Ω can be identified with an element pω1 , . . . , ωd q P t´1, 1ud , and ωγ “ pω1 γ1 , . . . ωd γd q. Thus, formally, ż ÿż F pγqdγ “ F pωγqdγ. pd R

p `d ωPΩ R

Since

ωPΩ

1

1{r a1{r ď ω bω

¸1{r1

¸1{r ˜

˜ ÿ

ÿ

ÿ

aω

ωPΩ

bω

,

ωPΩ

for 1 ă r ă 8 and aω , bω ě 0, we have the following regrouping inequality. p d q, G P Lr1 pR p d q. Proposition 3.10. Given 1 ă r ă 8 and suppose F P Lr pR Then, ˙1{r ˆż ˙1{r1 ÿ ˆż 1 |F pωγq|r dγ |Gpωγq|r dγ ď ||F ||r ||G||r1 . ωPΩ

p `d R

p `d R

4. Weighted Fourier transform norm inequalities 4.1. Generalizations of Plancherel’s theorem. The uncertainty principle inequalities on L2 pRd q, stated in Theorem 2.3, were statements about minimizing variance. However, in many applications, such as signal and image processing, as well as quantum mechanics itself, there are other optimization criteria that are of interest. Weighted uncertainty principle inequalities are one way of addressing this issue. For example, in linear system theory weights correspond to various filters in energy concentration problems, and in prediction theory weighted Lp - spaces arise for weights corresponding to power spectra of stationary stochastic processes [9]. Once a weighted uncertainty principle inequality is obtained, the goal is to determine a minimizer for this inequality, just as the Gaussian is a minimizer for the classical uncertainty principle inequality of Theorem 1.1. Plancherel’s theorem can be viewed as a specific example of a weighted norm inequality for the Fourier transform for the case of energy equivalence between space and spectral domains. Thus, an inequality of the form (4.1)

||fp||q,u ď C||f ||p,v ,

where u, v ě 0 are Borel measurable functions on Rd , can be viewed as a generalization of the Plancherel theorem with an eye towards applications, where the value of p is a relevant parameter and the weights u and v are relevant “filters” or impulse responses.

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JOHN J. BENEDETTO AND MATTHEW DELLATORRE

The main problems concerning p4.1q are characterizing the relationship between the weights u and v to ensure its validity, and in this case finding the smallest possible constant C so that (4.1) is true for all f P Lpv pRd q. Theorem 4.1 (Hausdorff-Young inequality). For all f P SpRd q, (4.2)

||fp||p1 ď Bd ppq||f ||p ,

where 1 ă p ď 2 and (4.3)

´ ¯ 1 d{2 Bd ppq “ p1{p pp1 q´1{p .

Remark 4.2. a. Theorem 4.1 can be extended to Lp pRd q since SpRd q is dense in Lp pRd q. In particular, the Fourier transform is well-defined for each f P Lp pRd q, 1 ă p ď 2. b. The optimal constants, Bd ppq, are due to Babenko [2](1961) and Beckner [5](1975), and represent an analytical tour de force. The extension of the HausdorffYoung inequality for Fourier series to the case of Fourier transforms is due to Titchmarsh [70] in 1924. 4.2. Ap - weights. Definition 4.3 (Ap - weights). Let 1 ă p ă 8, and let w ě 0 be a Borel measurable function on Rd . We call w an Ap -weight, written w P Ap , if ˆ ˙ˆ ˙p{p1 ż ż 1 1 ´p1 {p wpxqdx wpxq dx “ K ă 8, sup |Q| Q |Q| Q Q where Q is a compact cube with sides parallel to the axes and having non-empty interior, see [35] for a definitive treatise. Ap stands for Muckenhoupt weight classes. They are essential in characterizing the continuity of maximal functions and singular integral operators defined on weighted Lebesgue spaces, e.g., see [35], pages 411 ff., as well as the special case, Theorem 4.8, ahead, for the Riesz transform. More surprising is the role of Ap in establishing the continuity of the Fourier transform considered as an operator defined on weighted Lebesgue spaces. The basic relationship between the Fourier transform and Ap is found in [15], cf. [16]. The authors began their theory with the following result. Theorem 4.4. [15] Let w ě 0 be an even Borel measurable function on R, that is non-increasing on p0, 8q; and let 1 ă p ď 2. Then, there exists C ą 0 such that ż 8 @f P Cc pRq, |fppγq|p |γ|p´2 wp1{γqdγ ď C||f ||pp,w p R

if and only if w P Ap . Such inequalities naturally lead to subtle problems dealing with the proper definition of the Fourier transform on weighted Lebesgue spaces, see [17]. The extension of Theorem 4.4 to Rd is due to Heinig and Smith [44]. Theorem 4.5 ([44]). Let 1 ă p ď q ď p1 ă 8, and let w ě 0 be a Borel measurable function on Rd . Assume wp|t|q is increasing on p0, 8q. Then, there

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

65

exists C ą 0 such that (4.4)

@f P

˜ż

Cc8 pRd q,

pd R

1 |fppγq|q |γ|dpq{p ´1q w

ˆ

1 |γ|

¸1{q

˙q{p dγ

ď C||f ||p,w

if and only if w P Ap . Example 4.6. Theorems 4.4 and 4.5 are generalizations of classical results. Consider the case over R , where p “ q and w “ 1. Then p4.4q is the HardyLittlewood-Paley theorem (1931): ˙1{p ˆż ď C||f ||p . |fppγq|p |γ|p´2 dγ On the other hand, when q “ p1 and w “ 1, (4.4) is the Hausdorff-Young theorem; and if wptq “ |t|α , 0 ď α ă p ´ 1, then (4.4) becomes Pitt’s theorem (1937): ˙1{q ˆż ˙1{p ˆż q ´β p α p ďC |f ptq| |t| dt , |f pγq| |γ| dγ where β “ pq pα ` 1q ` 1 ´ q, cf. [4]. Definition 4.7. (Riesz transform) The d-dimensional Riesz transforms are the d singular integral operators R1 , . . . , Rd defined by the odd kernels kj pxq “ Ωj pxq{|x|d , j “ 1, . . . , d, where Ωj pxq “ cd xj {|xj | and cd “ Γ ppd ` 1q{2q {π pd`1q{2 . In fact, ż pRj f qpxq “

lim

T ´1 ,Ñ0 ď|t|ďT

f px ´ tqkj ptqdt

exists a.e. for each f P Lp pRd q, 1 ă p ă 8, and there is C “ Cppq such that @f P Lp pRd q,

||Rj f ||p ď C||f ||p ,

j “ 1, . . . , d. C “ Cppq does not depend on d. Also, we compute γj p kj pγq “ ´i , j “ 1, . . . , d. |γ| Theorem 4.8 (Hunt, Muckenhoupt, and Wheeden, 1973). Let 1 ă p ă 8, and suppose w P Ap . Then, Rj : Lpw pRd q ÝÑ Lpw pRd q is a continuous linear mapping for j “ 1, . . . , d. 4.3. Weighted Fourier transform norm inequalities. It is convenient to begin with the following definition. Definition 4.9. If 1 ă p, q ă 8 and if there is a constant K ą 0 such that ˜ż ¸1{q ˆż ˙1{p1 1{s s ´p1 {p (4.5) sup upγqdγ vptq dt “ K, są0

0

0

then we write pu, vq P F pp, qq. The following theorem, proved in 1982, is a weighted Hausdorff-YoungTitchmarsh inequality.

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JOHN J. BENEDETTO AND MATTHEW DELLATORRE

Theorem 4.10. [12] Let 1 ă p ď q ă 8, and let u, v ě 0 be even Borel p R, respectively, for which pu, vq P F pp, qq with measurable functions defined on R, constant K. Assume 1{u and v are increasing on p0, 8q. Then, there is a constant CpKq such that @f P SpRq X Lpv pRq,

(4.6)

||fp||Lqu ď CpKq||f ||Lpv .

Remark 4.11. If p “ 1 and q ą 1 then Theorem 4.10 is true for any positive Borel measurable function u. In this case the proof is routine and the constant CpKq is explicit [12], pages 272-273. If p ą 1, then the constant CpKq is less explicit, but it can be estimated by examining the proof of Calder´ on’s rearrangement inequality [21], that the authors also used in their proof of Theorem 4.10. The authors of [12] continued this program of understanding weighted Fourier transform norm inequalities in a series of papers through to [14] in 2003. We state two of more of their results. For the first inequality, u˚ : r0, 8q Ñ r0, 8q designates the decreasing rearrangement of any measurable function defined on a measure space. Theorem 4.12 ([14]). Let u, v ě 0 be Borel measurable functions on Rd , and suppose 1 ă p, q ă 8. There is a constant C ą 0 such that for all f P Lpv pRd q, the inequality, (4.7) ||fp||Lq ď C||f ||Lp , u

v

holds in the following ranges and with the following hypotheses on u and v: (i) 1 ă p ď q ă 8 and ˜ż ¸1{q ˆż ˙1{p1 1{s s 1 ˚ u ptqdt p1{vq˚ ptqp ´1 dt “ B1 ă 8; sup są0

0

0

(ii) for 1 ă q ă p ă 8, ¨ ˛1{r ˙r{q1 ż 8 ˜ż 1{s ¸r{q ˆż s 1 1 ˝ u˚ p1{vq˚pp ´1q p1{vq˚ psqp ´1 ds‚ “ B2 ă 8, 0

0

0

where 1r “ 1q ´ p1 . Moreover, the best constant C in ( 4.7) satisfies $ 1 1{p1 1{q ’ &B1 pq q q , if 1 ă p ď q, q ě 2, 1 C ď B1 pp1 q1{p p1{q , if 1 ă p ď q ă 2, ’ 1 % B2 pp1 q1{q q 1{q , if 1 ă p ă q ă 8. Take d ą 1. SOpdq is the non-commutative special orthogonal group of proper rotations. S P SOpdq is a real dˆd matrix whose transpose S t is also its inverse S ´1 p d is a radial if φpSγq “ φpγq and whose determinant detpSq is 1. A function φ on R for all S P SOpdq. Radial measures are defined in the following way. p d q, d ą 1, is radial if Sμ “ μ for all S P SOpdq, Definition 4.13. μ P M pR where Sμ is defined as p d q, xSμ, φy “ xμpγq, φpSγqy. @φ P Cc pR

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

67

p d q, then pSuqpγq “ If dμpγq “ upγqdγ, i.e., μ is identified with u P L1loc pR upS γq for S P SOpdq; in fact, ż ż ż pSuqpγqφpγqdγ “ upγqφpSγqdγ “ upS ´1 γqφpγqdγ, ´1

where the second equality follows since the Jacobian of any rotation is 1. p d q and assume μpt0uq “ 0. If μ is Proposition 4.14. [13] Given μ P M pR radial, then there is a unique measure ν P M p0, 8q such that for all radial functions p d q, φ P Cc pR ż (4.8) xμ, φy “ ωd´1 ρd´1 φpρqdνpρq, p0,8q

p d. where ωd´1 “ 2π {Γpd{2q is the surface area of the unit sphere Σd´1 of R 1 pd Formula ( 4.8) extends to the radial elements of Lμ pR q by Lebesgue’s theorem. d{2

Define M0 pdq “ tf P L1 pRd q : suppf is compact and fpp0q “ 0u, see Section 5 for more on moment spaces. Theorem 4.15. [13] Given radial ν P L1loc pRd q, ν ą 0 a.e., and radial μ P p d q, μpt0uq “ 0. Let v P M` pp0, 8qq denote the measure on p0, 8q correM` pR 1 sponding to μ (as in Proposition 4.14). Assume 1 ă p ď q ă 8 and ν 1´p P 1 d Lloc pR zBp0, yqq for each y ą 0. If ˜ż ¸ ˜ ¸1{p1 ´ ρ ¯ 1{q ż 1{y 1 1 ρd´1`q dν r d´1`p vprq1´p dr ă8 (4.9) B1 “ sup π yą0 p0,yq 0 and ˜ż (4.10)

ρd´1 dν

B2 “ sup yą0

¸ ˜ ´ ρ ¯ 1{q ż 8 π

py,8q

¸1{p1 1

r d´1 vprq1´p dr

ă 8,

1{y

then there is C ą 0 such that, for all f P M0 pdq X Lpν pRd q, ||fp||q,μ ď C||f ||p,ν .

(4.11)

Furthermore, C can be chosen as 1{q`1{p1

C “ 2ωd´1

1

π ´pd´1q{q ppq1{q pp1 q1{p pB1 ` B2 q.

The notation dνpρ{πq signifies p1{πqηpρ{πqdρ in the case dνpρq “ ηpρqdρ. 4.4. Weighted gradient inequalities. Theorem 4.16 ([65], Theorem 4.1). Let 1 ă q ă 8, and let u, v ě 0 be Borel measurable functions on Rd . (a) Then, (4.12)

D C ą 0,

@g P Cc8 pRd q,

||g||q,u ď C||t∇gptq||q,v

if and only if ˙1{q ˆż 8 ˙1{q1 ˆż 1 d´1 d ´q 1 {q ´1 sup upxsqx dx pvpxsqx q x dx “ K ă 8. sPRd

0

1

68

JOHN J. BENEDETTO AND MATTHEW DELLATORRE 1

The constants C and K satisfy the inequalities, K ď C ď Kq 1{q pq 1 q1{q . (b) Furthermore, D C ą 0,

@g P Cc8 pRd q for which gp0q “ 0,

||g||q,u ď C||t∇gptq||q,v

if and only if ˆż 8 ˙1{q ˆż 1 ˙1{q1 d´1 d ´q 1 {q ´1 upxsqx dx pvpxsqx q x dx “ K ă 8. sup sPRd

1

0

5. Uncertainty principle inequalities 5.1. Moment spaces. In this section we provide extensions and refinements of the classical uncertainty principle inequality by using the inequalities obtained in Sections 3 and 4. We introduced the moment space M0 pdq before Theorem 4.15 in Section 4. For all practical purposes, M0 pdq can be replaced by the following subspaces of the Schwartz space S pRd q: S0 pRd q “ tf P S pRd q : fpp0q “ 0u and S0,a pRd q “ tf P S pRd q : fppγ1 , . . . , γd q “ 0 if some γj “ 0u. In particular, f P S pRd q is an element of S0,a pRd q if fp “ 0 on the coordinate axes. where v ą 0 a.e., and Theorem 5.1. [13] Let v P Lrloc pRd q for some r ą 1, Ş choose p P p0, 8q. a. If h P Lpv pRd q1 annihilates S0 pRd q Lpv pRd q, then h is a constant function. Ş b. S0 pRd q Lpv pRd q “ Lpv pRd q or Lpv pRd q Ď L1 pRd q. Ş 1 c. If v 1´p R L1 pRd q, then S0 pRd q Lpv pRd q “ Lpv pRd q. Remark 5.2. a. The condition p ą 1 is necessary in Theorem 5.1. In fact, if p “ 1 and v “ 1, then by a standard spectral synthesis result [6], the L1 -closure of S0 pRd q is the closed maximal ideal tf P L1 pRd q : fpp0q “ 0u. b. Subsequent work dealing with S0 pRd q and weighted Lebesgue spaces is due to Carton-LeBrun [19]. Now consider S0 pRd q as a subspace of L21,1 pRd q. The following is not difficult to verify. Proposition 5.3. a. S0 pRd qK as a subspace of L21,1 pRd q1 is the set of constant functions on Rd . b. The closure of S0 pRd q in L21,1 pRd q is tf P L21,1 pRd q : fpp0q “ 0u. Proposition 5.3a and the inclusion L21,1 pRd q Ď L1 pRd q give Proposition 5.3b. 5.2. Weighted uncertainty principle inequalities on R. We begin with the following. Theorem 5.4 ([8], Proposition 2.1.2). Given 1 ă p ď 2. Then, (5.1)

@f P S pRq,

||f ||22 ď 4πB1 ppq||xf pxq||p ||γ fppγq||p ,

where B1 ppq was defined in ( 4.3).

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

69

The proof is similar to the proof of Theorem 1.1: the Lp pRq - version of H¨ older’s inequality is used instead of the L2 pRq - version, and the Hausdorff - Young inequality replaces the Plancherel theorem. Heinig and Smith strengthened Theorem 5.4: Theorem 5.5 ([44], Theorem 1.1). Given 1 ă p ď 2. Then, (5.2)

@f P S0 pRq,

||f ||22 ď 2πpB1 ppq||xf pxq||p ||γ fppγq||p .

The constant in (5.2) is sharper than that in (5.1) for 1 ă p ă 2. The proof of (5.2) is also similar to the proof of Theorem 1.1 but depends on Hardy’s inequality in the following way: ˇp1 ¸1{p1 ˆż 8 ˇ ˙1{p ˜ż 8 ˇ ż8ˇ ˇ ˇ ˇ ˇ1 ˇ p ˇ2 ˇ p ˇp ˇ fppγqˇ dγ ˇf pγqˇ dγ ď ˇγ f pγqˇ dγ ˇ ˇγ 0 0 0 ˇp1 ¸1{p1 ˆż 8 ˇ ˙1{p ˜ż 8 ˇ ˇ ˇ1 ˇ ˇ p ˇp 1 ˇ P1 ppfpq qpγqˇ dγ “ ˇγ f pγqˇ dγ ˇγ ˇ 0 0 ˆż 8 ˇ ˙1{p ˆż 8 ˇ ˇ ˇ 1 ˙1{p1 ˇ p ˇp ˇ p 1 ˇp ďp . ˇγ f pγqˇ dγ ˇpf q pγqˇ dγ 0

0

We can then prove the following weighted uncertainty principle inequality, see [7], page 408. Theorem 5.6. [7] Given 1 ă p ď q ă 8 and even Borel measurable functions v, w ě 0, that are increasing on p0, 8q. Assume p1{w, vq P F pp, qq with constant K (as in ( 4.5)). Then, there is a C “ CpKq ą 0 such that (5.3)

@f P S pRq,

||f ||22 ď 4πCpKq||xf pxq||p,v ||γ fppγq||q1 ,wq1 {q .

Proof. The proof is a consequence of the estimate, ż ˇ ˇ ˇ ˇ ||f ||22 “ ||fp||22 ď 2 ˇγ fppγqpfpq1 pγqˇ dγ ż ˇ ˇˇ ˇ ˇ ˇˇ ˇ “ 2 ˇγ fppγqwpγq1{q ˇ ˇpfpq1 pγqwpγq´1{q ˇ dγ ˆż ˇ ˙1{q1 ˆż ˇ ˙1{q ˇ ˇ1 ˇ p ˇq ˇ p 1 ˇq q 1 {q ´1 ď2 ˇγ f pγqˇ wpγq dγ ˇpf q pγqˇ wpγq dγ ˇˇ ˇˇ´ ˇˇ ˇˇ ¯_ ˇˇ ˇˇ ˇˇ ˇˇ ď 2C ˇˇ pfpq1 pxqˇˇ ˇˇγ fppγqˇˇ 1 q1 {q , p,v

q ,w

1 _

and the fact that ppfpq q pxq “ 2πixf pxq.



5.3. Weighted uncertainty principle inequalities on Rd . Combining Theorem 3.8 and the regrouping lemma (Proposition 3.10), we obtain the following. Theorem 5.7. Given 1 ă r ă 8 and Borel measurable functions v, w ě 0. 1 Suppose u “ w´r {r . Assume that for all ω P Ω, the weights upωγq and vpωγq p `d for p “ q “ r 1 and constants Kpωq, satisfy conditions ( 3.4), ( 3.5), ( 3.6) on R 1 C1 pp, ωq, C2 pp, ωq. If C “ supωPΩ KpωqC1 pp, ωq1{p C2 pp, ωq1{p , then (5.4)

@f P S0,a pRd q,

||f ||22 ď C||fp||r,w ||∇fp||r1 ,v .

70

JOHN J. BENEDETTO AND MATTHEW DELLATORRE

Remark 5.8. At this point, generalizations of Theorems 5.5 and 5.6 can be obtained by applying d´dimensional versions of Theorem 4.10 to the factor ||B1 , . . . Bd fp||r1 ,v on the right side of (5.4). We shall not look at all forms of rearrangements, but confine ourselves to the following. If v “ 1 and p “ q “ r 1 , then the supremum in Theorem 3.8 has the form, ˜ż ¸1{r1 upyqdy “K (5.5) sup ps1 . . . sd q1{r są0

xs,8y d

1

and (3.5) is satisfied for C1 pr q “ r . For this setting we have the following uncertainty principle inequality. Theorem 5.9. Given 1 ă r ď 2 and let the non-negative Borel measurable weight w be invariant under the action of Ω. Assume K ă 8 (in ( 5.5)) for u “ 1 w´r {r and that (5.6)

1

1

1

1

Pd1 pw´r {r pPd1 pw´r {r q´1{r q ď C2 pr 1 qpPd1 pw´r {r q1{r .

Then, for all f P S0,a pRd q, 1

||f ||22 ďp2πqd r d{r KC2 pr 1 q1{r Bd prq||t1 . . . td f ptq||r ||fp||r,w 1 ďp2πqd r d{r d´d{2 KC2 pr 1 q1{r Bd prq|||t|d f ptq||r ||fp||r,w . 1

The weight wpγq “ |γ1 . . . γd |r , 1 ă r ď 2, is Ω-invariant, K “ pr 1 ´ 1q´d{r in (5.5), and (5.6) is satisfied for C2 pr 1 q “ prpr 1 ´ 1qqd . Thus, we obtain the following d-dimensional generalization of Theorem 5.5. Theorem 5.10 ([8], Section 2.1.11). Given 1 ă r ď 2. Then, for all f P S0,a pRd q, ||f ||22 ď p2πrqd Bd prq||t1 . . . td f ptq||r ||γ1 . . . γd fppγq||r . Using Theorem 4.16, Theorem 4.8, and Minkowski’s inequality we obtain the following inequality. Theorem 5.11 ([8], [9], Theorem 7.62). Given 1 ă r ď 2 and a nonnegative radial weight w P Ar on Rd for which wp|t|q is increasing on p0, 8q. Assume (5.7) ˜ż ¸1{r1 ˜ż ¸1{r ˙´1 ˆ 1 1 8 wpxsq´r {r d´1 1 r ´1´pdr{r 1 q sup x dx w |xs| x dx “ K ă 8. |xs|r1 |xs| sPRd 0 1 Then, there is a constant C “ CpKq ą 0 such that @f P Cc8 pRd q,

||f ||22 ď C|||t|f ptq||r,w |||γ|fppγq||r,w .

Proof. For 1 ă r ă 8 we have (5.8) where

||f ||22 ď || |t|f ptq||r,w ||f ||r1 ,u , 1

1

uptq “ |t|´r wptq´r {r . The second factor on the right side of (5.8) is estimated by means of Theorem 4.16a, where q and v in (4.12) are q “ r 1 and ˆ ˙r1 {r 1 ´r 1 , vptq “ |t| w |t|

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

71

respectively. Thus, (5.9)

||f ||r1 ,u ď C1 ||t∇f ptq||r1 ,v

if and only if (5.7) holds. By Minkowski’s inequality the right side of (5.9) is bounded by ˜ż ˆ ˙r1 {r ¸1{r1 d ÿ 1 1 |pRj Gj q_ ptq|r w dt , (5.10) C1 |t| j“1 where G_ j ptq “ Bj f ptq. Combining (5.9) and (5.10), and applying Theorem 4.5 for the case p “ r and q “ p1 (so that 1 ă r ď 2), we obtain (5.11)

||f ||r1 ,u ď C1 C2

d ÿ

||Rj Gj ||r,w .

j“1

Finally, combining (5.8) and (5.11) and applying Theorem 4.8 to the right side of (5.11), we have the estimate ||f ||22 ď C1 C2 C3 |||t|f ptq||r,w

d ÿ

||pBj f q_ pγq||r,w

j“1 1{r 1

ď 2πd

˜ż C1 C2 C3 |||t|f ptq||r,w

˜ r

|fppγq|

d ÿ

¸1{r

¸ r

|γj |

wpγqdγ

j“1

ď 2πd1{2 C1 C2 C3 |||t|f ptq||r,w |||γ|fppγq||r,w .  Corollary 5.12. Given 1 ă r ď 2 and d ą r 1 , there is C ą 0 such that (5.12)

@f P SpRd q,

||f ||22 ď C|||t|f ptq||r |||γ|fppγq||r .

Remark 5.13. a. The constant C in Corollary 5.12 is of the form C “ 2πd1{2 C1 pr, dqBd prqC3 prq. Since it is of interest to measure the growth of C as d increases, we note that C1 pr, dq can be estimated in terms of K in (5.7) for any w. b. Theorem 4.16b gives rise to an analogue of Theorem 5.11 which, for w “ 1, yields (5.12) for d ă r 1 . See [14] for a summary of these and further results. 6. An uncertainty principle inequality for Hilbert spaces 6.1. An uncertainty principle inequality. We shall prove a well-known uncertainty principle inequality for Hilbert spaces [71], [8], [9], [32]. This result is also referred to as the Robertson uncertainty relation [62]. Definition 6.1. Let A, B be self-adjoint operators on a complex Hilbert space H. (A and B need not be continuous.) Define the commutator rA, Bs “ AB ´ BA, the expectation or expected value Ex pAq “ xAx, xy of A at x P DpAq, where DpAq denotes the domain of A, and the variance Δ2x pAq “ Ex pA2 q ´ tEx pAqu2 of A at x P DpA2 q.

72

JOHN J. BENEDETTO AND MATTHEW DELLATORRE

Theorem 6.2 ([9], Theorem 7.2). Let A, B be self-adjoint operators on a complex Hilbert space H (A and B need not be continuous). If x P DpA2 q X DpB 2 q X DpirA, Bsq and ||x|| ď 1, then tEx pirA, Bsqu2 ď 4Δ2x pAqΔ2x pBq.

(6.1)

Proof. By self-adjointness, we first compute Ex pirA, Bsq “i pxBx, Axy ´ xAx, Bxyq

(6.2)

“2ImxAx, Bxy.

Also note that DpA2 q Ď DpAq. Since ||x|| ď 1 and xAx, xy, xBx, xy P R by self-adjointness, we have ||pB ` iAqx||2 ´ |xpB ` iAqx, xy|2 ě 0

(6.3) and (6.4)

|xpB ` iAqx, xy|2 “ xBx, xy2 ` xAx, xy2 .

By the definition of || ¨ ||, we compute (6.5)

||pB ` iAqx||2 “ ||Bx||2 ` ||Ax||2 ´ 2Im xAx, Bxy.

Substituting (6.4) and (6.5) into (6.3) yields the inequality, (6.6)

||Ax||2 ´ xAx, xy2 ` ||Bx||2 ´ xBx, xy2 ě 2Im xAx, Bxy.

Letting r, s P R, so that rA and sB are also self-adjoint, (6.6) becomes ` ˘ ` ˘ r 2 ||Ax||2 ´ xAx, xy2 ` s2 ||Bx||2 ´ xBx, xy2 (6.7) ě 2rsIm xAx, Bxy. Setting r 2 “ ||Bx||2 ´ xBx, xy2 and s2 “ ||Ax||2 ´ xAx, xy2 , substituting into (6.7), squaring both sides and dividing, we obtain ˘` ˘ ` ||Ax||2 ´ xAx, xy2 ||Bx||2 ´ xBx, xy2 ě pImxAx, Bxyq2 . From this inequality and (6.2) the uncertainty principle inequality (6.1) follows.



6.2. Examples. Example 6.3. (The classical uncertainty inequality) The classical uncertainty principle inequality, Theorem 1.1, is a corollary of Theorem 6.2 for the case H “ L2 pRq, where the operators A and B are defined as Apf qptq “ pt ´ t0 qf ptq and

¯_ ´ Bpf qptq “ i 2πipγ ´ γ0 qfppγq ptq.

Straightforward calculations show that A and B are self-adjoint, and that ż Ef pAq “ pt ´ t0 q|f ptq|2 dt, ż Ef pBq “ ´2π pγ ´ γ0 q|fppγq|2 dγ,

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

Δ2f pAq Δ2f pBq

“ 4π

ż “ 2

˜ż

2

2

ˆż

|t ´ t0 | |f ptq| dt ´ 2

|γ ´ γ0 | |fppγq|2 dγ ´

2

˙2

pt ´ t0 q|f ptq| dt ˆż

73

,

pγ ´ γ0 q|fppγq|2 dγ

˙2 ¸ ,

tEf pirA, Bsqu2 “ ||f ||42 . Example 6.4 (Pauli and generalized Gell-Mann matrices). The following matrices, referred to as the Pauli matrices, are important in quantum mechanics, where they occur in the Pauli equation which models the interaction of a particle’s spin with external electromagnetic fields [64]: ˆ ˙ ˆ ˙ ˆ ˙ 0 1 0 ´i 1 0 A“ B“ C“ 1 0 i 0 0 ´1 Note that each Pauli matrix is Hermitian, and together with the identity matrix, the Pauli matrices span the vector space of 2 ˆ 2 Hermitian matrices. From a quantum mechanical point of view, Hermitian matrices are observables, and thus the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. The Pauli matrices may be generalized to the Gell-Mann matrices in dimension 3, and then these to the so-called generalized Gell-Mann matrices in any dimension d [36]. In dimension d, this is the following family of matrices. Let Ej,k denote the d ˆ d matrix with 1 in the jk´th entry. Define the following matrices: # for k ă j, Ek,j ` Ej,k , d Ak,j “ ´ipEj,k ´ Ek,j q, for k ą j, and

$ ’ for k “ 1, ’ & Id , d´1 d for 1 ă k ą d, hk ‘ 0, hk “ b ˘ ` d´1 ’ ’ 2 % ‘ p1 ´ dq , for k “ d. dpd´1q h1

Thus, for any dimension d, Theorem 6.2 can be applied to any pair of generalized Gell-Mann matrices, to obtain inequalities regarding the the components of vectors in the unit disc in Cd . For example, when d “ 2 and H “ C2 , we apply Theorem 6.2 to the operators A and B (as defined above). Straightforward calculations give the following inequality. Corollary 6.5. Let z “ pz1 , z2 q P C2 , |z| ď 1. Then, ˘2 ` ˘` ˘ ` 2 z1 z2 ´ z1 z¯2 q2 . |z1 | ´ |z2 |2 ď |z|2 ` pz1 z¯2 ´ z¯1 z2 q2 |z|2 ´ p¯ Similarly, using the pair B and C we obtain the following inequality. Corollary 6.6. Let z “ pz1 , z2 q P C2 , |z| ď 1. Then, ` ˘` ˘ pz¯1 z2 ` z1 z¯2 q2 ď |z|2 ` pz1 z¯2 ´ z¯1 z2 q2 |z|2 ´ p|z1 |2 ´ |z2 |2 q2 . Example 6.7 (Ornstein-Uhlenbeck operator). Let L2μ pRd q denote L2 pRd q with respect to the Gaussian measure dμ, where 2 1 dμpxq “ d{2 e|x| dx. π

74

JOHN J. BENEDETTO AND MATTHEW DELLATORRE

The Ornstein-Uhlenbeck operator, L, is the self-adjoint second-order differential operator defined as Lf pxq “ Δf pxq ´ x∇f pxq. Let Q be the position operator Qf pxq “ xf pxq. Taking d “ 1 and working on L2μ pRd q, we compute rL, Qsf pxq “ 2f 1 pxq ´ xf pxq, ż ` 1 ˘ 2 1 Ef prL, Qsq “ ? 2f pxqf pxq ´ xf pxq2 e´x dx, π R ż ż 2 2 1 1 Ef pLq “ ? pf 2 pxq ´ xf 1 pxqqf pxqe´x dx, Ef pQq “ ? xf pxq2 e´x dx, π R π R ż ` ˘ 2 1 Ef pL2 q “ ? f pxq f 4 pxq ´ 2xf 3 pxq ` xf 2 pxq ´ 2f 2 pxq ` xf 1 pxq e´x dx, π R and ż 2 1 2 Ef pQ q “ ? x2 f pxq2 e´x dx. π R Then, applying Theorem 6.2 and assuming sufficient differentiability, we have the following inequality. Corollary 6.8. ˇż ˇ ˇ ` 1 ˘ ˇ 2 ˇ 2f f ´ xf dμˇˇ ˇ R ˆ ˙1{2 ˆż ˙1{2 ż ` 4 ˘ 1 2 1 3 2 2 1 ? ď2 pf ´ xf qf dμ f f ´ 2xf ` xf ´ 2f ` xf dμ . π R R Remark 6.9. a. The theory that developed around the Ornstein-Uhlenbeck operator can be viewed as a model of harmonic analysis in which Lebesgue measure is replaced by a Gaussian measure. This theory has applications to quantum physics and probability. In an infinite dimensional setting, the theory leads to the Malliavin calculus [1], [53]. b. The Hermite polynomials form an orthogonal system with respect to the Gaussian measure in Euclidean space, and they are the eigenfunctions of the Ornstein-Uhlenbeck operator. 7. Epilogue In 1982, when the first named author began to travel (literally, driving from Toronto to Ottawa) with Hans Heinig on the path of weighted Fourier transform norm inequalities and uncertainty principle inequalities, he also had the great good fortune to begin a correspondence with John F. Price. This, combined with Fritz Carlson’s inequality (1934) and the Bell Labs inequalities of Henry J. Landau, David Slepian, and Henry Pollack [52], led to the exposition [8] in 1989 featuring local uncertainty principle inequalities, spearheaded by Faris [28], Cowling and Price [22], [23], and Price [60], and in the context of more classical work inspired by Carlson’s work. Subsequently, others have exposited the local theory, but there is an argument to update the current state of affairs, especially in light of the uncertainty principle

UNCERTAINTY PRINCIPLES AND WEIGHTED NORM INEQUALITIES

75

inequalities of Donoho and Stark [27] and Tao [69], and the advent of thinking in terms of sparsity, compressive sensing, and dimension reduction, as well as quantum inequalities emanating from the role of G˚ arding’s inequality, see, e.g., [30].

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[47] Hilbert, D., Grundz¨ uge einer allgemeinen Theorie der linearen Integralgleichungen, G¨ ottingen Nachr., 157-227, (1906). [48] J. Horv´ ath, A remark on the representation theorems of Frederick Riesz and Laurent Schwartz, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 403–416. MR634249 [49] Kennard, E. H., Zur quantentheoretischen einfacher Bewegungstypen, Zeit. Physik, 44, 326352, (1927). [50] Alois Kufner, Lech Maligranda, and Lars-Erik Persson, The prehistory of the Hardy inequality, Amer. Math. Monthly 113 (2006), no. 8, 715–732, DOI 10.2307/27642033. MR2256532 [51] Alois Kufner and Lars-Erik Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR1982932 [52] H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell System Tech. J. 40 (1961), 65–84. MR0140733 [53] Paul Malliavin, Integration and probability, Graduate Texts in Mathematics, vol. 157, Springer-Verlag, New York, 1995. With the collaboration of H´ el` ene Airault, Leslie Kay and G´ erard Letac; Edited and translated from the French by Kay; With a foreword by Mark Pinsky. MR1335234 [54] Vladimir Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR2777530 [55] Norman G. Meyers and James Serrin, H “ W , Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 1055–1056. MR0164252 [56] Joaquim Mart´ın and Mario Milman, Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces, J. Funct. Anal. 270 (2016), no. 9, 3307–3343, DOI 10.1016/j.jfa.2016.02.016. MR3475459 [57] Charles J. Mozzochi, On the pointwise convergence of Fourier series, Lecture Notes in Mathematics, Vol. 199, Springer-Verlag, Berlin-New York, 1971. MR0445205 [58] Umberto Neri, Singular integrals, Lecture Notes in Mathematics, Vol. 200, Springer-Verlag, Berlin-New York, 1971. Notes for a course given at the University of Maryland, College Park, Md., 1967. MR0463818 [59] B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR1069756 [60] John F. Price, Sharp local uncertainty inequalities, Studia Math. 85 (1986), no. 1, 37–45 (1987). MR879414 [61] Vicent¸iu D. R˘ adulescu, Qualitative analysis of nonlinear elliptic partial differential equations: monotonicity, analytic, and variational methods, Contemporary Mathematics and Its Applications, vol. 6, Hindawi Publishing Corporation, New York, 2008. MR2572778 [62] Robertson, H. P., The uncertainty principle, Phys. Rev., 34, 163-64, (1929). [63] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR1157815 [64] Schiff, L. I., Quantum Mechanics, Mcgraw-Hill College, 3rd edition, (1968). [65] Gordon Sinnamon, A weighted gradient inequality, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 3-4, 329–335, DOI 10.1017/S0308210500018606. MR1007530 [66] Robert S. Strichartz, Uncertainty principles in harmonic analysis, J. Funct. Anal. 84 (1989), no. 1, 97–114, DOI 10.1016/0022-1236(89)90112-2. MR999490 [67] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR1232192 [68] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR0304972 [69] Terence Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett. 12 (2005), no. 1, 121–127, DOI 10.4310/MRL.2005.v12.n1.a11. MR2122735 [70] Titchmarsh, E. C., A contribution to the theory of Fourier transforms, Proc. London Math. Soc., 23, 279-283, (1924).

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[71] von Neumann, J., Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, (1932). [72] Weyl, H., Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig, Revised English edition: The Theory of Groups and Quantum Mechanics, Methuen, London, 1931; reprinted by Dover, New York, 1950, (1928). Department of Mathematics, University of Maryland, College Park, Maryland 20742 Current address: Department of Mathematics, University of Maryland, College Park, Maryland 20742 E-mail address: [email protected] Department of Mathematics, University of Maryland, College Park, Maryland 20742 E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13936

The discrete Calder´ on reproducing formula of Frazier and Jawerth ´ ad B´enyi and Rodolfo H. Torres Arp´ Abstract. We present a brief recount of the discrete version of Calder´ on’s reproducing formula as developed by M. Frazier and B. Jawerth, starting from the historical result of Calder´ on and leading to some of the motivation and applications of the discrete version.

Alberto P. Calder´ on’s genius has produced a plethora of deep and highly influential results in analysis [7], and his reproducing formula certainly qualifies as a gem among them. Also commonly referred to as Calder´ on’s resolution of identity, Calder´ on’s Reproducing Formula (CRF) is a strikingly elegant relation, inspired by the simple idea of breaking down a function as a sum of appropriate convolutions or wave like functions (see (1.1) below). Moreover, mathematicians working on wavelets consider nowadays Calder´ on as one of the forefathers of the theory. The purpose of this expository note is to honor the memory of Bj¨ orn Jawerth by presenting a brief account of some aspects of one of his major contributions to mathematics: the development in collaboration with Michael Frazier of what they called “the φ-transform” and how it relates to Calder´on’s original formula. None of what is presented here is new and we will repeat some arguments commonly found in the literature; but some others that we shall present seem to be part of the folklore of the subject or are hard to locate in the references. We will try to follow a partially historical and partially formal approach to this beautiful formula as we learned it from the horse’s mouth, in particular from Bj¨orn Jawerth himself, but also from Michael Frazier, Richard Rochberg, Mitchell (Mitch) Taibleson, and Guido Weiss.1 The goal is to summarize here in a succinct way some simple motivations while pointing to both classical as well as some not so well-known references, including only some technical details for completeness purposes or to illustrate some concepts. Our hope is to convey, perhaps not to the experts but rather to a broader uninitiated audience, some of the profound contributions of Jawerth and collaborators, which are sometimes overlooked in the continuing proliferation and rediscovery of results in the area of function space decompositions. We also want to provoke the curiosity 2010 Mathematics Subject Classification. Primary 42B20; Secondary 42B15, 47G99. 1 In particular, in the development of some topics in this survey we have benefited a lot from unpublished lectures notes of courses taught by Frazier and Jawerth at Washington University in the 80’s. c 2017 American Mathematical Society

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of those who may have not read numerous original works we shall mention, and motivate them to further explore the rich existing literature. As it is today well-understood, the φ-transform is a way to represent functions and even distributions as linear combinations of translates and dilates of a fixed function. The representation is sometimes also referred to in the literature as the Frazier-Jawerth transform, almost orthogonal decomposition, or non-orthogonal wavelet expansion. Their theory was fully developed in two fundamental articles, [26] published in 1985, which dealt with decomposition of Besov spaces; and the more comprehensive treatise [28], which extended the decomposition techniques to the full scale of Triebel-Lizorkin spaces and presented many other applications.2 This second article took a long time to be published appearing in print in 1990, so the term φ-transform actually appeared first in [27], which was published in 1988 but was based on a lecture given by Jawerth in Lund in 1986. Like wavelets, the Frazier-Jawerth decomposition can be viewed as a discrete version of the CRF. Moreover, in appearance (see (2.1) below) there is no difference between discrete wavelets and the φ-transform, but their origins and initial motivations for their developments were somehow different. While Frazier’s and Jawerth’s were rooted in the analysis of function spaces, their intrinsic features and atomic decompositions, and the classical operators acting on them, the study of wavelets can be traced back to the rediscovery of CRF in an applied context and the construction of orthonormal bases with certain particular properties. We quote Frazier and Jawerth [28, p. 37]: “Wavelets are a collection of functions similar to the representing functions in our decomposition, but which are mutually orthogonal. In fact, wavelets form an unconditional basis for the usual function spaces in harmonic analysis listed above. Thus, unlike our theory, the wavelet theory is immediately connected to the vast literature on the construction of explicit unconditional bases for various function spaces. However, for the applications that we have considered (not related to bases), our more elementary decomposition has been sufficient. Thus, for reasons of simplicity (and perhaps stubbornness) we have presented our results without reference to the beautiful theory of wavelets. However, the reader will readily note that our conclusions generally apply as well to the wavelet decomposition.” Likewise, we will not attempt the almost impossible task to provide a comprehensive account of wavelet theory, but just implicitly point out to some common features with the Frazier-Jawerth decomposition and a few references along the way. (For a more authoritative account of the subject of wavelets, the reader is referred to the works of Y. Meyer [51], Hern´andez and Weiss [39], and the references therein.) Nor shall we describe in its fullness the connection of the φ-transform to the very active theory of frames; in particular, frames generated by the action of a group (like the ax + b group of translations and dilations) on a fixed collection of functions. In the derivation of the formula of Frazier and Jawerth one gives up the orthogonality of the analogous wavelet formula for simplicity in the construction. Such 2 Of main relevance is the fact that the Triebel-Lizorkin spaces encompass many classical spaces such as Lebesgue, Sobolev and Hardy spaces in a unified way.

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construction is astonishingly elementary, and this only adds to its beauty. At a theoretical level, this lack of perfect orthogonality is of no consequence in many uses. In addition, the redundancy and flexibility of frame representations is nowadays preferred in some numerical applications, while in others the perfect orthogonality of wavelets still becomes crucial. The potential of “almost orthogonal” or “quasiorthogonal” decompositions in applications has been observed early on in the development of all these and related expansions. For example, I. Daubechies, A. Grosmann, and Y. Meyer stated in [18, p. 1273]: “We believe that tight frames and the associated simple (painless!) quasiorthogonal expansions will turn out to be very useful in various questions of signal analysis, and in other domains of applied mathematics. Closely related expansions have already been used in the analysis of seismic signals” [the authors referred to the works in [32], [35], [36]]. While in their work they were focusing on the construction of frames based on the Weyl-Heinseberg group, the authors further mentioned in [18, p. 1273]: “...the construction of tight frames associated with the WeylHeisenberg group is essentially the same as that of tight frames associated with the ax+b group. Tight frames associated with the ax + b group were first introduced in a different context closer to pure mathematics. In Ref. 13(b) [Frazier-Jawerth [26]] one can find a definition of ‘quasiorthogonal families’ very close to our tight frames, and a short discussion of the similarities between a ‘quasiorthogonal family’ and an orthonormal basis. For the many miraculous properties of this orthonormal basis, see Ref. 14 [Meyer [48]].” As already mentioned, we will focus mainly on the relation of the φ-transform to the CRF and we want to keep this presentation not too extensive, hence leaving out many wonderful connections to other topics. For more details on the interplay between wavelets, frames, sampling, signal analysis, CRF, and much more, as well as historical accounts and relevant contributions, we refer to the delightful introduction by J. Benedetto to the compendium of articles in [38]. 1. The Continuous Calder´ on Reproducing Formula The formula in its “modern” form can be briefly stated as ˆ ∞ dt u ∗ φt ∗ φt , (1.1) u= t 0 for an appropriate function φ and with convergence also understood in an appropriate sense. Like several other of his profound contributions, the origins of the reproducing identity of Calder´ on are traced back to his 1964 paper [5] on complex interpolation. The abstractness and the generality of the presentation do not make this article the easiest of reads even for the experts. In his review of this work, J. Peetre says [52]: “The presentation is not very clear, mainly owing, in the reviewer’s opinion, to the unfortunate subdivision into two parts, the first giving the definitions and main results, the second giving

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the detailed proofs; the reader has to spend quite a lot of time just searching for the relevant passage for the proof of each particular statement.”, whereas C. Fefferman and E. Stein [20] eloquently summarize this masterpiece of Calder´ on as follows: “This important and lengthy paper of Calder´ on contains significant conceptual insights of broad interest, but at the same time requires a number of ingenious and tricky technical devices for executions.” Though highly appreciated as one of the foundation rocks on which modern interpolation theory was built upon, the article indeed contains many other results, including the reproducing formula. Frazier, Jawerth, and Weiss, point in [30] to [5, §34] as the birthplace of Calder´ on’s identity. And it is there, in §14 in fact, a hidden treasure waiting to be discovered by the inquisitive eye of the reader. At first, the formula in its great generality in [5] is not so easy to identify. Nonetheless, Calder´on’s idea, reduced to a particular case, is as follows. For a given function ϕ on Rn , and t > 0, let us write as usual ϕt (x) = t−n ϕ(t−1 x). Also τy ϕ(x) = ϕ(x − y) denotes the translation operator, and ∗ denotes the operation of convolution of two functions. Letting u ∈ L2 (R), and ϕ ∈ C ∞ (R) be a spherically symmetric function having vanishing moments up to a prescribed order, Calder´ on first introduces (in [5, §14]) the L2 (R)-valued function of t ˆ F (t) = T u = (τy u)t−1 ϕ(t−1 y) dy, R

where the integral is to be interpreted as a Riemann vector valued one. In other words, F (t)(u) = T u(t) = u ∗ ϕt . Letting now ψ1 , ψ2 ∈ C ∞ , spherically symmetric and with compact support, he further defines the operator S(F, u) = S(T u, u) = u ∗ ψ2 + Cu, ˆ

where Cu =

1

∞

ˆ R

τy F (t−1 )ψ1 (ty)dy dt.

Note now that, by selecting ψ1 = ϕ, and further writing φt = ϕt−1 we have in fact ˆ ∞ dt u ∗ φt ∗ φt . Cu = t 1 Calder´ on claims and later proves (in §34) that it is possible to select ψ1 and ψ2 to recover u through the operator S: u = S(T u, u). It follows that ˆ ∞ dt u ∗ φt ∗ φt , u = u ∗ ψ2 + t 1 which bears a strong resemblance to what is nowadays commonly referred to as Calder´ on’s reproducing formula. The formula, even in a more modern form, appears also to have been known to Calder´ on and others around him. But surprisingly, the next published version of a more detailed CRF that we are aware of did not appear until 10 years later in an article by N. J. H. Heideman in 1974, [37]. Therein the author writes in page 68: “We first prove a theorem of A.P. Calder´ on in [3] [here Calder´ on [5]] and a seminar at the University of Chicago...”

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The theorem he is referring to translates into the following. Theorem 1.1. Let μ be an L1 (Rn ) function so that its Fourier transform ´ μ &(ξ) = Rn μ(x)e−ix·ξ dx verifies that, as a function of t, μ &(tξ) is not identically zero for ξ = 0. Then, there exist a function φ in the Schwartz space S whose Fourier transform is compactly supported outside the origin and another function ψ ∈ S so that ˆ ∞ dt u ∗ μt ∗ φ t , (1.2) u= t 0 and ˆ 1 dt (1.3) u=u∗ψ+ u ∗ μt ∗ φ t , t 0 for each tempered distribution u ∈ S  . Actually Heideman stated the theorem more generally, the dilations by 1/t replaced by a certain one parameter group of operators Tt and μ a Borel measure with an appropriate Tauberian condition replacing the non-vanishing condition in the statement above. Heideman implicitly uses in his proof the existence of the improper integral in (1.2) as a distribution, allowing him to freely interchange the integration with the action of the Fourier transform. That this was the first explicit version in print of the formula in a form more similar to how it is used today seems to be corroborated by the comments in the 1981 article by S. Janson and M. Taibleson [42]. In fact, the authors state on page 29: “Il teorema di rappresentazione ´e stato formulato per la prima volta nel corso di un seminario alla University of Chicago tenuto da Alberto Calder´ on intorno al 1960 ed ´e stato usato (implicitamente) da Calder´ on in [5]. In forma esplicita compare nell’articolo di N. Heideman [2] [here [37]] e in molti altri successive.” Like those around Calder´ on in Chicago learning about his formula in seminars in the 60’s, those around Washington University in the 80’s (such as the second named author of this article) learned about Janson and Taibleson’s paper directly from Mitch. The authors in [42] made explicit the implicit weak convergence argument in Heideman’s paper looking at the limit of truncated integrals ˆ R dt u ∗ μt ∗ φ t t  in the sense of distributions. They rigorously showed, in particular, that given the function μ one can always construct the associated φ so that ˆ ∞ dt & =1 μ &(tξ)φ(tξ) t 0 for all ξ = 0, and ψ ∈ S, ˆ ∞ dt & = & ψ(ξ) μ &(tξ)φ(tξ) t 1 & for ξ = 0 and ψ(ξ) ≡ 1 in a neighborhood of the origin, such that (1.3) holds for u ∈ S  and in the sense of distributions. However, as observed by M. Wilson in [69] (where we also find a very useful survey of some of the history of CRF), Janson and

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Taibleson seem to be the first to state that the continuous homogeneous formula (1.2) cannot actually hold for an arbitrary distribution. The reason is simple: since φ& vanishes at the origin, convolution with φ is completely blind to distributions whose Fourier transforms are supported at the origin. Such distributions are the inverse Fourier transforms of finite linear combinations of the delta function and its derivatives at the origin, hence they are polynomials. Janson and Taibleson established then the validity of the homogeneous formula (1.2) for tempered distributions but with weak convergence modulo polynomials. An earlier study of this modulo polynomial convergence, but in a simpler discrete version of the CRF, is in the book by Peetre [53] (we will come back to this convergence in the next section). Presumably, some of the “several subsequent” works referred to by Janson and Taibleson in the above quote from their work implicitly include Calder´ on’s three articles on parabolic H p spaces, two in collaboration with A. Torchinsky, [8], [9] and [6]. These works were published in the period 1975-1977, hence before [42]. In particular, it is indicated in [9, pp.130-131] that the formula converges pointwise for smooth functions u with compact support, and in [6, pp. 219-220] that it converges when paired against test functions if u ∈ H p . The 80’s saw a proliferation of very important results in harmonic analysis that were proved taking advantage of the CRF. Some relevant appearances of the formula that should be mentioned are in the works of Chang-Fefferman [10] (where the formula is established by formally exchanging Fourier transform and integration), and Uchiyama [66] and Wilson [68], where the CRF plays a central role in constructing atomic decompositions. During this time, the formula started to be presented more frequently as ˆ ∞ dt u ∗ φt ∗ φt , u= t 0 for an appropriately normalized function φ (see Theorem 1.2 below). Another crucial use of the formula in the same decade is in the celebrated proof of the T 1-theorem of David and Journ´e in [19]. In that work, the authors also prove the convergence in L1 of the truncated integrals in the CRF for functions in L1 with mean zero. In their own proof of the T 1-theorem, Coifman and Meyer [13] made a very nice use of the so-called “Pt −Qt formula” (which goes back to their work with McIntosh [12]) where one can recognize traces of Calder´on’s own formula, except this time used to decompose an operator instead of a function. In an application to study the minimality of the Besov space B˙ 10,1 , Meyer [50] proved the convergence of the formula in such space norm. In the mid 80’s the CRF was rediscovered within the mathematical physics community; it is studied in the article of Grossman and Morlet [35] and other related works which helped jump start much of the wavelet revolution. In fact, the CRF can be written as ˆ ∞ˆ dydt u, φt,y φt,y (x) n+1 , (1.4) u= t Rn 0 where now φt,y (x) = t−n/2 φ((x − y)/t) and the last expression is nowadays referred to as the “continuous wavelet representation” of u. It turns out that the CRF converges in a very strong sense, namely in L2 . This has been folklore in the subject based on a formal computation using the Fourier transform, but we could not find a published explicit rigorous argument for this

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convergence until the book by Frazier, Jawerth and Weiss [30]. This work was published in 1991 but was based on lectures at a CBMS conference in Alabama in 1989. Within the wavelet literature, the L2 convergence can be traced to the book of I. Daubechies [17, pp. 24-30], where pointwise convergence for some continuous function is also presented. Though not commonly found in the literature, the CRF converges almost everywhere for arbitrary functions in L2 , as stated in the following theorem. Theorem 1.2. Assume that φ ∈ S(Rn ) is real valued, radial, with Fourier transform compactly supported away from the origin, and such that ˆ ∞ 2 dt & =1 (1.5) φ(tξ) t 0 for all ξ = 0. Then, for all u ∈ L2 (Rn ), we have ˆ ∞ dt (1.6) u= u ∗ φt ∗ φt , t 0 where the convergence ˆ u = lim+ u,R = lim+ →0 R→+∞

→0 R→+∞

R

u ∗ φt ∗ φt 

dt t

holds both in the L2 and the pointwise almost everywhere sense. Moreover, Lp and pointwise a.e. convergence also hold for u ∈ Lp , 1 < p < ∞. Proof. As mentioned, the statement for L2 appeared in this form in [30, §1], from where we repeat the proof. It is clearly not a challenge to construct a function in S, real valued, radial, and with Fourier transform compactly supported away from the origin. For any such function, it is easy to see that the integral in the righthand side of (1.5) is (a real) constant for any ξ = 0. Hence, normalizing any such function one obtains (1.5). Working formally, the identity (1.6) follows by simply comparing the Fourier & using the integral transforms of each side of (1.6). In particular, since φ&t (ξ) = φ(tξ), condition on φ, the Fourier transform of the right hand side simplifies to u &. The rigorous justification of this is a nice exercise in real analysis which uses the Fubini, Plancherel, and Lebesgue dominated convergence theorems. First, the argument for u ∈ L2 ∩ L1 goes as follows. One observes that in this case u ∗ φt ∗ φt ∈ L1 (Rn ) for all t > 0; in fact, u ∗ φt ∗ φt L1 ≤ uL1 φ2L1 . A similar reasoning shows that, for all 0 <  < R , we have u,R L1 ≤ ln(R/)uL1 φ2L1 , thus u,R ∈ L1 (Rn ). Now, using Fubini’s theorem we get ˆ u' &(ξ) ,R (ξ) = u 

R

2 dt & . φ(tξ) t

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Therefore, combining Plancherel and Lebesgue’s dominated convergence theorems leads to &2L2 u,R − u2L2 = (2π)−n u' ,R − u ˆ ˆ  −n 2 = (2π) |& u(ξ)| 1 − Rn

R 

2 dt & φ(tξ) t

2

dξ → 0

as  → 0 and R → ∞. Next, consider the case of a general function u ∈ L2 (Rn ) and let uj ∈ L2 ∩ L1 be such that uj − uL2 → 0 as j → ∞. Then (1.7)

u,R − uL2 ≤uj − uL2 + (uj ),R − uj L2  ˆ R dt   (uj − u) ∗ φt ∗ φt  2 . +  t L 

As explained in the L2 ∩ L1 case above, we see now that, since the function of ξ,  2 ´R 2 dt & 1 −  φ(tξ) , takes its values in [0, 1] and tends pointwise to zero as  → 0 t and R → ∞, we can write  (uj ),R − uj 2L2 = (2π)−n (u &j 2L2 j ),R − u ˆ ˆ R  2 2 dt &  |& uj (ξ)|2 1 − dξ φ(tξ) t Rn  ˆ R ˆ  2 2 dt &  & uj − u &2L2 + |& u(ξ)|2 1 − dξ φ(tξ) t Rn   uj − u2L2 + τ, for arbitrary small τ > 0 if  sufficiently small and R sufficiently large. Also, using Minkowksi’s inequality, we can bound the third term on the right hand side of (1.7) by ˆ R dt (uj − u) ∗ φt ∗ φt L2 t  ˆ R dt ≤ uj − uL2 φt 2L1  ln(R/)uj − uL2 . t  All in all, we have obtained that for  and 1/R sufficiently small depending on τ but fixed, u,R − uL2  uj − uL2 (1 + ln(R/)) + τ. Finally, for j large enough, we can then obtain u,R − uL2  τ, thus proving Calder´ on’s reproducing formula in this case as well. The pointwise almost everywhere convergence can be established in the following way. Following [19, p. 376], we observe that if φ ∈ S is as above, then there exists a function η ∈ S such that η&(0) = 1 and d (u ∗ ηt ) = u ∗ φt ∗ φt . dt We note that (1.8) can be viewed as an identity relating the families of operators d Qt (u) = u ∗ φt and Pt (u) = u ∗ ηt , namely Q2t = −t dt Pt . To verify the conditions

(1.8)

−t

´ THE DISCRETE CALDERON REPRODUCING FORMULA

on η simply set

ˆ

87

1

2 dt & . φ(tξ) t 0 We note immediately that η&(0) = 1. Then, since φ& ∈ S, for all multi-indices α and all N ∈ N, we can write for ξ = 0 and any M > max(|α|, N ), ˆ ∞ N α & 2 )(tξ)|t|α|−1 dt |ξ| |∂ η&(ξ)|  |ξ|N (∂ α |(φ) 1 ˆ ∞  |ξ|N |tξ|−M t|α|−1 dt

η&(ξ) = 1 −

1

 |ξ|N −M → 0 as |ξ| → ∞; thus proving that η& ∈ S and hence η ∈ S. ´∞ 2 ds & Moreover, using the fact that φ is radial we get that η&(tξ) = t φ(sξ) s , d 1 & 2 which implies dt η&(tξ) = − t (φt ) (ξ). The last equality gives the identity (1.8). If u ∈ L2 (Rn ), we simply need to recall that for such η ∈ S(Rn ) with η&(0) = 1, Pt forms an approximation to the identity. That is, we have the almost everywhere convergence of u ∗ ηt to u as t → 0+ . Note also that |u ∗ ηt (x)| ≤ uL2 ηt L2  t−n/2 uL2 ,

(1.9)

since ηt is L1 -normalized, and hence u ∗ ηt → 0 as t → ∞. Therefore, using now (1.8), we see that ˆ R ˆ R d dt = lim+ − (u ∗ ηt )(x)dt = u(x) u ∗ φt ∗ φt (x) lim+ t dt →0 →0   R→+∞

R→+∞

for a.e x. The method used for the pointwise convergence and again a simple limiting argument give also the convergence in Lp for 1 < p < ∞. Indeed, adapting the arguments from [64], we first note that for u ∈ L2 ∩ Lp , u ∗ ηt converges to u in Lp as t → 0+ and |u ∗ ηt (x)|  M u(x), where M is the Hardy-Littlewood maximal function. By dominated convergence, u ∗ ηt converges to 0 in Lp as t → ∞ and hence u,R converges to u. Now, for a general u ∈ Lp , we can approximate it by functions uj ∈ L2 ∩ Lp . Observe also that, by Minkowksi’s inequality, u,R makes sense in Lp and u,R − (uj ),R Lp ,R M (u − uj )Lp ,R u − uj Lp , while (u ∗ η − u ∗ ηR ) − (uj ∗ η − uj ∗ ηR )Lp  M (u − uj )Lp  u − uj Lp . It follows that we also have, for arbitrary u ∈ Lp and any 0 <  < R < ∞, u,R = u ∗ η − u ∗ ηR . Finally, notice that in place of (1.9) we have |u ∗ ηR (x)| ≤ uLp ηR Lp  R−n/p uLp , where 1/p + 1/p = 1, and so the same arguments used in the case p = 2 give the convergence for general p. 

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We want to remark that the conditions we have imposed on the function φ are to some extent excessive and they could be relaxed quite a bit (though it is easy to see that φ& must be zero at the origin and infinity). Also, as done in [30], it is possible and sometimes convenient to use φ with compact support instead of having φ& with such property. A lot has been and continues to be written about conditions on φ for the formula to hold and also about the converge of the CRF in many other senses and various function spaces. For example, S. Saeki [56] proved that for u ∈ Lp (Rn ), 1 < p < ∞, the integrals ˆ ∞ dt u ∗ φt ∗ φt (x) t y converge nontangentially to u(x) at (x, 0) ∈ R(n+1)+ , if x is a Lebesgue point of u. For 1 < p < ∞, M. Wilson [69] proved convergence in the norm of Lp (w), where w is an Ap weight, allowing also more general truncations of the improper integral in the formula. K. Li and W. Sun [47] established then pointwise almost everywhere convergence in Lp (w). Wilson also obtained in [70] convergence in H 1 and weak∗ in BM O (as the dual of H 1 ). More recently, the second named author of this article and E. Ward proved in the already cited article [64] the convergence of a natural analog of the CRF in mixed Lebesgue spaces Lp Lq (Rn × R). There is also a rich theory with versions of the CRF in more abstract group theoretic setting. One of the first works in this direction is the article by H. Feichtinger and K. Gr¨ ochenig [21] involving square integrable representations. More references to the literature dealing with wavelets generated through the action of other groups are given in the survey by E. Wilson and G. Weiss [67]. One may speculate that the continuous CRF converges in norm as a limit of the functions u,R in any (homogeneous) space which admits a Littlewood-Paley decomposition; in particular, in all the homogeneous Besov and Triebel-Lizorkin spaces B˙ pα,q and F˙ pα,q .3 However, we have not been able to locate an explicit reference for this fact in the literature. Such a general result does exist for the discrete formula in the next section, as proved by Frazier and Jawerth [26, 28] (see also [50] and the references therein for discrete orthonormal wavelets). Nonetheless, in many applications, the following very strong convergence in the appropriate common dense subspace for most B˙ pα,q and F˙ pα,q spaces suffices to justify many uses of the CRF. Proposition 1.3. Let S0 = {u ∈ S :

ˆ u(x)xα dx = 0 for all multi-indices α}.

Then, if u ∈ S0 , lim→0 u,1/ = u in the topology of S. Proof. Note that since φ& has compact support away from the origin, for each  > 0 we have ! ˆ 1/ dt 2 & φ(tξ) u &(ξ) 1 − =0 t  3 We will not need the definition of these spaces for this survey. The norm of some of these spaces, however, needs to be interpreted modulo polynomials of appropriate degrees, which is consistent with the convergence modulo polynomials of the CRF for arbitrary distributions.

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unless |ξ|   or |ξ|  1/. Also, for all α, M , we have |∂ α u &(ξ)| α,M (1 + |ξ|)−M because u & ∈ S. In addition, for all multi-indices α, β, |∂ α u &(ξ)| α,β |ξ||β| because ∂ α u &(0) = 0 for all α. With these estimates, it is easy to see that (1 + |ξ|)M |∂ α (& u(ξ) − u &,1/ (ξ))| α,M , which proves the required convergence.



Naturally, a simpler version of the reproducing formula also holds: supposing that ψ is such that ˆ ∞ dt & = 1, ψ(tξ) t 0 then ˆ ∞ dt u= u ∗ ψt . t 0 But the advantage of having the double convolution φt ∗ φt in Calder´on’s formula is that both convolution factors u ∗ φt and φt are with compact support on the Fourier side, while in the simpler version above only one of the factors, ψt , has this property. As it will become clear, having the double convolution proves to be a crucial ingredient in the discrete version of the formula by Frazier and Jawerth. 2. The Discrete Calder´ on Reproducing Formula A full discretization of Calder´on’s identity through appropriate Riemann sums may be used to motivate the Frazier-Jawerth φ-transform or discrete wavelet decompositions. We first work at a formal level. Suppose that φ is as in the statement of Theorem 1.2. For ν ∈ Z, k ∈ Zn , let Q = {Qνk }ν,k be the collection of dyadic cubes, where Qνk has side length l(Q) = 2−ν and lower left corner xQ = 2−ν k. By making the change of variable t = 2−μ in (1.6), we see that, modulo a multiplicative constant,  ˆ ν+1 ˆ u ∗ φ2−μ (y)φ2−μ (x − y)dy dμ. u(x) = ν∈Z

ν

Rn

The integrand in the last expression depends smoothly on μ so we can roughly approximate the integration in μ by the value of the integrand at say μ = ν. Hence, breaking the integral in Rn into dyadic cubes of side length 2−ν , we get   ˆ u ∗ φ2−ν (y)φ2−ν (x − y)dy. u(x) ≈ ν∈Z k∈Zn

Qνk

Note now that the Fourier transforms of both functions u ∗ φ2−ν and φ2−ν are supported around frequency of the order 2ν , and so the functions themselves cannot oscillate too much on a cube of side length 2−ν . We can then attempt to write the Riemann sum approximation   u(x) ≈ u ∗ φ2−ν (2−ν k)φ2−ν (x − 2−ν k)2−νn . ν∈Z k∈Zn

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Of course, we have ignored issues about convergence and have not precisely quantified any of the approximation, but this simple reasoning clearly suggests the possibility of a discrete φ-transform or wavelet expansion  (2.1) u= u, φνk φνk , ν,k 4

where the functions

φνk (x) = 2νn/2 φ(2ν x − k) = 2−νn/2 φ2−ν (x − 2−ν k)

(2.2)

are translated and dilated version of single function φ. A first very natural and rigorous discretization of the CRF, which appeared many times over the years and sometimes independently of Calder´on’s identity, has been extensively used in the context of function spaces on Rn and their LittlewoodPaley characterizations; see the books by Stein [57], Peetre [53], Triebel [65], and the many references therein for historical accounts. This discretization is sometimes stated in the following form. Let φ ∈ S be radial, real valued and such that φ& is & supported in the annulus π/4 < |ξ| < π. Assume also that |φ(ξ)| is bounded away from zero on a smaller annulus π/4 +  ≤ |ξ| ≤ π − , and  & −ν ξ)2 = 1, ξ = 0. φ(2 (2.3) ν∈Z

Such a function φ is sometimes referred to as “admissible”. The existence of a function ψ which satisfies all the above properties except possibly (2.3) is straightforward. Noticing that any such function satisfies for some 0 < c < C < ∞  & −ν ξ)2 < C, c< ψ(2 ν∈Z

one can define φ as desired by letting & = φ(ξ) 

& ψ(ξ) & −ν ξ)2 ν∈Z ψ(2

1/2 .

5

Then again, in various appropriate senses,  (2.4) u= u ∗ φ2−ν ∗ φ2−ν . ν∈Z

For example, we have the following result. Proposition 2.1. If u ∈ L2 , then the convergence of (2.4) holds in L2 sense, while, if u ∈ S0 , then the convergence is in S. In fact, the proof of the convergence in L2 is rather trivial while the one for functions in S0 can be obtained in a similar fashion to Proposition 1.3. Also, although the righthand side of the above formula makes sense for an arbitrary distribution u ∈ S  , there are issues with the convergence since, yet again, the functions φ&2−ν are completely blind to distributions supported at the frequency point ξ = 0. As mentioned earlier, Peetre [53, pp.52-54] seems to have been the first to rigorously address this issue. For the benefit of the reader, and since the 4 For

simplicity, we are assuming φ to be radial. that the conditions on the support of ψ make φ a smooth function!

5 Note

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reference [53] is not always readily available, we include here a sketch of Peetre’s result. Theorem 2.2. Let φ be an admissible function satisfying (2.3). Then, for each u ∈ S  , there exist a sequence of polynomials {PN }N of bounded degree (depending on u) and another polynomial P such that ! ! N −1   u = lim u ∗ φ2−ν ∗ φ2−ν + lim u ∗ φ2−ν ∗ φ2−ν − PN + P, N →∞

N →∞

ν=0

ν=−N

where the limits are taken in S  . Proof. Similar arguments to those in Proposition 1.3 show that for any g ∈ S ∞ 

& −ν ξ)2 g(ξ) φ(2

ν=0

converges in S, so if ·, · denotes the pairing of distributions with test functions, we have ) ( ∞ ) (N   (φ&2−ν )2 u &, g = u &, (φ&2−ν )2 g . lim N →∞

ν=0

ν=0

It follows that u∞ ≡ lim

N →∞

N 

! u ∗ φ2−ν ∗ φ2−ν

ν=0

exists in S  . Next, since u ∈ S  , there exists a continuous seminorm qL,M on S, qL,M (g) =

sup ξ∈Rn ,|α| 0 we can write         (1 + |x|2 )M α (1 + |x|2 )M  α,M f (k) (∂ g)(x − k) |f (k)|   (1 + |x − k|2 )M   |k|>N |k|>N  α,M |f (k)| (1 + |k|2 )M → 0 as N → ∞. |k|>N

In the estimate above, we have used Peetre’s inequality and the fast decay of the samples |f (k)| for f ∈ S.  As with the classical sampling theorem, there is of course a rescaled version of (2.8), with samples taken according to the Nyquist rate at a distance inversely proportional to the radius of the spectrum of the functions f and g. More precisely, if supp f&, supp g& ⊂ {ξ ∈ Rn : |ξ| < 2ν π}, then  f (2−ν k) g(x − 2−ν k). (2.9) (f ∗ g)(x) = 2−νn k∈Zn

Using now the formula (2.4), and then for each ν the formula (2.9) with f = u ∗ φ2−ν and g = φ2−ν , Frazier and Jawerth [26] arrived in a truly “painless” way to a simple yet deep and amazing discrete CRF: their φ-transform.7 Theorem 2.4. If φ is admissible, then   (2.10) u(·) = 2−νn (u ∗ φ2−ν )(2−ν k) φ2−ν (· − 2−ν k), ν∈Z

k∈Zn

where for u ∈ L the convergence is in L2 , for u ∈ S0 the convergence is in S, and for u ∈ S  the convergence is in S  modulo polynomials (in the sense of Theorem 2.2). 2

The just stated theorem above gives us, at last, a rigorous proof of the formula   (2.11) u= u, φνk φνk , ν∈Z k∈Zn

with φνk as in (2.2). It is remarkable that the Frazier-Jawerth approach produces so “easily” the above representation, which looks precisely like an expansion in an orthonormal basis of wavelets. By contrast, the construction of orthonormal wavelets with a generating function in S is equally beautiful but a rather intricate task. The first construction of an orthonormal basis of wavelets in L2 (Rn ) generated by Schwartz functions was published by Lemari´e and Meyer [46]; it is quite remarkable that 7 We were informed by the reviewer of this article that a verbal comment made by Jawerth during a lecture suggested that perhaps a germ of the formula was already present in Jawerth’s PhD thesis at Lund in Sweeden. It may have been used by him to prove the existence of unconditional bases for Besov spaces for 0 < p < 1. Unfortunately we have had no access to the thesis, but it would be of interest to research this topic further.

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such a construction is possible. For an earlier construction of spline wavelets see the work of Str¨ omberg [59], while for further compactly supported wavelets see Daubechies [16]. The method of Frazier and Jawerth, however, does not produce an orthonormal basis8 since not all the functions φνk are orthogonal to each other. Moreover, it does not even produce a basis but only a frame in L2 when φ ∈ S. Nonetheless, as mentioned earlier, this is of no technical consequence in many uses of the representation. Once again, the method allows for a lot of flexibility and one can use two different functions φ and ψ satisfying (2.5) instead of (2.3). This leads to the expansion   2−νn (u ∗ φ2−ν )(2−ν k) ψ2−ν (· − 2−ν k), (2.12) u(·) = ν∈Z

k∈Zn

or (2.13)

u=

 

u, φνk ψνk ,

ν∈Z k∈Zn

with ψνk defined in same way as φνk . Moreover, there is also an inhomogeneous version (as with discrete wavelets) in which all the low frequency supported functions φ2−ν and ψ2−ν are replaced by two single functions Φ and Ψ, so that  & −ν ξ) = 1 & −ν ξ)ψ(2 & Ψ(ξ) & φ(2 (2.14) Φ(ξ) + ν≥1

for all ξ ∈ R . This in turn leads to    u, Φ0k Ψ0k + u, φνk ψνk , (2.15) u= n

k∈Zn

ν≥1 k∈Zn

where Φ0k = Φ(· − k) and, similarly, Ψ0k = Ψ(· − k); see [28]. An advantage of this formula is that it now “honestly” converges in S and S  and it is best suited to study inhomogeneous Besov and Triebel-Lizorkin spaces. 3. Characterization of function spaces, atoms, and molecules The fact that the Fourier transform is no longer an isometry in Lp for p = 2 is a great example of how unfortunate constraints could sometimes be turned into fantastic opportunities (at least in the good hands of virtuoso mathematicians). Trying to overcome the impossibility of measuring Lp properties of functions directly from the Fourier transform promoted the creation of powerful mathematical tools, which have had and continue to have impressive efficacy in harmonic analysis, operator theory and partial differential equations. Littlewood-Paley theory breaks the Fourier transform of a function into mildly interfering frequency bands that allow the recovery of information related to function spaces beyond the Hilbert space context. Owing its origin to Littlewood and Paley in the 30’s, the theory took its full force in Rn with Stein’s introduction of his g-function in the late 50’s. Chapter IV of his book [57] has a marvelous and not only authoritative but also 8 It is actually possible to construct smooth orthonormal wavelets (“Shannon wavelets”) from  a characteristic function of an appropriate the sampling theorem approach if one starts with φ annulus. Moreover, such an approach can also be carried out in other groups beyond Rn ; see [24]. However this gives wavelets that, for example in dimension one, only decay like |x|−1 at infinity.

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still one of the most educational presentations of the topic. We quote from Stein’s presentation [57, p. 81] of the g-function, of which he says that “aside from its applications, it illustrates the principle that often the most fruitful way of characterizing various analytic situations (such as finiteness of Lp norms, existence of limits almost everywhere, etc.) is in terms of appropriate quadratic expressions.” Calder´ on’s reproducing formula is strongly connected to Littlewood-Paley theory. This is nicely illustrated in [30]. Let P denote the Poisson kernel for the upper half-space, P (x) = cn (1 + |x|2 )−(n+1)/2 , U (x, t) = (Pt ∗ u)(x), and consider a particular version of the g-function,

ˆ ∞ 1/2 ˆ ∞ 1/2 2 dt 2 dt g(u)(x) = |t∂t U (x, t)| = |φt ∗ u(x)| , t t 0 0 & = −|ξ|P&(ξ) = −|ξ|e−|ξ| . Littlewood-Paley theory says that, for 1 < p < where φ(ξ) ∞, we have uLp ≈ g(u)Lp , while by the CRF,

ˆ

∞

u(x) =

φt ∗ φt ∗ u(x)

0

dt . t

That is, we can view u → φt ∗ u as a bounded map from Lp (Rn ) into Lp (Rn , L2 ((0, ∞), dtdx t )) which is inverted by the CRF. Likewise, there is a discrete version of this fact: if φ is an admissible function, then  !1/2       2  , −ν uLp ≈  |φ ∗ u| 2    p  ν∈Z L

while other quadratic (or q-power) functions characterize numerous function spaces. It is perhaps not surprising then that the formulas (2.11), (2.13) or (2.15) could be related through Littlewood-Paley theory to the characterization of such spaces. It is quite incredible, however, that it is just the size of the coefficients in those expansions what gives the characterization. Indeed, Frazier and Jawerth showed in [26] and [28] that any space X in the Besov or Triebel-Lizorkin scales can be characterized by a corresponding space of coefficients S(X)9 . These discrete spaces are defined for sequences {sQ } indexed by the dyadic cubes and with norms in terms of {|Q|c(X) |sQ |}, where the power c(X) depends on the function space X. For some L2 -based spaces X, the spaces S(X) are simply weighted l2 spaces. For example, for X = L2 , the associated space 9 Frazier and Jawerth denote the sequence spaces associated with B ˙ pα,q , respectively F˙pα,q , by α,q ˙ , respectively f . b˙ α,q p p

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is S(L2 ) = l2 , and for L˙ 2α (= B˙ 2α,2 = F˙ 2α,2 ), the homogeneous Sobolev spaces of functions with their “derivatives of order α” in L2 , one has the weighted l2 norm ⎞1/2 ⎛ 2  {sQνk }S(L˙ 2α ) = ⎝ |Qνk |−α/n |sQνk | ⎠ . ν,k

For most spaces X based on L with p = 2, S(X) is more complicated than a weighted lp space, but it is still a Banach or quasi-Banach lattice with norm only depending on |sQ |; see [28]. In proving the characterizations p

uX ≈ {|u, φνk |}S(X) , Frazier and Jawerth [26, 28] developed also smooth atomic and molecular decomposition of the function spaces considered. There is a rich history in harmonic analysis related to atomic decompositions going back to the pioneering work of Coifman [11]. The atomic decomposition in [26, 28], however, seems most influenced by Uchiyama’s one for L2 and BM O in [66], which had been used by Frazier in his thesis-related paper [22]. Frazier and Jawerth also mention “an alternate direction” to decompositions (though somehow similar spirit) in the articles by Coifman and Rochberg [14] and Ricci and Taibleson [54]. A very interesting fact is the universality of the construction in [26, 28] of smooth atomic decompositions for Lp , Hardy and Sobolev spaces and, more generally, the whole scales of Besov or Triebel-Lizorkin spaces. Starting with the decomposition  θ2−ν ∗ φ2−ν ∗ u, u= ν∈Z

where θ is selected as in (2.5), having now compact support and a prescribed number of vanishing moments (which could be arbitrarily large but finite), one can rewrite a function u ∈ X as   sQ aQ . (3.1) u= ν∈Z k∈Zn n

Here Q = Qνk is a dyadic cube in R ,

sQ = sQνk = |Q|1/2 sup |φ2−ν ∗ u(y)| y∈Q

and

1

ˆ

θ −ν (x − y) φ2−ν ∗ u(y) dy. sQνk Qνk 2 The family {aQ } of smooth atoms, as defined in [26, 28], satisfies some desirable properties: the functions aQ are C ∞ , compactly supported on a dilate of Q, and have as many vanishing moments as θ. They are also normalized10 as the functions φνk and so |∂ γ aQ (x)| γ |Q|−1/2−|γ|/n . Moreover, if aQ have enough vanishing moments depending on the space X, it can be proved that {sQ }S(X)  uX . aQ = aQνk =

10 We note that, actually, the normalization in [26] is different from the one in [28], but consistent with appropriate modifications in the definitions of the spaces S(X) in those works.

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Likewise, Frazier and Jawerth showed that whenever   u= s Q φQ , ν∈Z k∈Zn

with φQ = φν,k as before, then also {sQ }S(X)  uX . The converse inequality needed for the characterization of the function spaces is actually established for a more general class of building blocks: a family {mQ } of smooth molecules. The molecules, for a given space X, are a rougher and less oscillating version of the wavelets φνk . They also have some limited decay away from the cube Q = Qνk and are normalized like the functions φνk . More precisely, they satisfy (3.2)

|∂ γ mQ (x)| 

|Q|−1/2−|γ|/n (1 +

|x−xQ | N l(Q) )

=

2ν(n/2+|γ|) (1 + |2ν x − k|)N

for all |γ| ≤ M and with N and M dependending on the function space X considered. In addition, the molecules have a number of vanishing moments also depending on X. The family of atoms and the family of functions φνk are hence molecules as well. It is important to note that, although the mQ satisfy estimates which behave as if the molecules were the translations and dilations of a single function m, they do not need to be so. Frazier and Jawerth showed that whenever   s Q mQ f= ν∈Z k∈Zn 

(most generally in S modulo polynomials), then f ∈ X and f X  {sQ }S(X) , completing the characterization of X in terms of the coefficients in the expansion (2.11). There are also inhomogeneous versions of all of the above characterizations. The theme of [28] (see also [29]) is to understand the properties of the sequence spaces S(X) in terms of duality, interpolation, traces, etc. and to then use them to derive in a functorial way analogous results for the corresponding function or distribution spaces X. Incidentally, the characterization of function spaces in terms of the coefficients proves also the convergence of the representation formula in the norm of such spaces. Regarding pointwise convergence, various results can be obtained. We point to the work on wavelets by Kelly, Kon, and Raphael [43, 44] where rates of convergence are also given, and to the article by Tao [61], where various summation methods for pointwise convergence are considered. Molecular characterizations of functions spaces have a long history as well; see, for example, the works of Taibleson and Weiss [60] and Coifman and Weiss [15]. It turns out that, as with other types of molecules in the literature, singular integrals and other important multiplier operators map smooth atoms into molecules (but not atoms into atoms). We will show a very simple application of this in the next section. What we have just briefly summarized about the characterization of functions spaces takes actually a tremendous arsenal of harmonic analysis tools to be rigorously achieved, as well as a wise use of them [26, 28]. Such tools include, in particular, the Plancherel-Polya inequality for functions of exponential type, the Peetre maximal function, and the Fefferman-Stein vector valued maximal function,

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which should warn the reader of the deep technical aspects of the proofs we have omitted. We should also point out that, though completely equivalent, the characterizations of functions spaces using the φ-transform or using wavelets as presented by Meyer in his books [51] (see also [46]) have some different features. The FrazierJawerth approach presents a unifying theory relying on the use of sophisticated machinery about function spaces starting from their Littlewood-Paley characterizations. The approach by Meyer is more direct, treating several different function spaces separately and exploiting also some of their own specific defining properties. The generating function in the non-orthogonal φ-transform decomposition is always a single function in S0 (Rn ) in every dimension, which is very convenient in some applications. The orthogonal wavelets can be constructed with functions in S0 but also with functions with minimal regularity properties mirroring the particular function space they are used to span and/or with compact support, which are also convenient properties in other situations. Both approaches are very instructive and, actually, the characterizations of the spaces in terms of the coefficients in the expansions can be obtained from each other essentially by a “change of basis”. Indeed, this is related to another crucial notion in this development, which is that of the “almost diagonal operator” succinctly described in the next section. 4. Almost orthogonality and almost diagonal operators From early on in their work, see [26, p. 797], Frazier and Jawerth observed the potential of reducing the study of an operator on a function space X to an associate matrix on the discrete space S(X). Let us recall in an informal way the philosophy behind the diagonalization of operators by a family of generating functions (essentially, linear algebra!). Let T be some linear operator acting on one of the spaces X. Starting with the φ-transform for u ∈ X,  u, φQ φQ , u= Q

and applying T (while ignoring for the moment convergence issues) one gets  u, φQ T (φQ ). T (u) = Q

Expanding now T (φQ ), we obtain    T (u) = u, φQ T (φQ ), φP φP = T (φQ ), φP u, φQ φP . Q

P

P

Q

It follows from the characterization of X in terms of the discrete space of coefficients S(X) that, to study the boundedness properties of T on X, it would be enough to study the boundedness properties of an infinite matrix AT = {AT (P, Q)}P,Q = {T (φQ ), φP }P,Q indexed by the dyadic cubes and acting on S(X). Ideally, if the family {φQ } is a basis of eigenvectors for T , then AT is just a diagonal matrix which is trivial to study. However, this requires constructing a generating family of functions for each operator T to be studied; something extremely unlikely to be done given all the constrains the functions φQ must satisfy to expand the space X. The notion of almost diagonalitzation consists then in

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relaxing the perfect diagonalization of each operator by looking instead at whole families of operators which come close to be diagonalized by the functions φQ . That is, instead of requiring T (φQ ) = cQ φQ , one is content with T (φQ ) ≈ φQ in some appropriate sense. Still wishfully thinking along these lines, one would then have T (φQ ), φP ≈ φQ , φP , and for the matrix AT , AT (P, Q) → 0

(4.1)

as the entry (P, Q) “moves away” from the diagonal of the matrix. Moreover, recall that the spaces S(X) are defined only in terms of the size of the coefficients |sQ |, so it may be possible to study the positive matrix |AT | instead of AT . Studying positive matrices is often a lot easier; in particular, there is the well-known criterion to study them given by Schur’s test. Frazier and Jawerth found a very precise quantification of (4.1), which implies the boundedness of a matrix on S(X) for different spaces X. For example, if Q = Qνk and P = Pμl , then a matrix A = {A(P, Q)}P,Q satisfying the “almost diagonal condition” (4.2)

|A(P, Q)| 

2−(μ−ν)(ε+n/2) (1 + 2ν |2−ν k − 2−μ l|)n+

for some  > 0 and ν ≤ μ, and symmetric estimates if ν > μ, is bounded on S(Lp ), 1 < p < ∞. Notice that 1+

|2−ν k −2−μ l| diam(Q ∪ P ) ≈1+ l(Q) 2−ν

so we can also rewrite (4.2) as

ε+n/2 l(P ) 1 . |A(P, Q)|  l(Q) (1 + l(Q)−1 diam(Q ∪ P ))n+ε We clearly see in this particular case that the entries of the matrix are small if the Euclidean distance between the cubes is large or if their sizes are very different. There are other similar size conditions for matrices acting on different sequence spaces S(X) (depending on the parameters defining the corresponding spaces X) which are sufficient to establish the boundedness of the matrix essentially through some Schur’s test argument. Moreover, Frazier and Jawerth showed that such matrices are essentially those of operators which map atoms into molecules or molecules into molecules. Instead of presenting the technical details of such results, we look at a very intuitive reasoning of why it may be possible to obtain something like (4.2). Assume that T (φQ ) = mQ , where mQ is a molecule, and consider the matrix AT = {T (φQ ), φP }P,Q = {mQ , φP }P,Q . We want to see that (4.1) holds true. Again, let Q = Qνk and P = Pμl . First, since the functions mQ and φP are respectively localized around Q and P and decay

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away from them (cf. (3.2)), each function “is very small where the other one is not” so ˆ mQ , φP = mQ (x)φP (x) dx → 0 Rn

as |xQ − xP | → ∞. Note that no cancelation of the wavelike functions is needed here. On the other hand, if xQ ≈ xP but, say, ν 0. For such spaces all that we need to verify are the conditions (3.2) for some N > n and all |γ| ≤ [α] + 1; no vanishing moments are required on these molecules [28]. If we verify such estimates, we will then be able to conclude that Ta (f )L2α  f L2α for all f ∈ S, and by density for all f ∈ L2α . Fix k ∈ Zn and ν ≥ 1 (the arguments for ν = 0 are similar but easier) and let ψQ = ψνk . We can compute ˆ νn/2 TQ (ψ)(2ν x − k), Ta (ψQ )(x) = eixξ a(x, ξ)ψ+ Q (ξ)dξ = 2 ˆ

where TQ (f )(x) =

eixξ a(2−ν (x + k), 2ν ξ)f&(ξ)dξ.

Furthermore, letting N > n/2, taking γ derivatives in x inside the integral sign, and then integrating by parts in ξ, we also compute (∂ γ TQ (ψ))(x) = ⎛ ⎞ ˆ N  (I − Δ ) ξ & ⎝ ⎠ dξ, (4.3) eixξ Cβ (iξ)β ∂xγ−β (a(2−ν (x + k), 2ν ξ)ψ(ξ)) (1 + |x|2 )N Rn β

where the sum runs over all multi-indices β such that βj ≤ γj for all j = 1, . . . , n. Note that when ρ of the (I − Δξ )N derivatives fall on the symbol a, we can estimate |∂ξρ ∂xγ−β (a(2−ν (x + k), 2ν ξ)|  2ν|ρ| 2−ν(|γ|−|β|) (1 + |2ν ξ|)|γ|−|β|−|ρ|  1,

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because the integral takes place on the support of ψ where |ξ| ≈ 1. By the same & are bounded by a constant. It follows reasoning, any derivatives falling on (iξ)β ψ(ξ) that |∂ γ TQ (ψ)(x)|  (1 + |x|2 )−N , and 2νn/2 2ν|γ| , |∂ γ Ta (ψQ )(x)| = |∂ γ (2νn/2 TQ (ψ))(2ν x − k)|  (1 + |2ν x − k|2 )N which is precisely (3.2). For L2 or L2α with α < 0, the molecules Ta (ψQ ) must have vanishing moments. This can be achieved if one imposes some cancelations on the transpose operator Ta∗ of the form Ta∗ (xγ ) = 0; see [62] for details. The approach also works for all the Besov and Triebel-Lizorkin spaces. Other pseudodifferential operators with homogeneous symbols (and of arbitrary order) were treated with the same arguments in [33]. In a multilinear setting, almost diagonal matrices (tensors) and applications to multilinear operators were investigated in this fashion in [34], [1] and [2]. 5. Concluding remarks There are many other applications of the reproducing formula and function space characterizations presented in [28]. Among others, there are results about interpolation, pointwise and Fourier multipliers, and traces involving several TriebelLizorkin spaces. It is impressive to see the outreach of the theory to many previously existing topics but also to an abundance of new results. Moreover, we believe that many new applications of these powerful techniques are yet to come. Frazier and Jawerth modestly state in [28, p. 36]: “In any case, we want to make clear that virtually all of our techniques already exist in some antecedent form. Nevertheless, their particular combination here leads to new conclusions and to sharpened versions of known results. Moreover, our presentation reveals an elementary discrete structure underlying a diverse range of topics in harmonic analysis.” Wavelets have become a great mathematical success story based in part on the discrete underlying structures referred to above. Their applications in image processing, signal analysis, compression of information, and so on, have impacted technology and our everyday life.11 It is stimulating to think that part of the genesis of this important mathematical contribution is in an obscure formula in an interpolation paper by Calder´ on. Bj¨ orn Jawerth’s original mathematical training was in function spaces and interpolation, areas in which he has made many other important contributions. But he also embraced the applications of wavelets theory and, during his late years, he did a lot of translational research, software and even hardware development, and founded a couple of companies. Yet many of the “miracles” of wavelets can only be fully explained and understood through the detailed harmonic analysis that preceded the applications and the arsenal of functional analytic tools of which Bj¨ orn was so fond of. The works [26, 28] are great contributions in this analysis. Personally, we will always be thankful to Bj¨ orn 11 Note Added in Proof: The exciting recent news of the awarding of the Abel Prize to Yves Meyer for his outstanding contributions to analysis in general and wavelets in particular also puts in evidence the tremendous impact of wavelets and related techniques to science and technology. See http://www.abelprize.no/nyheter/vis.html?tid=69588.

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and Michael for their work on discrete decompositions, for the mathematics they have shared with us, and for the influential role their ground breaking research has had in our own professional careers. Acknowledgement The authors would like to express their deepest gratitude to the referee for an extremely careful reading of our manuscript, making many valuable comments and ´ suggestions, and bringing to our attention some important additional references. A. B´enyi is supported in part by a grant from the Simons Foundation (No. 246024). References [1] [2] [3]

[4]

[5] [6] [7]

[8] [9] [10]

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[40] Lars H¨ ormander, Pseudo-differential operators of type 1, 1, Comm. Partial Differential Equations 13 (1988), no. 9, 1085–1111, DOI 10.1080/03605308808820569. MR946283 [41] Lars H¨ ormander, Continuity of pseudo-differential operators of type 1, 1, Comm. Partial Differential Equations 14 (1989), no. 2, 231–243, DOI 10.1080/03605308908820596. MR976972 [42] Svante Janson and Mitchell H. Taibleson, Calder´ on’s representation theorems (Italian, with English summary), Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), no. 2, 27–35 (1982). MR705574 [43] Susan E. Kelly, Mark A. Kon, and Louise A. Raphael, Pointwise convergence of wavelet expansions, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 87–94, DOI 10.1090/S0273-09791994-00490-2. MR1248218 [44] S. E. Kelly, M. A. Kon, and L. A. Raphael, Local convergence for wavelet expansions, J. Funct. Anal. 126 (1994), no. 1, 102–138, DOI 10.1006/jfan.1994.1143. MR1305065 [45] G. Kyriazis, Decomposition systems for function spaces, Studia Math. 157 (2003), 133-169. [46] P. G. Lemari´ e and Y. Meyer, Ondelettes et bases hilbertiennes (French), Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 1–18, DOI 10.4171/RMI/22. MR864650 [47] Kangwei Li and Wenchang Sun, Pointwise convergence of the Calder´ on reproducing formula, J. Fourier Anal. Appl. 18 (2012), no. 3, 439–455, DOI 10.1007/s00041-011-9211-4. MR2921079 [48] Y. Meyer, La transformation en ondellettes et les nouveaux paraproduits, Actes du Colloque d’Analyse Non-lineaire du Ceremade, Univ. de Paris-Dauphine (unknown publication date). [49] Yves Meyer, R´ egularit´ e des solutions des ´ equations aux d´ eriv´ ees partielles non lin´ eaires (d’apr` es J.-M. Bony) (French), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 293–302. MR636530 e des op´ erateurs d´ efinis [50] Y. Meyer, La minimalit´ e de l’espace de Besov B˙ 101,1 , et la continuit´ par des int´ egrales singuli` ers, Monografias de Matematicas, Vol 4, Univ. Antonoma de Madrid, 1985. [51] Yves Meyer, Ondelettes et op´ erateurs. I (French), Actualit´ es Math´ ematiques. [Current Mathematical Topics], Hermann, Paris, 1990. Ondelettes. [Wavelets]. MR1085487 [52] A.-P. Calder´ on, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR0167830 [53] Jaak Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. MR0461123 [54] Fulvio Ricci and Mitchell Taibleson, Boundary values of harmonic functions in mixed norm spaces and their atomic structure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 1, 1–54. MR713108 [55] T. Runst, Pseudo-differential operators in Hardy-Triebel spaces, Z. Anal. Anw. 2 (1983), 235-242. [56] Sadahiro Saeki, On the reproducing formula of Calder´ on, J. Fourier Anal. Appl. 2 (1995), no. 1, 15–28, DOI 10.1007/s00041-001-4020-9. MR1361540 [57] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR0290095 [58] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR1232192 [59] Jan-Olov Str¨ omberg, A modified Franklin system and higher-order spline systems on Rn as unconditional bases for Hardy spaces, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 475–494. MR730086 [60] Mitchell H. Taibleson and Guido Weiss, The molecular characterization of certain Hardy spaces, Representation theorems for Hardy spaces, Ast´ erisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 67–149. MR604370 [61] Terence Tao, On the almost everywhere convergence of wavelet summation methods, Appl. Comput. Harmon. Anal. 3 (1996), no. 4, 384–387, DOI 10.1006/acha.1996.0031. MR1420506 [62] Rodolfo H. Torres, Continuity properties of pseudodifferential operators of type 1, 1, Comm. Partial Differential Equations 15 (1990), no. 9, 1313–1328, DOI 10.1080/03605309908820726. MR1077277

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Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13935

A characterisation of the Besov-Lipschitz and Triebel-Lizorkin spaces using Poisson like kernels Huy-Qui Bui and Timothy Candy α and F˙ α Abstract. We give a complete characterisation of the spaces B˙ p,q p,q by using a non-smooth kernel satisfying near minimal conditions. The tools used include a Str¨ omberg-Torchinsky type estimate for certain maximal functions and the concept of a distribution of finite growth, inspired by Stein. In addition, our exposition also makes essential use of a number of refinements of the well-known Calder´ on reproducing formula. The results are then applied to obtain the characterisation of these spaces via a fractional derivative of the Poisson kernel. Moreover, our results offer an approach to deal with the calculus modulo polynomials in homogeneous function spaces, a subtle problem raised recently by Triebel.

1. Introduction and statements of main results α We aim to characterise the Besov-Lipschitz space B˙ p,q and the Triebel-Lizorkin α space F˙ p,q using a kernel ψ which satisfies near “minimal” conditions regarding cancellation, smoothness and decay. In particular, we give a direct characterisation that works in the full scale α ∈ R, 0 < p, q  ∞ in the important case of fractional derivatives of the Poisson kernel (in other words, via harmonic functions). In adα α dition, in the case α < np , we give a general characterisation of B˙ p,q and F˙ p,q as   subsets of S (rather than the more standard S /P); see Remark 1.4 below. The Besov-Lipschitz and Triebel-Lizorkin scales of spaces arise in many applications in mathematics. In particular, they are of crucial importance in interpolation theory [1, 15, 17, 22] and contain many well known function spaces in mathematical 0 analysis. For instance, F˙ p,2 is identified with the real-variable Hardy space H p of 0 Fefferman-Stein [10], while F˙ ∞,2 is identified with BM O, the space of functions of bounded mean oscillation [12]. To facilitate the discussion to follow, we begin by recalling the definition of the homogeneous Besov-Lispchitz and Triebel-Lizorkin spaces following Peetre [16, 17] (see also [22]). All functions and distributions are defined on the Euclidean space & = {1/2  |ξ|  2} and for Rn unless otherwise stated. Let ϕ ∈ S such that supp ϕ

2010 Mathematics Subject Classification. Primary 42B25, 46E35. Key words and phrases. Besov-Lipschitz space, Triebel-Lizorkin space, Calder´ on reproducing formula, maximal function, Poisson kernel. c 2017 American Mathematical Society

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every ξ = 0



(1)

ϕ(2 & −j ξ)ϕ(2 & −j ξ) = 1.

j∈Z

The function ϕ is fixed throughout this article. Given a function φ : Rn → C we let φj (x) = 2jn φ(2j x) denote the dyadic dilation of φ. Somewhat confusingly, for t ∈ R with t > 0, we let φt (x) = t−n φ(t−1 x) denote the standard dilation. It should always be clear from the context the type of dilation we are using. α Let 0 < p, q  ∞ and α ∈ R. The (homogeneous) Besov-Lipschitz space B˙ p,q is then defined as the class of all tempered distributions modulo polynomials f ∈ S  /P such that

 q  q1 = < ∞. (2) f B˙ p,q 2jα ϕj ∗ f Lp α j∈Z

Similarly, for 0 < p, q  ∞, p = ∞, and α ∈ R, the (homogeneous) Triebel-Lizorkin α space F˙ p,q is defined as the class of all f ∈ S  /P such that    q  q1    jα  < ∞.  2 |ϕj ∗ f | (3) f F˙ α =   p,q

Lp

j∈Z

α The definition of F˙ ∞,q is slightly different. The problem is that a naive extension of (3) to the case p = ∞ leads to spaces which are not independent of the choice 0 of kernel, and moreover the expected identification F˙ ∞,2 = BM O fails; see the discussion in Section 5 of [12]. Instead, following the influential work [12] we define  q1

   q 1 jα 2 |ϕj ∗ f (x)| dx f F˙∞,q = sup α |Q| Q Q j−(Q)

with the interpretation that when q = ∞,  1 (4) f F˙∞,∞ = sup sup 2jα |ϕj ∗ f (x)| dx, α Q j−(Q) |Q| Q where the sup is over all dyadic cubes Q, and (Q) = log2 ( side length of Q ). The above interpretation for the norm  · F˙∞,∞ is taken from [7] where one can also α α α α ˙ ˙ ˙ find the embedding F∞,q ⊂ F∞,∞ = B∞,∞ . α α Observe that elements of the quasi Banach spaces B˙ p,q and F˙ p,q are equivalence classes of distributions modulo polynomials. However we often make a common α α abuse of notation by regarding elements of B˙ p,q and F˙ p,q as distributions, rather than equivalence classes. α α The fundamental and central result in the study of the spaces B˙ p,q and F˙ p,q is the independence of these function spaces on the choice of the kernel function ϕ. Thus, given a kernel ψ ∈ S satisfying certain conditions, if we replace ϕ in (2) and (3) with ψ ∈ S we have an equivalent norm. This independence was initially established in the pioneering works of Peetre [16,17] for all Besov-Lipschitz spaces and for the Triebel-Lizorkin spaces when p < ∞, and the result applied in particular to band-limited kernels. The method used by Peetre was inspired by the real-variable theory for various maximal functions, developed in the seminal paper α was proved later in [12] (see [10] by Fefferman and Stein. The result for F˙ ∞,q also [7]). After further partial results by Triebel [21, 22], the essentially optimal

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conditions, at least in the Schwartz case ψ ∈ S, were developed in Bui, Paluszy´ nski and Taibleson in [5], [6] and [7] where it was shown that we have an equivalent norm for any f ∈ S  /P, provided that the kernel ψ ∈ S satisfies the following: (I) (Vanishing moments) The kernel ψ has [α] vanishing moments, thus  xκ ψ(x)dx = 0 Rn

for every |κ|  [α], with the understanding that no condition is required when α < 0. (II) (Tauberian condition) The kernel ψ satisfies the Tauberian condition; that is, for every ξ ∈ Sn−1 there exists a, b ∈ R (depending on ξ) with 0 < 2a ≤ b such that for every a ≤ t ≤ b & |ψ(tξ)| > 0. Here, given α ∈ R we let [α] denote the integer part of α. Note that the conditions (I) and (II) do not require that the kernel ψ is band-limited. Thus, for example, it is possible to characterise the Besov-Lipschitz and Triebel-Lizorkin spaces with 2 derivatives of the Gaussian kernel e−|x| . In this paper we consider the problem of removing the assumption ψ ∈ S. Our key goal is to give conditions on the kernel that are simply stated and easily checked, while still being as close as possible to be optimal. Moreover, they should apply in particular to the important case of fractional derivatives of the Poisson n+1 kernel (1 + |x|2 )− 2 . The first, and somewhat obvious, obstacle in the non-smooth case ψ ∈ S, is that to define the norms (2) and (3) we need to be able to define the convolution ψ ∗f for arbitrary f ∈ S  . This is clearly not possible unless ψ& is infinitely differentiable and all its derivatives are slowly increasing. As our main application, the Poisson kernel, does not satisfy the last conditions (its Fourier transform is not differentiable at the origin), we need to restrict the class of distributions slightly to a natural class of admissible distributions. To this end, inspired by Stein [19], we introduce the concept of distributions of bounded growth. Definition 1.1. [Distributions of growth ] We say f ∈ S  is a distribution of growth   0 if for any φ ∈ S we have φ ∗ f = O(|x| ) (as |x| → ∞). The importance of this definition is that it allows us to make sense of the convolution ψ ∗ f when ψ ∈ S. Definition 1.2. Assume f is a distribution of growth . Then if (1+|·|) ψ ∈ L1 we define the convolution ψ ∗ f ∈ S  as  ψ(x)(φ ∗ f )(x)dx, φ ∈ S, ψ ∗ f (φ) = Rn

 where φ(x) = φ(−x). This definition coincides with the pointwise definition for ψ ∗ f when f = O(|x| ) is locally integrable. We note that Stein used the concept of a bounded distribution (the case  = 0 in Definition 1.1) to characterise the Hardy spaces H p using the Poisson kernel (see [10, 19]). Before proceeding to state the main theorem we prove, we discuss the key conditions we require on our kernel. To this end, we take parameters Λ  0 and m, r ∈ R, and suppose ψ ∈ L1 (Rn ).

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(C1) (Cancelation) Let ψ& ∈ C n+1+[Λ] (Rn \ {0}) such that for every |κ|  n + 1 + [Λ] we have ∂ κ ψ& = O(|ξ|r−|κ| )

as |ξ| → 0.

(C2) (Tauberian condition) The kernel ψ satisfies the Tauberian condition (as in (II) above). (C3) (Smoothness) Take ψ& ∈ C n+1+[Λ] (Rn \ {0}) such that for every |κ|  n + 1 + [Λ] we have ∂ κ ψ& = O(|ξ|−n−m )

as |ξ| → ∞.

The parameters Λ and m roughly correspond to the decay and smoothness we require on a component of our kernel ψ. More precisely if φ ∈ S has Fourier support away from the origin, then ψ& ∈ C n+1+[Λ] (Rn \ {0}) implies that ψ ∗ φ(x) = O(|x|−n−1−[Λ] ) as |x| → ∞. Thus larger Λ requires more decay on the part of ψ with Fourier support away from the origin. Similarly, if (C3) holds, then ψ ∈ C [m] (Rn ). Thus the larger we take m, the smoother the kernel ψ is required to be. On the other hand, the parameter r and the cancellation condition (C1) are closely related to the vanishing moments condition (I). More precisely, if ψ ∈ S then it is easy to check that ψ has [α] vanishing moments (i.e. (I) holds), if and only if (C1) holds with α < r  [α] + 1. Of course if ψ ∈ S then the relationship between (I) and (C1) is somewhat complicated, but roughly speaking (I) requires more spatial decay, while (C1) requires more regularity of ψ& near the origin. It is worth pointing out that it is possible to prove the characterisations below with (C1) replaced with (I), but this requires more decay on the kernel ψ and leads to less optimal conditions. Instead we have chosen to use conditions on the Fourier transform of ψ, as firstly this matches up very well with our intended application to the Poisson kernel, and secondly, in the authors’ opinion the conditions (C1), (C2) and (C3) form an acceptable balance between the sharpness of our result, and the simplicity of its statement. It is natural to split the characterisation results into two theorems: “Necessary Conditions” and “Sufficient Conditions”. This is due to fact that, as noted in [5], each theorem requires a slightly different set of assumptions. The essential assumption for the former is the cancellation property of the kernel, expressed by the condition (C1), while for the latter the Tauberian condition (II) stated earlier in the introduction is critical. Other conditions, such as the decay at infinity of the kernel in the frequency domain expressed by the smoothness condition (C3), are needed to define the convolution with distributions of finite growth. In the necessary direction, the statement of our result is expressed using a maximal function introduced in the work of Peetre [16]. Given a kernel ψ, if f ∈ S  is such that each ψj ∗ f is a function, one defines the Peetre maximal function by (5)

∗ f (x) = sup ψj∗ f (x) = ψj,λ

y∈Rn

|ψj ∗ f (x − y)| , (1 + 2j |y|)λ

x ∈ Rn ,

where λ > n/p in the Besov-Lipschitz case and λ > max{n/p, n/q} in the Triebelα Lizorkin case (with λ > n for the space F˙∞,∞ ). Unless otherwise stated, the number λ satisfies these conditions throughout this work. In the rest of the paper we write A  B when there exists a positive constant C such that A  CB, where C may depend on the parameters such as n, α, p, q, . . .

CHARACTERISATION OF B-L AND T-L SPACES

113

but usually not on the variable quantities such as the distribution f . When j, k ∈ Z we write j  k to mean that j  k + c for some c ∈ Z independent of j and k. We can now state our main results. We start with the necessary direction. Theorem 1.1. Let α ∈ R and 0 < p, q  ∞. Let   0 with  > α − np . Assume (1 + | · |) ψ ∈ L1 and that ψ satisfies the cancellation condition (C1) and smoothness condition (C3) for parameters Λ  0, r > α, and m > Λ − α. α and Λ = n . Then there exists a polynomial ρ such that f − ρ (i) Let f ∈ B˙ p,q p

(6)

is a distribution of growth  and we have the inequalities  q  1q 2jα ψj∗ (f − ρ)Lp  f B˙ p,q α , j∈Z

(7)

and for any φ ∈ S      2jα  sup |φt ∗ ψj ∗ (f − ρ)|

Lp

t>0

j∈Z

q  q1

 f B˙ p,q α .

α (ii) Similarly if f ∈ F˙ p,q and we let Λ = max{ np , nq } (with Λ = n when p = q = ∞), then there exists a polynomial ρ such that f − ρ is a distribution of growth  and if p < ∞    q  q1    2jα ψj∗ (f − ρ) (8)   p  f F˙p,q α , L

j∈Z

(9)

(10)

and for p = ∞

 1 sup |Q| Q Q



Furthermore, if φ ∈ S, we have for p < ∞    q  1q    2jα sup |φt ∗ ψj ∗ (f − ρ)|   and p = ∞

 1 sup |Q| Q Q

 q1

 f F˙∞,q . α

j−(Q)

Lp

t>0

j∈Z

(11)

 jα ∗ q 2 ψj (f − ρ)(x) dx

 j−(Q)

 f F˙ α

 jα q 2 sup |φt ∗ ψj ∗ (f − ρ)(x)| dx t>0

p,q

 q1

 f F˙ α . ∞,q

Remark 1.1. A few remarks are in order. (i) When q = ∞, the inequality (11) is interpreted similarly to the definition α of the F˙ ∞,∞ -norm (4); i.e.,   jα  1 2 sup |φt ∗ ψj ∗ (f − ρ)(x)| dx  f F˙∞,∞ . sup sup α |Q| t>0 Q j−(Q) Q We adopt this interpretation hereafter in all the theorems and proofs. (ii) The assumptions on the kernel ψ ensure that each convolution ψj ∗ (f − ρ) is well-defined and moreover is a continuous function. This is a consequence of Theorem 2.2, Corollary 2.1 and Theorem 3.1.

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 (iii) Given φ ∈ S with φ = 0 and 0 < p  ∞, the characterisation of the real-variable Hardy space H p , defined by Fefferman and Stein in [10], gives      sup |φt ∗ g| p ≈ gH p . L

t>0

(Note that when 1 < p  ∞, H p = Lp with equivalent norms.) Thus it follows from (7) that  q  q1     2jα ψj ∗ (f − ρ) p (12)  f B˙ α . p,q

H

j∈Z

(iv) Since |ψj ∗ g| is clearly dominated pointwise by ψj∗ g we may replace (6) with  q  1q 2jα ψj ∗ (f − ρ)Lp  f B˙ α . (13) p,q

j∈Z

Similarly we may replace the Peetre maximal function ψj∗ (f − ρ) in (8) and (9) with the standard convolution |ψj ∗ (f − ρ)|. We next consider the converse to the above theorem; that is, to find sufficient conditions on the kernel ψ and the distribution f such that the reverse inequalities to those in Theorem 1.1 holds. We emphasise that the results in this sufficient part are the main contribution of this paper. As soon as the assumption ψ ∈ S is dropped, one immediately runs into the difficulty of defining the convolution ψj ∗ f when f ∈ S  . The situation is different from the necessary result in Theorem 1.1 α α where we already knew that f ∈ B˙ p,q or f ∈ F˙ p,q , and therefore the convolution given in Definition 1.2 can be seen to be a continuous bounded function. However, if f is a distribution of growth   0, then we have seen it is possible to define ψj ∗ f as a distribution under rather mild condition on ψ (see Definition 1.2). Our first result in the sufficient direction makes use of this observation. Theorem 1.2. Let α ∈ R and 0 < p, q  ∞. Assume f is a distribution of growth   0. Suppose (1 + | · |) ψ ∈ L1 satisfies the Tauberian condition (C2) and there exists m ∈ R such that the smoothness condition (C3) holds for Λ  0. (i) Let Λ = max{, np }. Then (14)

f B˙ p,q  α

 q  1q 2jα ψj ∗ f H p . j∈Z

(ii) Let Λ = max{, (15)

n n p, q}

and φ ∈ S with

φ(x)dx = 0. If p < ∞ then

   q  q1    2jα sup |φt ∗ ψj ∗ f | f F˙ α    p,q

and in the case p = ∞

 1 f F˙ α  sup ∞,q |Q| Q Q

Lp

t>0

j∈Z

(16)



 j−(Q)

,

 jα q 2 sup |φt ∗ ψj ∗ f (x)| dx t>0

(with Λ = max{n, } when p = q = ∞).

 1q

CHARACTERISATION OF B-L AND T-L SPACES

115

Remark 1.2. (i) Observe that we are free to choose the smoothness parameter m, thus an alternative way to state the smoothness condition on ψ is that we simply require ∂ κ ψ& to be slowly increasing for |κ|  n + 1 + [Λ]. (ii) Theorem 1.2 together with Theorem 1.1 give the following complete charα acterisation of B˙ p,q : Let  > α − np , r > α, m > np − α, and Λ = max{, np }. If the  α kernel (1 + | · |) ψ ∈ L1 satisfies the conditions (C1), (C2), and (C3), then f ∈ B˙ p,q 1   jα q  q if and only if f is a distribution of growth  and < ∞. j∈Z 2 ψj ∗ f H p A similar comment applies in the Triebel-Lizorkin case. If we want a version of Theorem 1.2 with H p replaced with Lp , we need to assume more on our kernel ψ to ensure that each ψj ∗ f is a measurable function (as apposed to just an element of S  ). It is worth noting that, under the assumptions of Theorem 1.2, if we assume the right hand side of (14) is finite, then since H p = Lp for p > 1, we may freely replace the H p norm with the Lp norm in (14). Thus at first glance, it may appear that the conditions on ψ in Theorem 1.2 are sufficient to also deduce an Lp version of (14). However this is slightly misleading, as we may only replace H p with Lp after making the a priori assumption that the right hand side of (14) is finite. Without this finiteness assumption, it is not possible to ensure that the distribution ψj ∗ f is in a fact a function. Thus in general, under the assumptions on ψ in Theorem 1.2, the norm ψj ∗ f Lp is not defined. Consequently, if our goal is to prove a direct characterisation without any auxiliary assumptions on the distribution f , to ensure that ψj ∗ f is a measurable function, we need to make further assumptions on our kernel ψ. One such condition is found in the next theorem. Theorem 1.3. Let α ∈ R and 0 < p, q  ∞. Assume f is a distribution of growth   0. Suppose (1 + | · |) ψ ∈ L1 satisfies the Tauberian condition (C2) and that for every m ∈ R, the smoothness condition (C3) holds with Λ  . Then for every j ∈ Z the convolution ψj ∗ f is a continuous function. Moreover, if Λ = max{, np } then  q  q1 2jα ψj ∗ f Lp (17) f B˙ α  . p,q

j∈Z

Similarly, if Λ = max{,

n n p, q}

f F˙ α

(18)

p,q

and p < ∞ then    q  q1    2jα |ψj ∗ f |   j∈Z

and in the case p = ∞ (19)



 sup f F˙∞,q α Q

1 |Q|





Lp

 jα q 2 |ψj ∗ f (x)| dx

 q1

Q j−(Q)

(with Λ = max{n, } when p = q = ∞). In the above two sufficiency theorems, we have restricted the class of distributions to those of growth . This condition is natural in light of Theorem 1.1 where α α it was shown that all elements of B˙ p,q and F˙ p,q are, perhaps modulo a polynomial, n distributions of growth α − p +  for every  > 0. Thus we do not lose anything by only considering distributions of some finite growth. Observe also that by making 

116

HUY-QUI BUI AND TIMOTHY CANDY

smaller, we weaken the condition on ψ, but unfortunately require a stronger growth condition on f . A good choice for , which is suggested by the necessary results, is to take  > α − n/p. Theorem 1.1, Theorem 1.2 and Theorem 1.3 provide necessary and sufficient conditions for a distribution to be in the Besov-Lipschitz space or in the TriebelLizorkin space. In other words, these theorems provide complete characterisations of the function spaces under study. We now come to our main application of the previous results. Namely we give α α and F˙ p,q via fractional derivatives of the Poisson kernel. a characterisation of B˙ p,q & Thus we consider the case ψ(ξ) = |ξ|β e−|ξ| ; that is, ψ = (−Δ)β/2 P , and P is the Poisson kernel on Rn , cn P (x) = . (1 + |x|2 )(n+1)/2 Note that the Poisson kernel case is one of the main motivations for this work. Theorem 1.4. Let α ∈ R, 0 < p, q  ∞. Let β  0, β > α, and define & = |ξ|β e−|ξ| . Let   0 such that ψ(ξ) , β β+1 n 2 ∈N α− 0, it is well-known that the characterisation of B˙ ∞,∞ via differences implies the stronger pointwise growth bound , |x|α log |x| α > 0 and α ∈ N |(f − p)(x)|  α > 0 and α ∈ N, |x|α

from which the Corollary follows. On the other hand, in the case α = 0, the growth bounds in Corollary 2.1 and Theorem 2.2 appear to be new. As is common in the study of function spaces via the Calder´on formula, we require some control over convolutions of the form ηk ∗ ψj (see for instance the work of Heideman [13]). The precise dilation estimate we need is a refined version of [5, Lemma 2.1] (see also [18, Lemma 1]). Lemma 2.3. Let m ∈ R, c > 0 and N ∈ N. Suppose η ∈ L1 with η& ∈ C N (Rn )   and supp η& ⊂ {a  |ξ|  b} for some 0 < a < b. Let ψ ∈ S  with ψ& ∈ C N Rn \{0} . & (i) Assume ∂ κ ψ(ξ) = O(|ξ|−m ) as |ξ| → ∞ for every |κ|  N . Then for any s  ct we have  s m−n t−n (21) |ηs ∗ ψt (x)|  . t (1 + t−1 |x|)N & = O(|ξ|m−|κ| ) as |ξ| → 0 for every |κ|  N . Then for any (ii) Assume ∂ κ ψ(ξ) t  cs we have  t m s−n (22) |ηs ∗ ψt (x)|  . s (1 + s−1 |x|)N Proof. Take c = 1 for simplicity of notation. The support assumption on η& implies that the convolution η ∗ ψ is well-defined (in fact is an L∞ function). Moreover, for every |κ|  N and any x ∈ Rn     /  0 sn−|κ| xκ ηs ∗ ψt (x)  xκ η ∗ ψ st (x)L∞  ∂ κ η&(ξ)ψ& st (ξ) L1   t |γ|   & t ξ)dξ. (23)  |∂ γ ψ( s s a|ξ|b γκ

CHARACTERISATION OF B-L AND T-L SPACES

121

In particular, if s  t, then assuming ∂ κ ψ& = O(|ξ|−m ) as |ξ| → ∞, and using the bound (23) we deduce that for every |κ|  N  |κ|   t |γ|    t −m  −1 κ  ξ  dξ (t x) ηs ∗ ψt (x)  s−n s s t s a|ξ|b γκ  s m−n   s |κ|−|γ|  s m−n  t−n . ≈ t−n t t t γκ

Applying this estimate for |κ| = 0 and |κ| = N we obtain (i). Similarly, if t  s, then assuming ∂ κ ψ& = O(|ξ|m−|κ| ) as |ξ| → 0 and again applying the bound (23) we have  m   t |γ|     t m−|γ| −|κ|  κ −n −n t   ξ x ηs ∗ ψt (x)  s s dξ  s s s s a|ξ|b γκ



which gives (ii).

To apply our characterisations to Poisson like kernels, we need to estimate the spatial decay of F −1 (|ξ|β e−|ξ| ). The required decay is a consequence of the following corollary of Lemma 2.3. Corollary 2.2. Let r >   0 and 1  p < ∞. Let ψ ∈ Lp and assume supp ψ& ⊂ {|ξ|  1}. Furthermore, suppose that ψ& ∈ C n+1+[] (Rn \ {0}) with & = O(|ξ|r−|κ| ) ∂ κ ψ(ξ)

as |ξ| → 0

for every |κ|  n + 1 + []. Then (1 + |x|) ψ ∈ L and moreover ψ has [] vanishing moments. 

1

Proof. We begin by observing that      |x| (ϕj ∗ ϕj ∗ ψ) 1  2−j (1 + 2j |x|) ϕj ∗ ψ L1 L j1

j1



(24)



2j(r−) < ∞

j1

where we used an application of (ii) in Lemma 2.3 (with t = 1 and s = 2−j ) to deduce that (1 + 2j |x|) ϕj ∗ ψL1  2jr 2jn (1 + 2j |x|)−(n+1+[]) L1  2jr . On the other hand, the assumption ψ ∈ Lp together with supp ψ& ⊂ {|ξ|  1} implies that we have the pointwise identity  (25) ψ(x) = ϕj ∗ ϕj ∗ ψ(x) j1

for a.e. x ∈ R (in fact as ψ is smooth, the identity holds for every x ∈ Rn ). Consequently, (24) implies that |x| ψ ∈ L1 . Therefore ψL1  ψLp + |x| ψL1 and so we deduce that (1 + |x|) ψ ∈ L1 . Finally, to check that ψ has [] vanishing moments, we simply note that ψ& ∈ C [] (Rn ), and hence the decay condition gives & ∂ κ ψ(0) = 0 for every |κ|  []. Together with the integrability (1 + | · |) ψ ∈ L1 , this implies that ψ has [] vanishing moments as claimed.  n

Corollary 2.2 has an immediate application to the Poisson kernel.

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HUY-QUI BUI AND TIMOTHY CANDY

& Corollary 2.3. Let β  0, and let ψ(ξ) = |ξ|β e−|ξ| . Then (1 + | · |) ψ ∈ L1 for every  < β. Proof. Let χ ∈ S such that supp χ ⊂ {|ξ|  1}, and χ = 1 in a neighbour& + (1 − χ)ψ& = θ& + μ hood of the origin. Write ψ& = ψχ &. Then μ ∈ S and θ satisfies the assumptions of Corollary 2.2 with r = β.  Finally we make use of the following elementary summation inequalities. Proposition 2.1. Fix 0 < p, q  ∞ and let fk be a sequence of measurable functions. If (aj )j∈Z ∈ min{p,q,1} (Z) then we have      q 1/q    q 1/q  fj  aj−k fk   . p

k∈Z

p

j∈Z

j∈Z

Similarly if (aj )j∈Z ∈ min{q,1} (Z) then       1/q  q 1/q      |aj−k fk | |fj |q    .  j∈Z

p

k∈Z

j∈Z

p

Proof. This proposition is a folklore result. The proof is based on Young’s inequality and the inequality  r  |bj |  |bj |r j∈Z

j∈Z

which holds whenever 0 < r  1. We omit the details.



3. Pointwise Definition of the Convolution The introduction of distributions of finite growth, together with Definition 1.2, makes it possible to define the convolution ψ ∗ f as a distribution. However, the characterisation results in Theorems 1.1 and 1.3 require a pointwise definition. In this section we give two sets of sufficient conditions to ensure that ψ ∗ f ∈ L1loc . The first is via what is essentially a duality argument exploiting the Calder´ on reproducing formula given in Theorem 2.2. This argument has the advantage that it requires very few assumptions on the kernel ψ. On the other hand it is only α applicable in the case f ∈ B˙ ∞,∞ , and thus is not helpful in Theorem 1.3. The second approach is much more general, and works for arbitrary distributions f ∈ S  , provided only that f has finite growth. However it correspondingly requires much stronger conditions on the the kernel ψ. α . The key result is the following. 3.1. The case f ∈ B˙ ∞,∞ α Theorem 3.1. Let α,  ∈ R with   0 and  > α. Let f ∈ B˙ ∞,∞ . Assume −α  1 ˙ ψ ∈ B1,1 with (1 + |x|) ψ ∈ L . Let p be the polynomial given by Theorem 2.2. Then the distribution ψ ∗ (f − p) is a bounded continuous function and we have the identity  ψ ∗ ϕj ∗ ϕj ∗ f (x) (26) ψ ∗ (f − p)(x) = j∈Z ∞

where the sum converges in L .

CHARACTERISATION OF B-L AND T-L SPACES

123

Proof. Let p be the polynomial given in Theorem 2.2. The decay assumption on ψ implies that the convolution ψ ∗ (f − p) ∈ S  . Define g as  g(x) = ψ ∗ ϕj ∗ ϕj ∗ f (x). j∈Z



The duality estimate j |ψ ∗ ϕj ∗ ϕj ∗ f (x)|  f B˙ α ψB˙ −α implies that g is a ∞,∞ 1,1 bounded continuous function. Thus the theorem would follow by showing that for every ρ ∈ S  g(x)ρ(x)dx. (27) ψ ∗ (f − p)(ρ) = Rn

To this end, by definition of the distribution ψ ∗ (f − p), together with the growth bound in Theorem 2.2, the Dominated Convergence Theorem, and the decay condition on ψ, we have for any ρ ∈ S  ψ ∗ (f − p)(ρ) = ρ ∗ (f − p)(x)ψ(x)dx Rn

 =

 ∞  lim ρ ∗ pN + ϕj ∗ ϕj ∗ f (x)ψ(x)dx

Rn N →−∞

j=N +1



=

lim

N →−∞

Rn

ρ(x)ψ ∗ pN (x)dx +

 ∞ 

j=N +1

Rn

 ψ ∗ ϕj ∗ ϕj ∗ f (x)ρ(x)dx .

We claim that  the assumptions on ψ imply that ψ has [α] vanishing moments, in other words xγ ψ = 0 for every |γ|  [α]. Accepting this claim for the moment, we have ψ ∗ pN = 0 for every N and hence      ψ ∗ ϕj ∗ ϕj ∗ f (x)ρ(x)dx = g(x)ρ(x)dx ψ ∗ (f − p) ρ = j∈Z

Rn

Rn

where the last equality follows by the uniform convergence of the sum. Therefore (27) follows as required. Thus it only remains to show that ψ has [α] vanishing moments. If α < 0 there is nothing to prove so we may assume that α  0. The decay assumption on ψ implies that ψ& ∈ C [α] (Rn ) and hence using the form of the Taylor series given in [11] we can write  ξγ  1  ξγ   γ& & & & ∂ ψ(0) + [α] (1 − t)[α]−1 ∂ γ ψ(tξ) − ∂ γ ψ(0) dt. ψ(ξ) = γ! γ! 0 |γ|[α]

|γ|=[α]

The continuity of ∂ γ ψ& at the origin then implies that & − ψ(ξ)

(28)

 ξγ   & ∂ γ ψ(0) = o |ξ|[α] . γ!

|γ|[α]

On the other hand, given any ξ = 0 we have  & & |ψ(ξ)|  |ξ|α 2−jα sup |ψ(ξ)|  |ξ|α jlog2 (|ξ|)

2j−1 |ξ|2j+1

 jlog2 (|ξ|)+1

2−jα ϕj ∗ ψL1

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HUY-QUI BUI AND TIMOTHY CANDY

  −α & = o |ξ|α as |ξ| → 0. Together and consequently as ψ ∈ B˙ 1,1 , we deduce that ψ(ξ) with the bound (28) we obtain  ξγ   & ∂ γ ψ(0) = o |ξ|[α] γ! |γ|[α]

& which is only possible if ∂ γ ψ(0) = 0 for every |γ|  [α]. Therefore ψ has [α] vanishing moments as claimed.  −α α and ψ ∈ B˙ 1,1 . It is well Remark 3.1. Let α ∈ R and suppose f ∈ B˙ ∞,∞ −α α known that B˙ ∞,∞ can be identified with the (topological) dual of B˙ 1,1 (see e.g., −α ˙ [2, 17]). Thus f represents a continuous linear functional on B1,1 and furthermore, by an application of Theorem 2.2 we have the identity  ϕj ∗ ϕj ∗ f (ψ). f (ψ) = j∈Z

Thus if we define a convolution ψ ∗d f (x) as  ψ ∗d f (x) = f (τx ψ), we immediately have the pointwise identity  ψ ∗d f (x) = ψ ∗ ϕj ∗ ϕj ∗ f (x) j∈Z

(the convolutions on the righthand side are the standard convolutions between S and S  ). Since the sum converges uniformly, we see that ψ ∗d f (x) is a continuous bounded function. Although this definition of the convolution almost immediately gives the result of Theorem 3.1, it has the drawback that it does not always agree with the standard definition of the convolution. In particular, if for instance ψ ∈ L1 and f ∈ L∞ is a constant, then ψ ∗d f = 0, however ψ ∗ f (x) = c ψ(y)dy. It is natural to ask when the convolution defined directly via duality, ψ ∗d f , agrees with the definition given in Definition 1.2. The solution is given by the previous theorem. More precisely, suppose we know in addition that (1+|x|) ψ ∈ L1 α for some  > α, then for every f ∈ B˙ ∞,∞ there exists a polynomial p such that we have the pointwise identity ψ ∗d f (x) = ψ ∗ (f − p)(x). α , 3.2. The general case f ∈ S  . We now drop the assumption f ∈ B˙ ∞,∞ and instead simply assume that f is a distribution of growth . Our goal is find conditions on ψ such that the convolution ψ ∗ f defined in Definition 1.2, which belongs to S  , is in fact an element of L1loc . One possible solution is to assume ψ ∈ S, as then ψ ∗ f ∈ C ∞ . However this is far to strong for our purposes, as we would like our characterisation, and thus the pointwise definition, to apply in α the case ψ ∈ S. The way forward, as in the case of f ∈ B˙ ∞,∞ , is to study the convergence of the Calder´on reproducing formula. The first step in this direction is the following lemma.

Lemma 3.2. Let   0 and assume f is a distribution of growth . Then there exists β  0 depending on f , such that for every φ ∈ S and k ∈ Z we have   |φk ∗ f (x)|  2|k|β 1 + |x| .

CHARACTERISATION OF B-L AND T-L SPACES

125

∞ Proof. Define the mapping T : S → L∞  by T (φ) = φ ∗ f where L denotes ∞ the weighted L space defined by − < ∞}. L∞  = {g : g(x)(1 + |x|) L∞ x

Since f is a distribution of growth , the linear mapping T is well-defined. We claim that T is continuous. To see this note that an application of the Closed Graph Theorem (see, for instance, Theorem 1 on page 79 of [25]) reduces the problem to proving that the graph of T     φ, T (φ)  φ ∈ S (j) is closed in S × L∞ converges to φ in S and T (φ(j) ) converges to  . Assume φ ∞ some g ∈ L . Then for some M > 0 we have  |T (φ(j) − φ)(x)| = |(φ(j) − φ) ∗ f (x)|  φ(j) (x − ·) − φ(x − ·)α,γ |α|,γM

 (1 + |x|)M



φ(j) − φα,γ

|α|,γM

and hence T (φ(j) ) converges to T (φ) pointwise. Therefore we must have T (φ) = g ∈ L∞  and so the graph of T is closed. Consequently T is continuous as claimed. by a finite number The continuity of T implies that we can bound T (φ)L∞ of Schwartz norms of φ (see, for instance, Corollary 1 on page 43 of [25]). Thus there exists M1 > 0 such that  (29) T (φ)L∞ = φ ∗ f L∞  φα,γ . |α|,γM1

To complete the proof, we observe that a simple computation shows that φk α,γ  2k(n+|α|−|γ|) and hence, using (29), we obtain |φk ∗ f (x)|  2|k|β (1 + |x|) for some (possibly large) β  0 as required.



We can now prove the following. Proposition 3.1. Let   0. Suppose (1 + | · |) ψ ∈ L1 such that ψ& ∈ C (Rn \ {0}) with ∂ κ ψ& rapidly decreasing for every |κ|  n + 1 + []. Let  f ∈ S be a distribution of growth . Then for every j ∈ Z the convolution ψj ∗ f is a well-defined continuous function. Moreover, there exists β = β(f ) > 0 such that for every x ∈ Rn n+1+[]

(30)

M,β (x, j) =

sup kj,y∈Rn

|ψk ∗ f (y)| 2β(j−k) < ∞. (1 + 2j |x − y|)

Proof. Fix j ∈ Z and let k  j. The assumptions on f and ψ imply that ψk ∗ f ∈ S  . Thus we can follow the standard proof of the Calderon reproducing formula to deduce the identity (31)

ψk ∗ f = φk ∗ ψk ∗ f +

∞  a=k+1

ϕa ∗ ϕa ∗ ψk ∗ f

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where the sum converges in the sense of S  (see (52) in the proof of Theorem 2.1). To show that ψk ∗ f is a continuous function it suffices to prove that the sum converges in L∞ loc . An application of Lemma 3.2 shows that there exists β  0 such that for every ρ ∈ S and k  j |ρk ∗ f (x)| j 2βk (1 + |x|) .

(32)

Note that by (i) in Lemma 2.3, the assumption that ψ& is rapidly decreasing together with the support of ϕ & implies that |ϕa ∗ψk (x)|  2(k−a)(β+1) 2kn (1+2k |x|)−(n+1+[]) . Therefore using an application of (32) we deduce the bound  |ϕa ∗ ψk (y)|(1 + |x − y|) dx  2k(β+1) 2−a (1 + |x|) |ϕa ∗ ϕa ∗ ψk ∗ f (x)| j 2aβ Rn

and hence the sum in (31) converges uniformly on compact sets. Consequently ψk ∗ f is a continuous function. Finally, to deduce the required bound, we note that after another application of (32) we have for every k  j  |ϕa ∗ ψk ∗ ϕa ∗ f (x)| |ψk ∗ f (x)|  |ψk ∗ φk ∗ f (x)| + a>k

j 2 (1 + |x|) + 2 kβ



k(β+1)



2−a (1 + |x|)  2kβ (1 + |x|)

ak



which then gives (30).

Remark 3.2. Lemma 3.2 assures us that for any distribution f of growth , there exists a β > 0 such that f satisfies the conditions of Proposition 3.1. Thus provided we have ψ ∈ L1 satisfying (1 + | · |) ψ(·) ∈ L1 and ψ& ∈ C n+1+ Rn \ {0} with, for every |κ|  n + 1 +  and some m > β, & = O(|ξ|−n−m ) ∂ κ ψ(ξ)

as |ξ| → ∞,

then the convolution ψ ∗ f is a continuous function. Unfortunately, we have no control over how large β is. Thus if we only assume that f is a distribution of (unspecified) finite growth, to ensure ψ ∗ f is a function, we need ψ to satisfy the conditions of Proposition 3.1 for every β. In particular we need ψ& to be rapidly decreasing. Moreover, some smoothness of ψ is required too. For example, for ψ ∗ f to be a well-defined function for every f ∈ S  of growth 0, a straightforward application of the Sobolev embedding theorem shows that ψ ∈ C ∞ . 4. Maximal Inequalities As in the seminal work of Fefferman and Stein [10], and Peetre [16, 17], the key step in the proof of our characterisation theorems is to obtain certain pointwise maximal inequalities relating ψj ∗ f and ϕj ∗ f . More precisely, assuming for the moment that the convolution ψk ∗ f ∈ L1loc , our goal in this section is to prove an inequality of the form   |ψk ∗ f (x − y)|r kn (33) (ϕ∗j f (x))r  2δ(j−k) 2 dy k λr Rn (1 + 2 |y|) kj

for some δ > 0, 0 < r < ∞, and λ is as in the definition of the Peetre maximal function (5). The argument used to prove (33) follows a strategy of Str¨ombergTorchinsky [20] together with a number of technical refinements. The first of which is the following extension of the Calder´on reproducing formula.

CHARACTERISATION OF B-L AND T-L SPACES

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Proposition4.1. Let   0. Suppose ψ ∈ L1 satisfies the Tauberian condition with ψ& ∈ C n+1+[] Rn \ {0} . There exists η&, φ& ∈ C n+1+[] (Rn ) such that for every g ∈ L1loc with g(x) = O(|x| ) we have for k ∈ Z and a.e. x ∈ Rn (34)

g(x) = φk ∗ g(x) +

∞ 

ηj ∗ ψj ∗ g(x).

j=k+1

Moreover supp φ& is compact, and supp η& is contained in some annulus about the origin. Proof. We start by observing that there exists an η ∈ L1 satisfying the required conditions, such that for all ξ = 0  & −j ξ) = 1 η&(2−j ξ)ψ(2 (35) j∈Z

The construction of η is standard and follows from the following observation: There exist positive numbers a, b, c with 0 < 2a  b such that for every ξ ∈ Rn there exists j ∈ Z satisfying a  2−j |ξ|  b and & −j ξ)|2  c. |ψ(2 We refer to [20, Chapter V, Lemma 6] for details of this contruction in the smooth case. The modification to the nonsmooth case has been carried out in the thesis [8] (see also [24]). Define , & −j ξ) &(2−j ξ)ψ(2 ξ = 0 j0 η & φ(ξ) = 1 ξ = 0. It is easy to check that φ satisfies the required conditions and that φ& = 1 in a neighbourhood of the origin. Moreover we have for any k, m ∈ Z with m > k (36)

φ m − φk =

m 

ηj ∗ ψj .

j=k+1

& ψ& &η ∈ C n+1+[] (Rn ) Take any g ∈ L1loc satisfying g(x) = O(|x| ). Note that as φ, −(n+1+[]) we have |φ|, |ψ ∗ η|  (1 + |x|) and hence the convolutions η ∗ ψ ∗ g and φ ∗ g are well defined. Moreover since φm forms an approximation to the identity we have limm→∞ φm ∗ g(x) = g(x) for a.e. x ∈ Rn (more precisely this holds at every Lebesgue point of g). Thus taking the convolution of g with both sides of (36) and letting m → ∞ proves the result.  To prove the maximal function inequality (33), we need to assume the boundedness of a particular auxiliary maximal function, namely, the following variation of the Peetre maximal function |ψk ∗ f (y)| 2(j−k)m . (37) Mλ,m (x, j) = sup j λ y∈Rn ,kj (1 + 2 |x − y|) Note that if Mλ,m (x0 , j) is finite for some x0 ∈ Rn , then we have Mλ,m (x, j) < ∞ for all x ∈ Rn . With these definitions at hand we now prove the following theorem which is essentially a non-smooth and discrete version of Theorem 2a in [20, page 61] (see also [6, Lemma 2]).

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Theorem 4.1. Let 0 < r  1, 0 < λ < ∞,   0, and m, β ∈ R. Assume (1 + | · |) ψ(·) ∈ L1 satisfies the Tauberian condition. Moreover, suppose that ψ& ∈ C max{n+1+[],[λ]+1} (Rn \ {0}) and for every |κ|  max{[], [λ]} + 1 we have   & = O |ξ|− max{m,β} as |ξ| → ∞ . ∂ κ ψ(ξ) Let f be a distribution of growth  such that for every j ∈ Z the distribution ψj ∗ f is a locally integrable function with M,m (x, j) < ∞. Then we have the pointwise inequality ∞ r   ∗ 2(j−k)(β−λ)r ψj f (x) 

(38)



k=j

Rn

|ψk ∗ f (x − y)|r kn 2 dy (1 + 2k |y|)λr

with constant independent of f , j, m,  and x. Proof. Fix u  j. The assumption M,m (x, j) < ∞ implies that ψu ∗ f = O(|x| ). Therefore the Tauberian condition and Proposition 4.1 give ψu ∗ f (x) = φu ∗ ψu ∗ f (x) +

(39)

∞ 

ηk ∗ ψk ∗ ψu ∗ f (x)

k=u+1

& η& ∈ C max{n+1+[],[λ]+1} (Rn ) and support of η& is contained in some annulus with φ, about the origin. An application of Lemma 2.3 gives (1 + 2u |x|)λ |ηk ∗ ψu (x)|  2−(k−u)β 2kn and thus we have the bound     |ηk ∗ ψk ∗ ψu ∗ f (x)|  (1 + 2u | · |)λ ηk ∗ ψu L∞ (1 + 2u |x − ·|)−λ ψk ∗ f L1    2kn 2−β(k−u) (1 + 2u |x − ·|)−λ ψk ∗ f L1 . On the other hand, the decay on φ shows     |φu ∗ ψu ∗ f (z)|  (1 + 2u | · |)λ φu L∞ (1 + 2u |z − ·|)−λ ψu ∗ f L1    2un (1 + 2u |z − ·|)−λ ψu ∗ f  1 L

and hence via (39) we obtain, for every z ∈ R and any u  j,  ∞  |ψk ∗ f (y)| 2(j−k)β 2kn dy |ψu ∗ f (z)|  2(u−j)β u λ Rn (1 + 2 |z − y|) n

k=u

where the constant depends only on ψ, β, and λ (in particular, it is independent of f , j, , and m). Now, since k  u  j, we have |ψk ∗ f (y)| 2(j−k)β (1 + 2u |z − y|)λ =

|ψk ∗ f (y)| 2(j−k)β (1 + 2j |x − y|)λ



|ψk ∗ f (y)| 2(j−k)β (1 + 2j |x − y|)λ

!r !r

|ψk ∗ f (y)| 2(j−k)β (1 + 2j |x − y|)λ

!1−r

Mλ,β (x, j)1−r (1 + 2j |x − z|)λ

(1 + 2j |x − y|)λ (1 + 2u |z − y|)λ

CHARACTERISATION OF B-L AND T-L SPACES

129

and hence using the elementary inequality (1 + 2j |y|)−1  2k−j (1 + 2k |y|)−1 we deduce that |ψu ∗ f (z)| 2(j−u)β (1 + 2j |x − z|)λ  ∞   Mλ,β (x, j)1−r 2(j−k)βr

|ψk ∗ f (y)|r 2kn dy j λr n (1 + 2 |x − y|) R k=u  ∞  |ψk ∗ f (y)|r 1−r (j−k)(β−λ)r  Mλ,β (x, j) 2 2kn dy. k λr Rn (1 + 2 |x − y|) k=j

Thus taking the supremum over z ∈ Rn and u  j yields,  ∞  |ψk ∗ f (y)|r 2(j−k)(β−λ)r 2kn dy. (40) Mλ,β (x, j)  Mλ,β (x, j)1−r k |x − y|)λr (1 + 2 n R k=j

If we had Mλ,β (x, j) < ∞, then noting that ψj∗ f (x)  Mλ,β (x, j), we obtain  ∞  ∗ r  |ψk ∗ f (y)|r (j−k)(β−λ)r (41) ψj f (x)  2 2kn dy. k λr Rn (1 + 2 |x − y|) k=j

Note that the constant in (41) is independent of f , j, m, , and x. Therefore it suffices to prove Mλ,β < ∞. To this end let m = max{m, β} and λ = max{, λ}. Note that by our assump  tion we have Mλ ,m  M,m < ∞. Moreover, we have (1+|·|)λ η, (1+|·|)λ φ ∈ L∞ and via Lemma 2.3 



2(k−u)m 2kn (1 + 2u |x|)λ |ηk ∗ ψu (x)| < ∞. Thus repeating the argument used to obtain (41) (with (λ, β) replaced by (λ , m )) we have (42)  ∞  |ψk ∗ f (z)|r r r (u−k)(m r−n) 2 2un dz. |ψu ∗ f (y)|  Mλ ,m (y, u) m u λ r Rn (1 + 2 |y − z|) k=u

Since the right hand side of (42) only gets larger if we decrease m and λ , we deduce that (42) in fact holds for λ = λ and m = β (but with a constant that depends on m, hence this argument cannot be used to prove (41) directly). Moreover, as 2(j−u)βr 2jn 2un (u−k)(βr−n) (j−k)(βr−n) × ×2  2 (1 + 2j |x − y|)λr (1 + 2u |y − z|)λr (1 + 2j |x − z|)λr we have for any u  j

 ∞  |ψk ∗ f (z)|r |ψu ∗ f (y)|r (j−u)βr (j−k)(βr−n) 2  2 2jn dz j λr (1 + 2j |x − y|)λr n (1 + 2 |x − z|) R k=u  ∞  |ψk ∗ f (z)|r  2(j−k)(β−λ)r 2kn dz. k λr Rn (1 + 2 |x − z|) k=j

Therefore, provided the right hand side of (41) is finite, we obtain Mλ,β < ∞ and so (38) follows. 

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The required maximal inequality (33) is now a corollary of the previous Str¨omberg-Torchinsky type estimate, Theorem 4.1, together with another application of the Calder´on reproducing formula in Proposition 4.1. Corollary 4.1. Let 0 < r, λ < ∞,   0, and m, β ∈ R. Assume (1 + | · |) ψ(·) ∈ L1 satisfies the Tauberian condition. Moreover, suppose that ψ& ∈ C n+1+max{[],[λ]} (Rn \ {0}) and for every |κ|  max{[], [λ]} + 1 we have   & = O |ξ|− max{m,β} as |ξ| → ∞ . (43) ∂ κ ψ(ξ) Let f be a distribution of growth  such that for every j ∈ Z the distribution ψj ∗ f is a locally integrable function with M,m (x, j) < ∞. Then we have the pointwise inequality   ∗ r  (j−k)(β−λ)r ϕj f (x)  2

Rn

kj

|ψk ∗ f (x − y)|r kn 2 dy (1 + 2k |y|)λr

with constant independent of f , j, m,  and x. Proof. Assume f is a distribution of growth . Then ϕj ∗ f = O(|x| ) and so we can apply Proposition 4.1 and obtain ∞  ϕj ∗ f (x) = φu ∗ ϕj ∗ f (x) + ηk ∗ ϕj ∗ ψk ∗ f (x). k=u+1 n where η&, φ& ∈ C n+1+max{[],[ p ]} (Rn ), supp φ& ⊂ {|ξ| < b}, and supp η& ⊂ {a < |ξ| < b} for some a, b > 0. Since supp ϕ & ⊂ {2−1  |ξ|  2}, by choosing u = j − s with s sufficiently large we have φu ∗ ϕj = 0. Similarly, perhaps choosing s slightly larger ηk ∗ ϕj = 0 for k > j + s. Therefore we have

|ϕj ∗ f (x)| 

(44)

j+s 

|ηk ∗ ϕj ∗ ψk ∗ f (x)|.

k=j−s

If r  1, we simply use an application of Holder’s inequality together with (44) to deduce that |ϕj ∗ f (x − y)| (1 + 2j |y|)λ  n |ψk ∗ f (x − z)| j n  2−j r (1 + 2j |z − y|)λ |ηk ∗ ϕj (z − y)|× 2 r dz (1 + 2j |z|)λ n R j≈k   (1 + 2j | · |)−λ ψk ∗ f (x − ·)2j nr  r  L j≈k

where we used the decay of ϕ. The require inequality now follows by taking the sup over y ∈ Rn and then taking r th powers of both sides. On the other hand, if 0 < r < 1, a similar application of (44) gives   ϕ∗j f (x)  (1 + 2j | · |)λ ηk ∗ ϕj L1 ψk∗ f (x)  ψk∗ f (x). j≈k th

j≈k

If we again take r powers of both sides, then result follows directly from an application of Theorem 4.1. 

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Remark 4.1. In the case r  1, the proof of Corollary 4.1 shows that the & (43), is not needed. In fact we only need the the smoothness decay condition on ψ, n+1+max{[λ],[]} & (Rn \ 0} to ensure that the η given by Proposition assumption ψ ∈ C 4.1 has sufficient decay. Moreover, a careful examination of the proof of Theorem 4.1 and Corollary 4.1, shows that we may replace the condition (43) with the slightly weaker condition     sup 2j max{β,m} (1 + |x|)max{λ,} ϕj ∗ ψ L∞ < ∞ j1

(c.f. the “poised spaces of Besov type” introduced by Peetre in [16]). 5. Proof of Characterisation Theorems In this section we give the proofs of our main results. We start with the sufficient direction, i.e. Theorems 1.2 and 1.3. The first step is the following preliminary version of Theorem 1.3. Theorem 5.1. Let 0 < p, q  ∞, α ∈ R. Assume λ > Λ  0 and   0. Let f be a distribution of growth  and (1 + | · |) ψ ∈ L1 satisfying the following: (S1) the kernel ψ satisfies the Tauberian condition and we have ψ& ∈ C n+1+max{[],[Λ]} (Rn \ {0}); (S2) there exists m  0 such that for every j ∈ Z the distribution ψj ∗ f is a locally integrable function with M,m (x, j) < ∞; (S3) there exists β > Λ − α such that for every |γ|  max{[Λ], []} + 1 ∂ γ ψ& = O(|ξ|− max{β,m} ) If Λ =

n p

as |ξ| → ∞.

then   q  1q q  1q 2jα ϕ∗j f Lp 2jα ψj ∗ f Lp  j∈Z

j∈Z

with constant independent of m and f . Similarly if Λ = max{ np , nq } and p < ∞ then       q  q1  q  1q      2jα ϕ∗j f 2jα |ψj ∗ f |  p   p  j∈Z

L

and in the case p = ∞  1   1   q1   jα ∗ q sup 2 ϕj f (x)) dx  sup |Q| Q |Q| Q Q Q j−(Q)

L

j∈Z



 q1  jα 2 |ψj ∗f (x)|)q dx

j−(Q)

where again the implied constant is independent of m and f . Note that when q = ∞, the previous inequality takes the form   1 1 jα ∗ sup sup 2 ϕj f (x)dx  sup sup 2jα |ψj ∗ f (x)|dx, Q j−(Q) |Q| Q Q j−(Q) |Q| Q where we require Λ = n.

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Proof. The proof is an adaptation of the arguments used in [5–7], with Theorem 4.1 replacing [6, Lemma 2]. We only prove the Triebel-Lizorkin case as the Besov-Lipschitz case is similar. As the lefthand side of the inequalities only gets larger if we decrease λ, we may assume max{ np , nq } = Λ < λ < min{α + β, [Λ] + 1}. Assume p < ∞. Choose 0 < r < min{p, q} with max{ np , nq } < nr < λ. An application of Corollary 4.1, together with a decomposition of Rn into annuli centred at x, gives the pointwise inequality  jα ∗ r  −k(β+α−λ)r (45) 2 ϕj f (x)  2 M ((2(k+j)α |ψk+j ∗ f |)r )(x) k1

 −n

where M (g) = supR>0 R |g(x − y)|dy denotes the Hardy-Littlewood maxi|y| nq . Therefore  jα q   2 |ψj ∗ f (x − y)| 2jn dy (1 + 2j |y|)λq a+ (Q) |y|≈2 j−(Q) a1     q 1  2jα |ψj ∗ f (y)| dy 2−a(λq−n) a+(Q) n (2 ) |x−y|2a+ (Q) a1 j−(Q) 

   jα q 1 2 |ψj ∗ f (x)| dx .  sup |Q | Q Q  j−(Q )

These two estimates imply the required inequality when q < ∞. The proof in the case p = q = ∞ is similar, in fact simpler, so we shall be brief. Fix a dyadic cube Q and let x ∈ Q as above. Let j  −(Q). Using Corollary 4.1 with r = 1 we get   2kα |ψk (x − y)| kn 2−(k−j)(α+β−λ) 2 dy. 2jα ϕ∗j f (x)  k λ Rn (1 + 2 |y|) kj

It follows that, by decomposing the y-integral as before and noting that λ > n in this case, one obtains 

  1 1 kα 2jα ϕ∗j f (x)dx  sup sup 2 |ψ ∗ f (x)|dx . k  |Q| Q Q k−(Q ) |Q | Q 

The proof of the theorem is thus complete.

The proof of the p = ∞ case in Theorem 1.1 requires the following corollary (c.f. the proof of Lemma 4 and 5 in the work of Rychkov [18]). Corollary 5.1. Let Then for any dyadic cube

  1 |Q| Q

0 < q < ∞, λ > Q we have  jα ∗ q 2 ϕj f (x) dx

j−((Q)+k)

1 |Q| j−((Q)+k)



sup

Q

n q,

and λ > n when q = ∞. Let k ∈ Z.

1/q

1   (1 + |k|) q f F˙ α , q < ∞ ∞,q

  2jα ϕ∗j f (x) dx  f F˙ α

∞,∞

.

Proof. Assume first that q < ∞. An application of Theorem 5.1 with ψ = ϕ (in which case the assumptions (S1), (S2), and (S3) clearly hold) gives 1/q

   q 1 jα ∗ 2 ϕj f (x) dx  f F˙ α . sup  ∞,q Q |Q | Q  j−(Q )

Thus it is enough to show that for j < −(Q),    ∗  ∗ q q 1 1 ϕ f (x) dx   ϕ f (x) dx (47) |Q| Q j |Q | Q j

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where Q is a dyadic cube with Q ⊂ Q and (Q ) = −j. To this end, note that if x, x ∈ Q , then |x − x |  2n−j and hence for any y ∈ Rn we have (1 + 2j |x − y|)λ  (1 + 2j |x − x| + 2j |x − y|)λ  (1 + 2j |x − y|)λ . Therefore the definition of ϕ∗j f implies that for any x, x ∈ Q we have ϕ∗j f (x)  ϕ∗j f (x ). Consequently ϕ∗j f (x) is essentially constant on cubes of side lengths < 2−j , in particular, we have (47). Thus the result for q < ∞ follows. The modification when q = ∞ is done in a similar manner to the proof of Theorem 5.1.  5.1. Sufficient Conditions. We now come to the proof of the sufficient direction of our characterisations which are now a straightforward consequence of Theorem 5.1. Proof of Theorem 1.3. We reduce to checking the conditions (S1), (S2), and (S3). The condition (S1) is clear. An application of Proposition 3.1 shows that there exists m  0 such that (S2) holds. Finally the rapid decay of ∂ κ ψ& implies that (S3) holds.  The proof of our characterisation with Lp replaced with H p , namely Theorem 1.2, again follows from Theorem 5.1.  Proof of  Theorem 1.2. Let f ∈ S be a distribution of growth  and take φ ∈ S with φ = 0. The idea is to show that there exists an s > 0 such that the kernel φs ∗ ψ satisfies the assumptions (S1), (S2), and (S3) of Theorem 5.1. To check (S1), note that by following the argument leading to (4.1), there exists 0 < 2a < b and c > 0 such that for every a  t  b and ξ ∈ Sn−1

& |ψ(tξ)|  c.  & = φ = 0, there exists r > 0 such that |φ(ξ)| & Since φ ∈ S and φ(0) > 0 for |ξ| < r. r + & Now as φs (ξ) = φ(sξ) we only need to choose s < a to ensure that φs ∗ ψ satisfies the Tauberian condition. Clearly the remaining conditions in (S1) are also satisfied. To verify (S2), observe that since φs ∗ f = O(|x| ), the convolution (φs ∗ ψ)j ∗ f is well-defined. Furthermore, an application of Lemma 3.2 shows that there exists m such that |φsk ∗ f (x)|  2|k|m (1 + |x|) which implies that for any x ∈ Rn , j ∈ Z, sup y∈Rn ,kj

|(φs ∗ ψ)k ∗ f (y)| (j−k)m (1 + |y|) 2  sup 2(j−k)m 2|k|m < ∞. j  (1 + 2j |x − y|) y∈Rn ,kj (1 + 2 |x − y|)

Thus (S2) holds. Finally, the rapid decay of ∂ κ φ& ensures that (S3) holds provided that ∂ κ ψ& is slowly increasing as |ξ| → ∞.

CHARACTERISATION OF B-L AND T-L SPACES

135

Therefore, we may apply Theorem 5.1 together with the pointwise bound |ϕj ∗ f (x)|  ϕ∗j f (x), to deduce that f B˙ α

p,q

 1  q q  jα   2 φsj ∗ ψj ∗ f Lp  j∈Z

 1  q q  jα   sup |φt ∗ ψj ∗ f | Lp  2 t>0

j∈Z

 1  q q  jα   2 ψj ∗ f H p  j∈Z

where the last line follow from the H p charaterisation of Fefferman-Stein [10]. Similarly, the Triebel-Lizorkin case follows via    q  q1    2jα |φsj ∗ ψj ∗ f |  f F˙p,q  p α L

j∈Z

   q  q1    2jα sup |φt ∗ ψj ∗ f |  

Lp

t>0

j∈Z

.

An identical computation gives the p = ∞ case. Thus the proof of Theorem 1.2 is complete.  Theorem 1.3 required ∂ κ ψ& to be rapidly decreasing to ensure that the convolution ψ ∗ f was a locally integrable function. One way to avoid this fairly strong assumption on the kernel ψ, was presented in Theorem 1.2 where we replaced the Lp norm with the Hardy norm H p which is defined for elements of S  . Consequently we only had to make sense of ψj ∗ f as an element of S  rather than L1loc . On the other hand, an alternative approach to finding a pointwise definition of the convolution is to instead make further assumptions on f . In particular, if we assume that f is a slowly increasing function of order , then the convolution ψ ∗ f is well defined as a function without the rapidly decreasing assumption. This leads to the following version of Theorem 5.1. Theorem 5.2. Let 0 < p, q  ∞, α ∈ R, and   0. Let Λ  0 and β > Λ − α. Assume (1 + | · |)− f ∈ L∞ . Suppose ψ ∈ L1 satisfies the Tauberian condition with (1 + | · |) ψ(·) ∈ L1 . Furthermore, assume that ψ& ∈ C n+1+max{[],[Λ]} (Rn \ {0}) with   & = O |ξ|− max{β,0} as |ξ| → ∞ ∂ κ ψ(ξ) for |κ|  max{[Λ], []} + 1. If Λ = f B˙ p,q  α

n p

then

 q  1q 2jα ψj ∗ f Lp j∈Z

Similarly if Λ = max{ np , nq } and p < ∞ then    q  q1    f F˙ α   2jα |ψj ∗ f |  p,q

j∈Z

Lp

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and in the case p = ∞ f F˙∞,q α

 1   sup |Q| Q Q

 q1  jα 2 |ψj ∗ f (x)|)q dx ,

 j−(Q)

with the usual interpretation when q = ∞ (in which case Λ = n). Proof. We begin by observing that for k  0 |ψk ∗ f (x)|  (1 + |x|) which implies that M,0 (x, 0) < ∞ and consequently M,0 (x, j) < ∞ for every j ∈ Z. Therefore result follows from Theorem 5.1.  α α Remark 5.1. Assume α > n/p. Then elements in B˙ p,q or in F˙ p,q are functions that satisfy the growth condition in Theorem 5.2 with  = α−n/p when α−n/p ∈ / N, and  > α − n/p when α − n/p ∈ N (see Remark 2.1). Hence this theorem readily gives the characterisation of these function spaces without the rapidly deacreasing & assumption on the Fourier transform of the kernel ψ.

5.2. Necessary Conditions. We now come to the necessary direction of our characterisation, namely the proof of Theorem 1.1. As in the proof of Theorem 5.1, we follow the maximal function arguments used in the work of Bui-PaluszynskiTaibleson [5–7]. In addition, in the case p = ∞, we rely also on an argument due to Rychkov [18]. n

−α

p . To this end, the Proof of Theorem 1.1. We first show that ψ ∈ B˙ 1,1 assumptions on ψ together with Lemma 2.3 imply that , 2−jm (1+|x|)1n+1+[Λ] j0 (48) |ϕj ∗ ψ(x)|  2jn jr 2 (1+2j |x|)n+1+[Λ] j  0.

It follows that   n n ψ ˙ np −α  2−j(α− p +m) + 2j(r−α+ p ) < ∞ (as α − n/p + m > 0, r > α). B1,1

j Λ − α). If we now use an application of the above Calder´ on formula we obtain   |ψk ∗ ϕj (y)| |ψk ∗ f (x − z)|  |ϕj ∗ f (x − z − y)|dy (1 + 2k |z|)λ (1 + 2k |z|)λ j∈Z R   (1 + 2j |z + y|)λ ∗ |ϕj f (x)| |ψk ∗ ϕj (y)| dy,  (1 + 2k |z|)λ R j∈Z

CHARACTERISATION OF B-L AND T-L SPACES

137

which implies the pointwise estimate 2kα ψk∗ f (x) 

(49)



aj−k 2jα ϕ∗j f (x),

j∈Z

where aj = 2−jα sup x∈Rn



j λ   ψ ∗ ϕj (y) (1 + 2 |x + y|) dy. λ (1 + |x|) Rn

Thus, provided (aj ) ∈ min{p,q,1} ( (aj ) ∈ min{q,1} in the Triebel-Lizorkin case), the first part of Theorem 1.1, (6) and (8), follows from Proposition 2.1 together with the maximal function characterisation of Peetre [16]. In fact, using (48), it is easy to see that , (50)

aj 

 2j(λ−m−α) Rn (1 + |y|)λ−n−1−[Λ] dy,  2j(r−α) Rn 2jn (1 + |2j y|)−n−1−[Λ] dy,

j0 j  0.

By our assumptions, λ > [Λ]  0, r > α, and λ − m − α < 0 by our choice of λ, we deduce that (aj ) ∈ β for all β > 0. Hence (6) and (8) are proved. It remains to consider the case p = ∞. We provide detail only in the case q < ∞ as the modification when q = ∞ is familiar by now. As in the case p < ∞, an application of (49) together with (50) shows that there exists δ > 0 such that  2−|j|δ 2(k−j)α ϕ∗k−j f (x). 2kα ψk∗ f (x)  j∈Z

Therefore, Corollary 5.1 gives for any dyadic cube Q    q 1 2kα ψk∗ f (x) dx |Q| Q k−(Q) q    1 −|j|δ (k−j)α ∗ 2 2 ϕk−j f (x) dx  |Q| Q j∈Z k−(Q) 

    kα ∗ q 1 2 ϕk f (x) dx 2−|j|δ min{1,q}  |Q| Q j∈Z



k−((Q)+j)

f qF˙ α

∞,q

,

where, in the second inequality, we also use the q-triangle inequality when q  1 and H¨ older’s inequality when q > 1. Thus (9) follows. We now turn to the proof of (7), (10), and (11). Let φ ∈ S. Since φt ∗ f satisfies α the same properties as f (φt ∗ f ∈ B˙ p,q ), we can repeat the proof of (49) to deduce that  2kα |φt ∗ ψk ∗ f (x)|  2kα ψk∗ (φt ∗ f )(x)  ak−j 2jα ϕ∗j (φt ∗ f )(x) j∈Z

with constant independent of t. If we now follow the arguments leading to (6), (8), and (9), it suffices to show that (51)

ϕ∗j (φt ∗ f )(x)  ϕ∗j f (x).

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To this end, let μ ∈ S with μ & = 1 for 2−1 < |ξ| < 2 and sup μ & ⊂ {2−2 < |ξ| < 4}. Then ϕj = ϕj ∗ μj and so  |ϕj ∗ φt ∗ f (x − y)| ∗  ϕj f (x) |μj ∗ φt (z)|(1 + 2j |z|)λ dz. (1 + 2j |y|)λ n R If 2j < 1t then (51) follows easily by changing the order of integration. On the other hand if 2j  1t , then since φ ∈ S we use part (i) of Lemma 2.3 to deduce that |φt ∗ μj (x)| 

 2−j n+1+[λ] t

 1+

t−n 2jn n+[λ]+1  (1 + 2j |x|)n+1+[λ] . |x| t

Therefore (51) follows and so we obtain (7), (10), and (11).  6. Appendix 6.1. Proof of Theorems 2.1 and 2.2.  & &2 (2−j ξ) for Proof of Theorem 2.1. Define φ ∈ S by letting φ(ξ) = j0 ϕ & & & ξ = 0 and φ(0) = 1. Then φ(ξ) = 1 for |ξ|  1, φ(ξ) = 0 for |ξ| > 2, and for any  N < 0 < M and f ∈ S , we have the identity (52)

M 

ϕj ∗ ϕj ∗ f (x) = φM ∗ f (x) − φN ∗ f (x).

j=N +1

Let f ∈ S  . By definition, there exists a > 0 such that for any κ we have    κ    ∂ φN ∗ f (x) = 2N |κ|  ∂ κ φ ∗ f (x) N      sup y α ∂ β ∂ κ φ N (x − y)  2N |κ| |α|,|β|a

y∈Rn

 (1 + |x|)a 2N (|κ|+n−a) . Define the polynomial pN (x) as (53)

pN (x) =

 |κ|a−n

  ∂ κ φN ∗ f (0) κ x κ!

(if a < n we can just take pN = 0 for every N ). By expanding φN ∗ f as a Taylor series about x = 0 and using the bound on ∂ κ (φN ∗ f ) obtained above, we have  1      κ  φN ∗ f (x) − pN (x)  |x|a+1−n ∂ φN ∗ f (tx)dt |κ|=a−n+1

0

 (1 + |x|)2a+1−n 2N . Consequently we see that φN ∗ f − pN → 0 in S  as N → −∞. On the other hand,  since φ = 1, we have φM ∗ f → f in S  as M → ∞. Therefore result follows from the identity (52).  A similar argument gives Theorem 2.2.

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139

Proof of Theorem 2.2. Let φ be as in the proof of Theorem 2.1. As previously, the key point is to study the convergence of φN ∗ f as N → −∞. Define the polynomials pN (x) as  xγ   ∂ γ φN ∗ f (0). pN (x) = γ! |γ|[α]

The form of the Taylor series remainder given in [11] implies that for any N < N        φN ∗ f −pN (x) − φN  ∗ f − pN  (x) N     xγ    ∂ γ (ϕj ∗ ϕj ∗ f )(0) ϕj ∗ ϕj ∗ f (x) − γ! |γ|[α]

j=N +1



 |x|[α]

N 

 

1

  γ ∂ (ϕj ∗ ϕj ∗ f )(tx) − ∂ γ (ϕj ∗ ϕj ∗ f )(0)dt.

0

j=N +1 |γ|=[α]

If we now observe that  γ  ∂ (ϕj ∗ ϕj ∗ f )(tx)−∂ γ (ϕj ∗ ϕj ∗ f )(0)    2j|γ| sup (∂ γ ϕ)(t2j x + ·) − (∂ γ ϕ)(·)L1 ϕj ∗ f L∞ 0 0} and the condition μ({ω ∈ Ω : λ1 (w) > 0}) < ∞ gives μ({ω ∈ Ω : λj (w) > 0}) < ∞ for all 1 ≤ j ≤ m. (3) Since Theorem 4.1 provides a formula for the error, it follows mthat the error for Problem 1, which is defined as E(F, ) := minM ∈C j=1 Fj −  m PM Fj 2 , coincides with the expression j=+1 Ω λj (w) dμ(w). Now we will consider an approximation problem for the Hilbert space H over an appropriate minimizing class. Even though the main reason for considering this problem is to solve Problem 2, the result turns out to be interesting by itself. 4.2. The approximation problem for an orthogonally decomposed H. In this subsection, we give a solution for an abstract version of the approximation problem for general Hilbert spaces. The main result is a generalization of [13, Theorem 6.1] where the problem is solved for 2 (Zd ) and a decomposition of it arising from a partition of Zd . Although our result is for general Hilbert spaces and arbitrary orthogonal decompositions, the proof follows the lines of that of [13, Theorem 6.1]. Let H be a separable Hilbert space and suppose that it is decomposed into an orthogonal sum as in (3.1), that is H = H1 ⊕ · · · ⊕ Hκ ,

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where κ ∈ N. Denote by PV the orthogonal projection of H onto the closed subspace V , and for a subspace S ⊆ H, let us abbreviate Si := PHi S for 1 ≤ i ≤ κ. The class of subspaces of H we shall work with is given by D := {S ⊆ H : S is a subspace, dim(S) ≤ , Si ⊆ S ∀ 1 ≤ i ≤ κ}. The reason for choosing this particular class of subspaces is not arbitrary. In fact, this is exactly the class we have to use to be able to relate the present situation to Problem 2. We shall explain this in more details when proving that Problem 2 has solution. The minimization problem we want to solve now is the following: given a set of data X = {x1 , . . . , xm } ⊆ H and  ∈ N we seek for a subspace S ∗ ∈ D satisfying m 

(4.6)

xj − PS ∗ xj 2 ≤

j=1

m 

xj − PS xj 2 ,

j=1

for any other S ∈ D . Note that, since x2 = PS x2 + x − PS x2 for all x ∈ H and any subspace S ⊆ H , then S ∗ satisfies (4.6) if and only if S ∗ satisfies m 

PS ∗ xj 2 ≥

j=1

m 

PS xj 2 ,

for every

S ∈ D .

j=1

On the other hand, for every subspace S of 1H it holds that S ⊆ the condition S ∈ D is equivalent to S = κi=1 Si . Then, we observe that for S ∈ D , m  j=1

PS xj 2 =

1κ i=1

Si . Hence,

m κ m κ m  κ       PSi xj 2 =  PSi PHi xj 2 = PSi PHi xj 2 . j=1

i=1

j=1

i=1

j=1 i=1

At this point we see calculation suggests the following strategy for  that the above m 2 2 P x  is as big as possible when finding S ∗ : since m S j j=1 j=1 PSi PHi xj  is as big as possible, for every 1 ≤ i ≤ κ, we will take an optimal Si –given by Theorem 4.1– minimizing 1 the data Xi := {PHi x1 , . . . , PHi xm } for every 1 ≤ i ≤ κ and then show that S ∗ := κi=1 Si is the optimal subspace we are looking for. In this process we will have to deal with an extra condition on the dimension of the Si s because ∗  ≥ dim(S ) = κi=1 dim(Si ). Before stating and proving the main result of this section about the existence of an optimal S ∗ solving (4.6) we need to set some notation. For X = {x1 , . . . , xm } ⊆ H and 1 ≤ i ≤ , Gi is the Gramian associated to the data set Xi = {PHi x1 , . . . , PHi xm }, that is the matrix in Cm × Cm given by (Gi )kj = i PHi xk , PHi xj H . By λi1 ≥ · · · ≥ λim we denote its eigenvalues and by y1i , . . . , ym ∈ m C the corresponding left-eigenvectors. Let Λ be the set of all eigenvalues of Gi for every 1 ≤ i ≤ , i.e. Λ = {λij : 1 ≤ i ≤ , 1 ≤ j ≤ m} and let μ1 ≥ · · · ≥ μ be the  biggest elements of Λ ordered decreasingly. Then, we have that for every s = 1, . . . , , μs = λijss for some 1 ≤ is ≤ , 1 ≤ js ≤ m. With this notation we define h1 , . . . , h ∈ H by (4.7)

hs := (λijss )−1/2

m  k=1

yjiss (k)PHis xk ,

AN APPROXIMATION PROBLEM IN MULTIPLICATIVELY INVARIANT SPACES

155

when μs = λijss = 0 and hs = 0 otherwise. Here, yjiss (k) is the k-th entry of yjiss . Note that hs is defined in terms of the s-th biggest eigenvalue of Λ, its associated eigenvector and the corresponding projection of the elements of X . Theorem 4.4. Let  ∈ N and X = {x1 , . . . , xm } ⊆ H. Keeping the above notation we have that there exists S ∗ ∈ D satisfying (4.6). Moreover, S ∗ = span{h1 , . . . , h } and {h1 , . . . , h } forms a Parseval frame for S ∗ , where hi is defined by (4.7) for all 1 ≤ i ≤ . Proof. We will follow the strategy describedabove. For this, we begin by κ defining Q := {α := (α1 , . . . , ακ ) : αi ∈ N ∪ {0}, i=1 αi ≤ }. Now, for a fixed α α ∈ Q, let Si be the optimal subspace that minimizes the expression m  PHi xj − PS PHi xj 2 j=1

over the class of subspaces of H with dimension at most αi . The existence of such the same result asserts that subspace Siα is guaranteed by Theorem 4.1. Moreover,  m α α i −1/2 i i , . . . , h where h := (λ ) Siα is generated by hα αi r r 1 k=1 yr (k)PHi xk if λr = 0 α α α and hr = 0 otherwise, and also asserts that {h1 , . . . , hαi } forms a Parseval frame for Siα . 1κ α α constructed Now for every α ∈ Q, we consider S α := i=1 Si where Si is  m β as above. Let β ∈ Q be such that S is the subspace that minimizes j=1 xj − PS α xj 2 over α ∈ Q – which exists because Q is finite –, and set S ∗ = S β . By construction, S ∗ ∈ D and it is easy to see that S ∗ satisfies (4.6). Furthermore, since {hβ1 , . . . , hββi } forms a Parseval frame for Siβ and {Siβ : 1 ≤ i ≤ κ} 2 are orthogonal subspaces, κi=1 {hβ1 , . . . , hββi } forms a Parseval frame for S ∗ . What 2κ is left is to observe that i=1 {hβ1 , . . . , hββi } = {h1 . . . , h }, where each hs is as in (4.7).  Remark 4.5. By Theorem m 4.4, for  4.1, if β ∈ Q is as in the2 proof of Theorem i P x − P P x  agrees with every 1 ≤ i ≤ κ, the error m β H j H j i i j=1 j=βi +1 λj . Si  m   κ m m Then, the error j=1 xj − PS ∗ xj 2 is i=1 j=βi +1 λij = j=+1 μj . 4.3. Optimal decomposable MI spaces. As we mentioned before, in the second version of our approximation problem we will minimize over the subclass of C , C{H1 ,...,Hκ }, (see (4.3)). Specifically, Problem 2 reads a follows: Problem 2: Let  ∈ N and {H1 , . . . , Hκ } a decomposition of H as in (3.1). Given F = {F1 , . . . , Fm } ⊆ L2 (Ω, H), find a D-MI space M ∗ ∈ C{H1 ,...,Hκ }, such that m m   Fj − PM ∗ Fj 2 ≤ Fj − PM Fj 2 , (4.8) j=1

j=1

for all M ∈ C{H1 ,...,Hκ }, . Recall that in the situation of Problem 2 we have that H is decomposed as in (3.1). Let F = {F1 , . . . , Fm } ⊆ L2 (Ω, H) be the data set we want to approximate using the class C{H1 ,...,Hκ }, where  ∈ N is fixed. For a.e. ω ∈ Ω, we will consider the optimal subspace J ∗ (ω) ∈ D for the data F(ω) = {F1 (ω), . . . , Fm (ω)} ⊆ H given by Theorem 4.4. These will give point-wise solutions that we will “paste

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up” to construct a solution for Problem 2. Roughly spiking, we will show that ω → J ∗ (ω) is a measurable range function and the associated MI space, M ∗ , given by Theorem 2.3 is the solution we are looking for. Note that since J ∗ (ω) ∈ D for a.e. ω ∈ Ω, PHi (J ∗ (ω)) ⊆ J ∗ (ω) for every 1 ≤ i ≤ κ, thus by Lemma 3.4, M ∗ will be decomposable with respect to {H1 , . . . , Hκ }. This justified the choice of the class D in Section 4.2. Using the notation of Section 4.2 we have that for a.e ω ∈ Ω, J ∗ (ω) is generated by {Φ1 (ω), . . . , Φ (ω)} where, for s = 1 . . . , , m    i (ω) i (ω) (4.9) Φs (ω) := λjss (ω) (ω)−1/2 yjss (ω) (ω) (k)Pis Fk (ω), k=1

if

i (ω) λjss (ω) (ω)

 0 and Φs (ω) = 0 otherwise. = Now, the solution for Problem 2 is provided by the upcoming theorem.

Theorem 4.6. Let  ∈ N and F = {F1 , . . . , Fm } ⊆ L2 (Ω, H). Then M ∗ = MD (Φ1 , . . . , Φ ) ∈ C{H1 ,...,Hκ }, with Φs defined a.e. ω ∈ Ω by (4.9), is a solution of Problem 2. Moreover, {Φ1 , . . . , Φ } is a uniform Parseval frame for J ∗ , where J ∗ is the measurable range function associated to M ∗ . Proof. The measurability of the functions Φ1 , . . . , Φ is a consequence of the i (ω) i (ω) measurability of the eigenvalues λjss (ω) (ω) and the eigenvectors yjss (ω) (ω), and this follows readily from the argument given in [13, Section 4]. Furthermore, the same calculations we used to obtain (4.5) shows that Φ1 , . . . , Φ ∈ L2 (Ω, H). Therefore, by Theorem 2.3, J ∗ (ω) = span{Φs (ω) : 1 ≤ s ≤ } is the measurable range function associated to the MI space M ∗ = MD (Φ1 , . . . , Φ ). As we mentioned at the beginning of this section, J ∗ (ω) ∈ D by construction and then, by Proposition 3.4, M ∗ ∈ C{H1 ,...,Hκ }, . To show that M ∗ solves Problem 2, we first observe that if M ∈ C{H1 ,...,Hκ }, with associated range function J, then by Proposition 3.4, J(ω) ∈ D for a.e. ω ∈ Ω. Thus, by construction and Theorem 4.4, we have that for a.e. ω ∈ Ω m m   Fj (ω) − PJ ∗ (ω) (Fj (ω))2H ≤ Fj (ω) − PJ(ω) (Fj (ω))2H . j=1

j=1

Then, integrating over Ω and using Lemma 2.5 we obtain m m   2 ∗ Fj − PM Fj  ≤ Fj − PM Fj 2 , j=1

j=1

which says that M ∗ is a solution for Problem 2. Finally, since by Theorem 4.4, {Φ1 (ω), . . . , Φ (ω)} is a Parseval frame for J ∗ (ω) for a.e. ω ∈ Ω, we conclude that {Φ1 , . . . , Φ } is a uniform Parseval frame for J ∗ .  Remark 4.7. (1) Taking into account Remark 4.5, we have that the approximation er i(ω) m κ m ror F − PM ∗ Fj 2 is exactly i=1 j=βi +1 Ω λj(ω) (ω) dμ(ω) =  mj=1 j j=+1 μj (ω) dμ(ω). Ω (2) Note that the strategies for finding a solution for Problems 1 and 2 are similar. In both cases, the idea is to find point-wise solutions using previous result for general Hilbert spaces and then paste them up together to construct the optimal subspaces which solve Problem 1 and 2.

AN APPROXIMATION PROBLEM IN MULTIPLICATIVELY INVARIANT SPACES

157

(3) Problem 2 can be solved using Problem 1 in different way. Indeed, given a data set F = {F1 , . . . , Fm } ⊆ L2 (Ω, H), we can consider the data set containing the elements of F but split according to the decomposition of H given by (3.1), which is F := {P1 F1 , . . . , P1 Fm , . . . , Pκ F1 , . . . , Pκ Fm } ⊆ L2 (Ω, H). It can be proven that the solution of Problem 1 constructed for the data F is a solution of Problem 2 for the data set F. 5. Application to SI spaces in LCA groups In this section we shall see how the result about MI spaces can be used to obtain optimal shift-invariant subspaces for a given set of data. We shall work in L2 (G), where G is a second countable locally compact abelian group (LCA group for short) with operation written additively and the measure involved is the Haar measure on G which we denote by mG . Furthermore, we consider a uniform lattice H on G, that is a countable discrete subgroup H of G such that G/H is compact, and the translation operators along elements of H, Th , defined by Th f (x) := f (x − h), for f ∈ L2 (G) and a.e. x ∈ G. We say that a closed subspace V ⊆ L2 (G) is H-invariant if for every f ∈ V , Th f ∈ V for all h ∈ H. Now, given any (at most countable) set of functions A ⊆ L2 (G) , the space V := SH (A) where SH (A) = span{Th φ : h ∈ H, φ ∈ A} is an H-invariant space that we call the H-invariant space generated by A. In this section we focus on finitely generated H-invariant spaces, that is, when A is a finite set of L2 (G). For a finitely generated H-invariant space V we define its length (V ) as the minimum number of functions we need to generate it, more precisely, (V ) := min{n ∈ N : ∃ φ1 , . . . , φn ∈ L2 (G), such that V = SH (φ1 , . . . , φn )}. 5.1. Optimal H-invariant spaces. The first approximation result we will prove concerns the optimal finitely generated H-invariant space of length at most  that best fits a given set of data. To properly state the result we define, for a given  ∈ N the class of H-invariant spaces V as V := {V ⊆ L2 (G) : V is a finitely generated H-invariant space with (V ) ≤ }. Theorem 5.1. Let  ∈ N and F = {f1 , . . . , fm } ⊆ L2 (G) be a set of data. Then there exists V ∗ ∈ V such that m m   fj − PV ∗ fj 22 ≤ fj − PV fj 22 for all V ∈ V . j=1

j=1

Moreover, there exists a generator set for V ∗ , {φ1 , . . . , φ } such that {Th φj : h ∈ H, 1 ≤ j ≤ } is a Parseval frame for V ∗ . To prove the above theorem we shall establish a one-to-one correspondence with MI spaces and use Theorem 4.2. For this, we first need to recall some notions and properties about LCA groups. & we denote the dual group of G, that is the set of continuous characters By G of G and by mG its Haar measure. We use the notation (x, γ) for the complex & → C is the value that the character γ takes at x. For every x ∈ G, ex : G & & character on G induced by x, i.e. ex (γ) := (x, γ) for all γ ∈ G. For a subgroup K & given by of G, we write K ∗ for its annihilator which is the closed subgroup of G

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& : (x, γ) = 1 ∀ x ∈ K}. In our case, since we have chosen the subgroup K ∗ = {γ ∈ G H to be such that G/H is compact, by [34, Lemma 2.1.3], its annihilator H ∗ is discrete. Moreover, since G/H is metrizable, H ∗ is countable, [26, Theorem 24.15]. & a measurable section of the quotient G/H & ∗ , whose existence is We then fix Ω ⊆ G provided by [32, Lemma 1.1]. Since additionally H is discrete, Ω can be chosen to be relatively compact and thus of finite measure (see [30, Lemma 2]). The Fourier  transform of f ∈ L1 (G) is defined by f&(γ) = G f (x)(−x, γ) dmG (x) and it can be extended to an operator from L2 (G) to L2 (G) which, for a proper normalization of mG and mG , turns out to be an isometric isomorphism. The upcoming proposition was proven in [14, Proposition 3.3] and provides an isometric isomorphism between L2 (G) and the vector-valued space L2 (Ω, 2 (H ∗ )). Proposition 5.2. The fiberization mapping T : L2 (G) → L2 (Ω, 2 (H ∗ )) defined by T f (ω) = {f&(ω + δ)}δ∈H ∗ is an isometric isomorphism and it satisfies T Th f (ω) = (−h, w)T f (ω) for all f ∈ L2 (G), all h ∈ H and a.e. ω ∈ Ω. The fiberization isometry of Proposition 5.2 not only gives that L2 (G) is isomorphic to L2 (Ω, 2 (H ∗ )) but also provides the correspondence between H-invariant spaces in L2 (G) and MI spaces of L2 (Ω, 2 (H ∗ )). Here the underlying determining set is D = {eh χΩ }h∈H , where χΩ is the characteristic function of Ω, [11, Corollary 3.6]. Then, since T Th f = e−h T f for all f ∈ L2 (G), we have that V ⊆ L2 (G) is an H-invariant space if and only is T V ⊆ L2 (Ω, 2 (H ∗ )) is an MI space with respect to D. This fact allows us to prove Theorem 5.1. Proof of Theorem 5.1. Let T be the isomorphism of Proposition 5.2 and consider the set {T f1 , . . . , T fm } ⊆ L2 (Ω, 2 (H ∗ )). As we discussed before, Hinvariant spaces are in one-to-one correspondence with D-MI spaces under T , where D = {eh χΩ }h∈H . Moreover, if A ⊆ L2 (G) is an at most countable set, then V = SH (A) if and only if T V = MD (T A) where T A := {T φ : φ ∈ A}. In particular this shows that V ∈ V if and only if T V ∈ C where C is as in (4.2). Let M ∗ = MD (Φ1 , . . . , Φ ) ∈ C be an optimal MI space for the data in L2 (Ω, 2 (H ∗ )), {T f1 , . . . , T fm }, which exists due to Theorem 4.2. Now, set V ∗ := T −1 M ∗ ∈ V and consider any V ∈ V . Then, by Theorem 4.2 and Proposition 5.2 we have m m   fj − PV ∗ fj 22 = T fj − T PV ∗ fj 2 j=1

j=1

=

m 

T fj − PM ∗ T fj 2

j=1

≤

m  j=1

T fj − PT V T fj 2 =

m 

fj − PV fj 22 .

j=1

Moreover, since M ∗ = MD (Φ1 , . . . , Φ ), V ∗ = SH (φ1 , . . . , φ ), where φj := T −1 Φj for all 1 ≤ j ≤ . Recall that by Theorem 4.2 {Φ1 , . . . , Φ } is a uniform Parseval frame for the range function associated to M ∗ , then, using [14, Theorem 4.1] we

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have that {Th φj : h ∈ H, 1 ≤ j ≤ } is a Parseval frame for V ∗ , and this completes the proof.  Remark 5.3. (1) It is well known that there are LCA groups which do not have any uniform lattice, (see [11, 30]). However, there is an easy way to construct groups that do have uniform lattices. For a given discrete and countable LCA group H and a compact group C, simply take the LCA group G = H × C with the product measure. Then H × {0} is a uniform lattice on G. (2) When instead of translations other operators – such as dilations – are involved, an analogous result to Theorem 5.1 can be obtained. To be more precise, when a discrete abelian group Γ is acting on a measure space X , one can consider closed subspaces of L2 (X ) that are invariant under unitary operators arising from the group action. In this situation, it was proven in [9] that the mapping that relates these invariant subspaces with MI spaces is the generalized Zak transform, introduced in [27] (see [9] for details). Under this identification it can be proven that the version of Theorem 5.1 adapted to this setting is also true. 5.2. Optimal H-invariant spaces with extra invariance. Keeping the notation and hypotheses on G and H described above, we consider a closed subgroup Γ of G containing H and H-invariant spaces V with extra invariance on Γ. This means that Ty V ⊆ V for all y ∈ Γ and we also say that V is Γ-invariant. These type of H-invariant spaces were completely characterized in [6] and they are in one-to-one correspondence with MI spaces in L2 (Ω, 2 (H ∗ )) that are decomposable with respect to the decomposition of 2 (H ∗ ) that we describe below. /Γ∗ . Then H ∗ = H ∗1 2 Let ∗N be an at most countable section for the quotient 2 ∗ 2 ∗ σ∈N Γ + σ where the union is disjoint and therefore  (H ) = σ∈N  (Γ + σ). For σ ∈ N we define the set Bσ as 3 (Ω + σ) + γ ∗ . (5.1) Bσ = Ω + σ + Γ ∗ = γ ∗ ∈Γ∗

& It is not difficult to see that {Bσ }σ∈N is a partition of G. Lemma 5.4. Let V ⊆ L2 (G) be an H-invariant space and Γ ⊆ G a closed subgroup containing H. If T is as in Proposition 5.2 and M := T V, then the following conditions are equivalent: (i) V is Γ-invariant. (ii) M ⊆ L2 (Ω, 2 (H ∗ )) is decomposable with respect to {2 (Γ∗ + σ)}σ∈N . Proof. For each σ ∈ N , let Pσ : L2 (Ω, 2 (H ∗ )) → L2 (Ω, 2 (Γ∗ + σ)) be the orthogonal projection onto L2 (Ω, 2 (Γ∗ + σ)). Now, for any f ∈ L2 (G) define f σ & given via its Fourier transform as f+σ = χBσ f&, where {Bσ }σ∈N is the partition of G by (5.1). Then we have that T f σ (ω) = Pσ (T f )(ω) for a.e. ω ∈ Ω. Therefore, item (ii) is equivalent to require that for every f ∈ V and each σ ∈ N , T f σ (ω) ∈ J(ω) for a.e. ω ∈ Ω, where J is the measurable range function associated to M through Theorem 2.3. Thus, the result follows from [6, Proposition 5.4]. 

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We now restrict ourself to subgroups Γ that are discrete. In that case, the section N of H ∗ /Γ∗ is finite (see [15, Remark 2.2]). We consider the subclass of V denoted by VΓ, and defined by VΓ, := {V ∈ V : V has extra invariance on Γ}. For this class we can obtain the analogous result to Theorem 5.1. Its proof is a consequence of Theorem 4.6 and Lemma 5.4 and follows the lines of the proof of Theorem 5.1. Thus, we omit it. Theorem 5.5. Let H ⊆ G be a uniform lattice and Γ a larger discrete subgroup of G containing H. Let  ∈ N and F = {f1 , . . . , fm } ⊆ L2 (G) be a set of data. Then there exists V ∗ ∈ VΓ, such that m 

fj − PV ∗ fj 22 ≤

j=1

m 

fj − PV fj 22

for all

V ∈ VΓ, .

j=1

Moreover, there exists a generator set for V ∗ , {φ1 , . . . , φ } such that {Th φj : h ∈ H, 1 ≤ j ≤ } is a Parseval frame for V ∗ . Remark 5.6. Theorems 5.1 and 5.5 provide extensions to the setting of LCA groups of [1, Theorem 2.1] and [13, Theorem 4.1] respectively. 6. Totally decomposable MI spaces and translation-invariant spaces In this section we shall work in the setting of Section 5 where G is a second countable LCA group and we consider H-invariant spaces of L2 (G) where H is a uniform lattice on G. Our main result here relates translation-invariant spaces, that are closed subspaces of L2 (G) invariant under any translation on G, with totally decomposable MI spaces (see Definition 3.1). For this, we need to consider translation-invariant spaces. In particular we shall prove a result which describes & associated to a translation-invariant space through the the measurable subset of G so-called Wiener Theorem. We begin by proving a version of Wiener’s Theorem in the setting of LCA groups. A version for G = Rd was proven in [35], with a beautiful proof which uses elementary theory. Although one can straightforwardly adapt it to the setting of LCA groups, we provide it here for the reader’s convenience. For a different proof in terms of range function we refer to [11, Corollary 3.9]. Proposition 6.1. Let V ⊆ L2 (G) be a closed subspace. Then, the following are equivalent: (i) V is translation invariant. & such that V = {f ∈ L2 (G) : f& = (ii) There exists a measurable set E ⊆ G 0 a.e. E}. Proof. (ii) ⇒ (i). This is a straightforward consequence of the fact that  (Tx f ) = e−x f& for every f ∈ L2 (G) and every x ∈ G. & onto V& = {f& : f ∈ V }. (i) ⇒ (ii). Let P be the orthogonal projection of L2 (G) 2 & Then, for f, g ∈ L (G) we have that f − P f, P g = 0. Since V is translation invariant, for every x ∈ G we have  (f − P f )(γ)P g(γ)e−x (γ) dmG = ((f − P f )P g)∧ (x), 0 = f − P f, P gex =  G

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& Swapping thus, (f − P f )P g = 0 which says that f P g = P f P g for all f, g ∈ L2 (G). the roles of f and g in the above equalities, we also obtain that gP f = P gP f for & Therefore, all f, g ∈ L2 (G). (6.1)

& f P g = P f g for all f, g ∈ L2 (G).

& be such that g0 > 0 a.e. G & (see Proposition A.1 in the Now, let g0 ∈ L2 (G) Appendix) and call Z the exceptional set of zero measure. Then, for φ := P g0 /g0 & \ Z, by (6.1) we have P f = φf for every f ∈ L2 (G). & Since P is defined a.e. on G an orthogonal projection & φf = P f = P 2 f = φ2 f for all f ∈ L2 (G), & \ Z. This implies that φ and choosing f = g0 we can conclude that φ2 = φ a.e. G & takes that values 0 and 1 a.e. G \ Z. & \ Z be the set on which φ = 0 a.e. Then, f ∈ V& if and only if Let E ⊆ G f = P f = φf and the latter holds if and only if f = 0 a.e. E.  Due to the separability of L2 (G), for every translation-invariant space V , which in particular is an H-invariant space, there exists an at most countable set A ⊆ L2 (G) such that V = SH (A). Using this description of V , we can describe the set E of Proposition 6.1 associated to V in terms of its generators as an H-invariant space. Corollary 6.2. Let V ⊆ L2 (G) be a translation-invariant space and A ⊆ L (G) an at most countable set such that V = SH (A). Then V = {f ∈ L2 (G) :  f& = 0 a.e. E} with E = φ∈A {φ& = 0} up to a measure zero set. 2

 ⊆ G & such that Proof. By Proposition 6.1, there exists a measurable set E 2 2 & &  V = {f ∈ L (G) : f = 0 a.e. E}. Let us call W = {f ∈ L (G) : f = 0 a.e. E},  =E where E = φ∈A {φ& = 0}. We will prove that V = W (or equivalently that E  up to a measure zero set). Note that V ⊆ W if and only if E ⊆ E. & First, since by definition of E, E ⊆ {φ = 0} for each φ ∈ A, we have A ⊆ W . Thus, span{Th φ : φ ∈ A, h ∈ H} ⊆ W because W is translation invariant. Finally, taking closure we find that V = SH (A) ⊆ W = W . For the other inclusion, we see that since φ ∈ A ⊆ V , then φ& must be zero a.e. &   ⊆ {φ& = 0} for every φ ∈ A and then E ⊆ in E. Thus E φ∈A {φ = 0} = E.(i.e. W ⊆ V.)  As the previous results show, translation-invariant spaces are in correspondence & One can also construct a translation-invariant space by with measurable sets of G. imposing an additional condition on its generators as an H-invariant space, as we shall see in the next proposition. For this, again as in Section 5, let Ω be & ∗ . The space Lp (Ω) is identified with a measurable section of the quotient G/H p & c {g ∈ L (G) : g = 0 a.e. Ω }, 1 ≤ p ≤ +∞. In this situation, the set {χΩ eh }h∈H defined by the characters induced by H, forms an orthogonal basis of L2 (Ω) (see [14, Proposition 2.16.]). Proposition 6.3. Let A ⊆ L2 (G) be an at most countable set such that φ& ∈ L2 (Ω) ∀ φ ∈ A, and consider the space V = SH (A). Then, V is translation

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 invariant. In particular, V = {f ∈ L2 (G) : f& = 0 a.e. E} with E = φ∈A {φ& = 0} ⊆ Ω. Proof. Let x ∈ G. Then, by [14, Lemma 4.4], there exists a sequence of trigonometric polynomials {pn }n∈N such that pn (ω) → e−x (ω) a.e. ω ∈ Ω with pn ∞ ≤ C for every n ∈ N. Now, for g ∈ V& , since |pn − e−x |2 |g|2 ≤ (C + 1)2 |g|2 on Ω, by Dominated Convergence Theorem, we have that pn g → e−x g in L2 (Ω). Therefore, since pn g ∈ V& for all n ∈ N, we can conclude that e−x g ∈ V& . As a consequence, V is translation invariant and by Corollary 6.2 the result follows.  We now state the main result of this section, where we characterize translationinvariant spaces in terms of MI spaces. We shall show that translation-invariant spaces are associated to MI spaces that are totally decomposable. Theorem 6.4. Let V ⊆ L2 (G) be an H-invariant space. Let T : L2 (G) → L (Ω, 2 (H ∗ )) be the fiberization mapping of Proposition 5.2, and {δk }k∈H ∗ be the canonical basis of 2 (H ∗ ). Then the following conditions are equivalent: 2

(i) M := T V is totally decomposable with respect to {span{δk }}k∈H ∗ . (ii) V is a translation-invariant space. Proof. (i) ⇒ (ii). We know that Mk ⊆ M for every k ∈ H ∗ where 1 Mk = Pk M and Pk (F )(ω) = Pspan{δk } (F (ω)) for a.e. ω ∈ Ω. Since, M = k∈H ∗ Mk , using that 1 T is an isometric isomorphism and denoting T Vk = Mk , we immediately have V = k∈H ∗ Vk . Therefore, if we prove that each Vk is translation invariant, we will be able to conclude that so is V . Fix k ∈ H ∗ . Denote by Ωk the translation by k of Ω, i.e. Ωk = Ω + k. For f ∈ V , fk is the function defined by its Fourier transform as f&k := χΩk f&. If F ∈ M is F = T f , then Fk (ω) = f&(ω + k)δk = T fk (ω) a.e. ω ∈ Ω and we have that −1 F ) ∈ (T −1 M ) = V &k . On the other hand, if f ∈ Vk , F = T f ∈ Mk and f&k = (T k k then Fk = F . Thus, fk = f and therefore Vk = {fk : f ∈ V }. Furthermore, since by Proposition 3.3, Mk is a D-MI space with D = {eh χΩ }h∈H , Vk is H-invariant. In particular, we obtain that for every f ∈ V , Wk := span{eh f&k : h ∈ H} ⊆ V&k . Now, by Proposition 6.3 (Wk )∨ is translation invariant. Hence, for x ∈ G and f ∈ V , Tx fk ∈ (Wk )∨ ⊆ Vk . Thus, Vk is translation invariant and so is V . (ii) ⇒ (i). We need to show that Mk ⊆ M for all k ∈ H ∗ which is equivalent to see that Vk ⊆ V for all k ∈ H ∗ . Thus, let us fix k ∈ H ∗ . Since V is translation-invariant, by Proposition 6.1 & measurable such that V = {f ∈ L2 (G) : f& = 0 a.e. E}. Then, there exists E ⊆ G if f ∈ V , f&k = χΩk f& satisfies that f&k = 0 a.e. E which implies that fk ∈ V .  Therefore, Vk ⊆ V as we wanted to prove.

7. Acknowledgments We are indebted to the anonymous referee for his/her meticulous report and fine comments. We would like to thank Michael Cwikel and Mario Milman for the invitation to be part of this homage.

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Appendix A In this Appendix we provide a proof for the existence of a function in L2 (G) which is never zero - up to, perhaps, a zero measure set. We believe that this is known, but since we did not find a precise reference, we construct it here. Proposition A.1. Let G be a second countable LCA group. Then, there exists a function f ∈ L2 (G) such that f > 0 a.e. on G. Proof. From [20, Lemma 4.19] one can conclude that for every compact set K ⊆ G there exits g ∈ L2 (G) such that g > 0 on K and g ≥ 0 on G. Since 2 G is second countable, in particular it is σ-compact. Then, we have that G = j∈N Kj where each Kj is compact. We now consider the sets {Fj }j∈N which 2j−1 are the disjoint versions of {Kj }j∈N , that 2 is, F1 := K1 and Fj := Kj \ ( i=1 Ki ) for j ∈ N. Note that we still have G = j∈N Fj where now the union is disjoint. For every j ∈ N, let gj ∈ L2 (G) be such that gj> 0 on Kj and gj ≥ 0 on G and consider fj := (1/2j )χFj (gj /gj 2 ). Then, f := j∈N fj is the function we are looking for. Indeed. First note that since {Fj }j∈N is a partition for G,   1 fj 22 ≤ < +∞, f 22 = 4j j∈N

j∈N

and therefore f ∈ L (G). Finally, since gj > 0 on Kj for each j ∈ N, fj > 0 on Fj for each j ∈ N, and thus, f > 0 a.e. on G.  2

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[36] Amos Ron and Zuowei Shen, Frames and stable bases for shift-invariant subspaces of L2 (Rd ), Canad. J. Math. 47 (1995), no. 5, 1051–1094, DOI 10.4153/CJM-1995-056-1. MR1350650 [37] Erhard Schmidt, Zur Theorie der linearen und nicht linearen Integralgleichungen Zweite Abhandlung (German), Math. Ann. 64 (1907), no. 2, 161–174, DOI 10.1007/BF01449890. MR1511432 [38] T. P. Srinivasan, Doubly invariant subspaces, Pacific J. Math. 14 (1964), 701–707. MR0164229 ˇ c and Edward N. Wilson, Lattice invariant subspaces and sampling, Appl. Comput. [39] Hrvoje Siki´ Harmon. Anal. 31 (2011), no. 1, 26–43, DOI 10.1016/j.acha.2010.09.006. MR2795873 ´tica, Facultad de Ciencias Exactas y Naturales, UniverDepartamento de Matema ´ n I, 1428 Buenos Aires, Argentina sidad de Buenos Aires, Ciudad Universitaria, Pabello – and – IMAS-CONICET, Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas, Argentina E-mail address: [email protected] ´tica, Facultad de Ciencias Exactas y Naturales, UniverDepartamento de Matema ´ n I, 1428 Buenos Aires, Argentina sidad de Buenos Aires, Ciudad Universitaria, Pabello – and – IMAS-CONICET, Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas, Argentina E-mail address: [email protected] ´tica, Facultad de Ciencias Exactas y Naturales, UniverDepartamento de Matema ´ n I, 1428 Buenos Aires, Argentina sidad de Buenos Aires, Ciudad Universitaria, Pabello – and – IMAS-CONICET, Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas, Argentina E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13931

Discrete decomposition of homogeneous mixed-norm Besov spaces G. Cleanthous, A. G. Georgiadis, and M. Nielsen Abstract. One of the most influential contributions of Frazier and Jawerth is the construction of the discrete decomposition transformation, the so-called ϕ-transform, on spaces of distributions. In this article we introduce and study homogeneous mixed-norm Besov spaces. Furthermore we obtain a mixed-norm extension of the ϕ-transform in this development. This discrete decomposition is new even for inhomogeneous mixed-norm Besov spaces and so we also adapt our results to the inhomogeneous case.

1. Introduction From the point of view of applicable harmonic analysis, the most useful smoothness spaces are the ones where smoothness is characterized by (or at least implies) some decay or sparseness of an associated discrete expansion. A well-known example is the O(1/N ) decay of Fourier coefficients for C 1 (T) functions. On the domain Rn , Besov and Triebel-Lizorkin spaces form two closely related families of smoothness spaces with numerous important applications in approximation theory and harmonic analysis, see [11–13, 35–37]. These spaces are constructed using a dyadic decomposition of the frequency space, and their proven usefulness for applications relies to a large degree on the fact that universal and stable discrete decomposition systems exist for the two families of spaces. It is now well-know that the theory of wavelets provides the machinery for easy construction of adapted unconditional bases for these spaces having desirable properties such as compact support, see [9, 14, 27]. A highly influential precurser for wavelet decompositions of Besov and TriebelLizorkin spaces is the so called ϕ-transform construction introduced by Frazier and Jawerth in their seminal papers [11–13]. The ϕ-transform can be viewed as a discretised sampled version of the Littlewood-Paley decomposition, and it provides a discrete convergent expansion of tempered distributions (modulo polynomials). The construction thereby provides a universal decomposition of the homogeneous s s and Triebel-Lizorkin spaces F˙ pq . Frazier and Jawerth successfully applied Besov B˙ pq the transform to an in-depth study of some of the finer properties of the properties 2010 Mathematics Subject Classification. 42B25, 42B35, 46F10. Key words and phrases. discrete decomposition, ϕ-transform, tempered distributions, Besov spaces, mixed-norms, homogeneous, embeddings, inhomogeneous. c 2017 American Mathematical Society

167

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G. CLEANTHOUS, A. G. GEORGIADIS, AND M. NIELSEN

s s of B˙ pq and F˙ pq . Later the ϕ-transform techniques were adapted for the wavelet setup, see [14, 27]. The influence of [11–13] on functional and harmonic analysis has been tremendous. Any reliable citation database will show a huge number of citations to the mentioned papers, but more importantly the papers have provided a de facto research guideline for many researchers interested in function spaces, wavelets, and approximation theory. A small sample of related work on Besov and TriebelLizorkin spaces on Rn is [5–7, 20, 21, 23, 25], see also references therein. For similar decompositions on other domains such as on the ball or on the sphere, see for example [8, 15, 19, 22, 24, 28, 31]. In this paper we pay homage to the ϕ-transform construction by considering the transform in a mixed-norm setting. Recently, there has been significant interest in the study of inhomogeneous Besov and Triebel-Lizorkin spaces with mixed Lebesgue norms, see [16–18] and references therein. We introduce and study homogeneous mixed-norm Besov spaces B˙ psq , for s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n and q ∈ (0, ∞]. The homogeneous spaces are in general spaces on the class S  /P of tempered distributions modulo the polynomials. After giving some first properties on B˙ psq spaces we will introduce the corresponding discrete space of sequences b˙ spq . The elements of the space b˙ spq will be complex-valued sequences {aQ }Q∈Q with indices Q on the countable set Q of the dyadic cubes. We then follow Frazier and Jawerth and consider two operators connecting B˙ psq , with the sequence space b˙ spq . The ϕ-transform Sϕ which maps a f ∈ B˙ psq , to a sequence {(Sϕ f )Q }Q∈Q and the so-called inverse ϕ-transform Tψ acting on the sequences (where ϕ, ψ are two “admissible” functions). The main result of this article is the following extension of the classical ϕtransform Theorem of Frazier and Jawerth [11].

Theorem 1.1. The ϕ-transform Sϕ : B˙ psq → b˙ spq and the inverse ϕ-transform Tψ : b˙ spq → B˙ psq are bounded and Tψ ◦ Sϕ is the identity on B˙ psq . In particular, f  ˙ s ∼ Sϕ f  ˙ s for every f ∈ B˙ s . Bp q

p q

bp q

Notation: Through the article, positive constants will denoted by c and they may vary at every occurrence.  The Schwartz class will be stated by S and the Fourier transform by fˆ(ξ) = Rn f (x)e−ix·ξ dx, for every f ∈ S. 2. Preliminaries In this section we present some background needed for the development of homogeneous mixed norm Besov spaces and their discrete analogous. 2.1. Schwartz functions and distributions. Let us recall briefly some basic facts about Schwartz functions and distributions. We denote by S = S(Rn ) the Schwartz space of rapidly decreasing, infinitely differentiable functions on Rn . A function ϕ ∈ C ∞ belongs to S, when for every k ∈ N ∪ {0} and every multi-index α ∈ (N ∪ {0})n , (2.1)

Pk,α (ϕ) := sup (1 + |x|)k |Dα ϕ(x)| < ∞. x∈Rn

We will need also the norms ∗ (2.2) PK,A (ϕ) := max{Pk,α (ϕ) : 0 ≤ k ≤ K, |α| ≤ A}, for every K, A ∈ N∪{0}.

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The dual S  = S  (Rn ) of S is the space of tempered distributions. Let M ∈ N, we set    xα ψ(x)dx = 0, |α| = α1 + · · · + αn ≤ M , SM := SM (Rn ) = ψ ∈ S : Rn

i.e. the Schwartz functions with vanishing moments of order M. We just mention & that an equivalent characterization of this class is SM = {ψ ∈ S : Dα ψ(0) =  α α& |α| x ψ(x)dx for every multi-index α ∈ (N ∪ 0, |α| ≤ M }, since D ψ(0) = (−i) {0})n . For consistency we will denote    n xα ψ(x)dx = 0, ∀α ∈ (N ∪ {0})n . S∞ := S∞ (R ) = ψ ∈ S : Rn

We note that S∞ is a Fr´echet space, because it is closed in S and its dual is  = S  /P, where P the family of polynomials on Rn . We will define homogeneous S∞ mixed-norm Besov spaces for elements of S  /P. 2.2. Dyadic cubes and frames. We start presenting our wavelet set-up. It is well known, see [4, Lemma 6.1.7], that we can choose ϕ, ψ satisfying the following conditions (2.3)

ϕ, ψ ∈ S(Rn ),

(2.4)

supp ϕ, ˆ ψˆ ⊆ {ξ ∈ Rn : 2−1 ≤ |ξ| ≤ 2},

(2.5)

ˆ |ϕ(ξ)|, ˆ |ψ(ξ)| ≥ c > 0 if 2−3/4 ≤ |ξ| ≤ 23/4 ,

and



(2.6)

ˆ ν ξ) = 1 if ξ = 0. ϕ(2 ˆ ν ξ)ψ(2

ν∈Z

We set ϕν (x) = 2 ϕ(2 x) and ψν (x) = 2νn ψ(2ν x), ν ∈ Z. For every τ ∈ N and multi-index γ of length |γ| := γ1 + · · · + γn , it holds νn

(2.7)

ν

|Dγ ϕν (x)|, |Dγ ψν (x)| ≤ cγ,τ 2ν(|γ|+n) (1 + 2ν |x|)−τ −|γ| .

Definition 2.1. Functions ϕ, ψ satisfying (2.3)-(2.5) will be called admissible. For ν ∈ Z and k ∈ Zn , we denote by Qνk the dyadic cube Qνk = {(x1 , . . . , xn ) ∈ Rn : ki ≤ 2ν xi < ki + 1, i = 1, . . . , n}. For every Q = Qνk we denote by xQ = 2−ν k the ”lower left-corner”and by (Q) = 2−ν the side length. For now on we will denote by Qν the set of all dyadic cubes of side length (Q) = 2−ν , ν ∈ Z and by Q the set of all dyadic cubes. Note that for every ν ∈ Z, the set of all the dyadic cubes of the same side-length Qν is a disjoint partition of Rn . Let Q = Qνk ∈ Qν . We denote (2.8)

ϕQ (x) := |Q|−1/2 ϕ(2ν x − k) = |Q|1/2 ϕν (x − xQ )

and we define ψQ similarly. Then it follows that (2.9)

supp ϕ &Q , ψ&Q ⊆ {ξ : 2ν−1 ≤ |ξ| ≤ 2ν+1 }

and for every τ ∈ N and multi-index γ of length |γ|, (2.10)

|Dγ ϕQ (x)|, |Dγ ψQ (x)| ≤ cγ,τ |Q|−1/2−|γ|/n (1 + (Q)−1 |x − xQ |)−τ −|γ| .

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2.3. Mixed norm Lebesgue spaces. Let p = (p1 , . . . , pn ) ∈ (0, ∞]n and f : Rn → C. We say that f ∈ Lp = Lp (Rn ) if (2.11)

⎞ p1n ⎛ ! pp3  pp2    2 1 f p := ⎝ · · · |f (x1 , . . . , xn )|p1 dx1 dx2 · · · dxn ⎠ < ∞, R

R

R

with the standard modification when pj = ∞ for some j = 1, . . . , n. The quasi-norm  · p , is a norm when min(p1 , . . . , pn ) ≥ 1 and turns (Lp ,  · p ) into a Banach space. Note that when p = (p, . . . , p), then Lp coincides with Lp . For additional properties of Lp , see for example [1–3, 10, 26, 34]. 2.4. Maximal operators. An important tool for our proofs is the maximal operator. Let 1 ≤ k ≤ n. We define  1 (2.12) Mk f (x) = sup |f (x1 , . . . , yk , . . . , xn )|dyk , I∈Ixk |I| I where Ixk is the set of all intervals I on Rk containing xk . We will use extensively the following iterated maximal function:  1/t (x), for every t > 0, x ∈ Rn . (2.13) Mt f (x) := Mn · · · M1 |f |t · · · Remark 2.2. If Q is a rectangle Q = I1 × · · · × In , it follows easily see that for every locally integrable f  (2.14) |f (y)|dy ≤ |Q|M1 f (x) = |Q|Mtt |f |1/t (x), for every x ∈ Rn . Q

We shall need the following variation of Fefferman-Stein vector-valued maximal inequality (see [2, 33]): If p = (p1 , . . . , pn ) ∈ (0, ∞)n and 0 < t < min(p1 , . . . , pn ) then Mt f p ≤ cf p .

(2.15)

Finally we state a Peetre’s-type maximal inequality [32]: For every t > 0, there exists a constant c = ct > 0, such that for every f with supp fˆ ⊂ [−2ν+1 , 2ν+1 ]n , for some ν ∈ Z |f (y)| ≤ cMt f (x), x ∈ Rn . (2.16) sup ν |y − x|)n/t n (1 + 2 y∈R 3. Homogeneous mixed-norm Besov spaces In this section we introduce an extension of the classical homogeneous Besov spaces (see Triebel [35–37] and Frazier-Jawerth [11]), using mixed-norms. Definition 3.1. For s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞] and ϕ admissible, we define the homogeneous mixed-norm Besov space B˙ psq , as the set of all f ∈ S  /P such that 1/q  (2νs ϕν ∗ f p )q < ∞, (3.17) f B˙ s := p q

ν∈Z

with the q -norm replaced by the supν if q = ∞.

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171

Remark 3.2. Several remarks regarding the homogeneous mixed-norm Besov spaces defined above are in order. s (α) When p = (p, . . . , p), then B˙ psq , coincides with B˙ pq , the standard homogeneous Besov space. (β) The homogeneous mixed-norm Besov space B˙ psq is quasi-Banach for all s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n and q ∈ (0, ∞]. Furthermore B˙ psq is Banach when p ∈ [1, ∞]n , q ∈ [1, ∞]. (γ) The triangle inequality does not hold in general in B˙ psq . Instead we have the sub-additivity f + grB˙ s ≤ f rB˙ s + grB˙ s , p q

p q

p q

where r := min(1, p1 , . . . , pn , q). (δ) The quasi-norm in the definition of B˙ psq depends on the choice of the admissible function ϕ and so we some times denote it by  · B˙ s (ϕ) . However in the p q next Proposition we prove that B˙ psq space is independent of the admissible function ϕ. Proposition 3.3. Let s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞]n and ϕ, Φ two admissible functions. Then  · B˙ s (ϕ) and  · B˙ s (Φ) are equivalent quasip q p q norms in B˙ psq . Consequently, the definition of B˙ psq is independent of the particular selection of the admissible function ϕ. Proof. Let s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞] and ϕ, Φ two admissible functions. Let also Ψ be admissible and such that  ˆ j ξ)Ψ(2 ˆ j ξ) = 1, if ξ = 0. Φ(2 j∈Z

We set Φj (ξ) := 2 Φ(2 ξ) and Ψj (ξ) := 2jn Ψ(2j ξ), j ∈ Z. Then for any f ∈ S  /P,  ˜ j ∗ Φj ∗ f, where the convergence is in S  /P, f= Ψ jn

j

j∈Z

˜ j (ξ) := Ψj (−ξ) for every ξ ∈ Rn . Hence for every ν ∈ Z, and Ψ    ν+1  ν+1           (ϕν ∗ Ψ ˜ ˜ j ) ∗ (Φj ∗ f )(x). ϕν ∗ Ψj ∗ Φj ∗ f (x) ≤ (3.18) |ϕν ∗ f (x)| =  j=ν−1  j=ν−1 Let 0 < t < min(1, p1 , . . . , pn , q) and τ > n + n/t. By (2.7) and for |j − ν| ≤ 1 we derive for every z ∈ Rn ,  ˜ ˜ j (w)|dw |ϕν (z − w)||Ψ |ϕν ∗ Ψj (z)| ≤ Rn  (1 + 2j |z − w|)−τ (1 + 2j |w|)−τ dw. ≤ c22jn Rn

We have that

1 + 2j |z| (1 + 2j |z − w|)(1 + 2j |w|)

τ

τ 1 1 + 1 + 2j |z − w| 1 + 2j |w|   τ ≤ 2 (1 + 2j |z − w|)−τ + (1 + 2j |w|)−τ . ≤



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By changing the variable in the integrals below,    ˜ j (z)| ≤ c22jn (1 + 2j |z|)−τ (1 + 2j |z − w|)−τ dw + (1 + 2j |w|)−τ dw |ϕν ∗ Ψ n Rn  R = c2jn (1 + 2j |z|)−τ 2 (1 + |x|)−τ dx ≤ c2 (1 + 2 |z|) jn

(3.19)

j

−τ

Rn

.

By (3.19), and the maximal inequality (2.16), we obtain since τ > n + n/t  ˜ ˜ j )(x − y)||(Φj ∗ f )(y)|dy (3.20) |(ϕν ∗ Ψ |(ϕν ∗ Ψj ) ∗ (Φj ∗ f )(x)| ≤ Rn  |Φj ∗ f (z)| ≤ c2jn sup (1 + 2j |x − y|)−τ +n/t dy j n/t n z∈Rn (1 + 2 |x − z|) R  −τ +n/t ≤ cMt (Φj ∗ f )(x) (1 + |z|) dz ≤ cMt (Φj ∗ f )(x). Rn

By (3.18) and (3.20) we deduce, ν+1 

2νs |ϕν ∗ f (x)| ≤ c

2js Mt (Φj ∗ f )(x).

j=ν−1

Passing now to the quasi-norm  · p , and by applying the maximal inequality (2.15), we obtain 2νs ϕν ∗ f p

 ν+1    js  ≤ c 2 Mt (Φj ∗ f )

p 

j=ν−1

≤ c

ν+1 

2js Mt (Φj ∗ f )p ≤ c

j=ν−1

ν+1 

2js Φj ∗ f p .

j=ν−1

Hence, f B˙ s

p q

(ϕ)

=



(2νs ϕν ∗ f p )q

1/q

≤c

ν∈Z

   ν+1  ν∈Z

1/q  ≤ c (2js Φj ∗ f p )q = cf B˙ s

2js Φj ∗ f p

q 1/q

j=ν−1

p q

(Φ) .

j∈Z

We switch the role of φ and Φ to obtain f B˙ s (Φ) ≤ cf B˙ s (ϕ) and the proof is p q p q complete.  Remark 3.4. All the constructions have been based on dyadic decomposition of the frequency space. However, we can use any other number b > 1 for all the constructions in Section 2.2, as well as in Definition of ( 3.1) Besov spaces (replace 2νs by bνs ) and one obtains the same spaces with equivalent norms. 3.1. Embeddings of mixed-norm Besov spaces. For two quasi-normed spaces X, Y we will denoted by X → Y the continuous embedding. Let s ∈ R, p ∈ (0, ∞)n and 0 < q < r ≤ ∞. The we have the embedding B˙ psq → B˙ psr ,

HOMOGENEOUS MIXED-NORM BESOV SPACES

173

coming from the well known embedding between the sequence spaces; q → r . We prove the basic proposition below, containing the embeddings between homogeneous  . Firstly we give a lemma with mixed-norm Besov spaces and the spaces S∞ and S∞ some necessary estimations. Lemma 3.5. Let ϕ, ψ ∈ S, M, N, K ∈ N such that K > M + N + n + 1. (i) If ψ ∈ SM , then there exists a constant C = CN,M,K > 0 such that (3.21)

∗ ∗ ν(−N +n+M +1) |ψ ∗ ϕν (x)| ≤ CPN,M (1 + |x|)−N +1 (ϕ)PK,0 (ψ)2

for every x ∈ Rn , and ν ≤ 0. (ii) If If ϕ ∈ SM , then there exists a constant C = CN,M,K > 0 such that ∗ ∗ −ν(−N +M +1) (1 + |x|)−N |ψ ∗ ϕν (x)| ≤ CPN,M +1 (ψ)PK,0 (ϕ)2

(3.22)

for every x ∈ Rn and ν ≥ 0. The reader can find the proof in the Appendix. We proceed to the proposition. Proposition 3.6. Let s ∈ R, p = (p1 , · · · , pn ) ∈ (0, ∞]n and q ∈ (0, ∞]. Then  . S∞ → B˙ psq and B˙ psq → S∞

Proof. Let s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞]n and q ∈ (0, ∞]. We start by proving S∞ → B˙ psq . Let ψ ∈ S∞ , then  ψqB˙ s = (2νs ψ ∗ ϕν p )q p q

(3.23)

ν∈Z

=

0 

(2νs ψ ∗ ϕν p )q +

ν=−∞

∞ 

(2νs ψ ∗ ϕν p )q =: Σ1 + Σ2 .

ν=1

  Pick N ∈ N such that N > max n, n/ min(p1 , . . . , pn ) , M ∈ N and K > N + M + 1 + n. We estimate first Σ1 . Since ψ ∈ S∞ , we have ψ ∈ SM . By (i) of Lemma 3.5

(3.24)

∗ ν(−N +M +1+n) |ψ ∗ ϕν (x)| ≤ cPK,M (1 + |x|)−N +1 (ψ)2 n * ∗ ν(−N +M +1+n) ≤ cPK,M +1 (ψ)2 (1 + |xj |)−N/n , j=1

∗ where we denote for simplicity c = CPK,M +1 (ϕ). Observe that for every 1 ≤ j ≤ n  ∞ (1 + |xj |)−N pj /n dxj ≤ c, −∞

since we have N pj /n > 1. So by (3.24) ∗ ν(−N +M +1+n) . ψ ∗ ϕν p ≤ cPK,M +1 (ψ)2

We use this in the sum Σ1 , and obtain (3.25)

0  ∗ q   ∗ q Σ1 ≤ c PK,M 2ν(s−N +M +1+n)q ≤ c PK,M , +1 (ψ) +1 (ψ) ν=−∞

provided s − N + M + 1 + n > 0.

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G. CLEANTHOUS, A. G. GEORGIADIS, AND M. NIELSEN

We proceed to estimate Σ2 . By the claim (ii) of Lemma 3.5, this time ∗ −ν(−N +M +1) ψ ∗ ϕν p ≤ cPK,M . +1 (ψ)2

Thus, considering −N + M + 1 − s > 0, ∞  ∗ q   ∗ q 2−ν(−N +M +1−s)q ≤ c PK,M . (3.26) Σ2 ≤ c PK,M +1 (ψ) +1 (ψ) ν=1

→ B˙ psq follows by (3.23), (3.25) and (3.26).

Now the embedding S∞

 . Let f ∈ B˙ psq and g ∈ S∞ . Then, by [21] We turn to prove B˙ psq → S∞     |f, g | =  ϕν ∗ f ∞ ψν ∗ g1 . ψ˜ν ∗ ϕν ∗ f, g  ≤ ν∈Z

ν∈Z ν( p1 +···+ p1n )

ϕν ∗ f p (see [17, Proposition 4]) and Furthermore ϕν ∗ f ∞ ≤ c2 1 thus  ν( 1 +···+ 1 ) pn 2 p1 ϕν ∗ f p ψν ∗ g1 |f, g | ≤ c ν∈Z

≤



cf B˙ s

p q

2ν( p1 +···+ pn −s) ψν ∗ g1 1

1

ν∈Z

≤

(3.27)

cf B˙ s

0  

p q

···+

ν=−∞

∞ 

···



=: cf B˙ s (Σ3 + Σ4 ), p q

ν=1

Estimation of Σ3 . Let ν ≤ 0. We consider M > 0, N > n such that 1 1 Θ := + ···+ − s − N + M + 1 + n > 0. p1 pn ∗ Then by (i) of Lemma 3.5 and denoting by c = PK,M +1 (ψ),  ∗ ν(−N +n+M +1) (g)2 (1 + |x|)−N dx ψν ∗ g1 ≤ cPK,M +1 Rn

≤

∗ ν(−N +n+M +1) cPK,M +1 (g)2

and then Σ3 =

0 

2ν( p1 +···+ pn −s) ψν ∗ g1 1

1

ν=−∞

2νΘ

ν=−∞ ∗ ≤ cPK,M +1 (g).

(3.28) 1 p1

0 

∗ ≤ cPK,M +1 (g)

Estimation of Σ4 . Let ν > 0. We restrict M satisfying s − N + M + 1 > + · · · + p1n . Then by (ii) of Lemma 3.5 we get

(3.29)

Σ4 =

∞ 

∗ 2ν( p1 +···+ pn −s) ψν ∗ g1 ≤ cPK,M +1 (g). 1

1

ν=1  By (3.27)-(3.29) the embedding B˙ psq → S∞ is established.



Another embedding between homogeneous mixed-norm Besov spaces of different smoothness is given in the sequel. The proof is the same with the one of the inhomogeneous case given by Johnsen and Sickel in [17] and therefore we omit it.

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175

Proposition 3.7. Let s, t ∈ R, p = (p1 , · · · , pn ), r = (r1 , . . . , rn ) ∈ (0, ∞)n and q ∈ (0, ∞] be such that t < s, p1 ≤ r1 , . . . , pn ≤ rn , and s −

1 1 1 1 − ··· − =t− − ··· − , p1 pn r1 rn

then B˙ psq → B˙ rt q . pm

Remark 3.8. Let s, t ∈ R, p = (p1 , · · · , pn ) ∈ (0, ∞]n and q ∈ (0, ∞]. We set := min(p1 , . . . , pn ) and pM := max(p1 , . . . , pn ), then B˙ pt m q → B˙ psq → B˙ pτM q ,

where t=s−

1 1 n n 1 1 + + + ··· + and τ = s − + ··· + . p1 pn pm p1 pn pM

For recent embedding results involving Besov spaces, see also V. Nguyen and W. Sickel [29]. Note that the above Remark (in its inhomogeneous version), in connection with Theorems 3.1 and 3.6 in [29], provide conditions for embeddings between the (inhomogeneous) mixed-norm Besov spaces and Sobolev spaces of dominating mixed smoothness. 4. The ϕ-transform This Section contains our main result; the discrete decomposition of homogeneous mixed-norm Besov spaces, in the sense of the ϕ-transform, introduced by Frazier and Jawerth. 4.1. Discrete Besov spaces. The discrete analogue of Besov spaces is the space of sequences introduced below. The set of all dyadic cubes is Q ∼ Zn+1 and so it is countable. The set Q will play the role of the domain of complex valued sequences {aQ }Q∈Q . Definition 4.1. For s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞] we define the sequence space b˙ spq = b˙ spq (Q), as the set of all complex-valued sequences a = {aQ }Q∈Q such that q 1/q     Q (·) (4.30) ab˙ s := |Q|−s/n |aQ | < ∞,   p q

ν∈Z

Q∈Qν

p 

 Q = |Q|−1/2 Q , the function Q is the characteristic of the cube Q and the where  q -norm is replaced by the supν if q = ∞. We are now ready to define the ϕ-transform Sϕ . Definition 4.2. Let ϕ be an admissible function. The ϕ-transform Sϕ , or the analysis operator, is the map sending each f ∈ S  /P (the set of tempered distribution modulo polynomials) to the complex-valued sequence Sϕ f = {(Sϕ f )Q }Q , with (Sϕ f )Q = f, ϕQ , for every Q ∈ Q. Let ψ be admissible. The so called inverse ϕ-transform  Tψ , or the synthesis operator, is the map taking a sequence a = {aQ }Q to Tψ a = aQ ψQ . Q∈Q

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A Lemma of fundamental importance coming from [11] is: Lemma 4.3. Suppose that ϕ, ψ are admissible satisfying ( 2.6) and f ∈ S  /P then  f (·) = f, ϕQ ψQ (·) = Tψ ◦ Sϕ f (·), where the convergence is in S  /P Q∈Q

and so Tψ ◦ Sφ is the identity on S  /P. The main theorem of this article is the discrete decomposition via the ϕtransform which is expressed in the following: Theorem 4.4. Let s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞] and ϕ, ψ admissible. The ϕ-transform Sϕ : B˙ psq → b˙ spq and the inverse ϕ-transform Tψ : b˙ spq → B˙ psq are bounded. Furthermore Tψ ◦ Sϕ is the identity on B˙ psq . In particular, f B˙ s ∼ Sϕ f b˙ s for every f ∈ B˙ psq . p q

p q

Theorem 4.4, generalizes the celebrated result of Frazier and Jawerth in [11], in the context of mixed-norms. For obtaining its proof, we shall need the next lemma: Lemma 4.5. Let 0 < t ≤ 1, τ > n/t and μ ∈ Z. Then for any complex-valued sequence a = {aP }P ∈Q , we have  P ∈Qμ

 |aP | 1 +

   −τ   |xP − xQ | ≤ c max 2(μ−ν)n/t , 1 Mt |aP |P (x), max((P ), (Q)) P ∈Qμ

for every Q ∈ Qν , for some ν ∈ Z and every x ∈ Q. The proof can be found in the Appendix. Remark 4.6. Let us state some consequencies of Lemma 4.5. (α) Let a = {aQ }Q be a complex-valued sequence. We define a∗ = {a∗Q }Q by a∗Q :=

(4.31)



|aP |(1 + 2ν |xP − xQ |)−τ ,

P ∈Qν

for every Q ∈ Qν , for some ν ∈ Z, and τ > n/t, for 0 < t ≤ 1. Lemma 4.5 applied for μ = ν gives that    |aP |P (x), (4.32) a∗Q ≤ Mt (P )=(Q)

for every x ∈ Q. (β) Since the set of all dyadic cubes with the same side-length is a disjoint partition of Rn , it follows from ( 4.32) (4.33)

 Q∈Qν

a∗Q Q (x) ≤ Mt

 

 |aP |P (x), x ∈ Rn .

P ∈Qν

We are now ready to present the proof of Theorem 4.4.

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177

4.2. Proof of Theorem 4.4. Proof. Let s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , and q ∈ (0, ∞). The other case, q = ∞, is similar. Let ϕ and ψ be two admissible functions satisfying (2.6). (α) We first prove the boundedness of the inverse ϕ-transform Tψ . (α.1) We will start with the case of a finitely supported sequence and we get by limit argument the result in the general case. Let a = {aQ } ∈ b˙ spq , be finitely supported sequence. We denote f := Tψ a =  Q aQ ψQ . Let ν ∈ Z, by (2.9) we have the representation ν+1 

ϕν ∗ f (x) =

(4.34)



aJ ϕν ∗ ψJ (x), x ∈ Rn .

μ=ν−1 J∈Qμ

We consider τ > n/t, 0 < t < min(1, p1 , . . . , pn , q). By (2.8) and (2.10) we derive  |ϕν ∗ ψJ (x)| ≤

|ϕν (x − y)||ψJ (y)|dy  = |Q|−1/2 |ϕQ (x − y + xQ )||ψJ (y)|dy Rn  (1 + 2μ |x − y|)−τ (1 + 2μ |xJ − y|)−τ dy. ≤ c|J|−3/2 Rn

Rn

We observe that (1 + 2μ |x − y|)−τ (1 + 2μ |xJ − y|)−τ ≤ 2τ

(1 + 2μ |x − y|)−τ + (1 + 2μ |xJ − y|)−τ (1 + 2μ |x − xJ |)τ

and by changing variables in the integrals below (4.35)

|ϕν ∗ ψJ (x)| ≤ cτ |J|−3/2 (1 + 2μ |x − xJ |)−τ ×   (1 + 2μ |x − y|)−τ dy + Rn

≤ c|J|

−1/2

(1 + 2 |x − xJ |) μ

Rn −τ

(1 + 2μ |xJ − y|)−τ dy



,

since τ > n. By (4.34) and (4.35) (4.36)

2νs |ϕν ∗ f (x)| ≤ c

ν+1 



|aJ ||J|−s/n−1/2 (1 + 2μ |x − xJ |)−τ .

μ=ν−1 J∈Qμ

For any x ∈ Rn and μ ∈ Z, there exists a unique dyadic cube Pμ := P (x, μ) ∈ Qμ containing x. Moreover 1 + 2μ |xPμ − xJ | ≤ 1 + 2μ (|x − xPμ | + |x − xJ |) ≤ √ 1 + n + 2μ |x − xJ | ≤ cn (1 + 2μ |x − xJ |) and thus from (4.36) (4.37)

2 |ϕν ∗ f (x)| ≤ c νs

ν+1 



μ=ν−1 J∈Qμ

|aJ ||J|−s/n−1/2 (1 + 2μ |xPμ − xJ |)−τ .

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G. CLEANTHOUS, A. G. GEORGIADIS, AND M. NIELSEN

We define the sequence b = {bJ }, by bJ := aJ |J|−s/n−1/2 , for every J ∈ Q. Then by (4.37), (4.31) and (4.33) we derive ν+1 

2νs |ϕν ∗ f (x)| ≤ c

b∗Pμ Pμ (x) ≤ c

μ=ν−1 ν+1 

≤ c

(4.38)

ν+1 



b∗J J (x)

μ=ν−1 J∈Qμ

Mt

μ=ν−1

 

 |bJ |J (x).

J∈Qμ

We use the estimate (4.38) in the definition of the Besov norm of f and obtain by the maximal inequality (2.15), 1/q  2νs (ϕν ∗ f )qp f B˙ s = p q

ν∈Z

≤

    ν+1     q 1/q   c Mt |bJ |J (·)  ν∈Z

μ=ν−1

  ν+1        q 1/q  |bJ |J (·) Mt

≤

c

≤

q 1/q     c |bJ |J (·) 

ν∈Z μ=ν−1

= ≤

p 

J∈Qμ

μ∈Z

J∈Qμ

μ∈Z

J∈Qμ

p 

J∈Qμ

p 

q 1/q     J (·) c |J|−s/n |aJ |   p 

cab˙ s

p q

and the boundedness of the inverse ϕ-transform has been proved for finitely supported sequences. (α.2) Let us now generalize to arbitrary sequences. We consider the sequence {aQ }Q∈Q ∈ b˙ spq . We pick an arbitrary ordering of the values of {aQ }Q∈Q in a sequence of indices in N. We denote by Q(j), the indices in Q of the first j ∈ N terms of the sequence. Since aQ ∈ b˙ spq , then {aQ }Q∈Q(j) → {aQ }Q∈Q , in b˙ spq , when j → ∞. Thus by the boundedness of Tψ on finitely supported sequences, the series   ˙ s . But since B˙ s → S∞ as we proved in Proposition 3.6 Q∈Q aQ ψQ converge in Bp q p q  the convergence holds true also in S∞ . So the operator Tψ is well defined in the whole of b˙ spq , (not only for finitely supported sequences). The boundedness now of Tψ follows by a standard limit argument from the case (α.1). (β) We pass to the proof of the boundedness of the ϕ-transform Sϕ . We set the admissible function ϕ(x) ˜ := ϕ(−x). Let now f ∈ B˙ psq , then for every Q ∈ Qν , ν ∈ Z we observe by (2.8) that |(Sϕ f )Q | = |f, ϕQ | = |Q|1/2 |ϕ˜ν ∗ f (xQ )|.

HOMOGENEOUS MIXED-NORM BESOV SPACES

179

√ Since for every x ∈ Q, it is 1 + 2ν |xQ − x| > 1 + n, we have    Q (x) = |Q|−s/n |(Sϕ f )Q | |Q|−s/n |ϕ˜ν ∗ f (xQ )|Q (x) Q∈Qν

Q∈Qν

≤ c ≤ c

(4.39)



Q∈Qν

2νs |ϕ˜ν ∗ f (xQ )| Q (x) (1 + 2ν |xQ − x|)n/t

Q∈Qν

2νs |ϕ˜ν ∗ f (y)|  (x), ν n/t Q y∈Q (1 + 2 |y − x|)



sup

where 0 < t < min(p1 , . . . , pn , q). We apply now the maximal inequality (2.16), since -by (2.9)- supp(ϕ ˜ν ∗ f ) = supp(ϕˆ ˜ν fˆ) ⊆ [−2ν+1 , 2ν+1 ]n and we derive     Q (x) ≤ c2νs Mt ϕ˜ν ∗ f (x), (4.40) |Q|−s/n |(Sϕ f )Q | Q∈Qν

for every x ∈ Rn . We estimate the discrete Besov norm of Sϕ f , and using (4.40) and the maximal inequality (2.15), we obtain q 1/q     Q (·) Sϕ f b˙ s = |Q|−s/n |(Sϕ f )Q |   p 

p q

ν∈Z

≤

c



Q∈Qν

  (2sν Mt ϕ˜ν ∗ f p )q

ν∈Z

≤

c



(2sν ϕ˜ν ∗ f p )q

1/q

1/q

≤ cf B˙ s , p q

ν∈Z

which proves the boundedness of the ϕ-transform. (γ) The operator Tψ ◦ Sϕ is the identity of B˙ psq and this follows immediately from Lemma 4.3.  5. Inhomogeneous mixed-norm Besov spaces Inhomogeneous mixed-norm Besov spaces have been extensively studied during the ongoing decade, see for example [16–18] and the references therein. Inspired by Frazier and Jawerth we dedicate our last section in this case. We will introduce the corresponding sequence spaces, after we first recall the definition of inhomogeneous mixed-norm Besov spaces. Let a function ϕ0 ∈ S(Rn ) satisfying (5.41)

supp ϕ +0 ⊆ {ξ ∈ Rn : |ξ| ≤ 2},

and (5.42)

|+ ϕ0 (ξ)| ≥ c > 0 if |ξ| ≤ 23/4 .

Let also ϕ satisfy (2.3)-(2.5) and define {ϕν }ν∈N as usual. The inhomogeneous mixed-norm Besov spaces are defined as follows: Definition 5.1. For s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞], ϕ0 as above and ϕ admissible, the inhomogeneous mixed-norm Besov space Bpsq , is the

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G. CLEANTHOUS, A. G. GEORGIADIS, AND M. NIELSEN

collection of all f ∈ S  such that ∞ 1/q  (5.43) f Bp sq := (2νs ϕν ∗ f p )q < ∞, ν=0

with the q -norm replaced by the supν if q = ∞. 2 Considering the set Q+ := ν≥0 Qν , we define the discrete version of the inhomogeneous mixed-norm Besov spaces. Definition 5.2. For s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞] the sequence space bspq = bspq (Q+ ), is the set of all complex-valued sequences a = {aQ }Q∈Q+ such that (5.44)

abps q :=

∞   q 1/q    Q (·) |Q|−s/n |aQ | < ∞,   ν=0

p 

Q∈Qν

 Q = |Q|−1/2 Q , the function Q is the characteristic of the cube Q and the where  q -norm is replaced by the supν if q = ∞. The relation between b˙ spq (Q) and bspq (Q+ ) spaces is expressed by the isometric imbedding V : bspq (Q+ ) → b˙ spq (Q) mapping every {aQ }Q∈Q+ to {(V a)Q }Q∈Q where (V a)Q = aQ for every Q ∈ Q+ and (V a)Q = 0 for Q ∈ Q \ Q+ . We define also W : b˙ spq (Q) → bspq (Q+ ) where (W a)Q = aQ for every Q ∈ Q+ and then W ◦ V is the identity on bspq (Q+ ) (see also Frazier and Jawerth [13]). We define the ϕ-transform Sϕ as well as the inverse ϕ-transform Tψ in the context of the inhomogeneous spaces as follows. Definition 5.3. Let ϕ, ψ be admissible functions and ϕ0 , ψ0 as above. The ϕ-transform Sϕ , or the analysis operator is the map sending each f ∈ S  to the complex-valued sequence Sϕ f = {(Sϕ f )Q }Q∈Q+ , with (Sϕ f )Q = f, ϕQ , if Q ∈ Qν , ν ≥ 1, and (Sϕ f )Q = f, (ϕ0 )Q , if Q ∈ Q0 . The inverse ϕ-transform Tψ , or the synthesis operator is the map taking a sequence a = {aQ }Q∈Q+ to Tψ a =

 Q∈Q0

aQ (ψ0 )Q +

∞  

aQ ψQ .

ν=1 Q∈Qν

We now have the inhomogeneous analogue of Theorem 4.4. The proof is similar with the one of Theorem 4.4 and therefore we omit it. Theorem 5.4. Let s ∈ R, p = (p1 , . . . , pn ) ∈ (0, ∞)n , q ∈ (0, ∞]. Let also ϕ, ψ be admissible functions and ϕ0 , ψ0 as above. The ϕ-transform Sϕ : Bpsq → bspq and the inverse ϕ-transform Tψ : bspq → Bpsq are bounded. Furthermore Tψ ◦ Sϕ is the identity on Bpsq . In particular, f Bp sq ∼ Sϕ f bps q for every f ∈ Bpsq .

HOMOGENEOUS MIXED-NORM BESOV SPACES

181

6. Appendix 6.1. Proof of Lemma 3.5. Proof. (i) Let ϕ ∈ S, ψ ∈ SM , N > 0 and ν ≤ 0, then  ψ ∗ ϕν (x) = 2νn ψ(x − y)ϕ(2ν y)dy n R 5 4  Dα ϕ(2ν ·)(x) νn (y − x)α dy. = 2 ψ(x − y) ϕ(2ν y) − α! Rn |α|≤M

Then



(6.45)

|ψ ∗ ϕν (x)| ≤ 2νn

where

Rn

|ψ(x − y)||x − y|M +1 Φ(x, y)dy,

 1  β D ϕ(2ν ·)(x + ε(y − x)) β! 1 ν|β| ≤ max sup 2 PN,β (ϕ)(1 + 2ν |x + ε(y − x)|)−N |β|=M +1 0 q2 .

Conversely, we have s2 ,α d d M ps11,β ,q1 (R ) → M p2 ,q2 (R )

if and only if p1 ≤ p2 and , s2 ≤ s1 + d · s(1) ,    s2 < s1 + d · s(1) + (1 − β) q1−1 − q2−1 ,

if q1 ≤ q2 , if q1 > q2 .

Using the theory of embeddings between decomposition spaces from Subsection 3.3, one can explain the main geometric properties of the α coverings Q(α) which lead to the preceding theorem: The main point is that the covering Q(α) is almost subordinate to (“finer than”, cf. equation (3.5)) the covering Q(β) for α ≤ β. Furthermore, the precise conditions depend on the number of “smaller sets” that are needed to cover the “bigger” sets, cf. [70, Lemma 6.1.5] and Theorems 15 and 16 from below. We finally remark that the theorems in [70] apply for the full range (0, ∞] of the exponents. But in the present paper, we restrict ourselves to the range [1, ∞] for simplicity. 3.3. Embeddings between different decomposition spaces. In this subsection, we consider embeddings between decomposition spaces with respect to different coverings, i.e. of the form (3.4)

D(Q, Lp1 , qu1 ) → D(P, Lp2 , qv2 )

for p1 , p2 , q1 , q2 ∈ [1, ∞] and two almost structured coverings Q = (Qi )i∈I = (Ti Qi + bi )i∈I

and

P = (Pj )j∈J = (Sj Pj + cj )j∈J

of two (possibly different) subsets O, O of the frequency space Rd . We assume the weights u = (ui )i∈I and v = (vj )j∈J to be moderate w.r.t. Q and P, respectively. As seen in the previous subsection, our setting includes embeddings between α-modulation spaces for different values of α. As the main standing requirement for this subsection, we assume that Q is almost subordinate to P. Very roughly, this means that the sets Qi are “smaller” than the sets Pj , or that Q is “finer” then P. Rigorously, it means that there is some fixed n ∈ N0 such that for every i ∈ I, there is some ji ∈ J satisfying 3 (3.5) Qi ⊂ Pjn∗ := P . i ∈jin∗

Note that this assumption implies O ⊂ O . Even more importantly, this assumption destroys the symmetry between Q, P in equation (3.4), so that in addition to (3.4), we will also consider the “reverse” embedding (3.6)

D(P, Lp2 , qv2 ) → D(Q, Lp1 , qu1 ).

FROM FRAZIER-JAWERTH CHARACTERIZATIONS TO DECOMP. SPACES

201

Under the assumption that Q is almost subordinate to P, the object which describes those features of the coverings Q, P which are relevant for us is the family of intersection sets given by Ij := {i ∈ I | Qi ∩ Pj = ∅}

for j ∈ J.

Of course, Ij = ∅ if and only if Pj ∩ O = ∅. Since these sets will be irrelevant for our purposes, we additionally define JO := {j ∈ J | Pj ∩ O = ∅} . In the remainder of this subsection, we will state sufficient conditions and necessary conditions for the existence of the embeddings (3.4) and (3.6), respectively. In general, these two conditions will only coincide for a certain range of p1 or p2 , while there is a gap between the two conditions outside of this range. Under suitable additional assumptions, however, more strict necessary conditions can be derived; in fact, a complete characterization of the existence of the embeddings can be achieved. For this to hold, we will (occasionally, but not always) assume that the following properties are fulfilled: Definition 14. (1) We say that the weight u = (ui )i∈I is relatively P-moderate, if ui < ∞. sup sup u j∈J i,∈Ij  (2) The (almost structured) covering Q = (Ti Qi + bi )i∈I is called relatively P-moderate, if the weight (| det Ti |)i∈I is relatively P-moderate. Roughly speaking, this means that two (small) sets Qi , Q have essentially the same measure if they intersect the same (large) set Pj . Although these assumptions might appear rather restrictive, they are fulfilled in many practical cases; in particular if Q and P are coverings associated to αmodulation spaces, and if u, v are the usual weights for these spaces. We can now analyze existence of the embedding (3.6): Theorem 15. (cf. [70, Theorem 5.4.1])  Define p 2 := max{p2 , p2 } and for r ∈ [1, ∞], let      7 !       1 1 (r) − p p    C1 :=  , vj   | det Ti | 2 1 · ui i∈Ij  q ·(r/q )  1 (Ij )   1 j∈J q1 ·(q2 /q1 ) (J) ⎛      ⎞   1 1 1 1 − p1 − 1  ui j  p2 − q1 −  q 1 1 p2 p2  ⎝ + +⎠ C2 :=  · | det T | · | det S | ij j  vj   

j∈JO q1 ·(q2 /q1 ) (JO )

where for each j ∈ JO some ij ∈ I with Qij ∩ Pj = ∅ can be chosen arbitrarily. Then the following hold: (p 2 )

(1) If C1 (3.7)

< ∞ and p2 ≤ p1 , then the map

ι : D(P, Lp2 , qv2 ) → D(Q, Lp1 , qu1 ), f → f |F (Cc∞ (O)) is well-defined and bounded.

,

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H. G. FEICHTINGER AND F. VOIGTLAENDER

(2) Conversely, if

  θ : F −1 (Cc∞ (O)), ·D(P,Lp2 ,qv2 ) → D(Q, Lp1 , qu1 ), f → f

(3.8)

(p )

is bounded, then C1 2 < ∞ and p2 ≤ p1 . (3) Finally, if Pj ⊂ O holds4 for all j ∈ JO and if additionally the covering Q and the weight u = (ui )i∈I are both relatively P-moderate, we have the following equivalence: ι is bounded

⇐⇒ ⇐⇒

θ is bounded   (p ) C1 2 < ∞ and p2 ≤ p1

⇐⇒

(C2 < ∞ and p2 ≤ p1 ) .

Remark. (1) We achieve a complete characterization of the existence of the embedding (3.6) if Q and u are relatively P-moderate, but also in case (p )

2 of p2 ∈ [2, ∞], since in this case, p = C1 2 . 2 = p2 and hence C1 (2) Even for well-understood special cases like α-modulation spaces, the above theorem yields new results, since even in the most general previous result [52], only the case (p1 , q1 ) = (p2 , q2 ) was studied. (3) Considering ι as an embedding is not always justified. For example, if O \ O has nonempty interior, then every f ∈ F −1 (Cc∞ (O \ O)) satisfies ιf = 0, although f = 0 is possible.

(p )

The result for the embedding (3.4) is similar: Theorem 16. (cf. [70, Theorem 5.4.4])  Let (ϕi )i∈I be a BAPU for Q. Define p 2 := min{p2 , p2 } and for r ∈ [1, ∞], let  !           1 1 (r) −1 p1 − p2     , C1 :=  vj ·  ui · | det Ti |   i∈Ij r·(q1 /r) (I )  j j∈J q2 ·(q1 /q2 ) (J) ⎛      ⎞   1 1 1 1 1 1 − − − −  vj  p1 q1 p2 q1 p p 2 2   ⎝ ⎠ + + C2 :=  · | det Tij | · | det Sj | ,   ui j  j∈JO q2 ·(q1 /q2 ) (JO )

where for each j ∈ JO some ij ∈ I with Qij ∩ Pj = ∅ can be chosen arbitrarily. Then the following hold: (p 2)

(1) If C1 (3.9)

< ∞ and if p1 ≤ p2 , then the map ι = ιΦ : D(Q, Lp1 , qu1 ) → D(P, Lp2 , qv2 ), f →



F −1 (ϕi · f&)

i∈I

is well-defined and bounded. Here, ιf acts as follows:  ιf, γ = f, F(ϕi · F −1 γ) for γ ∈ F(Cc∞ (O )), i∈I

with absolute convergence of the series for f ∈ D(Q, Lp1 , qu1 ). Furthermore, ιf, γ = f, γ holds for all γ ∈ F(Cc∞ (O)), so that ιf ∈ D(P, Lp2 , qv2 ) ⊂ Z  (O ) extends f ∈ D(Q, Lp1 , qu1 ) ⊂ Z  (O). 4 The

main case in which this holds is if O = O  .

FROM FRAZIER-JAWERTH CHARACTERIZATIONS TO DECOMP. SPACES

(2) If the map (3.10)

203

  θ : F −1 (Cc∞ (O)), ·D(Q,Lp1 ,qu1 ) → D(P, Lp2 , qv2 ), f → f (p )

is bounded, then p1 ≤ p2 and C1 2 < ∞. (3) Finally, if Pj ⊂ O holds for all j ∈ JO and if additionally the covering Q and the weight u = (ui )i∈I are both relatively P-moderate, we have the following equivalence: ι is bounded

⇐⇒ ⇐⇒

θ is bounded.   (p ) C1 2 < ∞ and p1 ≤ p2

⇐⇒

(C2 < ∞ and p1 ≤ p2 ) .

Remark. (1) We achieve a complete characterization of the existence of the embedding (3.4) for p2 ∈ [1, 2]. If Q and u are relatively P-moderate, we get a complete characterization for arbitrary p2 ∈ [1, ∞]. (2) In contrast to Theorem 15, ι is always injective in the present setting. (3) Note that the definition of ι is independent of p1 , p2 , q1 , q2 and u, v. In fact, if O = O , then ιf = f for all f ∈ D(Q, Lp1 , qu1 ) ⊂ Z  (O) = Z  (O ). For one concrete application of Theorems 15 and 16, we refer the reader to the characterization of embeddings between different α-modulation spaces in Theorem 13. Further applications will be given in Section 4. There are also results which apply if neither Q is almost subordinate to P, nor vice versa. In fact, it suffices if one can write O ∩ O = A ∪ B, such that Q is almost subordinate to P near A and vice versa near B. For a precise formulation of this condition, and the resulting embedding results, we refer to [70, Theorem 5.4.5]. Finally, there are also results for embeddings of the form D(Q, Lp , Y ) → W k,q (Rd ) for the classical Sobolev spaces W k,q . These results are easy to apply, since no subordinateness is required. As shown in [71], existence of the embedding can be completely characterized for q ∈ [1, 2] ∪ {∞}, while for q ∈ (2, ∞), certain sufficient and certain necessary criteria are given; but in general, these do not coincide. 3.4. Banach frames for Decomposition spaces. Borup and Nielsen [6] gave a construction of Banach frames for decomposition spaces which applies in a very general setting. In particular, their construction applies to (α)-modulation spaces and Besov spaces. Since this construction makes the power of Banach frames available for decomposition spaces, we could not resist discussing their results. Furthermore, their results fit well into the present context: As Borup and Nielsen write themselves: “[our] frame expansion should perhaps be considered an adaptable variant of the ϕ-transform of Frazier and Jawerth” (cf. [6, Section 3.2]). To describe their construction, we first introduce structured coverings. An almost structured covering Q = (Qi )i∈I = (Ti Qi + bi )i∈I of O = Rd is called structured if Qi = Q for all i ∈ I, i.e., if all Qi are affine images of a fixed set. The idea is to transfer the orthonormal basis (e2πik,· )k∈Zd of L2 ([−1/2, 1/2]d ) (a) to each of the sets Qi = Ti [−a, a]d +bi ⊃ Qi for certain a > 0. Then, one truncates these periodic functions using a certain (quadratic) partition of unity subordinate

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H. G. FEICHTINGER AND F. VOIGTLAENDER

to Q. Thus, one obtains a tight frame for L2 (Rd ). The nontrivial part is to show that one also obtains Banach frames for the whole range of decomposition spaces. The construction proceeds as follows: By [6, Proposition 1] on finds a family (θi )i∈I such that: (1) (2) (3) (4)

supp  θi2⊂ Qi for all i ∈dI, i∈I θi ≡ 1 on O = R , supi∈I F −1 θi L1 < ∞, supi∈I ∂ α [θi (Ti · +bi )]sup < ∞ for all α ∈ Nd0 .

Given such a family (θi )i∈I , we choose a cube Qa ⊂ Rd of side-length 2a satisfying Q ⊂ Qa . Finally, for i ∈ I and n ∈ Zd , define en,i : Rd → C by −1

en,i (ξ) := (2a)−d/2 · | det Ti |−1/2 · χQa (Ti−1 (ξ − bi )) · ei a n·Ti π

(ξ−bi )

for ξ ∈ Rd ,

and set ηn,i := F −1 (θi · en,i ). Sincethe family (en,i )n∈Zd forms an orthonormal basis of L2 (Ti Qa +bi ) and because of i∈I θi2 ≡ 1, it follows (cf. [6, Proposition 2]) that the family (ηn,i )n∈Zd ,i∈I forms a tight frame for L2 (Rd ). Of course, we are not simply interested in (tight) frames for L2 (Rd ) with a given form of frequency localization—we want to obtain a Banach frame for the decomposition space D(Q, Lp , qu ). Thus, we define the Lp -normalized version ηn,i := | det Ti | 2 − p · ηn,i (p)

1

1

for i ∈ I and n ∈ Zd .

Then, Borup and Nielsen showed (cf. [6, Proposition 3, Definition 8, Lemma 4 and Theorem 2]) that there is a suitable solid BK space d(Q, p , qu ) such that the coefficient operator (p)

C : D(Q, Lp , qu ) → d(Q, p , qu ), f → (f, ηn,i )n∈Zd ,i∈I is bounded. As familiar by now, there is also a bounded reconstruction operator R : d(Q, p , qu ) → D(Q, Lp , qu ) which satisfies R ◦ C = idD(Q,Lp ,qu ) . Thus, the (p)

family (ηn,i )i∈I,n∈Zd forms a Banach frame for D(Q, Lp , qu ). 4. Abstract and Concrete Wavelet Theory s s As noted in Section 2, the description of the spaces Bp,q and Fp,q via the ϕtransform—or via wavelets—can be viewed (at least in part) as an application of the general theory of coorbit spaces to the affine group which acts on L2 (Rd ) via translations and isotropic dilations. To be precise, this group action yields the ˙ s and F˙ s , not the inhomohomogeneous Besov- and Triebel-Lizorkin spaces B p,q p,q geneous ones. One well known characterization of (homogeneous) Besov spaces shows that these spaces are obtained by placing certain integrability conditions on the (continuous) wavelet transform

Wϕ f : Rd × R∗ → C, (x, a) → f, Tx Da ϕ of a function or distribution f and a certain analyzing window ϕ. Other applications of the wavelet transform include the characterization of the wave-front set of

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a distribution using the decay of the transform [63]. However, due to the isotropic nature of the dilations Da f (x) = a−d/2 · f (a−1 x), such a single wavelet characterization is only valid in dimension d = 1 (cf. [7] or [35, Lemma 4.4, Lemma 4.10]), where smoothness is an “undirected property”. Even beyond this specific problem of characterizing the wave-front set, it was noted in recent years that the isotropic, directionless nature of the wavelet transform is a limitation for many applications. To overcome this problem, a vast number of “directional” variants of wavelets were invented: In particular, curvelets [7, 8] and shearlets [14, 54]. Among these two systems, shearlets have the special property that there is—as in the case of wavelets—an underlying dilation group through which the family of shearlets can be generated from a single “mother wavelet”, see also Section 2.3. In view of these two very different dilation groups—the affine group and the shearlet group—it becomes natural to consider the bigger picture: Given any (closed) subgroup H ⊂ GL(Rd ), one can form the group G = Rd  H of all affine mappings generated by arbitrary translations and all dilations in H. The multiplication on G is given by (x, h)(y, g) = (x + hy, hg) and the Haar measure is (4.1)

d(x, h) = | det h|−1 dx dh,

where dh is the Haar measure of H. The group G from above acts unitarily on L2 (Rd ) via translations and dilations, i.e., by the quasi-regular representation (4.2)

π : G → U(L2 (Rd )), (x, h) → Tx Dh .

This representation comes with an associated (generalized) wavelet transform Wϕ f : G → C, (x, h) → f, π(x, h)ϕ

for

f, ϕ ∈ L2 (Rd ),

where the (fixed) function ϕ is called the analyzing window. In the general description of coorbit theory in Section 2, this was called the voice transform. Given this transform, it is natural to ask which properties of f can be easily read off from Wϕ f . As for wavelets (for d = 1 [63]) or for shearlets (for d = 2 [47, 54]), it turns out [35] that for large classes of dilation groups, the wave-front set of a (tempered) distribution f can be characterized via the decay of Wϕ f . Another important property of a generalized wavelet system (like shearlets) are its approximation theoretic properties. Here, the question is: Which classes of functions can be approximated well by linear combinations of only a few elements of the wavelet system? For “ordinary” wavelets, this question leads to the theory of Besov spaces and their atomic decompositions, as explored by Frazier and Jawerth. For a general dilation group, these approximation theoretic properties are (at least in part) encoded by the associated wavelet type coorbit spaces, which we will now discuss in greater detail. One particular problem which is of interest to us is the following: If a function/signal f can be well approximated by one wavelet system, can it also be well approximated using a different wavelet system? Of course, the answer to this question will depend on the precise nature of the two wavelet systems and on the way in which the statement “f can be well approximated by . . . ” is made mathematically precise.

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4.1. General wavelet type coorbit spaces. As long as π acts irreducibly on L2 (and if the representation is (square) integrable), we can apply the general coorbit theory as described in Section 2 to form the coorbit spaces Co(G, Y ) = {f ∈ R | Wϕ f ∈ Y } ,

(4.3)

where R = RY is a suitable reservoir, which plays the role of S  (Rd ) for the usual Besov- or Triebel-Lizorkin spaces. Furthermore, ϕ ∈ L2 (Rd ) has to be a suitable analyzing window. Formally, this means that ϕ must fulfill ϕ ∈ Av0 (cf. equation (2.2)) for a so-called control weight v0 : G → (0, ∞). As we will see (cf. Definition 18), this condition is closely related to the usual “vanishing moments condition” for ordinary wavelets. In this section, instead of the general coorbit spaces Co(Y ) = Co(G, Y ), we will consider the more restrictive case of the (weighted) mixed Lebesgue space Y = Lp,q m (G) for p, q ∈ [1, ∞] and a weight m : H → (0, ∞). Precisely, the space Lp,q m (G) is the space of all measurable functions f : G → C for which the norm   f Lp,q := h → m(h) · f (x, h)Lp (Rd , dx)  q L (H, dh/| det h|)

m

is finite. This normalization—in particular the measure dh/| det h| on H—is chosen p such that we have Lp,p m (G) = Lm (G), cf. equation (4.1). Recall from Section 2 that the space Y needs to be a solid BF space on G which is invariant under left- and right translations. Clearly, Y = Lp,q m (G) satisfies all of these properties, except possibly for invariance under left- and right translations. To ensure this, we assume that m is v-moderate for some (measurable, locally bounded, submultiplicative) weight v : H → (0, ∞), i.e., we assume m(xyz) ≤ v(x)m(y)v(z)

∀x, y, z ∈ H.

Under these assumptions, it is shown in [44, Lemma 1 and Lemma 4] that there is a control weight v0 : G → (0, ∞) for Y = Lp,q m (G) which is (by abuse of notation) of the form v0 (x, h) = v0 (h) and measurable, submultiplicative and locally bounded. In the present setting, coorbit theory can be seen as a theory of nice wavelets and nice signals, cf. [42]. Nice wavelets are those belonging to the class Av0 of analyzing windows, while a nice signal f (with respect to an analyzing wavelet ϕ) p,q is one for which Wϕ f ∈ Lp,q m , i.e., for which f ∈ Co(Lm ). Now, coorbit theory—if it is applicable—yields two main properties: • A consistency statement: Nice wavelets agree upon nice signals, i.e. if ϕ, ψ ∈ Av0 \ {0} (with v0 depending on p, q, m), then Wϕ f ∈ Lp,q m

⇐⇒

Wψ f ∈ Lp,q m .

We even get a norm-equivalence, so that the coorbit space from equation (4.3) with Y = Lp,q m (G) is well-defined. • An atomic decomposition result: As seenin Theorem 3, we can guarantee atomic decompositions of the form f = g∈G0 αg (f ) · π(g)ψ for elements f ∈ Co(Lp,q m ) and coefficients (αg (f ))g∈G0 lying in a suitable sequence space, if ψ is a better vector (in comparison to just being an analyzing vector), i.e. if ψ ∈ Bv0 .

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Despite these pleasant features, the theory of (generalized) wavelet type coorbit spaces raises several questions: (Q1) For which dilation groups H is the quasi-regular representation from equation (4.2) irreducible and (square)-integrable, so that coorbit theory is applicable in principle? (Q2) Is coorbit theory applicable, i.e. are there “nice wavelets”? Precisely, do we have Av0 = {0} and Bv0 = {0} and are there convenient sufficient criteria for a function ϕ ∈ L2 (Rd ) to belong to Av0 or to Bv0 ? (Q3) How are the resulting coorbit spaces Co(Rd  H, Lp,q m ) related to classical s s function spaces like Bp,q of Fp,q (or their homogeneous counterparts)? Furthermore, how are coorbit spaces with respect to different dilation groups related to each other? This connects to the question posed above: If a given function/signal can be well approximated using one wavelet system, does the same also hold for a different system? Even for the special case of the shearlet dilation group, these questions are nontrivial and triggered several papers [10, 11, 14, 16]. Nevertheless, they also admit satisfactory answers in the present generality: As we will see, each of these questions is linked to the dual action  : H × Rd → Rd , (h, ξ) → h−T ξ of the dilation group H on the frequency space Rd . To see the relevance of the dual action, note that the Fourier transform of Wϕ f (·, h) is given by (4.4)

& T ξ). (F[Wϕ f (·, h)]) (ξ) = | det h|1/2 · f&(ξ) · ϕ(h

Thus, if ϕ & has support in U ⊂ Rd , then Wϕ f (·, h) is bandlimited to h−T U = (h, U ). The remaining subsections deal with the three questions listed above. 4.2. Question 1: Irreducibility and square-integrability of π. As shown in [4, 39, 40], the quasi-regular representation π is irreducible and square-integrable if and only if the following two properties are satisfied: (1) There is some ξ0 ∈ Rd \ {0} such that the orbit  . O := H T ξ0 = hT ξ0  h ∈ H c is open and of full measure (i.e. null set).  TO is a Lebesgue .  (2) The stabilizer Hξ0 := h ∈ H h ξ0 = ξ0 is compact.

In this case, we call O the (open) dual orbit of the dilation group H, whereas the null-set Oc is called the blind spot of H. Finally, a dilation group H fulfilling the two properties above is called admissible. In the following, we fix ξ0 ∈ O. To give the reader an idea of the richness of admissible dilation groups, we mention the following admissible dilation groups in dimension d = 2: (1) The diagonal group H1 := {diag(a, b) | a, b ∈ R \ {0}} 2

with dual orbit O = (R \ {0}) . (2) The similitude group H2 := (0, ∞) · SO(R2 ), with O = R2 \ {0}.

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(3) The family of shearlet type groups  

 a b  (c) a ∈ (0, ∞), b ∈ R, ε ∈ {1, −1} . (4.5) H3 := ε · 0 ac  Here, the anisotropy parameter c ∈ R can be chosen arbitrarily. Regardless of this choice, the dual orbit is always O = (R \ {0}) × R. 4.3. Question 2: Existence of “nice” wavelets. Here, we are given an admissible dilation group H and we are interested in conditions which guarantee Av0 = {0} or Bv0 = {0}, where v0 : H → (0, ∞) (interpreted by abuse of notation as a weight on G = Rd  H) is a (locally bounded) control weight for Y = Lp,q m . As we saw above, such a control weight always exists under our general assumptions on the weight m : H → (0, ∞). In the present setting, “nice wavelets” and “better wavelets” exist in abundance: Theorem 17. (cf. [44, Theorem 9]) Let v0 : H → (0, ∞) be measurable and locally bounded. Then, every function ϕ ∈ F −1 (Cc∞ (O)) ⊂ S(Rd ) satisfies ϕ ∈ Bv0 ⊂ Av0 and the map (4.6)

 : Cc∞ (O) → L1v0 (G), g → Wϕ (F −1 g)

is well-defined and continuous. This theorem, however, does not yield existence of compactly supported “nice wavelets”. In the case of “traditional” wavelets, it is well-known that a certain & = 0 for all |α| ≤ r) makes a wavelet amount of “vanishing moments” (i.e. ∂ α ϕ(0) “nice”. This generalizes to the present setting: Definition 18. ([43, Definition 1.4]) Let r ∈ N. We say that ϕ ∈ L1 (Rd ) has & ∈ C r (Rd ) and if vanishing moments on Oc of order r if ϕ & Oc ≡ 0 (∂ α ϕ)|

for |α| < r.

As shown in [43, Theorems 1.5(c), 2.12], given the weight v0 : H → (0, ∞), one can (in most cases) explicitly compute a number  = (H, v0 ) ∈ N such that every & , < ∞ and with vanishing moments of order  on function ψ ∈ L1 (Rd ) with ψ Oc satisfies ψ ∈ Bv0 ⊂ Av0 . Here, we employed the usual Schwartz-type norm f , :=

max

sup (1 + |x|) · |∂ α f (x)| ∈ [0, ∞].

d α∈Nd 0 ,|α|≤ x∈R

Finally, in the setting of ordinary wavelets, one can obtain a wavelet with enough vanishing moments by taking derivatives of a given smooth function of sufficient decay. In the present general setting, [41, Lemmas 3.1 and 4.1] yield a similar “algorithm” for obtaining functions with suitably many vanishing moments on O c . For the sake of brevity, we refrain from giving more details and instead refer the interested reader to [41, 43]. 4.4. Question 3: Relation of generalized wavelet coorbit spaces to other spaces. Here, we are interested in the relation of the coorbit space Co(Lp,q m ) to the classical Besov- or Triebel-Lizorkin spaces, but also in the relation between Co(Lpm11,q1 , Rd  H1 ) and Co(Lpm22,q2 , Rd  H2 ) for two different admissible dilation groups H1 , H2 ≤ GL(Rd ). In view of the embedding theory for decomposition spaces (cf. Subsection 3.3), this question would be solved (at least to a significant extent) if we knew that the

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coorbit space Co(Lp,q m ) is (canonically isomorphic to) a decomposition space. But the main result of [44] is precisely such an isomorphism. In more detail, [44] shows p q d ∼ Co(Lp,q m (R  H)) = D(QH , L , u )

(4.7)

for a so-called induced covering QH and a so-called decomposition weight u. Below, we provide a concrete example indicating how this isomorphism and the above embedding results can be used to obtain novel embedding results for shearlet coorbit spaces. In the above isomorphism, the induced covering QH of the dual orbit O is given by QH = (h−T i Q)i∈I for certain hi ∈ H and a suitable Q ⊂ O. Its precise construction is described in the following paragraph. But once Q = (h−T i Q)i∈I is known, the decomposition weight u = (ui )i∈I from above (cf. [44, Lemma 35 and the ensuing remark]) is given by ui = | det hi | 2 − q · m(hi ) 1

(4.8)

1

∀i ∈ I.

Let us reconsider the induced covering QH . The family (hi )i∈I has to be well spread in H, i.e. there are compact unit neighborhoods K1 , K2 ⊂ H such that 3 H= h i K1 and hi K2 ∩ hj K2 = ∅ for i, j ∈ I with i = j. i∈I

Finally, the set Q ⊂ O is an arbitrary open, bounded set such that the closure Q ⊂ Rd is contained O and such that there is a smaller open set P ⊂ Rd with 2 in−T P ⊂ Q and O = i∈I hi P . As shown in [44, Theorem 20], such sets P, Q always exist if (hi )i∈I is well-spread in H. The same theorem also shows that the resulting covering QH = (h−T i Q)i∈I is an (almost) structured admissible covering of O. In particular, there is a BAPU (ϕi )i∈I subordinate to QH , cf. Theorem 12. Finally, [44, Lemmas 22 and 23] show that the decomposition weight from equation (4.8) is QH -moderate. This implies (cf. Subsection 3.1) that the decomposition space D(QH , Lp , qu ) on the right-hand side of equation (4.7) is well-defined. As an illustration of the concept of an induced covering, we consider the shearlet (c) type group H3 from equation (4.5). A picture of (a part of) the associated induced covering for different values of the anisotropy parameter c ∈ R is shown in Fig. 1. For the explicit description of the isomorphism from equation (4.7), we note that it is a (relatively) easy consequence of equation (4.6) that ι : Z(O) = F(Cc∞ (O)) → Hv10 , f → f , with Hv10 as in equation (2.1) is well-defined, antilinear and continuous. By duality, this implies that (4.9)

∠  ιT : Hv10 → Z  (O), θ → θ ◦ ι

is well-defined, continuous and linear. Recall from Section 2 that Co(Lp,q m ) is a ∠  for a certain control weight v0 . Thus, subspace of the reservoir R = Hv10 ιT f ∈ Z  (O) is well-defined for every f ∈ Co(Lp,q m ). The claim of equation (4.7) is p q ) → D(Q, L ,  ) is an isomorphism of Banach spaces. precisely that ιT : Co(Lp,q u m A proof of this fact can be found in [44, Theorem 43].

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Figure 1. The figure shows (a part of) the induced covering for (c) the group H3 for c = − 12 , c = 0 and c = 12 . These choices show the qualitatively different behaviour of the covering for different values of c. For c1 < c2 , the covering S (c1 ) is “larger/coarser” near the y-axis, whereas the covering S (c2 ) is “larger/coarser” away from the y-axis. Note: The images are taken from [70, Figure 1].

This representation of Co(Lp,q m (G)) as a decomposition space has several important consequences, both mathematically and conceptually: • As we saw above, the coorbit space Co(Lp,q m (G)) with its original definition is heavily tied to the group G = Rd × H and thus to the dilation group H. 1 ∠ In particular, Co(Lp,q m (G)) is a subspace of the reservoir R = (Hv0 (G)) . p1 ,q1 d This makes it difficult to consider an element f ∈ Co(Lm1 (R H1 )) as an element of another coorbit space Co(Lpm22,q2 (Rd  H2 )), or as an s . element of more classical function spaces like Bp,q In contrast, as we saw in Section 3, it is (at least in principle) possible to compare decomposition spaces D(Q, Lp1 , qu1 ) and D(P, Lp2 , qv2 ) which are defined using two different coverings Q, P of the sets O, O ⊂ Rd . Thus, using the decomposition space view, it becomes possible to compare wavelet coorbit spaces defined by different dilation groups.

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• Even ignoring the issue of the different reservoirs (e.g. by restricting to Schwartz functions), it is not at all obvious how the decay or integrability condition Wϕ f ∈ Lpm11,q1 (Rd  H1 ) relates to another decay condition Wϕ f ∈ Lpm22,q2 (Rd H2 ), even if the same analyzing window is used in both cases. One of the reasons is that it is difficult to compare the two actions of the dilation groups on ϕ, as well as the two distinct Haar measures. In comparison, the decomposition space point of view translates these two elusive properties into (more or less) transparent quantities, namely (1) The induced covering QH = (h−T i Q)i∈I for some well-spread family (hi )i∈I in H and a suitable set Q ⊂ O, 1 1 (2) The decomposition weight ui = | det hi | 2 − q · m(hi ). Using the methods from Subsection 3.3, it is then (comparatively) easy to establish embeddings D(QH1 , Lp1 , qu11 ) → D(QH2 , Lp2 , qu22 ) between the associated decomposition spaces and thus of the two coorbit spaces Co(Lpm11,q1 (Rd  H1 )) and Co(Lpm22,q2 (Rd  H2 )). • Similarly, one can use the methods from Subsection 3.3 to establish embeddings between generalized wavelet coorbit spaces and classical smoothness spaces like Sobolev- and Besov spaces. • Conceptually, all these considerations show that the approximation theoretic properties of the wavelet system generated by a dilation group H are completely determined by the way in which (the dual action of ) H covers/partitions the frequencies. Theorem 19 below is an example of results that can be obtained by combining the embedding results from Subsection 3.3 with the isomorphism p q d ∼ Co(Lp,q m (R  H)) = D(QH , L , u ).

Precisely, we consider embeddings between shearlet coorbit spaces and inhomogeneous Besov spaces. For the sake of brevity, we only consider embeddings of the shearlet coorbit space into inhomogeneous Besov spaces. Results for the reverse direction are also available (cf. [70, Theorem 6.3.14]), but are omitted here. We only consider the case c ∈ (0, 1]. This ensures that the induced covering S (c) = QH (c) is almost subordinate to the inhomogeneous dyadic covering. See 3 [70, Lemma 6.3.10] for a formal proof and Figure 1 for a graphical illustration. Note, however, that S (c) is not relatively moderate with respect to the inhomogeneous dyadic covering. This limits sharpness of our results to a certain range of p2 . Theorem 19. ([70, Theorem 6.3.14]) Let c ∈ (0, 1], p1 , p2 , q1 , q2 ∈ [1, ∞] and  α, β, γ ∈ R. Set p 2 := min{p2 , p2 }, define the weight u(α,β) : H3 → (0, ∞), h → h−1 α · | det h|β (c)

and set α

(1)

γ (1)

 1 1 1 1 − − + +β , p1 p2 q1 2 



 1 1 1 1 1 1 := −(1 + c) · − − + + β + (c − 1) · − . p1 p2 q1 2 p q1 + 2 1+c := c



212

,

H. G. FEICHTINGER AND F. VOIGTLAENDER

If p1 ≤ p2 as well as γ ≤ α + γ (1) , γ < α + γ (1) ,

if q1 ≤ q2 , if q1 > q2

, and

 max

1 p 2

−

1 q1 ,

 α < α(1) ,

max {0, α} ≤ α(1) ,

if q1 > p 2, if q1 ≤ p 2

hold, then (4.10)

(c)

,q1 Co(Lpu1(α,β) (R2  H3 )) → B γp2 ,q2 (R2 ).

A necessary condition for existence of this embedding is obtained by replacing p 2 by p2 everywhere (also in the definition of γ (1) ).

Remark. (1) Existence of the embedding (4.10) has to be interpreted suitably. Precisely, (4.10) means that there is a bounded linear map (c)

,q1 ι : Co(Lpu1(α,β) (R2  H3 )) → B γp2 ,q2 (R2 ) (c)

,q1 (R2  H3 )). which satisfies ιf = f for all f ∈ L2 (R2 ) ∩ Co(Lpu1(α,β) (2) The preceding theorem is superficially similar to [16, Theorem 4.7]. But the two results are very different, since Dahlke et al. consider embeddings (1/2) of the strict subspace SCC p,r  Co(Lp,p (R2  H3 )) into a sum of u(0,−2r/3) 1 2 ˙ σp,p ˙ σp,p (R2 ) + B (R2 ) for certain σ1 , σ2 . homogeneous Besov spaces B In contrast, the preceding theorem investigates embeddings of the whole shearlet coorbit space into a single, inhomogeneous Besov space. (3) The preceding theorem achieves a complete characterization of the embedding (4.10) for p2 ∈ [1, 2], since we have p 2 = p2 in this range.

As a conclusion, we remark that the embedding results and the isomorphism between generalized wavelet coorbit spaces and decomposition spaces can also be (c) 2 used to derive embeddings between the coorbit spaces Co(Lp,q m (R  H3 )) for different values of c, see [70, Theorem 6.3.9]. They can also be used to derive (non)boundedness of certain operators—e.g. dilation operators—acting on coorbit spaces. For example, in [70, Theorem 6.5.9], the set of matrices which act boundedly by dilation simultaneously on all coorbit (c) spaces of the shearlet type group H3 (for a fixed c ∈ (0, 1)) is characterized completely. Acknowledgments Both authors want to thank HIM—the Hausdorff Institute of Mathematics— where we both spent some time during the preparation of this manuscript. FV was funded by the Excellence Initiative of the German federal and state governments, and by the German Research Foundation (DFG), under the contract FU 402/5-1. References [1] J. Arazy, S. D. Fisher, and J. Peetre, M¨ obius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145, DOI 10.1007/BFb0078341. MR814017 [2] R. Balian, Un principe d’incertitude fort en th´ eorie du signal ou en m´ ecanique quantique (French, with English summary), C. R. Acad. Sci. Paris S´er. II M´ ec. Phys. Chim. Sci. Univers Sci. Terre 292 (1981), no. 20, 1357–1362. MR644367 [3] J. J. Benedetto, C. Heil, and D. F. Walnut. Differentiation and the Balian-Low theorem. J. Fourier Anal. Appl., 1(4):355–402, 1995.

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H. G. FEICHTINGER AND F. VOIGTLAENDER

Faculty of Mathematics, University of Vienna E-mail address: [email protected] ¨r Mathematik, RWTH Aachen University Lehrstuhl A fu E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13940

Traces and extensions of weighted Sobolev and potential spaces Michael Frazier and Svetlana Roudenko In memory of Bj¨ orn Jawerth, my collaborator, mentor, and friend. –M.F. Abstract. Let u be an Ap weight on Rn+1 and v a doubling weight on Rn .    n Define the trace, or restriction, operator T rf  (x ) =  f (x , 0), where x ∈ R and f is a function on Rn+1 . If α >

1 p

+n

1 p

−1

+

+

β−n , p

where β is the

doubling exponent of v, then the trace operator is bounded from the weighted Bessel potential space Lpα (u) (which coincides with the weighted Sobolev space α−1/p, p (v) if and only if Lpk (u) if α = k ∈ N) into the weighted Besov space Bp there exists C > 0 such that   1 1 v dx  ≤ C u dx |I| I |Q(I)| Q(I) for all dyadic cubes I ⊆ Rn with side length less than or equal to 1, where Q(I) = I×[0, (I)]. If u and v satisfy the converse inequality, then there exists a α−1/p, p (v) → Lpα (u). If both inequalities hold, continuous linear map Ext : Bp α−1/p, p

T r ◦ Ext is the identity on Bp (u). More generally, the results hold with Lpα (u) replaced by the inhomogeneous Triebel-Lizorkin space Fpα, q (u) for any 0 < q ≤ ∞.

Contents 1. Introduction 2. Decomposition of Weighted Function Spaces 3. Traces and Extensions References

2010 Mathematics Subject Classification. Primary 42B35, 42B37, 42B25, 42C40. Key words and phrases. Trace, restriction, extension, weight, Ap class, Sobolev space, doubling measure. The second author was partially supported by NSF CAREER grant # 1151618. c 2017 American Mathematical Society

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MICHAEL FRAZIER AND SVETLANA ROUDENKO

1. Introduction A weight w on Rn is a non-negative, locally integrable function which is not almost everywhere 0. We consider weighted function spaces, such as the weighted Lebesgue spaces , 8

  1/p

Lp (w) =

f : Rn → C : f Lp (w) =

Rn

|f (x)|p w(x) dx

0 and 1 < p < ∞, where ∧ denotes the Fourier transform and ∨ denotes the inverse Fourier transform. We will also consider the weighted Besov spaces Bpα,q (w) and Fpα,q (w), defined in §2. When w is identically 1, we denote the unweighted spaces simply by Lp , Lpk , Lpα , Bpα,q , and Fpα,q , or sometimes Lp (Rn ), etc., when we want to emphasize the dimension of the underlying space. For a continuous function on Rn+1 , one defines T r(f ), the trace of f , on the hyperplane {(x  , 0) : x  ∈ Rn }, which we identify with Rn , by T r(f )(x  ) = f (x  , 0). In some cases, T r can be extended linearly and continuously from the continuous functions in some function or distribution space X on Rn+1 to the entire space X, into some function or distribution space Y in Rn , so that T r : X → Y . Classical results for the unweighted spaces state that (see the original references [2], [21], and the text [23]) (1.1)

T r : Lpα (Rn+1 ) → Bpα−1/p,p (Rn ),

(1.2)

T r : Bpα,p (Rn+1 ) → Bpα−1/p,p (Rn ),

and both of these maps are onto, for α > 1/p + n(1/p − 1)+ , where x+ = max(x, 0). The fact that the distinct spaces Lpα and Bpα,p have the same trace can be generalized to the result that (1.3)

T r : Fpα,q (Rn+1 ) → Bpα−1/p,p (Rn ),

for any 0 < q ≤ ∞, and this map is also onto. This last result includes the previous two because Fpα,2 = Lpα and Fpα,p = Bpα,p . Our goal is to obtain weighted versions of (1.1) and (1.3). In [11], the result (1.2) was extended to the matrix weight case, which includes the case of scalar weights considered here. We use the notation w(E) to denote E w dx, when E is measurable. We always assume that w satisfies a doubling condition (or, we say, w is a doubling weight), which means that there exists a constant C > 0 such that for all cubes Q ⊆ Rn , (1.4)

w(2Q) ≤ Cw(Q),

where in general cQ is the cube with the same center as Q, but with side length c times the side length (Q) of Q. If C = 2β is the smallest constant such that (1.4)

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219

holds for all cubes Q, we call β the doubling exponent of w. We always have β ≥ n, with equality for Lebesgue measure. We say that w ∈ Ap , or w is an Ap -weight, for 1 < p < ∞, if

1/(p−1)   1 1 1 − p−1 w dx · w dx < ∞, [w]Ap = sup |B| B B |B| B where the supremum is taken over all balls B = B(x, r) ⊆ Rn , for x ∈ Rn and r > 0. The Ap condition is necessary and sufficient for the Hardy-Littlewood maximal function M to be bounded on Lp (w) ([18]), and it is the right condition for Calder´ on-Zygmund operators to be bounded on Lp (w) ([15]). As a consequence, we obtain that if α = k ∈ N, and w ∈ Ap , the weighted Sobolev space Lpk (w) coincides with the weighted potential space Lpα (w), for 1 < p < ∞. Another consequence, in the context of vector-valued Calder´on-Zygmund operators, is that if w ∈ Ap , then Lp (w) ≈ Fp0,2 (w) if w ∈ Ap . Then applying the Riesz potential yields (1.5)

Lpα (w) ≈ Fpα,2 (w),

with equivalent norms, for w ∈ Ap , when 1 < p < ∞ and α > 0. The Lebesgue measure of any measurable set E is denoted |E|. Dyadic cubes in Rn are of the form Ij,k = 2−j [0, 1]n + 2−j k for j ∈ Z and k ∈ Zn . For I a dyadic cube in Rn , let Q(I) = I × [0, (I)]; note that Q(I) is a dyadic cube in Rn+1 . Theorem 1.1. Suppose u is an Ap weight on Rn+1 and v is a doubling  weight on

Rn with doubling exponent β. Suppose 0 < p < ∞ and α >

1 p

+n

1 p

−1

+ +

β−n p .

Then the trace operator T r extends to be a continuous map (1.6)

T r : Lpα (u) → Bpα−1/p, p (v)

if and only if there is a constant C > 0 such that   1 1   v(x ) dx ≤ C u(x) dx (1.7) |I| I |Q(I)| Q(I) for all dyadic cubes I in Rn with (I) ≤ 1. Note that (1.7) can be written more concisely in the form (1.8)

(I)v(I) ≤ Cu(Q(I)).

The formulation (1.7) has the advantage of emphasizing the role of averages, in particular, making it obvious that (1.7) is satisfied in the unweighted case. We define the extension operator Ext as in [11], using Daubechies’ wavelets. n Given nonnegative integers K and N , let Φ be the scaling function and let {ψ (i) }2i=1−1 be the generators of Daubechies’s DL wavelet system in Rn , where L is taken sufficiently large so that Φ, ψ (i) ∈ C K for 1 ≤ i ≤ 2n − 1, and each ψ (i) satisfies  α (i) x ψ (x) dx = 0 for all multi-indices α such that |α| ≤ N . Then there exists Rn r > 0, depending on L, such that supp ψ (i) , supp Φ ⊆ B(0, r) for all i. Let Dn = {Ij,k }k∈Zn ,j=0,1,2,... be the set of dyadic cubes in Rn with side length (I) ≤ 1. For I = I0,k , define (1.9)

ΦI (x  ) = Φ(x  − k),

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MICHAEL FRAZIER AND SVETLANA ROUDENKO

and for I = Ij,k for j > 1, let xI = 2−j k and define ψI (x  ) = |I|−1/2 ψ (i) ((x  − xI )/(I)) . (i)

(1.10)

(i)

Then {ΦI }I∈Dn ,(I)=1 ∪{ψI }1≤i≤2n −1,I∈Dn ,(I)

1 p

+n

1 p

−1

+ +

β−n p .

If there is a constant C > 0 such that   1 1 u(x) dx ≤ C v(x  ) dx  (1.12) |Q(I)| Q(I) |I| I for all dyadic cubes I in Rn with (I) ≤ 1, then the extension operator Ext extends to be a continuous map Ext : Bpα−1/p, p (v) → Lpα (u).

(1.13)

α−1/p, p

If we also assume ( 1.7), then T r ◦ Ext is the identity map on Bp

(v).

Because of (1.5), Theorems 1.1 and 1.2 are consequences of the following result for the weighted Triebel-Lizorkin spaces Fpα,q , which exhibits the characteristic property that the trace and extension properties of these spaces are independent of the index q. Theorem 1.3. Suppose u is doubling weight on Rn+1 and v is a doubling weight on Rn with  doubling exponent β. Suppose 0 < p < ∞, 0 < q ≤ ∞ and 1 1 α > p + n p − 1 + β−n p . +

(i) The trace operator T r extends to be a continuous map (1.14)

T r : Fpα,q (u) → Bpα−1/p, p (v)

if and only if there exists C > 0 such that ( 1.7) holds for all dyadic cubes I in Rn with (I) ≤ 1. (ii) If there is a constant C > 0 such that ( 1.12) holds for all dyadic cubes I in Rn with (I) ≤ 1, the extension operator Ext extends to be a continuous map (1.15)

Ext : Bpα−1/p, p (v) → Fpα,q (u).

(iii) If we assume ( 1.7) and ( 1.12), then T r ◦ Ext is the identity map on α−1/p, p (v). Bp

Notice that the Ap condition on u is not required in Theorem 1.3. The Ap condition is needed in Theorems 1.1 and 1.2 in order to have Lpα (u) ≈ Fpα,2 (u). α−1/p, p (v) There is an analogue of Theorem 1.3 for the homogeneous spaces B˙ p α,q and F˙ p (u), via slight modifications of the arguments below. In this case, condition (1.7) is required for all dyadic cubes. The trace problem in the weighted setting has been previously considered in [13], where a class of radial weights was considered.

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In §2, we discuss the function space decomposition results that are needed in the proof of Theorem 1.3. Earlier results along these lines occur, for example, in [3, 4], [12] (see further discussion on that in §2). In §3, we present the proof of Theorem 1.3. The methods of this paper are an extension to the weighted case of the approach in [9, Section 11]. We would like to mention that Professor Jawerth’s interest in traces of function spaces originated very early in his career: one of his first papers was “The trace of Sobolev and Besov spaces if 0 < p < 1” [17]. 2. Decomposition of Weighted Function Spaces To define the weighted Besov and Triebel-Lizorkin spaces on Rn , we select ˆ functions Φ, ϕ ∈ S(Rn ) (the usual Schwartz space) satisfying Φ(ξ) = 0 if |ξ| ≥ 2, ˆ |Φ(ξ)| ≥ c if |ξ| ≤ 5/3, ϕ(ξ) ˆ = 0 if |ξ| ≤ 1/2 or if |ξ| ≥ 2, and |ϕ(ξ)| ˆ ≥ c if 3/5 ≤ |ξ| ≤ 5/3, for some c > 0. Let ϕ0 = Φ, and, for j ≥ 1, let ϕj (x) = 2jn ϕ(2j x). Note that ϕ +j (ξ) = ϕ(2 ˆ −j ξ). If w is a doubling weight on Rn , α ∈ R, and 0 < p, q ≤ ∞, we define the weighted Besov spaces ⎧ ⎫ ⎞1/q ⎛ ⎪ ⎪ ∞ ⎨ ⎬   jα q α,q  n ⎠ ⎝ α,q 2 ϕj ∗ f Lp (w) βp + n(1 − p1 )+ . Suppose {mQ }Q∈Dn is a family of functions satisfying  (M1) xγ mQ (x) dx = 0, for |γ| ≤ N and (Q) < 1,

− max (M,M −α) |x − xQ | (M2) |mQ (x)| ≤ |Q|−1/2 1 + , l(Q) and

−M |x − xQ | γ if |γ| ≤ [α] + 1, (M3) |D mQ (x)| ≤ |Q|−1/2−|γ|/n 1 + l(Q)  for each Q ∈ Dn . Suppose s = {sQ }Q∈Dn ∈ bαq p (w). Then Q∈Dn sQ mQ ∈ αq Bp (w), and there exists c > 0 independent of s and {mQ }Q such that         (2.2) s Q mQ  ≤ c sbαq . p (w)   Q∈Dn  αq Bp (w)

It is understood that (M1) is void if N = −1. No vanishing moment condition is assumed for mQ when (Q) = 1. In the case of Daubechies’ wavelets, one can obtain norm equivalence, rather than the one-sided inequality (2.2), as follows. n Let Φ be the scaling function and let {ψ (i) }2i=1−1 be the mother wavelets for Daubechies’ DL wavelets on Rn with compact support (see [6]), where L is taken large enough that Φ, ψ i ∈ C K for K = [α] + 1 and each ψ i has N (as defined in Theorem 2.1) vanishing moments. Recall the definitions (1.9) of ΦI and (1.10) of (i) (i) (i) ψI . Note that each ψI satisfies supp ψI ⊆ (2r + 1)I (recall that supp ψ (i) , Φ ⊆ (i) B(0, r)) and ψI ∈ C K , with (2.3)

|Dγ ψI (x )| ≤ C|I|−1/2−|γ|/n , (i)

TRACES OF WEIGHTED SPACES

223

for all |γ| ≤ K and all x ∈ Rn . Hence, for some constant C > 0, the functions (i) ΦI /C and ψI /C satisfy (M1)-(M3). Theorem 2.2. Suppose α ∈ R, 0 < p < ∞, 0 < q ≤ ∞ and let w be a doubling weight on Rn with doubling exponent β. Suppose f ∈ Bpαq (w). Then f=



f, ΦI ΦI +

n 2 −1



(i)

(i)

f, ψI ψI ,

i=1 I∈D n ,(I) 0, define gj,λ ∗ (x) = sup gj,λ

y∈Rn

|g(x − y)| . (1 + 2j |y|)λ

If g ∈ C 1 (Rn ), define (∇g)∗ν,λ (x) = sup

y∈Rn

|∇g(x − y)| . (1 + 2j |y|)λ

We begin with a standard lemma about functions of exponential type. Lemma 2.3. Suppose g ∈ S  (Rn ) and supp gˆ ⊆ B(0, 2j+1 ) for some ν ∈ Z. Then g ∈ C ∞ (Rn ) and there exists C > 0 depending only on n and λ such that ∗ (x), (∇g)∗j,λ (x) ≤ C2j gj,λ

for all x ∈ Rn . Proof. [14], p. 21). Then γ&j = 1 with respect obtain

The Paley-Wiener theorem for S  gives that g ∈ C ∞ (Rn ) (see, e.g., Select γ ∈ S(Rn ) such that γˆ = 1 on B(0, 2). Let γj (x) = 2jn γ(2j x). on B(0, 2j+1 ). Hence, g = g ∗ γj . Let ∂ denote the partial derivative to x . Since ∂ (γj ) = 2j (∂ γ)j , where (∂ γ)j (x) = 2jn (∂ γ)(2j x), we 

(∂ g)(x − y) = 2j Hence,



Rn

(∂ γ)j (x − y − z)g(z) dz = 2j

Rn

(∂ γ)j (t − y)g(x − t) dt.



|g(x − t)| |(∂ γ)j (t − y)|(1 + 2j |t − y|)λ (1 + 2j |y|)λ dt (1 + 2j |t|)λ Rn  ∗ (x) |(∂ γ)j (t − y)|(1 + 2j |t − y|)λ dt ≤ 2j (1 + 2j |y|)λ gj,λ Rn  ∗ = 2j (1 + 2j |y|)λ gν,λ (x) |(∂ γ)(s)|(1 + |s|)λ ds

|(∂ g)(x − y)| ≤ 2j

Rn j

∗ (x). = C,γ,λ 2 (1 + 2 |y|)λ gj,λ j

Dividing by (1 + 2j |y|)λ and taking the supremum over all y ∈ Rn yields the result.  The next lemma is the weighted generalization of the Peetre maximal function characterization of the Triebel-Lizorkin spaces (see [19]). The proof for the weighed case requires only a few modifications (as noted for a similar situation in [9, p. 165]) of Peetre’s original proof. Lemma 2.4. Let α ∈ R, 0 < p < ∞, 0 < q ≤ ∞, and let w be a doubling   weight on Rn . Let β be the doubling exponent of w. Choose λ > β max p1 , 1q . Let Φ, ϕ ∈ S(Rn ) be as in the definition of Fpα,q (w) above, and let ϕ0 = Φ, and ϕj (x) = 2jn ϕ(2j x). Then there exists C > 0 depending only on α, q, p, λ, and w such that for f ∈ S  (Rn ), ⎛ ⎞1/q   ∞      q   jα ∗   2 (ϕj ∗ f )j,λ ⎠  ≤ Cf Fpα,q (w) . f Fpα,q (w) ≤ ⎝    j=0  p L (w)

TRACES OF WEIGHTED SPACES

225

    Proof. The first inequality is trivial because |ϕj ∗ f | ≤ (ϕj ∗ f )∗j,λ . To prove the second, let r = β/λ. We claim that there exists C > 0 depending only on β, λ, and n such that for all g ∈ S  (Rn ) with supp gˆ ⊆ B(0, 2j+1 ), ∗ gj,λ ≤ C (Mw (|g|r ))

1/r

(2.7)

,

where Mw is the maximal function with respect to w, defined in (2.5). By a standard regularization (see [23], p. 22, or [9], p. 149), we can assume g ∈ S, so ∗ < ∞ at every point. Let x, y ∈ Rn . We will choose δ ∈ (0, 1) with δ that gj,λ independent of j and g. For each z ∈ B(x − y, 2−j δ), the mean-value theorem gives |g(x − y)| ≤ |g(z)| + 2−j δ

sup t∈B(x−y,2−j δ)

|∇g(t)|.

Taking the r th power and averaging over B(x − y, 2−j δ) with respect to the weight w gives |g(x − y)|r ≤ cr (I + II), where  1 |g(z)|r w(z) dz I= w(B(x − y, 2−j δ)) B(x−y,2−j δ) and sup |∇g(t)|r , II = 2−jr δ r t∈B(x−y,2−j δ)

where cr = max(1, 2 ). We enlarge the region of integration in I using B(x − −j δ ≤ 2k . Since y, 2−j δ) ⊆ B(x, |y| + 2−j δ). Let k ∈ Z be such that 2k−2 < |y|+2 2−j δ the doubling exponent is defined with cubes, we denote by Q(z, s) the cube with center z and side √ B(z, s) ⊆ √ length 2s, with sides parallel to the axes. Observe that Q(z, s) ⊆ B(z, ns). Let cn be the smallest integer greater than log2 n. Then √ B(x, |y|+2−j δ) ⊆ Q(x, |y|+2−j δ) ⊆ Q(x−y, 2k−j+1 δ) ⊆ Q(x−y, 2k+cn −j+1 δ/ n). r−1

Since w is doubling with doubling exponent 2β , we have

√ w(B(x, |y| + 2−j δ)) ≤ w(Q(x − y, 2k+cn −j+1 δ/ n)) √ ≤ 2β(k+cn +1) w(Q(x − y, 2−j δ/ n)) ≤ Cβ,n 2βk w(B(x − y, 2−j δ))

and

: kβ

2

≤4

β

|y| + 2−j δ 2−j δ

;β

≤ 4β δ −β (1 + 2j |y|)β ,

since δ < 1. Hence,  δ −β (1 + 2j |y|)β I ≤ cβ,n

1 w(B(x, |y| + 2−j δ))

 B(x,|y|+2−j δ)

|g(z)|r w(z) dz

 ≤ cβ,n δ −β (1 + 2j |y|)β Mw (|g|r )(x).

To estimate II, we again use B(x − y, 2−j δ) ⊆ B(x, |y| + 2−j δ) and write sup t∈B(x−y,2−j δ)

|∇g(t)| ≤

≤

sup t∈B(x,|y|+2−j δ)

sup |z|≤|y|+2−j δ

|∇g(t)| =

(1 + 2j (|y| + 2−j δ))λ

sup |z|≤|y|+2−j δ

|∇g(x − z)|

|∇g(x − z)| (1 + 2j |z|)λ

∗ ≤ 2 (1 + 2j |y|)λ (∇g)∗j,λ (x) ≤ Cn,λ 2j (1 + 2j |y|)λ gj,λ (x), λ

226

MICHAEL FRAZIER AND SVETLANA ROUDENKO

using the assumption that δ < 1 and Lemma 2.3. Hence,  ∗ r r II ≤ δ r Cn,λ (1 + 2j |y|)rλ gj,λ (x) . Combining these estimates, taking the r th root, and recalling that λ = β/r gives ∗ (x). |g(x − y)| ≤ cr 4λ δ −λ (1 + 2j |y|)λ (Mw (|g|r ))1/r (x) + cr δCn,λ (1 + 2j |y|)λ gj,λ

Dividing by (1 + 2j |y|)λ and taking the supremum over y ∈ Rn shows that ∗ (x) ≤ cr 4λ δ −λ (Mw (|g|r )) gj,λ

1/r

∗ (x) + cr δCn,λ gj,λ (x).

∗ Since we assumed g ∈ S, we have gj,λ (x) < ∞. We can choose δ < 1 such that  cr δCn,λ ≤ 1/2, where δ does not depend on j or g. Then subtracting the term on the far right yields (2.7). We apply (2.7) to each ϕj ∗ f , which satisfy supp (ϕj ∗ f ) ⊆ B(0, 2j+1 ) by our conditions on ϕj . We obtain ⎛ ⎡ ⎞1/q  ⎞1/q   ∞    ∞       0 0 ⎝ /   ⎣ jα  ∗  q⎠ jαr r q/r ⎠ 2 (ϕj ∗ f )j,λ Mw (2 |ϕj ∗ f | ) ≤C         j=0  p  p  j=0 L (w)

L (w)

⎛ ⎞r/q  1/r  ∞   / 0q/r  ⎠  Mw (2jαr |ϕj ∗ f |r ) = C ⎝     p/r  j=0 L

(w)

⎛ ⎞r/q   ∞ 1/r  /  0 q/r  ⎠  2jαr |ϕj ∗ f |r ≤ C ⎝     j=0  p/r (w) L ⎛ ⎞1/q   ∞   /  0q   2jα |ϕj ∗ f | ⎠  = C ⎝ = Cf Fpα,q (w) ,   j=0   p L (w)   by (3.1) with indices p/r, q/r > 1, since r = β/λ and λ > β max p1 , 1q .



The proof of the smooth atomic decomposition of Fpα,q (w) follows the same lines as in the unweighted case. Theorem 2.5. Let α ∈ R, 0 < p < ∞, 0 ≤ q ≤ ∞, let w be a doubling weight on Rn , and let N be a given non-negative integer. Suppose f ∈ Fpα,q (w). Then there exists a sequence s = {sQ }Q∈Dn ∈ fpα,q (w) and C ∞ functions {aQ }Q∈Dn , and constants cγ , satisfying (A1) supp aQ ⊆ 3Q, (A2)  |Dγ aQ (x)| ≤ cγ |Q|−1/2−|γ|/n for each multi-index γ, (A3)

xγ aQ (x) dx = 0, for |γ| ≤ N and (Q) < 1,

such that (2.8)

f=

 Q∈Dn

sQ aQ ,

TRACES OF WEIGHTED SPACES

227

where the sum converges in norm in q = ∞ and in S  if q = ∞, with sfpα,q (w) ≤ Cf Fpα,q (w) ,

(2.9) with C independent of f .

Proof. As in [7, p. 783], we can find real, radial functions Φ, Θ, ϕ, θ ∈ S(Rn ) α,q such that Φ, ϕ satisfy all of  theγconditions in the definition of Fp (w), supp Θ ⊆ B(0, 1), supp θ ⊆ B(0, 1), Rn x θ(x) dx = 0 for all |γ| ≤ N , and ˆ Θ(ξ) ˆ Φ(ξ) +

∞ 

ˆ −j ξ) = 1, ϕ(2 ˆ −j ξ)θ(2

j=1

for all ξ ∈ R . Let ϕ0 = Φ, θ0 = Θ, and, for j ≥ 1, ϕj  (x) = 2jn ϕ(2j x) and jn j θj (x) = 2 θ(2 x). Then we have the Calder´ on formula f = ∞ j=0 ϕj ∗ θj ∗ f , with convergence in S  . If Q ∈ Dn and (Q) = 2−j , we let n

sQ = |Q|1/2 sup |ϕj ∗ f (y)|. y∈Q

If sQ = 0, we set aQ = 0, whereas if sQ = 0, we set  1 θj (x − y)ϕj ∗ f (y) dy. aQ (x) = sQ Q  on Then Q∈Dn :(Q)=2−j sQ aQ = ϕj ∗ θj ∗ f , so we obtain (2.8) from the Calder´ formula. Then the support condition of Θ and θ yield (A1). Since Θ, θ ∈ S, elementary estimates show that (A2) holds. The moment condition on θ implies (A3). Finally,  (Q)−α−n/2 |sQ |χQ (x) ≤ C2jα sup √ |ϕj ∗ f (x − y)| |y|≤2−j

Q∈Dn :(Q)=2−j

n

≤ C2jα (ϕj ∗ f )∗j,λ (x). Because the dyadic cubes of fixed side length are a.e. disjoint, the exponent q can be interchanged with the sum over Q ∈ Dn : (Q) = 2−j . Hence taking the q th power, summing on j, taking the 1/q power, then the Lp (w) norm and applying Lemma 2.4, we obtain (2.9).  Functions satisfying (A1), (A2), and (A3) are called smooth atoms for Q. The proof shows that we could have worked with sup √ |ϕj ∗ f (x − y)|

|y|≤2−j

n

f )∗j,λ (x),

but the proof in Lemma 2.4 only simplifies a little, and instead of (ϕj ∗ the stronger statement is often useful. Next we need the analogue of Theorem 2.1 for the Triebel-Lizorkin spaces. The following lemma is the weighted analogue of [8, Lemma 2.4]. Lemma 2.6. Let w be a doubling weight on Rn with doubling exponent β. Suppose 0 < A ≤ 1, λ > β/A and m ∈ Z with m ≥ 0. For any sequence {sQ }Q∈Dn , ⎛ ⎛ ⎞⎞1/A   |sQ | ≤ C2mβ/A ⎝Mw ⎝ |sQ |A χQ ⎠⎠ (x), k−m |x − x |)λ (1 + 2 Q −k −k (Q)=2

where C depends only on n, β, A, and λ.

(Q)=2

228

MICHAEL FRAZIER AND SVETLANA ROUDENKO

Proof. Let A0 = {Q ∈ Dn : (Q) = 2−k , Q ⊆ B(x, 2m−k )}, and for i ∈ N, let Ai = {Q ∈ Dn : (Q) = 2−k , Q ⊆ B(x, 2i+m−k ), Q ⊆ B(x, 2i+m−k−1 )}. Then  Q∈Ai

|sQ | ≤ C2−iλ (1 + 2k−m |x − xQ |)λ ⎛ = C2−iλ ⎝



Q∈Ai

1 w(Q)



⎛ |sQ | ≤ C2−iλ ⎝

Q∈Ai



⎞1/A |sQ |A ⎠

Q∈Ai

⎞1/A



|sQ |A χQ w dx⎠

.

Since Q ⊆ B(x, 2i+m−k ) and (Q) = 2−k , it follows that B(x, 2i+m−k ) ⊆ 2m+i+2 Q, and hence w(B(x, 2i+m+k ) ≤ 2β(m+i+2) w(Q). Thus, the last displayed expression is bounded by ⎛ ⎞1/A   1 |sQ |A χQ w dx⎠ C2−iλ 2β(m+i+2)/A ⎝ w(B(x, 2i+m−k )) Q∈Ai

⎛

⎛

≤ C2−iλ 2β(m+i+2)/A ⎝Mw ⎝ ⎛ −i(λ−β/A) mβ/A

≤ C2

2

⎛

⎝Mw ⎝



⎞

⎞1/A

|sQ |A χQ ⎠ (x)⎠

Q∈Ai



⎞

⎞1/A

|sQ | χQ ⎠ (x)⎠ A

.

(Q)=2−k



The result follows by summing on i.

Theorem 2.7. Suppose α ∈ R, 0 < p < ∞, 0 < q ≤ ∞ and let w be a doubling weight on Rn with doubling exponent β. Let N > −n − 1 − α + β/(min(1, p, q)) and λ > β/ min(1, p, q). Suppose {mQ }Q∈Dn is a family of functions satisfying  (Mo1) xγ mQ (x) dx = 0, for |γ| ≤ N and (Q) < 1,

− max(λ,N +n+1) |x − xQ | (Mo2) |mQ (x)| ≤ |Q|−1/2 1 + , l(Q) and

−λ |x − xQ | γ (Mo3) |D mQ (x)| ≤ |Q|−1/2−|γ|/n 1 + if |γ| ≤ [α] + 1, l(Q)  for each Q ∈ Dn . Suppose s = {sQ }Q∈Dn ∈ fpαq (w). Then Q∈Dn sQ mQ ∈ αq Fp (w), and there exists c independent of s and {mQ }Q such that          (2.10) s m ≤ c sfpαq (w) . Q Q  Q∈Dn  αq Fp (w)

Proof. Select  A ∈ (0, 1] such that β/(min(1, p, q)) < β/A < min(λ, N + n + 1 + α). Let f = Q∈Dn sQ mQ . Let ϕj be as in the definition of Fpα,q (w) for j ≥ 0. Let K = max([α] + 1, 0). By [9, Lemmas B.1-2], if Q ∈ Dn with (Q) = 2−k , then |ϕj ∗ mQ (x)| ≤ C|Q|−1/2 2−(j−k)K (1 + 2k |x − xQ |)−λ for k ≤ j,

TRACES OF WEIGHTED SPACES

229

and |ϕj ∗ mQ (x)| ≤ C|Q|−1/2 2−(k−j)(N +1+n) (1 + 2j |x − xQ |)−λ for j ≤ k, where C depends on the fixed parameters but not j, k, Q or x. Let ⎛ ⎞1/A ⎛ ⎞A  α 1 ⎜ ⎟ Bk (x) = ⎝Mw ⎝ |Q|− n − 2 |sQ |χQ ⎠ (x)⎠ . (Q)=2−k

Then by the estimates for |ϕj ∗ mQ (x)| and Lemma 2.6 (with m = 0 for the first sum and m = k − j for the second) 2jα |ϕj ∗

 Q∈Dn

s Q mQ | ≤ C

j 

2−(j−k)σ Bk (x) + C

k=0

∞ 

2−(k−j)τ Bk ,

k=j+1

where σ = K − α > 0 and τ = N + 1 + n + α − β/A > 0. Hence, by Young’s inequality a ∗ bq ≤ a1 bq if q > 1, or by q → 1 and a ∗ b1 ≤ a1 b1 if q ≤ 1, we obtain ⎛ ⎛ ⎞1/q  ⎞1/q   ∞  ∞              f Fpαq (w) = ⎝ (2jα |ϕj ∗ sQ mQ |)q ⎠  ≤ C ⎝ Bjq ⎠      Q∈Dn  j=0  j=0  p  p L (w)

L (w)

⎛ ⎞ 1/A  ⎛ ⎛ ⎞A ⎞q/A A/q    ∞  ⎜  ⎟ α 1  ⎜ ⎜ ⎟ = C ⎝ |Q|− n − 2 |sQ |χQ ⎠ ⎠ ⎟ ⎝Mw ⎝ ⎠      j=0 (Q)=2−j  p/A  L

. (w)

Applying (2.6) to remove Mw (since p/A, q/A > 1) and using the a.e. disjointness of the dyadic cubes of a fixed side length, gives the result.  3. Traces and Extensions The following lemma is a weighted analogue of [9, Proposition 2.7]. It will be used to see that the trace of Fpα,q (w) is independent of the parameter q, just as in the unweighted case in [9, §11]. Lemma 3.1. Suppose that for each Q ∈ Dn , EQ is a measurable subset of Q such that EQ contains at least one of the 2n dyadic subcubes of Q of length (Q)/2. Then there exists C = C(w, n, α, q, p) > 0 such that ⎛ ⎞1/q        q   (Q)−α−n/2 |sQ |χEQ ⎠  sfpα,q (w) ≤ C ⎝ ≤ Csfpα,q (w) .    Q∈Dn  p L (w)

Proof. The second inequality is trivial because χEQ ≤ χQ . For each Q ∈ Dn , let S(Q) be a dyadic subcube of Q with (S(Q)) = (Q)/2 such that S(Q) ⊆ EQ . Then Q ⊆ 4S(Q). Hence, if C˜ is the doubling constant of w, then w(Q) ≤ w(4S(Q)) ≤ C˜ 2 w(S(Q)) ≤ C˜ 2 w(EQ ). Hence, for x ∈ Q,  1 1 w(EQ ) (3.1) Mw χEQ (x) ≥ ≥ χE w dx = . ˜ w(Q) Q Q w(Q) C2

230

MICHAEL FRAZIER AND SVETLANA ROUDENKO

Therefore, for any A > 0,  1/A (x) . χQ (x) ≤ C˜ 2/A Mw χA EQ Select A such that p/A > 1 and q/A > 1. Then ⎛ ⎞A/q   1/A  

  A q/A  ⎠  sfpα,q (w) ≤ C˜ 2/A ⎝ Mw (Q)−α−n/2 |sQ |χEQ     Q∈Dn  p/A ⎛ ⎞1/q        q ⎝  2/A −α−n/2 ˜ (Q) ≤ C Cp/A,q/A  |sQ |χEQ ⎠     Q∈Dn 

L

(w)

,

Lp (w)



by (2.6).

Proof of Theorem 1.3 (i) First suppose that (1.7) holds for all I ∈ Dn . Let f ∈ Fpα,q (u). By Theorem 2.5, there exists a family of smooth atoms  {aQ }Q∈Dn+1 and a sequence of complex numbers s = {sQ }Q∈Dn+1 such that f = Q∈Dn+1 sQ aQ and sfpα,q (u) ≤ Cf Fpα,q (u) . + consist of those dyadic cubes in Rn+1 which are of the form Q = Let Dn+1 + Q(I) = I × [0, (I)] for some I ∈ Dn ; cubes in Dn+1 sit on top of the hyperplane −  n {x = (x , xn+1 ) : xn+1 = 0} = R . Let Dn+1 consist of those dyadic cubes in − Rn+1 which are of the form I × [−(I), 0] for some I ∈ Dn ; cubes in Dn+1 sit just + − n ∗ ∗ underneath R . Let Dn+1 = Dn+1 \ (Dn+1 ∪ Dn+1 ); cubes in Dn+1 do not intersect Rn . Then    f= sQ aQ + sQ aQ + sQ aQ = f1 + f2 + f3 . − Q∈Dn+1

+ Q∈Dn+1

∗ Q∈Dn+1

∗ , then aQ (x  , 0) = 0 for all x  , since aQ is supported in 3Q. Therefore, If Q ∈ Dn+1 T r(f3 ) = 0. We consider f1 . Define a sequence s+ = {s+ truncating s: let s+ Q }Q∈Dn+1 by Q =  + + + sQ if Q ∈ Dn+1 and let sQ = 0, otherwise. Hence f1 = Q∈Dn+1 sQ aQ . By the + monotinicity of the norm, s+ fpα,q (u) ≤ Csfpα,q (u) . For each Q = Q(I) ∈ Dn+1 , let T (Q) = I × [(I)/2, (I)] be the top half of Q. Using Lemma 3.1, we have ⎛ ⎞1/q       q   ⎜ ⎟  + α,q −α−(n+1)/2  s fp (u) ≤ C ⎝ (Q) |sQ |χT (Q) ⎠  .    Q∈D+   n+1 p L (u)

+ Dn+1

overlap at most on their boundaries. Then, for almost The sets T (Q) for Q ∈ every x, the sum in the last expression has only one non-zero term. Therefore, the exponents q and 1/q cancel, and the norms s+ fpα,q (u) are equivalent for any two choices of q. In particular, we can take q = p to obtain (3.2)

s+ fpα,q (u) ≈ s+ fpα,p (u) = s+ bα,p , p (u)

using (2.1). Thus, the problem is essentially reduced to the weighted version of the Besov case (1.2), which is covered in [11], as follows (reduced to the scalar

TRACES OF WEIGHTED SPACES

231

+ case). We can index the cubes Q = Q(I) in Dn+1 by I ∈ Dn . For I ∈ Dn , define −1/2  1/2  tI = (I) sQ(I) and let bI (x ) = (I) aQ(I) (x , 0). Then   sQ aQ (x  , 0) = tI bI (x  ). T rf1 (x  ) = f1 (x  , 0) = Q∈Dn+1

I∈Dn

Note that bI is supported in 3I, since aQ is supported in 3Q. Moreover, |Dγ bI (x  )| = |Dxγ  aQ(I) (x  , 0)| ≤ cγ (I)−|γ|−(n+1)/2 (I)1/2 = cγ (I)−|γ|−n/2 , for all |γ| ≤ [α] + 1. Hence, for some constant C, we have that {bI /C}I∈Dn is a α−1/p,p sequence of smooth atoms for Bp (v), because the vanishing moment  condition (M1) in Theorem 2.1 is void because of the condition α > p1 + n p1 − 1 + β−n p . +

Hence, by Theorem 2.1, T rf1 p α−1/p,p Bp

(v)

≤ Ctpα−1/p bp

(v)

=C

p   (I)−(α−1/p)−n/2 v(I)1/p |tI | . I∈Dn

−1

Applying our condition v(I) ≤ C(I) u(Q(I)) and the definition of TI , and in+ dexing over Dn+1 again, we have p   (Q)−α−(n+1)/2 u(Q)1/p |sQ | T rf1 p α−1/p,p ≤ C Bp

(v)

+ Q∈Dn+1

= Cs+ pbα,p ≤ Cs+ pfpα,q (u) ≤ Cspfpα,q (u) ≤ Cf pFpα,q (u) , p (u) using (3.2). The estimate for T rf2 is virtually the same, once we note that if Q− (I) = I × [−(Q), 0], then we have v(I) ≤ C(I)−1 u(Q− (I)) for all I ∈ Dn . This fact follows from (1.7) and the doubling condition on u, because u(Q(I)) ≤ u(3Q− (I)) ≤ Cu(Q− (I)). Since T rf = T rf1 + T rf2 , we obtain T rf B α−1/p,p (v) ≤ Cf Fpα,q (u) . p

For the converse, fix I ∈ Dn with (I) < 1. Let f (x) = (I)α+n/2 Ext (ψI )(x) = (I)α+n/2 ψI (x  )h(xn+1 /(I)),  where h is chosen to satisfy tj h(t) dt = 0 for j = 0, 1, . . . , N , where N is as in (1) Theorem 2.7. Since h(0) = 1, we have T rf (x ) = (I)α+n/2 ψI (x  ). Hence, by Theorem 2.2, p  T rf p α−1/p,p ≥ C (I)−(α−1/p)−n/2 v(I)1/p (I)α+n/2 = (I)v(I), (1)

Bp

(1)

(v)

where C is independent of I. If we set aQ(I) (x) = (I)−1/2 ψI (x  )h(xn+1 /(I)), (1)

then up to a constant multiple which is independent of I, the function aQ(I) is a smooth atom for Q(I). Then f = (I)α+(n+1)/2 aQ(I) satisfies, by Theorem 2.7, f pFpα,q (u) ≤ C(I)−α−(n+1)/2 (I)α+(n+1)/2 χQ pLp (u) = Cu(Q). α−1/p,p

Hence, the boundedness of T r : Fpα,q (u) → Bp

(I)v(I) ≤ cu(Q(I)),

(v) implies that

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MICHAEL FRAZIER AND SVETLANA ROUDENKO

which is the required condition. If I ∈ Dn with (I) < 1, the same argument with ψ (1) replaced by Φ yields the needed estimate in this case. This completes the proof of (i). (ii) Suppose (1.12) holds, or, equivalently, u(Q(I)) ≤ C(I)v(I) for all I ∈ Dn , with C independent of I. Let 



f (x ) =



sI ΦI (x ) +

n 2 −1



sI ψI (x  ), (i)

(i)

i=1 I∈Dn ,(I) 0 such that if aQ(I) = CΦ(x  )h(xn+1 ) (1)

for (I) = 1, and aQ(I) = C(I)−1/2 ψI (x  )h(xn+1 /(I)) for (I) < 1, then each (i)

(i)

(i)

aQ(I) is a smooth atom for Q(I), where we choose h to have sufficiently many van(i)

ishing moments. For 2 ≤ i ≤ n and (I) = 1, we let aQ(I) = 0. By the definition of Ext and linearity, we have Ext f = C

n 2 −1



(i) (i)

(I)1/2 sI aQ(I) .

i=1 I∈Dn

For Q = Q(I), we (3.4)

(i) sQ

=

Ext f 

(i) sQ(I)

Fpα,q (u)

=

(i) sI .

≤C

By Theorem 2.7,

n 2 −1

(i)

{(I)1/2 sQ(I) }I∈Dn fpα,q (u) .

i=1

For each i, we apply Lemma 3.1 to obtain (i)

{(I)1/2 sQ(I) }I∈Dn fpα,q (u)

⎛ ⎞1/q         q ⎝  −α−(n+1)/2 1/2 (i) ⎠ (Q) = (Q) |sQ |χQ     Q=Q(I),I∈Dn  p L (u) ⎛  ⎞1/q      q    (i) (Q)−α−n/2 |sQ |χT (Q) ⎠  ≤ C ⎝ ,    Q=Q(I),I∈Dn  p L (u)

where T (Q) = I × [(I)/2, (I)], for Q = Q(I), as above. Since the sets {T (Q(I))}I∈Dn overlap only on a set of measure 0, the sum has only one nonzero term for almost every x. Therefore q and 1/q cancel, and we can evaluate the Lp (u) norm to obtain ⎞1/p ⎛  p  (i) (Q)−α−n/2 |sQ | u(T (Q(I)))⎠ . C⎝ Q=Q(I),I∈Dn

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233

We have u(T (Q(I))) ≤ u(Q(I)) ≤ C(I)v(I), by our assumption. Hence, the last expression is bounded by ! p 1/p   −α−n/2 1/p 1/p (i) C (I) v(I) |sI | = s(i) bα−1/p,p (v) . (I) p

I∈Dn

By (3.3) and (3.4), we obtain Ext f Fpα,q (u) ≤ Cf Bpα,p (v) . Using this estimate for finite sums of wavelets, we can now extend Ext by density to define a bounded operator from Bpα,p (v) to Fpα,q (u). (iii) From the definition, T r ◦ Ext is the identity operator on (finite) linear combinations of wavelets. Therefore, by parts (i) and (ii) of the theorem and density, T r ◦ Ext is the identity on Bpα,p (v). 

References [1] Kenneth F. Andersen and Russel T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980/81), no. 1, 19–31. MR604351 (82b:42015) [2] O. V. Besov, Investigation of a class of function spaces in connection with imbedding and extension theorems, Trudy. Mat. Inst. Steklov. 60 (1961), 42–81. MR0133675 (24 #A3501) [3] Marcin Bownik, Anisotropic Triebel-Lizorkin spaces with doubling measures, J. Geom. Anal. 17 (2007), no. 3, 387–424. MR2358763 (2009d:42042) [4] Marcin Bownik and Kwok-Pun Ho, Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1469–1510. MR2186983 (2006j:42027) [5] Ronald R. Coifman, A real variable characterization of H p , Studia Math. 51 (1974), 269–274. MR0358318 (50 #10784) [6] Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR951745 (90m:42039) [7] Michael Frazier and Bj¨ orn Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), no. 4, 777–799. MR808825 (87h:46083) [8] Michael Frazier and Bj¨ orn Jawerth, The φ-transform and applications to distribution spaces, Function spaces and applications (Lund, 1986), 1988, pp. 223–246. MR942271 (89g:46064) [9] Michael Frazier and Bj¨ orn Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170. MR1070037 (92a:46042) [10] Michael Frazier and Svetlana Roudenko, Matrix-weighted Besov spaces and conditions of Ap type for 0 < p ≤ 1, Indiana Univ. Math. J. 53 (2004), no. 5, 1225–1254. MR2104276 (2005f:42035) [11] Michael Frazier and Svetlana Roudenko, Traces and extensions of matrix-weighted Besov spaces, Bull. Lond. Math. Soc. 40 (2008), no. 2, 181–192. MR2414777 (2009d:46063) [12] Dorothee D. Haroske and Iwona Piotrowska, Atomic decompositions of function spaces with Muckenhoupt weights, and some relation to fractal analysis, Math. Nachr. 281 (2008), no. 10, 1476–1494. MR2454945 (2009i:46066) [13] Dorothee D. Haroske and Hans-J¨ urgen Schmeisser, On trace spaces of function spaces with a radial weight: the atomic approach, Complex Var. Elliptic Equ. 55 (2010), no. 8-10, 875–896. MR2674870 (2011k:46046) [14] Lars H¨ ormander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-G¨ ottingen-Heidelberg, 1963. MR0161012 (28 #4221) [15] Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR0312139 (47 #701) [16] B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), no. 3, 231–270. MR779906 (86k:42034)

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[17] Bj¨ orn Jawerth, The trace of Sobolev and Besov spaces if 0 < p < 1, Studia Math. 62 (1978), no. 1, 65–71. MR0482141 (58 #2227) [18] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR0293384 (45 #2461) [19] Jaak Peetre, On spaces of Triebel-Lizorkin type, Ark. Mat. 13 (1975), 123–130. MR0380394 (52 #1294) [20] Svetlana Roudenko, Matrix-weighted Besov spaces, Trans. Amer. Math. Soc. 355 (2003), no. 1, 273–314 (electronic). MR1928089 (2003k:42037) [21] Mitchell H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean nspace. I. Principal properties, J. Math. Mech. 13 (1964), 407–479. MR0163159 (29 #462) [22] Mitchell H. Taibleson and Guido Weiss, The molecular characterization of certain Hardy spaces, Representation theorems for Hardy spaces, 1980, pp. 67–149. MR604370 (83g:42012) [23] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkh¨ auser Verlag, Basel, 1983. MR781540 (86j:46026) Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996 E-mail address: [email protected] Department of Mathematics, George Washington University, Washington, DC 20052 E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13933

Compact embeddings of weighted smoothness spaces of Morrey type: An example Dorothee D. Haroske and Leszek Skrzypczak Dedicated to the memory of Bj¨ orn Jawerth. Abstract. In this short note we consider some embedding of the weighted Besov space Bps11 ,q1 (Rd , |x|α ) into a space of Besov-Morrey type Nps22,u2 ,q2 (Rd ) and study necessary and sufficient conditions for its continuity and compactness. This strengthens and continues our earlier results. In addition, we obtain a parallel outcome dealing with (weighted) spaces of Triebel-Lizorkin(-Morrey) type. The main trick is to apply appropriate Franke-Jawerth embeddings in the new settings, which also demonstrates the power and elegance of that approach.

Introduction We study embeddings between weighted function spaces of Besov and Triebels s Lizorkin type, Bp,q and Fp,q , and smoothness spaces related to Morrey spaces, in s s and Triebel-Lizorkin-Morrey spaces Ep,u,q . particular Besov-Morrey spaces Np,u,q While the first-named spaces are indispensable components in harmonic analysis meanwhile and thoroughly studied for long, the latter spaces attracted more attention in recent years only. The classical Morrey spaces Mp,u (Rd ), 0 < u ≤ p < ∞, were introduced by Morrey in [26] and are part of the wider class of MorreyCampanato spaces, cf. [27]. They refine the scale of Lp spaces, since Mp,p (Rd ) = Lp (Rd ). However, on the one hand the Morrey spaces with u < p consist of locally u-integrable functions, but on the other hand the spaces scale with d/p instead of d/u, that is, f (λ ·)|Mp,u (Rd ) = λ−d/p f |Mp,u (Rd ),

λ > 0.

This property is very useful for some partial differential equations. Built upon this basic family Mp,u (Rd ), different spaces of Besov-Triebel-Lizorkin type were defined in the last years. Kozono and Yamazaki [23] and Mazzucato [24] used them in the theory of Navier-Stokes equations. The construction is made in such a way that, as before, the spaces coincide with classical Besov and TriebelLizorkin spaces if u = p. In the recent past some properties of the spaces including their atomic and wavelet characterisations were described in the papers by Sawano 2010 Mathematics Subject Classification. Primary 46E35. The author was supported by the National Science Center, 2014/15/B/ST1/00164.

Poland,

Grant No.

c 2017 American Mathematical Society

235

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK

[32, 33, 35], Sawano and Tanaka [37, 38], Tang and Xu [45], Rosenthal [28] and Rosenthal and Triebel [29]. The most systematic and general approach to the spaces of this type can be found in the monograph [57] of Yuan, Sickel and Yang and in the recent survey papers by Sickel [41, 42]. In contrast to this approach Triebel followed the original Morrey-Campanato ideas to develop local and hybrid spaces in [51, 52]. He also considered their applications in heat equations and Navier-Stokes equations (see also [58]). We contributed to this field with some embedding results in [15–17] and studies concerning the local regularity of distributions belonging to s s and Ep,u,q , see [11, 12]. spaces Np,u,q The technique used in this paper has its origin in two crucial papers by Frazier and Jawerth, cf. [8, 9], where the wavelet method was very fruitfully applied to function spaces of Besov and Triebel-Lizorkin type. In particular, in the second paper the authors gave the characterisation of Triebel-Lizorkin spaces for p = ∞ that inspired the research of smoothness Morrey spaces, cf. [57] for further remarks on the subject. Another important ingredient of our technique can also be traced back to Jawerth’s work: In his seminal contribution [21] he proved the inequality that is called the Jawerth inequality nowadays. We hope that our paper makes it evident that Jawerth’s contributions had a great impact on the theory of function spaces and his ideas are still important today. In [15] we already considered the embedding idα : Bps11 ,q1 (Rd , |x|α ) → Nps22,u2 ,q2 (Rd ), where α ≥ 0, s1 ≥ s2 , 0 < p1 < ∞, 0 < u2 ≤ p2 < ∞, 0 < q1 , q2 ≤ ∞ and obtained a criterion for its continuity. As expected, this extended previous results in [13, 14] for those embeddings when the target space Nps22,u2 ,q2 is replaced by the slightly smaller space Bps22 ,q2 . However, nothing has been done so far concerning compactness of this embedding. Now we are able to show that idα is compact if, and only if,

 1 d d α 1 − s2 + > > d max − ,0 . s1 − p1 p2 p1 p2 p1 Moreover, due to our proof technique, based on wavelet decompositions, we restricted ourselves in [15] to the Besov situation only. But meanwhile, having adapted Franke-Jawerth embeddings available, see [17] and, in a more general setting, [54], we may easily obtain necessary and sufficient conditions for the TriebelLizorkin case, too. The paper is organised as follows. In Section 1 we collect basic facts about the function spaces, in Section 2 we recall some known embedding results needed later on, whereas our main outcome can be found in Section 3. 1. Function spaces First we fix some notation. By N we denote the set of natural numbers, by N0 the set N ∪ {0}, and by Zd the set of all lattice points in Rd having integer components. The positive part of a real function f is given by f+ (x) = max(f (x), 0), the integer part of a ∈ R by #a$ = max{k ∈ Z : k ≤ a}. For two positive real sequences {αk }k∈N and {βk }k∈N we mean by αk ∼ βk that there exist constants c1 , c2 > 0 such that c1 αk ≤ βk ≤ c2 αk for all k ∈ N; similarly for positive functions. Given two (quasi-) Banach spaces X and Y , we write X → Y if X ⊂ Y and the natural embedding of X in Y is continuous.

COMPACT EMBEDDINGS OF WEIGHTED SMOOTHNESS MORREY SPACES

237

All unimportant positive constants will be denoted by c, occasionally with subscripts. 1.1. Weighted Besov spaces. We assume that the reader is familiar with Muckenhoupt Aρ weights, 1 ≤ ρ < 2 ∞, and refer to [10, 44, 46] for the definition and properties. Let w ∈ A∞ = ρ>1 Aρ be a Muckenhoupt weight. Then the weighted Lebesgue space Lp (Rd , w), 0 < p < ∞, contains all measurable functions such that

 1/p d p |f (x)| w(x) dx f |Lp (R , w) = Rd

is finite. Note that for p = ∞ one obtains the classical (unweighted) Lebesgue space, L∞ (Rd , w) = L∞ (Rd ), w ∈ A∞ . We thus mainly restrict ourselves to p < ∞ in weighted cases. As usual, we use the abbreviation  w(x) dx, (1.1) w(Ω) = Ω

where Ω ⊂ R is some bounded, measurable set. d

Example 1.1. In [13, 14] we dealt with Muckenhoupt weights of type , |x|α , |x| < 1, wα,β (x) = |x|β , |x| ≥ 1, where α, β > −d. For simplicity we shall restrict ourselves in this paper to the situation when α = β, i.e., we merely consider the weight (1.2)

wα (x) = |x|α ,

x ∈ Rd ,

α > −d.

The Schwartz space S(Rd ) and its dual S  (Rd ) of all complex-valued tempered distributions have their usual meaning here. Let ϕ0 = ϕ ∈ S(Rd ) be such that . and ϕ(x) = 1 if |x| ≤ 1 , supp ϕ ⊂ y ∈ Rd : |y| < 2 and for each j ∈ N let ϕj (x) = ϕ(2−j x) − ϕ(2−j+1 x). Then {ϕj }∞ j=0 forms a  d smooth dyadic resolution of unity. Given any f ∈ S (R ), we denote by Ff and F −1 f its Fourier transform and its inverse Fourier transform, respectively. Let f ∈ S  (Rd ), then the Paley-Wiener-Schwartz theorem implies that F −1 (ϕj Ff ) is an entire analytic function on Rd . Definition 1.2. Let 0 < q ≤ ∞, 0 < p < ∞, s ∈ R and {ϕj }j a smooth dyadic resolution of unity. Assume w ∈ A∞ . s (Rd , w) is the set of all distributions f ∈ S  (Rd ) (i) The weighted Besov space Bp,q such that

  ∞  q 1/q   s f |Bp,q (Rd , w) = 2jsq F −1 (ϕj Ff )|Lp (Rd , w) j=0

is finite. In the limiting case q = ∞ the usual modification is required.

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK

s (ii) The weighted Triebel-Lizorkin space Fp,q (Rd , w) is the set of all distributions f ∈ S  (Rd ) such that    ∞  1/q      s d jsq −1 q d   f |Fp,q (R , w) =  2 |F (ϕj Ff )(·)| |Lp (R , w)   j=0 

is finite. In the limiting case q = ∞ the usual modification is required. s Convention. We adopt the nowadays usual custom to write Asp,q instead of Bp,q or s , when both scales of spaces are meant simultaneously in some context. Fp,q

Remark 1.3. The spaces Asp,q (Rd , w) are independent of the particular choice of the smooth dyadic resolution of unity {ϕj }j appearing in their definitions. They are quasi-Banach spaces (Banach spaces for p, q ≥ 1), and S(Rd ) → Asp,q (Rd , w) → S  (Rd ), where the first embedding is dense if q < ∞; cf. [3]. Moreover, for w ≡ 1 ∈ A∞ we obtain the usual (unweighted) spaces Asp,q (Rd ); we refer, in particular, to the series of monographs by Triebel [47–49] for a comprehensive treatment of the unweighted spaces. The above spaces with weights of type w ∈ A∞ have been studied systematically by Bui in [3–6]. Rychkov extended in [31] the above class of weights in order to incorporate locally regular weights Aloc p . Recent works are due to Roudenko [30], and Bownik [1, 2]. We rely on our approach in [13, 14]. 1.2. Morrey and Smoothness Morrey spaces. We start by recalling the definition of Morrey spaces. Let B(x, R) = {y ∈ Rd : |x − y| < R} stand for the ball centred at x ∈ Rd with radius R > 0. Definition 1.4. Let 0 < u ≤ p < ∞. The Morrey space Mp,u (Rd ) is the set d of all locally u-integrable functions f ∈ Lloc u (R ), such that !1/u  d d − sup R p u |f (y)|u dy < ∞. f |Mp,u (Rd ) = x∈Rd ,R>0

B(x,R)

Remark 1.5. The spaces Mp,u (R ) are quasi-Banach spaces (Banach spaces for u ≥ 1). They were introduced by Morrey in [26] and are part of the wider class of Morrey-Campanato spaces, cf. [27]. As a matter of fact, Mp,p (Rd ) = Lp (Rd ), 0 < p < ∞. To extend this relation we put M∞,∞ (Rd ) = L∞ (Rd ). One can easily see that for 0 < u2 ≤ u1 ≤ p < ∞ d

Lp (Rd ) = Mp,p (Rd ) → Mp,u1 (Rd ) → Mp,u2 (Rd ). In an analogous way one can define the spaces M∞,u (Rd ), 0 < u < ∞, but using the Lebesgue differentiation theorem one can easily prove that M∞,u (Rd ) = L∞ (Rd ). Now we define smoothness spaces of Besov and Triebel-Lizorkin type modelled on Mp,u (Rd ). Definition 1.6. Let 0 < u ≤ p < ∞ or p = u = ∞. Let 0 < q ≤ ∞, s ∈ R and {ϕj }j a smooth dyadic resolution of unity. s (i) The Besov-Morrey space Np,u,q (Rd ) is the set of all distributions f ∈ S  (Rd ) such that

  ∞   1/q   s d  jsq  −1 d q f |Np,u,q (R ) = 2 F (ϕj Ff )|Mp,u (R ) j=0

COMPACT EMBEDDINGS OF WEIGHTED SMOOTHNESS MORREY SPACES

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is finite. In the limiting case q = ∞ the usual modification is required. s (Rd ) is the set of all distributions (ii) The Triebel-Lizorkin-Morrey space Ep,u,q f ∈ S  (Rd ) such that  ∞  1/q      s d  jsq −1 q d   f |Ep,u,q (R ) =  2 |F (ϕj Ff )(·)| |Mp,u (R ) j=0

is finite. In the limiting case q = ∞ the usual modification is required. s s Convention. Again we adopt the custom to write Asp,u,q instead of Np,u,q or Ep,u,q , when there is no need to distinguish between the spaces.

Remark 1.7. The spaces Asp,u,q (Rd ) are independent of the particular choice of the smooth dyadic resolution of unity {ϕj }j appearing in their definitions. They are quasi-Banach spaces (Banach spaces for u, q ≥ 1), and S(Rd ) → Asp,u,q (Rd ) → S  (Rd ). Moreover, for u = p we re-obtain the usual Besov and Triebel-Lizorkin spaces, Asp,p,q (Rd ) = Asp,q (Rd ). It turned out that many of the results from the classical situation have their counterparts, e.g., (1.3)

d s d As+ε p,u,r (R ) → Ap,u,q (R )

if

ε > 0,

0 < r ≤ ∞,

and Asp,u,q1 (Rd ) → Asp,u,q2 (Rd ) if q1 ≤ q2 . Moreover, Sawano proved in [32] that for s ∈ R, 0 < u < p < ∞, (1.4)

s s s Np,u,min(u,q) (Rd ) → Ep,u,q (Rd ) → Np,u,∞ (Rd ),

where for the latter embedding r = ∞ cannot be improved – unlike in case of u = p, when (1.5)

s s s Bp,min(p,q) (Rd ) → Fp,q (Rd ) → Bp,max(p,q) (Rd ),

where again 0 < p < ∞, 0 < q ≤ ∞, s ∈ R. More precisely, s s Ep,u,q (Rd ) → Np,u,r (Rd ) if, and only if, r = ∞ or

u = p and r ≥ max(p, q).

Mazzucato has shown in [25, Prop. 4.1] that 0 (Rd ) = Mp,u (Rd ), Ep,u,2

1 < u ≤ p < ∞,

in particular, 0 0 Ep,p,2 (Rd ) = Lp (Rd ) = Fp,2 (Rd ),

1 < p < ∞.

1.3. Wavelet characterisation of the function spaces. Finally, we briefly describe the wavelet characterisation of Besov spaces with A∞ weights proved in [13], cf. also [19], and the wavelet characterisation of Besov-Morrey spaces proved in [32]. For m ∈ Zd and ν ∈ Z we define a d-dimensional dyadic cube Qν,m with sides parallel to the axes of coordinates,by  d : * mi mi + 1 Qν,m = , , m = (m1 , . . . , md ) ∈ Zd , ν ∈ Z . ν ν 2 2 i=1

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK (p)

For 0 < p < ∞, ν ∈ Z and m ∈ Zd we denote by χν,m the p-normalised characteristic function of the cube Qν,m , , νd νd 2p for x ∈ Qν,m , (p) χν,m (x) = 2 p χν,m (x) = 0 for x∈ / Qν,m , (p)

(p)

hence χν,m |Lp  = 1 and χν,m |Mp,u  = 1. Let φ be a scaling function on R with compact support and of sufficiently high regularity. Let ψ be an associated wavelet. Then the tensor-product ansatz yields a scaling function φ and associated wavelets ψ1 , . . . , ψ2d −1 , all defined now on Rd . We suppose φ ∈ C N1 (R) and supp φ ⊂ [−N2 , N2 ] for certain natural numbers N1 and N2 . This implies φ, ψi ∈ C N1 (Rd ) and

(1.6)

supp φ, supp ψi ⊂ [−N3 , N3 ]d ,

for i = 1, . . . , 2d − 1. We use the standard abbreviations φν,m (x) = 2νd/2 φ(2ν x − m) and

ψi,ν,m (x) = 2νd/2 ψi (2ν x − m).

To formulate the result we introduce some sequence spaces with weights. For 0 < p < ∞, 0 < q ≤ ∞, σ ∈ R, and w ∈ A∞ , let , λ = {λν,m }ν,m : λν,m ∈ C ,

bσp,q (w) :=

   d      λ |bσp,q (w) =  2ν(σ− p )  λν,m χ(p) ν,m |Lp (w) m∈Zd

and p (w) :=

, λ = {λm }m : λm

ν∈N0

8   |q  < ∞

8      ∈ C, λ|p (w) =  λm χ0,m |Lp (w) < ∞ . m∈Zd

We will write bσp,q and p if w ≡ 1. In view of the definition of Qν,m one can easily verify that   1/p     νσ    σ p    (1.7) |λν,m | w(Qν,m ) |q  λ|bp,q (w) ∼  2  ν∈N0

m∈Zd

and

1/p     λ|p (w) = |λm |p w(Q0,m ) , m∈Zd

using notation (1.1). For smooth weights and compactly supported wavelets it makes sense to consider the Fourier-wavelet coefficients of tempered distributions f ∈ S  (Rd ) with respect to such an orthonormal basis. In [13, Thm. 1.13] we proved the following result. We denote ρw := inf{ρ ≥ 1 : w ∈ Aρ },

w ∈ A∞ .

Proposition 1.8. Let 0 < p, q ≤ ∞ and s ∈ R. Let φ be a scaling function and let ψi , i = 1, . . . , 2d − 1, be the corresponding wavelets satisfying (1.6). We assume

COMPACT EMBEDDINGS OF WEIGHTED SMOOTHNESS MORREY SPACES

241

  that N1 > max s, d( ρpw − 1)+ − s . Then a distribution f ∈ S  (Rd ) belongs to s (Rd , w), if, and only if, Bp,q

    s  f |Bp,q (Rd , w) =  {f, φ0,m }m∈Zd |p (w) +

d 2 −1 

    {f, ψi,ν,m }ν∈N0 ,m∈Zd |bσp,q (w)

i=1 s Furthermore,  f |Bp,q (Rd , w) may be used as an is finite, where σ = s + s d equivalent (quasi-) norm in Bp,q (R , w). d 2.

We come to the counterpart of smoothness Morrey spaces and need the following sequence space. For 0 < u ≤ p < ∞, 0 < q ≤ ∞ and σ ∈ R, let , nσp,u,q :=

λ = {λν,m }ν,m : λν,m ∈ C ,  λ |nσp,u,q 

   d     =  2ν(σ− p )  λν,m χ(p) |M p,u  ν,m

ν∈N0

m∈Zd

8   |q  < ∞ .

Remark 1.9. We introduced in [15] an equivalent norm in the sequence spaces nσp,u,q that is sometimes more convenient to use. Let σ ∈ R, 0 < q ≤ ∞, 0 < u ≤ p < ∞ or u = p = ∞. Then for a sequence {λj,m }j,m , j ∈ N0 , m ∈ Zd ,     u1  d 1 1   |λj,m |u |q , λ|nσp,u,q  ∼ 2j(σ− p ) sup 2d(j−ν)( p − u ) ν : ν ≤ j, k ∈ Zd

m: Qj,m ⊂ Qν,k

with the usual modification if u = p = ∞. Remark 1.10. In particular, we would like to refer to the fundamental and pioneering papers of Frazier and Jawerth [8, 9] in this context. In [9] TriebelLizorkin spaces with p = ∞ have been studied in great detail and also sequence spaces similar to the above ones can already be found there. The following characterisation was proved in [28, Thm. 4.5, Cor. 4.17], see also [32]. Proposition 1.11. Let 0 < u ≤ p < ∞ or u = p = ∞, 0 < q ≤ ∞ and let s ∈ R. Let φ be a scaling function and let ψi , i = 1, . . . , 2d −> 1, be the corresponding ? wavelets satisfying (1.6). We assume that max (1 + #s$)+ , d( u1 − 1)+ − s ≤ N1 . s (Rd ), if, and only if, Then a distribution f ∈ S  (Rd ) belongs to Np,u,q f

s |Np,u,q (Rd )

d −1    2      =  {f, φ0,m }m∈Zd |p  +  {f, ψi,ν,m }ν∈N0 ,m∈Zd |nσp,u,q 

i=1 s Furthermore,  f |Np,u,q (Rd ) may be used as an is finite, where σ = s + s d equivalent (quasi-) norm in Np,u,q (R ). d 2.

Remark 1.12. It follows from Proposition 1.11 that the mapping   (1.8) T : f → {f, φ0,m }m∈Zd , {f, ψi,ν,m }ν∈N0 ,m∈Zd ,i=1,...,2d −1

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK

 2d −1 σ  s is an isomorphism of Np,u,q (Rd ) onto p ⊕ ⊕i=1 np,u,q , σ = s+ d2 , cf. [28, Thm. 4.5, s (Rd ) in the Cor. 4.17] and [32]. There is a counterpart for spaces of type Ep,u,q literature given above. Together this covers the characterisation of spaces Asp,q (Rd ) by Daubechies wavelets, cf. [50, p.15/16] and the references given there, see also [9]. Likewise Proposition 1.8 implies that the mapping T from (1.8) is an isomor d  s phism of Bp,q (Rd , w) onto p (w) ⊕ ⊕2i=1−1 bσp,q (w) . 2. Continuous embeddings We briefly collect what is known about unweighted embeddings in smoothness Morrey spaces on Rd with a special focus laid on embeddings between the different scales of Besov and Triebel-Lizorkin spaces. Limiting embeddings of the latter type are nowadays usually called Franke-Jawerth embeddings. 2.1. Embeddings in the unweighted setting on Rd . In [15,17] we studied necessary and sufficient conditions for the boundedness of the embeddings of type id : Asp11 ,u1 ,q1 (Rd ) → Asp22 ,u2 ,q2 (Rd ), where this time it should be understood in such a way that both spaces are either of Besov-Morrey, or of Triebel-Lizorkin-Morrey type. Theorem 2.1. Let si ∈ R, 0 < qi ≤ ∞, 0 < ui ≤ pi < ∞, i = 1, 2. (i) There is a continuous embedding Nps11,u1 ,q1 (Rd ) → Nps22,u2 ,q2 (Rd )

(2.1) if, and only if,

p1 ≤ p2

(2.2) and (2.3)

u2 u1 ≤ p2 p1

and

⎧ ⎨s1 −

d p1

> s2 −

d p2 ,

⎩s1 −

d p1

= s2 −

d p2

or q1 ≤ q2 .

and

(ii) There is a continuous embedding (2.4)

Eps11,u1 ,q1 (Rd ) → Eps22,u2 ,q2 (Rd )

if, and only if, (2.2) holds and ⎧ s1 − pd1 > s2 − pd2 , ⎪ ⎪ ⎨ (2.5) s1 − pd1 = s2 − pd2 ⎪ ⎪ ⎩ s1 = s 2 , p1 = p2

or and

p1 = p2 ,

and

q1 ≤ q2 .

or

The embeddings (2.1) and (2.4) are never compact. Remark 2.2. The results can be found in [15] and [17]. In case of ui = pi , i = 1, 2, this recovers the well-known results for spaces of type Asp,q , including the phenomenon of the different influence of the fine parameters qi , i = 1, 2, in the limiting situations as can be observed above. A partial forerunner in case of homogeneous spaces can be found in [36, Thm. 2.2].

COMPACT EMBEDDINGS OF WEIGHTED SMOOTHNESS MORREY SPACES

243

Remark 2.3. Let us explicitly point out the following consequence of Theorem 2.1: Let s ∈ R, 0 < q ≤ ∞, 0 < u < p < ∞. Then Asp,u,q  Aσr,

for any σ ∈ R, 0 < r < ∞ and any 0 <  ≤ ∞ .

Moreover, this immediately leads to the fact that spaces of type Asp,u,q (Rd ) with 0 < u < p < ∞, 0 < q ≤ ∞, s ∈ R are never embedded in any space Lr (Rd ) for 1 ≤ r < ∞. However, when it comes to r = ∞, that is, when the target space Lr (Rd ) is replaced by L∞ (Rd ) or C(Rd ), the situation changes completely. If 0 < u ≤ p < ∞, 0 < q ≤ ∞, and s ∈ R, then , 0 < q ≤ ∞, if s > dp , s d d Np,u,q (R ) → C(R ) if, and only if, 0 < q ≤ 1, if s = dp , s that is, if and only if, Bp,q (Rd ) → C(Rd ), cf. [16]. If 0 < u < p < ∞, 0 < q ≤ ∞, s ∈ R, then d s Ep,u,q (Rd ) → C(Rd ) if, and only if, s> , p

cf. [17]. Note that this is different from the situation when u = p in view of the limiting case s = dp . A partial forerunner can be found in [34, Prop. 1.11] dealing with the sufficiency part, see also [42]. In [36, Cor. 3.2] there is a parallel observation for homogeneous spaces. We also refer to the recent paper [56]; it is d mainly devoted to spaces of type As,τ p,q (R ) and their embeddings within the scale, d d or to C(R ) and bmo(R ). 2.2. Embeddings of Franke-Jawerth type. We come to continuous embeddings between spaces of the (in general) different scales of Besov and TriebelLizorkin type, also for later use. We begin with the classical background. In contrast to (1.5) where we fixed the smoothness s ∈ R we deal with embeddings along constant differential dimension s − dp = const. The following is well-known. Let 0 < p1 < p < p2 ≤ ∞, s, s1 , s2 ∈ R, with (2.6)

s1 −

d d d = s− = s2 − , p1 p p2

and 0 < q, r1 , r2 ≤ ∞. Then (2.7)

s Bps11 ,r1 (Rd ) → Fp,q (Rd ) → Bps22 ,r2 (Rd )

if, and only if, (2.8)

0 < r1 ≤ p ≤ r2 ≤ ∞.

In particular, (2.9)

s Bps11 ,p (Rd ) → Fp,q (Rd ) → Bps22 ,p (Rd ).

For a proof we refer to [43, Sect. 5.2]. Note that the “if”-part of the right-hand embedding of (2.7) is due to Jawerth [21], whereas the “if”-part of the left-hand embedding of (2.7) was proved by Franke [7]. The embedding (2.9) is nowadays usually called Franke-Jawerth embedding. For a particularly elegant proof of (2.9) we refer to [53] by Vyb´ıral. In [41, 42] Sickel posed the (so far open) question what happens with embeddings of Franke-Jawerth type in the setting of smoothness

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK

Morrey spaces which we could answer almost completely in case of spaces Asp,u,q (Rd ) in [17]. For convenience we state the results for the two embeddings in (2.10)

s Nps11,u1 ,q1 (Rd ) → Ep,u,q (Rd ) → Nps22,u2 ,q2 (Rd )

separately and begin with the latter one. Theorem 2.4. Let si ∈ R, 0 < qi ≤ ∞, 0 < ui ≤ pi < ∞, i = 1, 2. Assume that d d = s2 − , p1 p2 (i) If u1 = p1 and u2 = p2 , then s1 −

(2.11)

and

s1 > s2 .

Eps11,u1 ,q1 (Rd ) → Nps22,u2 ,q2 (Rd )

(2.12) if, and only if,

q2 ≥ p1 .

(ii) If u1 < p1

and

u2 p2

≤

u1 p1 ,

then (2.12)

(iii) If u1 < p1

and

u2 p2

>

u1 p1 ,

then Eps11,u1 ,q1 (Rd ) → Nps22,u2 ,q2 (Rd ).

holds

if, and only if,

q2 = ∞.

Remark 2.5. We refer to [39, Prop. 2.3] for a special case related to the situation when u2 = p2 = q2 = ∞. As for the second embedding in (2.10) we obtained in [17] the following. Theorem 2.6. Let si ∈ R, 0 < qi ≤ ∞, 0 < ui ≤ pi < ∞, i = 1, 2, and assume that (2.11) is satisfied. (i) If u1 = p1 and u2 = p2 , then (2.13)

Nps11,u1 ,q1 (Rd ) → Eps22,u2 ,q2 (Rd )

if, and only if, q1 ≤ p2 . (ii) Assume that u1 < p1 and that u2 satisfies d d (2.14) = s2 − s1 + . u2 u1 Then (2.15)

Nps11,u1 ,u2 (Rd ) → Eps22,u2 ,q2 (Rd ).

Remark 2.7. Note that by simple monotonicity arguments one immediately obtains the extension of (2.15) to (2.16)

Nps11,u1 ,u (Rd ) → Eps22,u2 ,q2 (Rd )

for all u satisfying d d > s2 − s1 + . u u1 Hence discussing the sharpness of the above result leads to the so far open question whether u2 given by (2.14) is the largest possible number such that (2.15) remains true. In [54] we studied Franke-Jawerth embeddings in the context of d spaces As,τ p,q (R ). Finally, we recall the weighted counterparts of the above embeddings in the special situation when both source and target space are weighted with the same w ∈ A∞ . Here we shall only need the following basic observations, see [13, 14].

COMPACT EMBEDDINGS OF WEIGHTED SMOOTHNESS MORREY SPACES

245

Proposition 2.8. Let 0 < q ≤ ∞, 0 < p < ∞, s ∈ R and w ∈ A∞ . (i) Let ε > 0, 0 < r ≤ ∞, and 0 < q0 ≤ q1 ≤ ∞, then d s d As+ε p,r (R , w) → Ap,q (R , w)

and

Asp,q0 (Rd , w) → Asp,q1 (Rd , w).

(ii) We have s s s Bp,min(p,q) (Rd , w) → Fp,q (Rd , w) → Bp,max(p,q) (Rd , w).

(2.17)

(iii) Assume that there are numbers c > 0, r > 0 such that for all cubes, w (Qν,m ) ≥ c2−νr ,

ν ∈ N0 ,

m ∈ Zd .

Let 0 < p0 < p < p1 < ∞, −∞ < s1 < s < s0 < ∞ satisfy r r r s0 − = s − = s1 − . p0 p p1 Then Bps00 ,q (Rd , w) → Bps11 ,q (Rd , w),

(2.18) and (2.19)

s (Rd , w) → Bps11 ,p (Rd , w). Bps00 ,p (Rd , w) → Fp,q

Remark 2.9. These embeddings are natural extensions from the unweighted case w ≡ 1, see [47, Prop. 2.3.2/2, Thm. 2.7.1] and [43, Thm. 3.2.1]. The above result essentially coincides with [3, Thm. 2.6] and can be found in [13, Prop. 1.8]. Example 2.10. For our model weight wα given by (1.2) Proposition 2.8 can be exemplified as follows. Let 0 < p0 < p < p1 < ∞, −∞ < s1 < s < s0 < ∞, 0 < q ≤ ∞, α ≥ 0. Assume that α+d α+d α+d = s1 − s0 − =s− . p0 p p1 Then (2.18) and (2.19) hold for w = wα . 3. Compact embeddings of weighted spaces: a model case We have mentioned in Theorem 2.4 explicitly that there is no compact embedding when dealing with smoothness Morrey spaces on Rd . When dealing with spaces defined on a bounded domain Ω ⊂ Rd instead, we proved compactness criteria for the corresponding spaces in [16]. A different approach to shrink (or enlarge) the involved spaces sufficiently to generate compactness (and not only continuity) of the embedding is usually related with weighted settings. This leads to weighted spaces of Morrey type which is itself an interesting matter. First basic observations in this direction can be found in [20, 22, 40], see also [18], but we shall not follow this general idea here. We prefer to deal with the situation only where the source space is a weighted Besov space and the target space an unweighted smoothness space of Morrey type, that is, we consider (3.1)

idw : Bps11 ,q1 (Rd , w) → Nps22,u2 ,q2 (Rd ),

w ∈ A∞ .

We obtained in [13] necessary and sufficient conditions such that idw becomes continuous. Now we want to simplify the setting further and restrict ourselves to our typical weight wα given by (1.2), only. Let s1 ≥ s2 , 0 < q1 , q2 ≤ ∞, 0 < p1 < ∞, 0 < u2 ≤ p2 < ∞, wα (x) = |x|α , α > −d, and (3.2)

idα : Bps11 ,q1 (Rd , wα ) → Nps22,u2 ,q2 (Rd ).

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK

Observe that in case of p2 = u2 the embedding (3.2) was studied in detail in [13]; it is known that (3.3)

idα : Bps11 ,q1 (Rd , wα ) → Bps22 ,q2 (Rd )

is continuous if, and only if,

and

,

,

α≥0 α d p1 > p∗ δ≥ δ>

if if

α p1 α p1

if if

where we used the abbreviations

 1 1 1 = − , p∗ p2 p1 +

p∗ = ∞, p∗ < ∞, q ∗ = ∞, q ∗ < ∞, 1 = q∗



1 1 − q2 q1

 +

and

d d − s2 + . p1 p2 The embedding (3.3) is compact if, and only if, δ = s1 −

(3.4)

δ>

α d > ∗. p1 p

Now we deal with (3.2) in the case u2 < p2 and obtain the following extension of our results in [13]. Theorem 3.1. Let s1 ≥ s2 , 0 < q1 , q2 ≤ ∞, α > −d. (i) The embedding idα from (3.2) is continuous , δ ≥ pα1 d α ≥ ∗ and (3.5) p1 p δ > pα1

0 < p1 < ∞, 0 < u2 < p2 < ∞, if, and only if, if if

q ∗ = ∞, q ∗ < ∞.

(ii) The embedding idα from (3.2) is compact if, and only if, (3.6)

δ>

d α > ∗, p1 p

s1 d s2 d that is, if and only if, idB α : Bp1 ,q1 (R , wα ) → Bp2 ,q2 (R ) is compact.

Proof. Step 1. As already mentioned, part (i) coincides with [15, Prop. 3.10], whereas the equivalence of condition (3.6) and the compactness of s1 d s2 d idB α : Bp1 ,q1 (R , wα ) → Bp2 ,q2 (R )

is covered by [13]. In view of the embedding Bps22 ,q2 (Rd ) → Nps22,u2 ,q2 (Rd ) for all u2 ≤ p2 , and the coincidence of (3.4) and (3.6), it remains to prove that the compactness of idα from (3.2) implies (3.6). We use the wavelet decompositions recalled in Propositions 1.8 and 1.11. By Remark 1.12 it is thus sufficient to show that the compactness of the corresponding sequence space embedding idα : bsp11 ,q1 (wα ) → nsp22 ,u2 ,q2

COMPACT EMBEDDINGS OF WEIGHTED SMOOTHNESS MORREY SPACES

247

implies (3.6). Note that since wα (Qj,m ) ∼ 2−j(d+α) |m|α , where |m|α = max (1, |m|)α , m ∈ Zd , j ∈ N0 , we have ⎛ ! pq1 ⎞ q11 ∞ 1    s  d α jq1 (s1 − p − p ) p1 λ|bp1 ,q (wα ) ∼ ⎝ ⎠ . 1 1 (3.7) 2 |λ | |m| j,m α 1 1 m

j=0

If pα1 < pd∗ or δ < pα1 , then the embedding (3.2) is not continuous by (i). Moreover if α = 0, then the embedding is not compact, recall Theorem 2.1. So we are left to disprove compactness in the two cases: (a) pα1 = pd∗ and u2 < p2 < p1 < ∞, (b) δ = pα1 . First we assume that pα1 = pd∗ and u2 < p2 < p1 < ∞. We define a  (ν)  family of finite sequences λ(ν) = λj,m j,m , ν ∈ N, putting , d+α 2−ν p1 , if j = 0 and Q0,m ⊂ Q−ν,0 , (ν) λj,m = 0, otherwise. Step 2.

Since



(3.8)

|m|α ∼ 2ν(d+α) ,

m:Q0,m ⊂Q−ν,0

(3.7) implies for any ν that λ(ν) |bsp11 ,q1 (wα ) ∼ 1 .

(3.9)

Let μ < ν; we show that the norm λ(ν) − λ(μ) |nsp22 ,u2 ,q2  can be estimated from below by a positive constant. We have  (ν) (μ) |λ0,m − λ0,m |u2 m:Q0,m ⊂Q−ν,0



=

(ν)

(μ)

|λ0,m − λ0,m |u2 +

m:Q0,m ⊂Q−μ,0

μ(d− α+d u2 )  p

u2 (μ−ν) α+d p



(ν)

|λ0,m |u2

m:Q0,m ⊂Q−ν,0 \Q−μ,0 α+d

1 1−2 + 2ν(d− p1 u2 ) (1 − 2(μ−ν)d )   α+d α+d  α+d u = 2ν(d− p1 u2 ) 2(μ−ν)(d− p1 u2 ) 1 − 2(μ−ν) p1 2 + 1 − 2(μ−ν)d .

=2

But

α p1

=

d p∗

1

implies d − 

α+d p1 u2 (ν)

 u2 > 0 such that   1 1 ≥ 2νd u2 − p2 u2 (1 − 2−d ).

 = d u12 − (μ)

|λ0,m − λ0,m |u2

1 p2

m:Q0,m ⊂Q−ν,0

This results in (3.10)

1

λ(ν) − λ(μ) |nsp22 ,u2 ,q2  ≥ (1 − 2−d ) u2

if ν = μ. Thus the sequence (λ(ν) )ν , ν = 1, 2, . . ., cannot contain any convergent subsequence.

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK

Step 3. Now we assume that δ = pα1 . Again we define special sequences  (ν)  λ(ν) = λj,m j,m , ν ∈ N, by , α+d 2ν( p1 −s1 ) , if j = ν and m = 0, (ν) λj,m = 0, otherwise. In view of wα (Qν,0 ) ∼ 2−ν(α+d) , ν ∈ N, (1.7) yields λ(ν) |bsp11 ,q1 (wα ) ∼ 1, whereas plainly λ(ν) − λ(μ) |nsp22 ,u2 ,q2  = 2, μ =  ν. 

This completes the proof of (ii).

Remark 3.2. If δ > pα1 = pd∗ > 0 and u2 < p2 , then the embedding (3.2) is continuous but not compact and the increase of smoothness of the source space does not improve the properties of the embedding. In the limiting case u2 = p2 the embedding does not hold. On the other hand, if δ = pα1 > pd∗ , then the embedding (3.2) is continuous if and only if it is continuous in the classical case, i.e., if (3.3) holds. As already mentioned, we dealt in [15] with the situation described in (3.1) for general weights w ∈ A∞ and proved a criterion for the continuity of idw , but no counterpart for compactness so far (in this generality). In [55] we studied a similar question, that is, when the target space Nps22,u2 ,q2 (Rd ) is replaced by a space s,τ (Rd ). There we obtained a complete characterisation for continuity and of type Bp,q compactness assertions. Finally we combine Theorem 3.1 with the Franke-Jawerth embeddings in Theorems 2.4, 2.6, and Proposition 2.8, in particular, Example 2.10. Corollary 3.3. Let s1 ≥ s2 , 0 < q1 , q2 ≤ ∞, 0 < p1 < ∞, 0 < u2 ≤ p2 < ∞, α > −d. (i) The embedding idα : Fps11,q1 (Rd , wα ) → Eps22,u2 ,q2 (Rd )

(3.11) is continuous if (3.12) and (3.13)

α d ≥ ∗ p1 p ⎧ ⎨δ >

α p1

⎩δ =

α p1

or, and

s1 > s2

and

p 1 ≤ u2 < p 2 .

Conversely, the continuity of (3.11) implies (3.14)

α d ≥ ∗ p1 p

and

δ≥

α . p1

(ii) The embedding idα from (3.11) is compact if, and only if, (3.15)

δ>

α d > ∗, p1 p

s1 d s2 d that is, if and only if, idF α : Fp1 ,q1 (R , wα ) → Fp2 ,q2 (R ) is compact.

COMPACT EMBEDDINGS OF WEIGHTED SMOOTHNESS MORREY SPACES

249

Proof. Step 1. The compactness in (ii) is a direct consequence of Theorem 3.1 and elementary embeddings (1.4) and (2.17): if idα from (3.11) is compact, then via (3.16) Bps11 ,min(p1 ,q1 ) (Rd , wα ) → Fps11,q1 (Rd , wα ) → Eps22,u2 ,q2 (Rd ) → Nps22,u2 ,∞ (Rd ) and Theorem 3.1(ii) – which is independent of the fine parameters – we conclude (3.6) which coincides with (3.15). Conversely, (3.15) implies the compactness of Bps11 ,max(p1 ,q1 ) (Rd , wα ) → Nps22,u2 ,min(u2 ,q2 ) (Rd ) and thus via (1.4) and (3.17) Fps11,q1 (Rd , wα ) → Bps11 ,max(p1 ,q1 ) (Rd , wα ) → Nps22,u2 ,min(u2 ,q2 ) (Rd ) → Eps22,u2 ,q2 (Rd ) the compactness of idα . Step 2. The necessity of (3.14) for the continuity of idα given by (3.11) can also be obtained by (3.16) and Theorem 3.1(i), which refers to the case q ∗ = ∞ there. Step 3. It remains to show the sufficiency of (3.12), (3.13) for the continuity of idα given by (3.11). We always assume (3.12) now and begin with the non-limiting situation in (3.13), that is, δ > pα1 . Choose ε such that 0 < ε < 12 (δ − pα1 ), that is, α d d < (s1 − ε) − − (s2 + ε) + . p1 p1 p2 Then Proposition 2.8(i), (1.3), and Theorem 3.1(i) imply the continuity of idα in view of (3.18)

−ε +ε (Rd , wα ) → Nps22,u (Rd ) → Eps22,u2 ,q2 (Rd ). Fps11,q1 (Rd , wα ) → Bps11 ,q 1 2 ,q2

Let now δ = pα1 and p1 ≤ u2 < p2 . If α = 0, then this is well-known, see also Theorem 2.1(ii) (with u1 = p2 ), so we may assume δ = pα1 > 0. Choose numbers σi and ri , i = 1, 2, such that s2 < σ2 < σ1 < s1 , r1 > p1 , r2 < p2 , and (3.19)

s1 −

α+d α+d = σ1 − , p1 r1

σ2 −

d d = s2 − . r2 p2

Thus Proposition 2.8(iii), in particular, Example 2.10, implies (3.20)

Fps11,q1 (Rd , wα ) → Brσ11,p1 (Rd , wα ).

2 = u12 − p12 + r12 > Furthermore, u2 < p2 yields u1 := u12 − s2 −σ d to Theorem 2.6(ii) (with u1 = u, p1 = r2 , s1 = σ2 ),

(3.21)

1 r2 .

Then according

Nrσ22,u,u2 (Rd ) → Eps22,u2 ,q2 (Rd ),

since the latter part of (3.19) coincides with (2.11) in our setting. It remains to verify that (3.22)

Brσ11,p1 (Rd , wα ) → Nrσ22,u,u2 (Rd )

to complete the argument. We want to apply Theorem 3.1(i) and first observe that α 1 1 d r1 > d( r2 − r1 )+ = r ∗ as required. This can be seen as follows. Assumption σ1 > σ2 and (3.19) lead to rα1 > d( r12 − r11 ) since δ = pα1 . Together with α > 0 this

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DOROTHEE D. HAROSKE AND LESZEK SKRZYPCZAK

yields rα1 > rd∗ . Next we see that the differential dimension δr corresponding to the embedding (3.22) satisfies δ r = σ1 −

d d α+d α d α α α − σ2 + = s1 − + − s2 + =δ− + = r1 r2 p1 r1 p2 p1 r1 r1

by (3.19) and δ = pα1 . In addition, our assumption p1 ≤ u2 – referring to the case q ∗ = ∞ in Theorem 3.1(i) applied to the embedding (3.22) – ensures its continuity. But now the composition of the continuous embeddings (3.20), (3.22), and (3.21)  yields the boundedness of idα given by (3.11). Remark 3.4. Note that the parallel trick to apply the Franke-Jawerth embedding Theorem 2.4 for the necessity as well does not work here in view of part (ii) of that theorem: the best possible fine index sticks always at ∞. Dealing with the sufficiency condition for the continuity of idα in case of δ = pα1 one could also use (3.17) and assume max(p1 , q1 ) ≤ min(u2 , q2 ) in addition. If s1 > s2 this is obviously more restrictive than p1 ≤ u2 < p2 , but could it give some sufficient condition for s1 = s2 in that case? Unfortunately not, since for s1 = s2 , (3.12) leads to δ = pα1 = d( p12 − p11 ) = pd∗ ≥ 0 in that case. Consequently max(p1 , q1 ) ≤ min(u2 , q2 ) is only satisfied in the trivial case when p1 = q1 = p2 = u2 = q2 and α = 0. Note that we do not even have a complete characterisation for the continuity of the embedding idF α yet, that is, (3.11) with u2 = p2 , only for its compactness. References [1] Marcin Bownik, Atomic and molecular decompositions of anisotropic Besov spaces, Math. Z. 250 (2005), no. 3, 539–571, DOI 10.1007/s00209-005-0765-1. MR2179611 [2] Marcin Bownik and Kwok-Pun Ho, Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1469–1510, DOI 10.1090/S0002-9947-05-03660-3. MR2186983 [3] Bui Huy Qui, Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (1982), no. 3, 581–605. MR676560 [4] Huy-Qui Bui, Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures, J. Funct. Anal. 55 (1984), no. 1, 39–62, DOI 10.1016/0022-1236(84)90017-X. MR733032 [5] H.-Q. Bui, M. Paluszy´ nski, and M. H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math. 119 (1996), no. 3, 219– 246. MR1397492 [6] H.-Q. Bui, M. Paluszy´ nski, and M. Taibleson, Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q < 1, Proceedings of the conference dedicated to Professor Miguel de Guzm´ an (El Escorial, 1996), J. Fourier Anal. Appl. 3 (1997), no. Special Issue, 837–846, DOI 10.1007/BF02656489. MR1600199 [7] Jens Franke, On the spaces Fspq of Triebel-Lizorkin type: pointwise multipliers and spaces on domains, Math. Nachr. 125 (1986), 29–68. MR847350 [8] Michael Frazier and Bj¨ orn Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), no. 4, 777–799, DOI 10.1512/iumj.1985.34.34041. MR808825 [9] Michael Frazier and Bj¨ orn Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170, DOI 10.1016/0022-1236(90)90137-A. MR1070037 [10] Jos´ e Garc´ıa-Cuerva and Jos´e L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matem´ atica [Mathematical Notes], 104. MR807149 [11] Dorothee D. Haroske and Susana D. Moura, Some specific unboundedness property in smoothness Morrey spaces. The non-existence of growth envelopes in the subcritical case, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 2, 137–152, DOI 10.1007/s10114-016-5104-4. MR3441298

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Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13952

Tracking the structural deformation of a sheared biopolymer network Louise M. Jawerth and David A. Weitz This paper is dedicated to Bj¨ orn Jawerth Abstract. Biopolymer networks provide mechanical integrity in many important environments in vivo ranging from the cytoskeleton within a cell to the structural support of cells themselves in tissues and tendons. Rheological studies have shown that they exhibit many unique material properties. Modelling these properties requires a precise knowledge of how the individual filaments in the network deform locally during a global deformation. Here, we present an image processing method to track the three-dimensional motion of a biopolymer network as a simple shear deformation is applied. We track the structure of the network from one shear position to the next by determining the displacement of each branch point using a cross-correlation. To illustrate the use of this algorithm, we apply it to a fluorescently labelled fibrin network.

Introduction Biopolymer networks provide mechanical integrity in many important environments in vivo ranging from the cytoskeleton within a cell to the structural support of cells themselves in tissues and tendons. These networks are composed of protein sub-units that polymerize into long fibers which self-associate to form the network itself. To understand their behavior in the complex environment of a living organism, the material properties of biopolymers are routinely studied in the absence of other proteins in vitro. Rheological studies characterizing the force required to deform a biopolymer network have shown that they exhibit many unique material properties, such as strain stiffening([SPM05] [BM14]). For many of these measurements, the networks are subjected to a simple shear strain: the network is held between two parallel plates and the top plate is translated while the bottom plate remains fixed. Modelling the material properties of biopolymer networks requires a precise knowledge of how the individual filaments in the network deform locally during such a global deformation ([BM14]); however, there currently exist no experimental methods that quantitatively track the motion of individual fibers as a network is deformed. Such a method is essential to directly test the validity of current models and for the development of new ones. Confocal fluorescence microscopy is a powerful technique to resolve the threedimensional structure of many biopolymer networks ([JMV10]). Networks in 2010 Mathematics Subject Classification. Primary 92C05, 74-04, 68U10. Key words and phrases. Image processing, fibrin, shear. c 2017 American Mathematical Society

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Figure 1. A) A fluorescently labelled fibrin network before any deformation is applied. B) The same region with an applied shear strain.

which the average spacing between fibers is larger than a micron, such as those found in bundled actin systems, collagen and fibrin are particularly amenable to this technique ([BKC08] [MJL13], [JMV10]). Typically for this technique, some of the protein sub-units are tagged with fluorophores that emit light when excited. The acquired images are gray-scale with pixel values corresponding to the emitted light intensities; thus, fibers appear as bright filaments on a dark background. To image a three-dimensional region, a series of images is taken at increasing z positions where each image represents a cross-section of the network in xy; each such series of images is referred to as an image stack. Such a procedure can be repeated to capture the behavior of a network over time or in response to a deformation ([MJL13]). Previous studies have developed methods to extract the position of fibers taken in a single image stack. Typically, they result in a one-pixel thick line that follows along the axis of each fiber in the network. Amongst other terms, the positions of the fibers along their axes is referred to alternatively as the skeleton, medial axis, backbone or centerline ([KML12] [SVJ08] [MMJ] [XVH]). In this paper, we refer to the results of such a method as the skeleton of the network. Yet, despite the ability to quantitatively identify the structure of a biopolymer network, no image processing methods exist to quantitatively track how the structure of a network changes from one image stack to another. Such a method is imperative to extract how the individual fibers in a network deform when sheared. From this we could discern the physical principles underlying the unique properties exhibited by biopolymer networks in vitro, thus broadening our understanding of their behavior in the more complex environment of a living organism. Here, we present an image processing method to track the three-dimensional motion of a biopolymer network as it is sheared. The data is assumed to be in the form of image stacks that are taken at several time points as the network is increasingly sheared (see Fig. 1).We first extract the three-dimensional skeleton of the biopolymer network using existing techniques and subsequently apply our method. We find that network branch points represent unique features and utilize this to track the structure of the network from one shear position to the next. We demonstrate this method for a fluorescently labelled fibrin network.

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Figure 2. An illustration of typical variations that occur during skeletonization. Intersecting fibers are shown in light gray, the identified branch points are shown as red dots and the skeleton is indicated as a black line. A) One possible skeletonization where one branch point is identified. B) A second example illustrating how two branch points can easily be identified instead of one. C) A skeletonization where the branch point is located at a slightly different position as compared to the branch point in A.

The Method Overview of the Tracking Algorithm One approach to tracking a network would be to identify the skeleton of the network at every time point and then determine how one skeleton must deform to assume the position and shape of the subsequent one. In practice, such a method is difficult to implement because the results of a skeletonization algorithm are subject to some variations when performed on two datasets. For instance, a node representing the junction of four fibers may sometimes be skeletonized properly as the junction of four fibers, but is also likely to be skeletonized into two T-junctions connected by a small fiber (see Fig. 2B). Similarly, the exact location of a node may change slightly each time the data is skeletonized (see Fig. 2C). These types of artifacts make the process of finding the proper correspondence between a network skeletonized at two different time points very difficult. Instead of using such an approach, we utilize the fact that under most circumstances associated fibers remain bound, do not break and do not associate with new fibers as a network is deformed ([MJL13]). This entails that the network topology remains fixed. Therefore, our algorithm is not concerned with extracting the three-dimensional network structure; rather, once a structure has been identified, this algorithm determines how the structure deforms over time. We achieve this by skeletonizing the network in the first time point. After which, we find the displacement of the network to each subsequent time point using a method based on cross-correlations of the grayscale values of three-dimensional regions centered around feature points. We track nodes that are the junctions between three or more fibers during the deformation; physically, these nodes represent branch points in the network. Our approach can be summarized as follows (see Fig. 3). We identify the initial three-dimensional skeleton of the network and all its node positions. We denote each node by its linear index, i. The gray-scale voxel region around each node at the initial time point represents the feature to be tracked (denoted, fi ). For

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Figure 3. Algorithm flow chart each time point t and node i, we select a voxel search region where we expect the node to be located, si (t). We cross-correlate fi with the search region, si (t) . The location at which the value of the cross-correlation (CC) is maximal, represents the three-dimensional displacement of the feature from the one frame to the next. The CC is deemed successful if one or more quality measures exceed a threshold level. If the CC is successful, we record the corresponding displacement as the position of the node at that time point. If the CC fails, we replace fi with a voxel region taken from the previous time point, t − 1, in which the node was successfully located. We cross-correlate this updated feature region with the search region. If the CC again fails, the feature is

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Figure 4. A) The original image rendered to show the threedimensional structure of a small region in a typical network B) The corresponding area with the skeleton represented by gray tubes and the branch points marked with small yellow spheres. considered ”lost” and will not be tracked through further timepoints. If this CC is, instead, successful, we record the node position at this time point and continue with this new feature region for further time points. We repeat this procedure for all nodes, i in the network and then proceed to the next time point. In the next sections we review the details of this procedure. 1. Initial Processing: Network Extraction There are many algorithms that can be used to find the skeleton of a network (for a good overview of such methods please see [XVH]). To use the algorithm described here it is important that the branch points or intersections between fibers are identified as part of the algorithm (see Fig. 4). 2. Selection of features and search regions 2.1. Selection of initial feature regions. Typically, a network is composed of fibers that are relatively uniform in intensity and width. A small section of one fiber is generally indistinguishable from another one (see Fig. 6). It is therefore difficult to track fiber segments. By contrast, the points where fibers branch are excellent features to track: Each such node is the junction between several fibers with relative angles that are distinct when compared to those of other nodes (see Fig. 5 and Fig. 6). Moreover, these characteristics do not change significantly for small network deformations. From the initial network extraction, we know the locations of all the nodes in the network in the first time point. We use a small voxel region of the gray-scale image stack taken from this time point centered about each node, i, as the feature, fi , that will be tracked. We found that a voxel region large enough to enclose the node and the fibers but small enough to avoid most of the length of the fibers was ideal (see Fig. 5A). 2.1.1. Replacement of feature regions. As a fibrin network is deformed, we found that the fibers that join at some nodes will gradually rotate or buckle from one time point to the next. This can lead to a persistent deformation that, once great enough, the node may no longer closely resemble itself at the initial time

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(A) Feature Region

(B) Search Region

Figure 5. Maximum projections of a region taken from a deforming fibrin network. A) The feature region is outlined in white (size: 32x32 pixels) B) The corresponding search region taken at the following time point outlined in white (size: 64x64 pixels). point. In our algorithm, the node displacement is given by matching the feature region within a search region. If the node is too deformed in the search region as compared to the feature region, it no longer becomes possible to find a match. When this occurs, we replace the current feature region with a similarly sized region taken from the last time point in which the node was successfully located. With this procedure, as long as the deformation of a node is small between time points, the node position can still be successfully tracked. 2.2. Selection of a search region. A search region is a region in the network where we expect a specific node to be located. In our initial approach, we chose a search region centered around the node position in the previous time point; however, we found that in a sheared network, nodes located closer to the translated upper plate move significantly more than those close to the stationary bottom surface. In fact, this motion can be so large that the node has moved out of the neighborhood of its previous location. We therefore found that in practice it is more accurate to choose a search region centered about the predicted position of a node. Since we are applying a shear to the network, we assume that the predominate deformation of the network is that of an affine, shear deformation. This implies that for a node with a position x, y, z, its predicted position xp , yp , zp is given by, ⎡ ⎤ ⎡ ⎤⎡ ⎤ xp 1 0 0 x ⎣ yp ⎦ = ⎣0 1 γ ⎦ ⎣y ⎦ (2.1) zp 0 0 1 z where γ is the amount of shear imposed in the experiment; this value is defined as γ ≡ −d/h where d is the displacement of the top glass plate during one shear step and h is the height between the top and bottom plates (see Fig. 6 and Fig. 7). In our datasets, we sheared in the −y direction and γ is negative. However, the deformation tensor can easily be modified for any shear direction. For every node, i, we calculate its expected position and define a voxel region, si (t), centered at that expected position from the image stack representing the time point, t. Generally, we found that a region larger than the feature region was needed to accurately locate a node during our tracking (see Fig. 5).

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Figure 6. A) The light blue structure is the initial skeleton and the dark blue structure is a skeleton found at a later time point. For this image, we have tracked the structure using CCs as described in this paper with the modification that all possible feature points along the fibers were cross-correlated, not just the branch points of the network. The blue arrows indicate the expected fiber positions based on a simple shear deformation. The red arrows mark the deviation from the expected position and the displacement found from the cross-correlation. B) An example of the errors that occur when fiber segments are cross-correlated. The red arrows show that a fiber segment is being matched with an incorrect segment in the later time point.

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Figure 7. An illustration of two branch points in a network. The initial branch point positions are shown in bold colors and the branch point positions in the next time point are shown in lighter colors. The shear, γ, on the network is given by γ = −d/h, where d is the distance the top plate has moved and h is the distance between the two plates. The farther a branch point is away from the bottom plate, the more it is expected to move; this is illustrated by the shear profile shown with blue arrows on the right. For the two branch points illustrated in this diagram, their expected motion is shown with blue arrows, and the correction from this expected motion to their actual displacement is shown with red arrows.

3. Finding the displacement of a feature region within the corresponding search region In this section we describe how the position of one node, i, is located at a time, t. Since we are just treating one node and one time point, for a cleaner notation we have neglected to write the index i and specify the time t. Specifically, in this section we will refer to si and fi as s and f , respectively. All search regions are taken from the current time point, t, and all feature regions are taken either from the initial time point or from their most recently updated time point (see section 2.1.1) Moreover, for a feature region, we refer to f (x, y, z) to denote the value of f for the voxel located at the point (x, y, z) within this region. Voxels within s and our cross-correlation C are denoted similarly.

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Figure 8. One XY plane through C for the cross-correlation between the feature region and search region shown in Fig. 5

3.1. Cross-correlation method to find one node displacement. To locate the node position, we calculate the cross-correlation C(x, y, z) between s(x, y, z) and f (x, y, z). The cross-correlation is defined as

(3.1) C(x, y, z) =  u,v,w (f (u + x, v + y, w + z) − f (x, y, z))(s(u, v, w) − s(x, y, z)) # # 2 2 (f (u + x, v + y, w + z) − f (x, y, z)) u,v,w u,v,w (s(u, v, w) − s(x, y, z)) where f is the mean of f and s is the mean of s each in a box the size of f centered at (x, y, z).The denominator corresponds to the auto-correlation of each function. It normalizes the function so that if the two functions correlate perfectly the result is 1 and -1 if they are perfectly anti-correlated. The position (x, y, z) with the maximum value of C corresponds to the relative shift between the centers of f and s (see Fig. 8). Briefly, this approach moves a normalized feature region around in a normalized search region. For every translated position (x, y, z), the product of each voxel in the normalized feature region with the corresponding underlying voxel in the normalized search region is calculated; this product is the value C(x, y, z). The position with the greatest value (the maximum of C) corresponds to the translated position with the best match. 3.1.1. Quality Measures. There are two measures we used to examine the quality of the cross-correlation. The first is the magnitude of the cross-correlation at its maximum. For a normalized cross-correlation, as in equation 3.1, a value of 1 corresponds to a perfect match and a value of −1 corresponds to a perfect mismatch. A threshold value can be set as a quality measure for the cross-correlation. However, we found that in practice a different quality measure was often more accurate in identifying a poorly matched point: When a node did not cross-correlate well its displacement was very large. This was due to the dark areas around the fibers matching well; therefore, a good second quality measure was to only allow displacements smaller than a threshold displacement. In summary, a cross-correlation is considered successful if the maximum of C is larger than a threshold value and/or the displacement is smaller than a threshold displacement; otherwise the crosscorrelation is considered failed.

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Application of algorithm to a fibrin network undergoing shear 4. Details of data acquisition and application of the tracking algorithm To illustrate the use of this algorithm, we apply it to a fibrin network as it is sheared. We polymerize a fluorescently labelled fibrin network between two glass plates on a confocal microscope (Leica SP5, 63x 1.2NA equipped with a water immersion objective). Once polymerization is complete, we acquire a stack of images representing a three-dimensional region of the network (approximately 250μmx250μmx100μm) (see Fig. 1). By translating the upper-glass plate parallel to the bottom plate, we impose a simple shear deformation on the fibrin network. We translate the upper-glass plate in increments corresponding to approximately 1% shear steps. After each movement, we acquire a new stack of images corresponding to the increasingly deformed fibrin network. To track the deformation of the network, we use the following parameters. (1) Pre-processing: We smoothed the image stack corresponding to the first time point using a Gaussian filter to suppress acquisition noise. Subsequently, we determined the skeleton using the commercial software package AMIRA (FEI Software, Ver 5.3). (2) Size of feature region: 32x32x32 voxels (see Fig: 5A) (3) Size of search region: 64x64x64 voxels (see Fig: 5B) (4) Cross-correlation quality measure: For displacements larger than 8 voxels, we considered a cross-correlation unsuccessful. We track the branch points in the network for 50 time points. We take a maximum projection of one small region and overlay this with the tracked positions in this region (see Fig. 5). We carefully inspect the accuracy of the algorithm. We find that many of the branch points are accurately located from one time point to the next. When a branch point is unsuccessfully tracked it is often the result of an aberrant branch point found in the initial skeletonization. These branch points occur when two fibers cross close to each other without adhering to one another. During skeletonization, this can appear to be the junction or branch point between four fibers.

5. Discussion In this paper, we have presented an image processing method that tracks the structure of a biopolymer network as it is sheared. As an initial step, we have taken the skeleton of a network using an existing skeletonization technique. In general, however, our algorithm does not require the existence of a network skeleton. Since the algorithm cross-correlates regions from the original gray scale image stacks, it just requires knowledge of the position of network branch points. We, however, did not know of other algorithms that identify fiber branch points and therefore relied on skeletonization routines. Using a skeleton in the initial time step is also advantageous because it determines how individual branch points are connected. Together with the motion of the network, the connectivity of a branch point has been an important consideration in many models of network mechanics ([BMLM11] [LMS15]). Moreover, since branch points are connected via a fiber,

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γ = 0%

γ = 5%

γ = 10%

γ = 15%

265

γ = 20%

Figure 9. A small region of a fibrin network with different amounts of shear strain applied. For clarity, in these images, we have transformed the sheared images using a simple shear deformation (see the Materials and Methods section). The motion of the network that remains is movement around the expected shear deformation. The branch points are indicated in red in the upper panels. These same points are outlined in the middle panels in yellow with the magenta outlines indicating the original undeformed branch point positions. The lower panels are the same region without the branch points outlined The z-direction corresponds to the vertical direction in these images.

by measuring the relative displacement between the two fiber ends, one can calculate values such as the elongational strain across individual fibers. This is also an important parameter in many models of network mechanics ([SPM05][MKJ95]). Our algorithm is based on using cross-correlations to track the positions of branch points in a network. We perform the cross-correlations on the original gray-scale images. The exact location of a branch point may vary slightly when a skeletonization algorithm is applied to a network (see Fig. 2). By tracking

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the original gray-scale images, we ensure that even if the branch point position determined by the skeletonization routine is slightly inaccurate, the tracked motion of the node can still be of high quality. Using cross-correlations to find a deformation field is not new. One prominent use is in particle image velocimetry (PIV) ([WEA13] [SMS16]). Typically in this approach, feature regions are selected at regularly spaced intervals that cover the whole image stack at an initial time point and at a later time point. The features from the initial time point are then cross-correlated with a search region at a later time point to determine the relative displacement of each region. This may seem very similar to the approach we report here, but there are important differences. To successfully find the deformation field using PIV, each region must encompass a distinct feature that can be accurately located in the next time point; therefore, PIV techniques are particularly poor at tracking highly oriented structures in which the correlation function shows very broad peaks along the primary direction of alignment. Moreover, since features in PIV are defined at regularly spaced intervals, for a dilute biopolymer network these regions need to be relatively large to ensure that they contain enough fibers and branch points to be properly located. The resulting deformation field is, therefore, inherently very coarse. By instead crosscorrelating regions centered around branch points as we do here, we can use a much smaller region that only encompasses a single feature itself. We can thereby track the structural deformation on a much finer level and with less computational effort. Moreover, a cross-correlation determines the relative translated movement between two images or image regions. It cannot, for instance, find rotations or deformations smaller than the region. Since the region used for PIV is large, as the fibers in the network deform by bending or buckling the cross-correlation of a region often fails. We found that in a deforming biopolymer network, the branch points typically deform much less than the surrounding fibers. Therefore, by using a small region centered around a branch point, we can still successfully track the motion of the network. Our method was designed to track the deformation of a network as it undergoes a simple shear deformation. To track the deformation, we place our search region at the expected location of a network undergoing a simple shear deformation. This informed placement allows us to track the motion of a branch point even if its displacement is very large as long its actual position is a small displacement from its expected position. Tracking such absolutely large displacements would be difficult in other techniques such as PIV. In this paper, we assume that the network primarily deforms according to an affine shear deformation; however, we expect that this technique could be easily extended to other deformations. Moreover, if the overall network deformation does not lead to a significant amount of motion from one image stack to the next, we could center our search region around a branch point’s position in the previous image stack. Thus, our approach should be applicable to a wide range of questions.

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6. Material and Methods For a complete overview of materials and methods please see [Jaw13]. 6.1. Polymerization of fibrin networks. Fibrinogen (FIB3, Enzyme Research Labs, South Bend, IN) is diluted into buffer (150mM N aCl, 20mM HEP ES, and 20mM CaCl2 pH7.4). A small amount of fluorescently labelled fibrinogen (final ratio 6 : 1) is added to allow the sample to be visualized using fluorescence microscopy. Polymerization into a network is initiated by the addition of thrombin (H-T 1002a, Enzyme Research Labs, South Bend, IN) with a final concentration of 0.1U/ml. 6.2. Application of shear and acquisition of fluorescent images. Briefly, the method can be described as follows. Two parallel glass plates with a spacing of 800μm are constructed on a fluorescent confocal microscope (Leica SP5). Quickly, after the addition of thrombin, a fibrinogen solution is pipetted between these plates. Mineral oil is added around the plates to avoid evaporation and the sample is incubated for at least six hours to allow polymerization to complete. Once polymerization is complete the network is sheared by moving the top glass plate in approximately 8μm steps leading to an approximate shear, γ, of −1%. Between each step, a set of images corresponding to the three-dimensional volume corresponding to a field-of-view of approximately 250μmx250μmx100μm is acquired. For a more complete description, please see [Jaw13] [MJL13]. 6.3. Details of application of tracking algorithm. To find the skeleton, the image stack representing the undeformed network is smoothed using a 3D Gaussian filter (size: 5x5x5, sigma: 0.5) and then thresholded by eye. We have used the commercial software AMIRA (FEI Software, Ver 5.3) to extract the skeleton of the network. The skeleton is saved as a text file that is imported as coordinates into MATLAB (The MathWorks, Inc.) The algorithm we describe here was implemented using MATLAB (The MathWorks, Inc.). Fourier transforms were used for the implementation of the crosscorrelations to increase processing speed. We define the XY directions to be parallel to the imaging plane. The −y direction is the direction of applied shear. 6.4. Image production for this paper. All grayscale images in this paper represent a maximum intensity projection along the x-axis of a region in threedimensions taken from a fibrin network (Figures 5, 9, 1). For Fig. 9, we have applied the StackReg plugin with an affine transformation using FIJI to a movie of the maximum intensity projection of a whole network as it is sheared ([SACF12] [SRE12]). The images shown in Fig. 9 are a small region from this movie that have been scaled by a factor of 3 for clarity. 7. Acknowledgements We would like to thank Stefan M¨ unster and Mona Rozovich for careful reading of the manuscript. We would also like to thank Bj¨ orn Jawerth for useful discussions and guidance. We acknowledge support from the National Science Foundation (DMR-1310266), the Harvard Materials Research Science and Engineering Center (DMR- 1420570).

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References [BKC08]

Poul M. Bendix, Gijsje H. Koenderink, Damien Cuvelier, Zvonimir Dogic, Bernard N. Koeleman, William M. Brieher, Christine M. Field, L. Mahadevan, and David A. Weitz, A quantitative analysis of contractility in active cytoskeletal protein networks, Biophysical Journal 94 (2008), no. 8, 3126 – 3136. [BM14] C. P. Broedersz and F. C. MacKintosh, Modeling semiflexible polymer networks, Rev. Mod. Phys. 86 (2014), 995–1036. [BMLM11] Chase P. Broedersz, Xiaoming Mao, Tom C. Lubensky, and Frederick C. MacKintosh, Criticality and isostaticity in fibre networks, Nat Phys 7 (2011), no. 12, 983–988. [Jaw13] L.M. Jawerth, The mechanics of fibrin networks and their alterations by platelets, Ph.D. thesis, Harvard University, 2013. [JMV10] Louise M Jawerth, Stefan M¨ unster, David A Vader, Ben Fabry, and David A Weitz, A blind spot in confocal reflection microscopy: the dependence of fiber brightness on fiber orientation in imaging biopolymer networks, Biophysical Journal 98 (2010), no. 3, L1–L3. [KML12] Patrick Krauss, Claus Metzner, Janina Lange, Nadine Lang, and Ben Fabry, Parameter-free binarization and skeletonization of fiber networks from confocal image stacks, PLoS ONE 7 (2012), no. 5, 1–8. [LMS15] Albert J Licup, Stefan M¨ unster, Abhinav Sharma, Michael Sheinman, Louise Jawerth, Ben Fabry, David A Weitz, and Fred C. MacKintosh, Stress controls the mechanics of collagen networks, Proceedings of the National Academy of Sciences 112 (2015), no. 31. [MJL13] Stefan M¨ unster, Louise M Jawerth, Beverly A Leslie, Jeffrey I Weitz, Ben Fabry, and David A Weitz, Strain history dependence of the nonlinear stress response of fibrin and collagen networks, Proceedings of the National Academy of Sciences 110 (2013), no. 30, 12197–12202. [MKJ95] MacKintosh, K¨ as, and Janmey, Elasticity of semiflexible biopolymer networks., Phys Rev Lett 75 (1995), no. 24, 4425–4428. [MMJ] Walter Mickel, Stefan M¨ unster, Louise M Jawerth, David A Vader, David A Weitz, Adrian P Sheppard, Klaus Mecke, Ben Fabry, and Gerd E Schr¨ oder-Turk, Robust pore size analysis of filamentous networks from three-dimensional confocal microscopy, Biophysical Journal 95 (2008), no. 12, 6072–6080. [SACF12] Johannes Schindelin, Ignacio Arganda-Carreras, Erwin Frise, Verena Kaynig, Mark Longair, Tobias Pietzsch, Stephan Preibisch, Curtis Rueden, Stephan Saalfeld, Benjamin Schmid, Jean-Yves Tinevez, Daniel James White, Volker Hartenstein, Kevin Eliceiri, Pavel Tomancak, and Albert Cardona, Fiji: an open-source platform for biological-image analysis., Nat Methods 9 (2012), no. 7, 676–682. [SMS16] Julian Steinwachs, Claus Metzner, Kai Skodzek, Nadine Lang, Ingo Thievessen, Christoph Mark, Stefan Munster, Katerina E Aifantis, and Ben Fabry, Threedimensional force microscopy of cells in biopolymer networks, Nat Meth 13 (2016), no. 2, 171–176. [SPM05] Cornelis Storm, Jennifer J. Pastore, F. C. MacKintosh, T. C. Lubensky, and Paul A. Janmey, Nonlinear elasticity in biological gels., Nature 435 (2005), no. 7039, 191–194. [SRE12] Jerry Westerweel, Gerrit E. Elsinga, and Ronald J. Adrian, Particle image velocimetry for complex and turbulent flows, Annual review of fluid mechanics. Volume 45, 2013, Annu. Rev. Fluid Mech., vol. 45, Annual Reviews, Palo Alto, CA, 2013, pp. 409–436, DOI 10.1146/annurev-fluid-120710-101204. MR3026125 [SVJ08] Andrew M. Stein, David A. Vader, Louise M. Jawerth, David A. Weitz, and Leonard M. Sander, An algorithm for extracting the network geometry of threedimensional collagen gels, J. Microsc. 232 (2008), no. 3, 463–475, DOI 10.1111/j.13652818.2008.02141.x. MR2649407 [WEA13] Jerry Westerweel, Gerrit E. Elsinga, and Ronald J. Adrian, Particle image velocimetry for complex and turbulent flows, Annu. Rev. Fluid Mech., vol. 45, 2013, pp. 409–436, DOI 10.1146/annurev-fluid-120710-101204. MR3026125 [XVH] Ting Xu, Dimitrios Vavylonis, and Xiaolei Huang, 3d actin network centerline extraction with multiple active contours, Medical Image Analysis 18, no. 2, 272–284.

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Department of Physics, Harvard University, Cambridge, Massachusetts 02138 Current address: Max Planck for the Physics of Complex Systems, 01187 Dresden Germany E-mail address: [email protected] Department of Physics and the School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13937

Extrapolation, a technique to estimate L´aszl´o Lempert Abstract. We introduce a technique to estimate a linear operator by embedding it in a family At of operators, t ∈ (σ0 , ∞), with suitable curvature properties. One can then estimate the norm of each At by bounds that hold in the limit t → σ0 , respectively, t → ∞. We illustrate this technique on an extension problem that arises in complex geometry.

1. Introduction This paper grew out of a joint paper with Berndtsson, that used Berndtsson’s theorem on the curvature of direct images to prove an Ohsawa–Takegoshi type extension result for holomorphic functions, [BL, B, OT]. Here we distill from [BL] an abstract theorem on estimating Hilbert and Banach space operators by a technique we call extrapolation, and discuss a dual theorem of similar nature as well as an application. The related notion of interpolation is a well established method in harmonic analysis to estimate Banach space operators. Loosely stated, given two pairs of Banach spaces and operators A 0 : E 0 → F0 ,

A 1 : E 1 → F1

among them, one can construct interpolating families Et , Ft of Banach spaces and operators At : Et → Ft , t ∈ (0, 1), whose norms can be bounded in terms of the norms of A0 and A1 . There are several inequivalent ways to construct the interpolation spaces. In the complex method, going back to M. Riesz and Thorin, [R, T], the estimate on At  is obtained from Hadamard’s Three Circles Theorem, i.e., from properties of subharmonic functions. In extrapolation—again loosely stated—we will be given only one operator, through which we will estimate an entire family of operators At : Et → Ft . This again depends on subharmonicity and convexity, derived from curvature properties of the bundles that the Et , Ft form. To be concrete, consider two hermitian holomorphic vector bundles (E, h) and (F, k) over the same base S, and a holomorphic homomorphism A : E → F . We assume the metrics h, k are of class C 2 . The fibers of E and F could be Hilbert spaces, but finite rank bundles will already illustrate the main idea. By the norm A : S → [0, ∞) of A we mean the function A(s) = operator norm of As , where 2010 Mathematics Subject Classification. Primary 32L05, 32W05, 46B70. Research in part supported by NSF grant DMS–1464150. c 2017 American Mathematical Society

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As = A|Es : (Es , h) → (Fs , k). In [L] we have defined what it means for A to decrease curvature; this definition will be reproduced in section 2 and Theorem 3.1. Theorem 1.1. (i) If A : E → F decreases curvature, then log A is plurisubharmonic. (ii) If in addition S is a half plane S = {s ∈ C : Re s > σ0 }, A is bounded, and A(s) = A(Re s) for s ∈ S, then A(t) is a decreasing function of t ∈ (σ0 , ∞). So, in the setting of (ii), if we can estimate A(t) for some t = t0 , the same estimate will hold for t ≥ t0 , an instance of extrapolation. For this conclusion we had to know in advance that supS A < ∞. Thus (ii) has the flavor of an a priori estimate: assuming a crude bound on all A(t), a sharper bound follows. For the application we have in mind we will need an analogous result but with a homomorphism A that increases curvature. We could formulate a clean theorem for quite special bundles E, F . First off, both bundles will be trivial, E = F = S × V → S. In this case their metrics h, k, can be thought of as families hs , ks of, say, Hilbertian norms on V . While from the theorem below (F, k) has disappeared, it is really about the norm of the identity homomorphism A : (E, k) → (E, h), when ks is independent of s (and so (E, k) = (F, k) is flat). Theorem 1.2. Suppose that S = {s ∈ C : Res > σ0 } and E = S × V → S is a trivial Hilbert bundle, with   the norm on V . Suppose further that a C 2 metric h on E has semipositive curvature, hs = hRes , and with some c > 0 inf hs (v) ≥ cv

s∈S

for all v ∈ V.

Then ht (v) is an increasing function of t ∈ (σ0 , ∞) for any v ∈ V , in particular, ht (v) ≤ lim hτ (v). τ →∞

In section 3 we will formulate and prove results more general than Theorems 1.1 and 1.2, that apply to metrics and norms that are not necessarily Hilbertian, see Theorems 3.1, 3.2. Various problems in analysis of course boil down to estimating a norm on a space V . The way to do this by extrapolation is to include the norm in a family hs of norms, s ∈ S, that satisfies the assumptions of say, Theorem 1.2, and estimate hs in the limit Re s → ∞. Whatever estimate one can prove in the limit will then hold for the norm we started with. The success of this approach depends on whether hs gets easier to estimate as Re s → ∞; in the application in section 4 this will be so. The method is reminiscent of the continuity method. In the continuity method one attacks a problem P first by connecting it, through a family Pt , t ∈ [0, 1], P0 = P , of problems with a problem P1 that one knows how to solve, and then by studying the parameter values t for which Pt can be solved. In this approach the choice of Pt connecting P and P1 is largely arbitrary, and the success of the method depends on proving suitable estimates for the solutions of Pt . By contrast, it is estimates that extrapolation furnishes. One still needs to connect problem P through a family Pt with a limiting problem limt→∞ Pt one knows how to solve, but the family Pt should have suitable curvature properties and in its choice is quite restricted. In harmonic analysis the notion of extrapolation is not new. Its first appearance is in Yano’s paper [Y] from 1951 that studies what certain Lp → Lp estimates

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of an operator for all p > 1 imply in the limit p → 1. In the 1980s Rubio de Francia introduced a different notion of extrapolation of operators and estimates, see [Ru1,Ru2]. Knowing that a certain operator is bounded in weighted Lp spaces, for fixed p ∈ [1, ∞) but simultaneously for a large class of weights, he concludes that the operator is bounded in correspondingly weighted Lq spaces for all q ∈ [1, ∞). Originally motivated by Yano’s theorem, in the 1990s Jawerth and Milman proposed a general theory of extrapolation. As Mart´ın and Milman subsequently pointed out, even Rubio de Francia’s results fit in this framework, see [JM,MM]. However, apart from formal similarity, that theory of extrapolation seems to be unrelated to the results in this paper. I am grateful to Richard Rochberg who pointed me to relevant literature, and suggested that the paper be included in this volume in memory of Bj¨ orn Jawerth. 2. Background A holomorphic Banach bundle is a holomorphic map π : E → S of complex Banach manifolds, with each fiber Es = π −1 {s} endowed with the structure of a complex vector space. It is required that for each s0 ∈ S there be a neighborhood U ⊂ S, a complex Banach space W , and a biholomorphic map Φ : U × W → π −1 U (a local trivialization) that maps {s} × W to Es linearly, s ∈ U . If W can be chosen a Hilbert space, we speak of a holomorphic Hilbert bundle. A metric on E is a locally uniformly continuous function p : E → [0, ∞) that restricts on each fiber Es to a norm ps inducing the topology of the fiber. We measure the curvature of a metric as follows, see also [L]. First, if D ⊂ C is open, z0 ∈ D, and u : D → R is upper semicontinuous, we let   1 (2.1) Λu(z0 ) = lim sup r −2 u(z0 + re2πiθ )dθ − u(z0 ) ∈ [−∞, ∞]. r→0

0

In particular, Λu = ∂ u/∂z∂z when u ∈ C 2 (D). Second, if S is a complex manifold, u : S → R is upper semicontinuous, and ξ ∈ T 1,0 S, we let 2

ξξu = inf Λ(u ◦ f )(0) ∈ [−∞, ∞], where the inf is taken over holomorphic maps f of some neighborhood of 0 ∈ C into S that send ∂/∂z ∈ T01,0 C to ξ. Third, returning to a holomorphic Banach bundle E → S endowed with a metric p, we define the Kobayashi curvature Kξ (v) of p, for ξ ∈ Ts1,0 S, v ∈ Es \ {0}, by (2.2)

Kξ (v) = − inf ξξ log p(ϕ)(s),

the inf taken over sections ϕ of E, holomorphic near s, such that ϕ(s) = v. For example, by [L, Lemma 2.3] Kξ (v) ≤ 0 for all ξ, v if and only if log p(ϕ) is plurisubharmonic for all local holomorphic sections of E. (The above definition (2.2) of curvature differs from [L, (2.6)] by a factor of 2.) If E is a Hilbert bundle, p is the metric associated with a hermitian metric h of class C 2 , and R is the curvature operator of h (so it is an End E valued (1, 1) form on S), then Kξ (v) =

h(v, R(ξ, ξ)v) . 2h(v, v)

The duals Es∗ of the fibers Es of a holomorphic Banach bundle also form a holomorphic Banach bundle, denoted E ∗ . If E was locally isomorphic to U × W ,

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then E ∗ will be locally isomorphic to U × W ∗ . If E is endowed with a metric p, the norms p∗s on Es∗ , dual to the norms ps , form a metric on E ∗ , denoted p∗ . Let K ∗ stand for the Kobayashi curvature of p∗ . From now on we assume dim S < ∞. Lemma 2.1. If the Kobayashi curvature of (E ∗ , p∗ ) is semipositive (K ∗ ≥ 0), then the Kobayashi curvature of (E, p) is seminegative (K ≤ 0). When p is a Hilbertian metric, the converse is also true, but we do not know whether this converse holds in general. We also do not know whether in general K ∗ ≤ 0 is equivalent to K ≥ 0.—For the proof of the lemma we need a characterization of plurisubharmonicity: Lemma 2.2. Suppose w : S → R is upper semicontinuous, and for every ξ ∈ T 1,0 S there is a holomorphic map f of a neighborhood of 0 ∈ C into S, sending ∂/∂z ∈ T01,0 C to ξ, such that Λ(w ◦ f )(0) ≥ 0.

(2.3) Then w is plurisubharmonic.

In [L] we already used a similar result. In Lemma 2.3 there the assumption was that (2.3) holds for all holomorphic maps f , which allowed us to quote a theorem of Saks, [Sa]. It turns out that Saks’s proof, through a maximum principle, can be tweaked to prove the stronger Lemma 2.2. Proof. We can assume that S is an open subset of some Cm . It follows from (2.1) that if w ∈ C 2 (S) then Λ(w ◦ f )(0) = ξξw = ∂∂w(ξ, ξ), and the lim sup there is a limit. Of course, the point of Lemma 2.2 is that w is just assumed upper semicontinuous. However, upon replacing w(s) by w(s) + ε|s|2 , with ε > 0, we can assume without loss of generality that whenever ξ ∈ T 1,0 S is nonzero, in (2.3) the strict inequality Λ(w ◦ f )(0) > 0 holds. Consider now a u ∈ C 2 (S) such that for every s ∈ S there is a nonzero ξ ∈ Ts1,0 S with ξξu = ∂∂u(ξ, ξ) = 0. If f is chosen as in the lemma, then Λ(w ◦ f − u ◦ f ) > 0, which shows that w − u cannot have a local maximum in S. To prove that w is plurisubharmonic, we need to take a closed disc Δ ⊂ S, which we assume to be Δ = {s ∈ Cm : |s1 | ≤ r,

s2 = · · · = sm = 0};

a function h ∈ C(Δ), harmonic on the open disc, such that u < h on the boundary of Δ; and prove that this implies u < h on all of Δ. Choose δ > 0 so that P = {s ∈ Cm : |s1 | ≤ r,

|s2 |, . . . , |sm | ≤ δ} ⊂ S,

and define u(s) = h(s1 , 0, . . . , 0) + c(|s2 |2 + · · · + |sm |2 ),

s ∈ P,

where c > 0 is so large that w < u on ∂P . Then ξξu = 0 when ξ = ∂/∂s1 . By what we have said above, this implies w − u has no local maximum in int P , and so it is everywhere negative. In particular, w < u = h on Δ, as needed. 

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Proof of Lemma 2.1. We need to show that log p(ϕ) is plurisubharmonic for any local holomorphic section ϕ of E. Again we can assume S is an open 2 subset of some Cm , and upon replacing p by peε|s| , ε > 0, that Kξ∗ (l) > 0 when ξ = 0, l = 0. Fix s ∈ S and assume ϕ(s) = v = 0. By the Banach–Hahn theorem there is a linear form l ∈ Es∗ of norm 1 such that l, v = p(v). Let ξ ∈ Ts1,0 S \ {0}. As Kξ∗ (l) > 0, there are a section ψ of E ∗ , holomorphic near s, such that ψ(s) = l, and a holomorphic map f of some neighborhood of 0 ∈ C into S, sending ∂/∂z ∈ T01,0 C to ξ, these two satisfying Λ log p∗ (ψ ◦ f )(0) ≤ 0.

(2.4) At the same time

log |ψ ◦ f, ϕ ◦ f | ≤ log p∗ (ψ ◦ f ) + log p(ϕ ◦ f ), with equality at 0 ∈ C. Since the left hand side is harmonic near 0, by (2.4) Λ log p(ϕ ◦ f )(0) ≥ 0. Applying Lemma 2.2 with w = log p(ϕ) we obtain that log p(ϕ) is plurisubharmonic wherever it is finite; and this implies it is plurisubharmonic in fact everywhere.  We will need one more lemma, familiar in the context of hermitian vector bundles. Lemma 2.3. Suppose F → S is a holomorphic Banach bundle with reflexive fibers and q is a metric on F . Letting E → S be another holomorphic Banach bundle, B : F → E a holomorphic epimorphism, and p the induced metric on E: (2.5)

p(v) = inf{q(w) : Bw = v},

v ∈ E,

the curvatures of p and q are related by (2.6)

Kξp (v) ≥ inf{Kξq (w) : Bw = v}.

In particular, K p ≥ 0 if K q ≥ 0. Proof. The point is that in (2.5) the inf is attained. Indeed, if s ∈ S and v ∈ Es is a nonzero vector, the kernel of B|Fs being proximal (as any subspace of Fs , see [C, V.4.6]), B −1 (v) contains a shortest vector, that we will denote w. Given now a holomorphic map f of a neighborhood of 0 ∈ C into S, sending ∂/∂z ∈ T01,0 C to some ξ ∈ Ts1,0 S, and a section ψ of F , holomorphic near s, satisfying ψ(s) = w, let ϕ = B ◦ ψ. Then p(ϕ ◦ f ) ≤ q(ψ ◦ f ), with equality at 0 ∈ C. Hence Λ log p(ϕ ◦ f ) ≤ Λ log q(ψ ◦ f ) at 0, and so Kξp (v) ≥ Kξq (w) by (2.2).



We do not know if the lemma holds without the assumption of reflexivity. For bundles of finite rank and for smooth metrics S. Kobayashi and Rochberg were the first to put forward the notions and results discussed above, see [K, Ro]. In particular, Rochberg was the first to realize the connection between curvature and interpolation.

´ ´ LEMPERT LASZL O

276

3. The main results The following results generalize Theorems 1.1 and 1.2: Theorem 3.1. (i) Consider a holomorphic Hilbert bundle E → S, endowed with a hermitian metric h of class C 2 , and a holomorphic Banach bundle F → S endowed with a metric p. If a holomorphic homomorphism A : E → F decreases curvature in the sense that Kξp (Av) ≤ Kξh (v) for all s ∈ Es , ξ ∈ Ts1,0 S, and v ∈ Es , Av = 0, then log Ais plurisubharmonic. (ii) In particular, if S is a half plane S = {s ∈ C : Re s > σ0 }, A is bounded, and A(s) = A(Re s), then A(t) is a decreasing function of t ∈ (σ0 , ∞). Proof. Part (i) was proved, in a somewhat greater generality, in [L, Theorem 2.4]; see also the discussion at the end of that paper. When rk E < ∞, this was proved earlier in [CS], again in greater generality. Part (ii) immediately follows: under the assumptions log A(s) is subharmonic and independent of Im s, hence log A(t) is convex. If it is also bounded above, it must decrease.  Somewhat related convexity theorems have already occurred in the framework of interpolation; the earliest seem to be in [H, St]. Theorem 3.2. Consider a trivial Banach bundle E = S × V → S over a half plane S = {s ∈ C : Re s > σ0 }, endowed with a metric p such that ps , viewed as norms on V , depend only on Re s. Assume that the second dual p∗∗ of p has semipositive Kobayashi curvature K ∗∗ ≥ 0. If for all l ∈ V ∗ , or at least for l in a dense subset of V ∗ , . (3.1) sup p∗t (l) : t ∈ (σ0 , ∞) < ∞, then pt (v) is an increasing function of t, for all v ∈ V . In particular, pt (v) ≤ lim pτ (v). τ →∞

When p is a Hilbertian metric, and more generally, when V is reflexive, (E ∗∗ , p∗∗ ) and (E, p) are isometrically isomorphic, so that the assumption K ∗∗ ≥ 0 is the same as K ≥ 0. But for a general V we could not prove the theorem just assuming K ≥ 0. Proof. By Lemma 2.1 the dual metric p∗ has seminegative curvature. Hence for any l ∈ V ∗ the function s → log p∗s (l) is subharmonic and log p∗t (l) is convex in t ∈ (σ0 , ∞). Knowing for a dense set of l’s that the latter function is bounded above implies it decreases, and then by continuity it in fact decreases for all l ∈ V ∗ . Let now v ∈ V and t > σ0 . By the Banach–Hahn theorem there is a nonzero l ∈ V ∗ such that l, v = p∗t (l)pt (v). Thus |l, v | ≤ p∗τ (l)pτ (v) p∗τ (l)

for all τ ∈ (σ0 , ∞), p∗t (l)

≤ when τ > t, we must have pτ (v) ≥ with equality when τ = t. Since  pt (v). This proves that pt (v) is an increasing function of t. 4. Application. Extending holomorphic sections We illustrate the technique of extrapolation, Theorem 3.2, by deriving an Ohsawa–Takegoshi type extension theorem. Ohsawa–Takegoshi type refers to extending from a submanifold Y of a complex manifold X holomorphic sections of

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certain vector bundles to all of X; the L2 norm of the extension should be controled by the L2 norm of the data, and the constant in this estimate should depend on crude geometric properties of X and Y . Following the original work of Ohsawa and Takegoshi [OT] a great many instances of such theorems were discovered—the list would be too long to reproduce here. In this section we will discuss a version and its proof that is a variant of what is done in [BL]. The reader will be able to trace back to [BL] most of the ideas in our current proof. If there is improvement it is that some of the ideas in [BL] are no longer needed here. The advantage of this approach is that it is quite painless, and produces arguably sharp estimates. In the setting we treat here we recover the estimates that B locki and Guan–Zhou obtain in [Bl, GZ] by a different approach. We will consider the problem to extend sections of the canonical bundle KX of an m dimensional Stein manifold X. We could deal in the same way with bundles obtained by twisting KX by a Nakano semipositive vector bundle, but to keep notation simple we refrain from doing so. For sections f of KX , i.e., for (m, 0)  forms, there is a natural L2 norm | X f ∧ f |; however, for sections of the restriction of KX to a submanifold Y an L2 norm, which will depend on the geometry of Y , has to be defined. We start with an oriented smooth manifold X of dimension a < ∞, and a submanifold Y ⊂ X of dimension b. We assume given a function r : X → [0, ∞), of class C 3 , whose zero set is Y , and whose critical points on Y are transversely nondegenerate. Let K → X denote the bundle of real valued forms of (maximal) degree a. If ω ∈ K, we denote by |ω| either ω or −ω, whichever is a nonnegative multiple of the orientation form. Given any compactly supported continuous section ϕ of K|Y , we extend it to a compactly supported continuous section ψ of K, and define an L1 –type norm of ϕ by  b−a |ψ|. (4.1) ϕ1 = lim ε ε→0

{x∈X:r(x) 0. t→∞ t > 0, for t > 0, Proposition 4.1. If ψ is a compactly supported continuous section of K and ϕ its restriction to Y (so a section of K|Y ), then  (4.3) ϕ1 = lim e(a−b)t/2 e−χ(t+log r) |ψ|. t→∞

X

Proof. Writing X(t) = {x ∈ X : r(x) < e−t },   (a−b)t/2 −χ(t+log r) (a−b)t/2 e |ψ| = e e X(t)

|ψ| → ϕ1

X(t)

as t → ∞ by (4.1). To estimate the contribution of X \ X(t) to the integral in (4.3) we can assume, as before, that ψ is supported in a coordinate chart and in this chart r = x2b+1 + · · · + x2a . With a suitable M ∈ R and by a change of coordinates y = et/2 x   (a−b)t (a−b)t 2 2 e 2 e−χ(t+log r) |ψ| ≤ M e 2 e−χ(t+log(xb+1 +···+xa )) dxb+1 · · · dxa x2b+1 +···+x2a ≥e−t

X\X(t)



≤ M

e−χ(t+t log(yb+1 +···+ya )) dyb+1 · · · dya → 0 2

2

2 2 ≥1 yb+1 +···+ya

as t → ∞ by dominated convergence (the point being that when t is large, the last 2 integrand is ≤ (yb+1 + · · · + ya2 )b−a by (4.2)).  Now let X ⊃ Y be complex manifolds and KX the canonical bundle of X. A function r ∈ C 3 (X) as above defines an L2 norm for continuous sections ϕ of KX |Y , (4.4)

2

ϕ2Y = in ϕ ∧ ϕ1 ≤ ∞,

n = dimC Y.

Theorem 4.2. Let X be a Stein manifold and Y ⊂ X a complex submanifold. Suppose r : X → [0, 1] is of class C 3 , vanishes precisely on Y , and its critical points on Y are nondegenenate in directions transverse to Y . If log r is plurisubharmonic, any holomorphic section f of KX |Y can be extended to a holomorphic section g of KX satisfying     g ∧ g  ≤ f 2Y . (4.5)  X

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Proof. We will include this extension problem in a family of extension problems Pt to which extrapolation can be applied and which becomes trivial in the limit t → ∞. The extension problems will be of the same nature as what the theorem is claiming to solve, with the difference that on the left of (4.5) the L2 norm will be replaced by weighted L2 norms, with the weights concentrating near Y more and more sharply as t → ∞. Before constructing this family, though, we exhaust X by a sequence Xν of relatively compact pseudoconvex subsets. It will then suffice to find gν ∈ O(KXν ) extending f |Y ∩ Xν such that     gν ∧ g ν  ≤ f 2Y ,  Xν

because once gν found, a subsequence will converge locally uniformly to a g ∈ O(KX ) that extends f and satisfies (4.5). We pick a convex function χ : R → R satisfying (4.2). With ν fixed, for any s ∈ C we define a Hilbert norm on a subspace W ⊂ O(KXν ). If dimC X = m and dimC Y = n, let  1/2  (m−n)Re s/2  e−χ(Re s+log r) ψ ∧ ψ  ≤ ∞, ψ ∈ O(KXν ); qs (ψ) = e  Xν

the Bergman space W = Wν consists of ψ ∈ O(KXν ) for which qs (ψ) < ∞ for some (and then for all) s ∈ C. On W all the norms qs are equivalent, and  1/2  (m−n)Re s/2  ψ ∧ ψ  , when Re s ≤ 0. qs (ψ) = e  Xν

The norms qs together define a metric q on the bundle F = C × W → C. Since the function C × Xν % (s, x) → (n − m)Re s + χ(Re s + log r(x)) is plurisubharmonic, by Berndtsson’s direct image theorem the Kobayashi curvature of q is semipositive. [B, Theorem 1.1] deals with pseudoconvex subdomains of Cm instead of Stein Xν , in which case the canonical bundle is trivial; but the proof carries over to Stein manifolds. Another difference is that [B, Theorem 1.1] claims Nakano semipositivity of the direct image bundle. However, for one dimensional bases like C Nakano semipositivity is the same as Griffiths and Kobayashi semipositivity (the latter two are equivalent for arbitrary bases as long as the metrics involved are hermitian and of class C 2 ). Consider next the closed subspace I ⊂ W consisting of sections ψ that vanish on Y ∩ Xν , and the quotient W/I = V = Vν . The trivial bundle C × V → C is a quotient of F by the subbundle C × I → C, and inherits a Hilbertian metric p, ps (v) = inf{qs (ψ) : ψ ∈ v},

v ∈ V = W/I.

Here inf can be replaced by min; the minimum will be attained by ψ ∈ v that is perpendicular to I, when measured in qs . As a quotient of a semipositively curved metric, p itself will be semipositively curved, see Lemma 2.3. Thus K p∗∗ = K p ≥ 0. To apply Theorem 3.2 we need to check one more assumption, (3.1). We take an arbitrary σ0 < 0, let S = {s ∈ C : Re s > σ0 } and E = S × V → S. So suppose l : V → C is a linear form. Composing it with the projection W → V we obtain a form L ∈ W ∗ whose kernel contains I. Examples of such forms come from sections of K Y ∩Xν ⊗ T m−n,0 Xν |Y as follows (K Y ∩Xν stands for

´ ´ LEMPERT LASZL O

280

the bundle of (0, n) forms on Y ∩ Xν ). Given y ∈ Y , λ = α ⊗ ξ ∈ K Y |y ⊗ Tym−n,0 X, and ψ ∈ KX |y , set λ  ψ = α ∧ ιξ ψ. If now λ is a compactly supported continuous section of K Y ∩Xν ⊗ T m−n,0 Xν |Y , we define Lλ ∈ W ∗ by  λ  ψ, ψ ∈ W. Lλ (ψ) = Y ∩Xν

Clearly, Lλ vanishes on I, and linear forms of this type are dense among forms in W ∗ that vanish on I, because ψ ∈ W must be in I if Lλ (ψ) = 0 for all λ. In checking (3.1) we can restrict ourselves to l ∈ V ∗ induced by such Lλ . Since p∗s (l) = qs∗ (Lλ ), it suffices to check sup qs∗ (Lλ ) < ∞ s∈S

for every compactly supported continuous λ. Even better, we may assume that λ is supported in a coordinate patch on Xν , where Y is given in local coordinates by xn+1 = · · · = xm = 0. We take a ψ ∈ W and apply the submean value theorem to the coefficient in ψ ∧ ψ on balls x1 = const, . . . , xn = const,

|xn+1 |2 + · · · + |xm |2 < εe−Re s ,

with ε > 0 small but fixed independently of s ∈ S. Letting ξ = ∂/∂xn+1 ∧ · · · ∧ ∂/∂xm , this gives     ιξ ψ ∧ ιξ ψ  ≤ Cqs (ψ)2 ,  Y ∩ supp λ

C independent of ψ and s ∈ S. Hence |Lλ (ψ)| ≤ C  qs (ψ) holds, and so sups∈S p∗s (l) = sups∈S qs∗ (Lλ ) < ∞. Therefore Theorem 3.2 applies: for any v ∈ V and t > σ0 pt (v) ≤ lim pτ (v).

(4.6)

τ →∞

This implies the theorem as follows. By Cartan’s Theorem B we can extend f ∈ O(KX |Y ) to ψ ∈ O(KX ). For each ν let vν ∈ Vν be the class of ψ|Xν ∈ Wν . Choose a compactly supported continuous function θ : X → [0, 1] that is 1 on Xν . By (4.6), Proposition 4.1, and (4.4) pt (vν )2

≤

lim pτ (vν )2 ≤ lim sup qτ (ψ|Xν )2 τ →∞     ≤ lim sup e(m−n)τ  e−χ(τ +log r) θψ ∧ θψ ≤ f 2Y . τ →∞

τ →∞

X

This means that for every t there is a gν ∈ vν , i.e., gν ∈ Wν extending f |Y ∩ Xν , such that qt (gν ) ≤ f Y . In particular, setting t = 0     gν ∧ g ν  ≤ f 2Y ,  Xν

which, as we have seen, implies the theorem.



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[Bl] [C] [CS]

[GZ]

[H] [JM] [K] [L]

[MM]

[OT] [R] [Ro] [Ru1] [Ru2] [Sa] [St] [T]

[Y]

Bo Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, Ann. of Math. (2) 169 (2009), no. 2, 531–560, DOI 10.4007/annals.2009.169.531. MR2480611 Bo Berndtsson and L´ aszl´ o Lempert, A proof of the Ohsawa–Takegoshi theorem with sharp estimates, J. Math. Soc. Japan 68 (2016), no. 4, 1461–1472, DOI 10.2969/jmsj/06841461. MR3564439 Zbigniew Blocki, Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math. 193 (2013), no. 1, 149–158, DOI 10.1007/s00222-012-0423-2. MR3069114 John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR768926 Ronald R. Coifman and Stephen Semmes, Interpolation of Banach spaces, Perron processes, and Yang-Mills, Amer. J. Math. 115 (1993), no. 2, 243–278, DOI 10.2307/2374859. MR1216432 Qi’an Guan and Xiangyu Zhou, A solution of an L2 extension problem with an optimal estimate and applications, Ann. of Math. (2) 181 (2015), no. 3, 1139–1208, DOI 10.4007/annals.2015.181.3.6. MR3296822 Isidore I. Hirschman Jr., A convexity theorem for certain groups of transformations, J. Analyse Math. 2 (1953), 209–218. MR0057936 Bj¨ orn Jawerth and Mario Milman, Extrapolation theory with applications, Mem. Amer. Math. Soc. 89 (1991), no. 440, iv+82, DOI 10.1090/memo/0440. MR1046185 Shoshichi Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), 153–166. MR0377126 L´ aszl´ o Lempert, A maximum principle for Hermitian (and other) metrics, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2193–2200, DOI 10.1090/S0002-9939-2015-12472-0. MR3314125 Joaquim Mart´ın and Mario Milman, Extrapolation methods and Rubio de Francia’s extrapolation theorem, Adv. Math. 201 (2006), no. 1, 209–262, DOI 10.1016/j.aim.2005.02.006. MR2204755 Takeo Ohsawa and Kensh¯ o Takegoshi, On the extension of L2 holomorphic functions, Math. Z. 195 (1987), no. 2, 197–204, DOI 10.1007/BF01166457. MR892051 Marcel Riesz, Sur les maxima des formes bilin´ eaires et sur les fonctionnelles lin´ eaires (French), Acta Math. 49 (1927), no. 3-4, 465–497, DOI 10.1007/BF02564121. MR1555250 Richard Rochberg, Interpolation of Banach spaces and negatively curved vector bundles, Pacific J. Math. 110 (1984), no. 2, 355–376. MR726495 Jos´ e L. Rubio de Francia, Vector-valued inequalities for operators in Lp -spaces, Bull. London Math. Soc. 12 (1980), no. 3, 211–215, DOI 10.1112/blms/12.3.211. MR572104 Jos´ e L. Rubio de Francia, Factorization theory and Ap weights, Amer. J. Math. 106 (1984), no. 3, 533–547, DOI 10.2307/2374284. MR745140 Stanislaw Saks, On subharmonic functions. Acta Litt. Sci. Szeged. 5 (1932) 187–193. Elias M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482– 492. MR0082586 G. Olaf Thorin, Convexity theorems generalizing those of M. Riesz and Hadamard with some applications, Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 9 (1948), 1–58. MR0025529 Shigeki Yano, Notes on Fourier analysis. XXIX. An extrapolation theorem, J. Math. Soc. Japan 3 (1951), 296–305. MR0048619

Department of Mathematics, Purdue University, West Lafayette, Indiana 479072067 E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13932

On a dual property of the maximal operator on weighted variable Lp spaces Andrei K. Lerner Abstract. L. Diening [5] obtained the following dual property of the maximal operator M on variable Lebesque spaces Lp(·) : if M is bounded on Lp(·) , then  M is bounded on Lp (·) . We extend this result to weighted variable Lebesque spaces.

1. Introduction Given a measurable function p : Rn → [1, ∞), denote by Lp(·) the space of functions f such that for some λ > 0,  |f (x)/λ|p(x) dx < ∞, Rn

with norm f Lp(·)

  = inf λ > 0 : Rn

|f (x)/λ|

p(x)

 dx ≤ 1 .

p(x) and p+ ≡ ess sup p(x). Set p− ≡ ess inf n x∈R

x∈Rn

Let M be the Hardy-Littlewood maximal operator defined by  1 |f (y)|dy, M f (x) = sup Qx |Q| Q where the supremum is taken over all cubes Q ⊂ Rn containing the point x. In [5], L. Diening proved the following remarkable result: if p− > 1, p+ < ∞  p(x) . and M is bounded on Lp(·) , then M is bounded on Lp (·) , where p (x) = p(x)−1 Despite its apparent simplicity, the proof in [5] is rather long and involved. In this paper we extend Diening’s theorem to weighted variable Lebesgue spaces p(·) Lw equipped with norm f Lp(·) = f wLp(·) . w

We assume that a weight w here is a non-negative function such that w(·)p(·) and  p(·) w(·)−p (·) are locally integrable. The spaces Lw have been studied in numerous works; we refer to the monographs [3, 6] for a detailed bibliography. 2010 Mathematics Subject Classification. 42B20, 42B25, 42B35. Key words and phrases. Maximal operator, variable Lebesgue spaces, weights. This research was supported by the Israel Science Foundation (grant No. 953/13). c 2017 American Mathematical Society

283

284

ANDREI K. LERNER

Recall that a non-negative locally integrable function v satisfies the Muckenhoupt Ar , 1 < r < ∞, condition if 

r−1

  1 1 1 v dx v − r−1 dx < ∞. sup |Q| Q |Q| Q Q Set A∞ = ∪r>1 Ar . Our main result is the following. Theorem 1.1. Let p : Rn → [1, ∞) be a measurable function such that p− > 1 p(·) and p+ < ∞. Let w be a weight such that w(·)p(·) ∈ A∞ . If M is bounded on Lw ,  p (·) then M is bounded on Lw−1 . The relevance of the condition w(·)p(·) ∈ A∞ in this theorem will be discussed in Section 6 below.  p(·)  p (·) p(·) p (·) = Lw−1 (see Notice that Lw−1 is the associate space of Lw , namely, Lw Sections 2.1 and 2.2). Hence, it is desirable to characterize Banach function spaces X with the property that the boundedness of M on X implies the boundedness of M on X  . In Section 3, we obtain such a characterization in terms of an A∞ -type p(·) property of X. However, a verification of this property in the case of X = Lw is not as simple. In doing so, we use some ingredients developed by L. Diening in [5] (Lemmas 5.1 and 5.2). We slightly simplified their proofs and we give them here in order to keep the paper essentially self-contained. 2. Preliminaries 2.1. Banach function spaces. Denote by M+ the set of Lebesgue measurable non-negative functions on Rn . Definition 2.1. By a Banach function space (BFS) X over Rn equipped with Lebesque measure we mean a collection of functions f such that f X = ρ(|f |) < ∞, where ρ : M+ → [0, ∞] is a mapping satisfying (i) ρ(f ) = 0 ⇔ f = 0 a.e.; ρ(αf ) = αρ(f ), α ≥ 0; ρ(f + g) ≤ ρ(f ) + ρ(g); (ii) g ≤ f a.e. ⇒ ρ(g) ≤ ρ(f ); (iii) fn ↑ f a.e. ⇒ ρ(fn ) ↑ ρ(f ); (iv) if E ⊂ Rn is bounded, then ρ(χ  E ) < ∞; (v) if E ⊂ Rn is bounded, then E f dx ≤ cE ρ(f ). Note that it is more common to require that E is a set of finite measure in (iv) and (v) (see, e.g., [1]). However, our choice of axioms allows us to include weighted  p(·) variable Lebesque spaces Lw (with the assumption that w(·)p(·) , w(·)−p (·) ∈ L1loc ) in a general framework of Banach function spaces. Moreover, it is well known that all main elements of a general theory work with (iv) and (v) stated for bounded sets (see, e.g., [13]). We mention only the next two key properties that are of interest for us. The first property says that if X is a BFS, then the associate space X  consisting of f such that  sup |f g| dx < ∞ f X  = g∈X:gX ≤1

Rn

ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

285

is also a BFS. The second property is the Lorentz-Luxemburg theorem saying that X = X  and f X = f X  . The definition of f X  implies that  (2.1) |f g|dx ≤ f X gX  , Rn

and the fact that f X = f X  yields  f X =

(2.2)

sup g∈X  :gX  ≤1

Rn

|f g| dx.

2.2. Variable Lp spaces. It is well known (see [3] or [6]) that if p : Rn →  [1, ∞), then Lp(·) is a BFS. Further, if p− > 1 and p+ < ∞, then (Lp(·) ) = Lp (·) and 1 f Lp (·) ≤ f (Lp(·) ) ≤ 2f Lp (·) 2

(2.3)

(see [6, p. 78]). Assume now that p : Rn → [1, ∞) and w is a weight such that w(·)p(·) and  p(·) w(·)−p (·) are locally integrable. The weighted space Lw consists of all f such that f Lp(·) = f wLp(·) < ∞. w

p(·)

It is easy to see that Lw is a BFS. Indeed, axioms (i)-(iii) of Definition 2.1 follow immediately from the fact that the unweighted Lp(·) is a BFS. Next, (iv) follows from that w(·)p(·) ∈ L1loc . Finally, applying (2.1) with X = Lp(·) along with (2.3) yields  f dx ≤ 2f wLp(·) w−1 χE Lp (·) , E

and this proves (v) with cE = 2w−1 χE Lp (·) < ∞ (here we have used that  w(·)−p (·) ∈ L1loc ). Since f w−1  p(·)  = f  p(·)  , we obtain from (2.3) that if p− > 1 and L Lw  p(·)  p (·) = Lw−1 and p+ < ∞, then Lw 1 f Lp (·) ≤ f  p(·)  ≤ 2f Lp (·) . Lw 2 w−1 w−1 Denote (f ) = (see [3, p. 25]).

 Rn

|f (x)|p(x) dx. We will frequently use the following lemma

Lemma 2.2. Let p : Rn → [1, ∞) and p+ < ∞. If f Lp(·) > 1, then (f )1/p+ ≤ f Lp(·) ≤ (f )1/p− . If f Lp(·) ≤ 1, then (f )1/p− ≤ f Lp(·) ≤ (f )1/p+ .

286

ANDREI K. LERNER

2.3. Dyadic grids and sparse families. The standard dyadic grid in Rn consists of the cubes 2−k ([0, 1)n + j),

k ∈ Z, j ∈ Zn .

Following its basic properties, we say that a family of cubes D is a general dyadic grid if (i) for any Q ∈ D its sidelength Q is of the form 2k , k ∈ Z; (ii) Q ∩ R ∈ {Q, R, ∅} for any Q, R ∈ D; (iii) for every k ∈ Z, the cubes of a fixed sidelength 2k form a partition of Rn . Given a dyadic grid D, consider the associated dyadic maximal operator M D defined by  1 D |f (y)|dy. M f (x) = sup Qx,Q∈D |Q| Q On one hand, it is clear that M D f ≤ M f . However, this inequality can be reversed, in a sense, as the following lemma shows (its proof can be found in [10, Lemma 2.5]). Lemma 2.3. There are 3n dyadic grids Dα such that for every cube Q ⊂ Rn , there exists a cube Qα ∈ Dα such that Q ⊂ Qα and |Qα | ≤ 6n |Q|. We obtain from this lemma that for all x ∈ Rn , n

(2.4)

M f (x) ≤ 6

n

3 

M Dα f (x).

α=1

Given a cube Q0 , denote by D(Q0 ) the set of all dyadic cubes with respect to Q0 , that is, the cubes from D(Q0 ) are formed by repeated subdivision of Q0 and each of its descendants into 2n congruent subcubes. Consider the local dyadic d defined by maximal operator MQ 0  1 d sup |f (y)|dy. MQ0 f (x) = Qx,Q∈D(Q0 ) |Q| Q  1 f . The following lemma is a standard variation of the Denote fQ = |Q| Q Calder´ on-Zygmund decomposition (see, e.g., [8, Theorem 4.3.1]). We include its proof for the reader convenience. Lemma 2.4. Suppose D is a dyadic grid. Let f ∈ Lp (Rn ), 1 ≤ p < ∞, and let γ > 1. Assume that Ωk = {x ∈ Rn : M D f (x) > γ k } = ∅

(k ∈ Z).

Then Ωk can be written as a union of pairwise disjoint cubes Qkj ∈ D satisfying (2.5)

|Qkj ∩ Ωk+l | ≤ 2n (1/γ)l |Qkj |

(l ∈ Z+ ).

The same property holds in the local case for the sets d f (x) > γ k |f |Q0 } Ωk = {x ∈ Q0 : MQ 0

Qkj

(f ∈ L1 (Q0 ), k ∈ Z+ ).

Proof. Consider the case of Rn , the same proof works in the local case. Let be the maximal cubes such that |f |Qkj > γ k . Then, by maximality, they are

pairwise disjoint and |f |Qkj ≤ 2n γ k . Also, Ωk = ∪j Qkj . Therefore,   k+l k k+l |Qi | < (1/γ) |f | ≤ 2n (1/γ)l |Qkj |. |Qj ∩ Ωk+l | = Qk+l ⊂Qk i j

Qk j



ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

287

Definition 2.5. Let D be a dyadic grid, and let 0 < η < 1. We say that a family of cubes S ⊂ D is η-sparse if for every cube Q ∈ S, there is a measurable subset E(Q) ⊂ Q such that η|Q| ≤ |E(Q)| and the sets {E(Q)}Q∈S are pairwise disjoint. Lemma 2.6. Let D be a dyadic grid, and let 0 < η < 1. For every non-negative f ∈ Lp (Rn ), 1 ≤ p < ∞, there exists an η-sparse family S ⊂ D such that for all x ∈ Rn , 2n  M D f (x) ≤ fQ χE(Q) (x). 1−η Q∈S



Proof. For k ∈ Z, set Ωk = x ∈ Rn : M D f (x) > ∪j Qkj ,

|Qkj



2n 1−η

k  . Then, by Lemma

η)|Qkj |.

2.4, Ωk = and ∩ Ωk+1 | ≤ (1 − Therefore, setting E(Qkj ) = k k k Qj \ Ωk+1 , we obtain that η|Qj | ≤ |E(Qj )|, and the sets {E(Qkj )} are pairwise disjoint. Further,  2n   2n k MDf ≤ (M D f )χΩk \Ωk+1 ≤ χΩk \Ωk+1 1−η 1−η k∈Z k∈Z 2n  ≤ fQkj χE(Qkj ) , 1−η j,k

which completes the proof with S =

{Qkj }.



2.4.  Ap weights. Given a weight w and a measurable set E ⊂ Rn , denote w(E) = E wdx. Given an Ap , 1 < p < ∞, weight, its Ap constant is defined by 

p−1

  1 1 [w]Ap = sup wdx w−1/(p−1) dx . |Q| Q |Q| Q Q Every Ap weight satisfies the reverse H¨ older inequality (see, e.g., [9, Theorem 9.2.2]), namely, there exist c > 0 and r > 1 such that for any cube Q,

1/r   1 1 r (2.6) w dx ≤c w dx. |Q| Q |Q| Q It follows from this and from H¨older’s inequality that for every Q and any measurable subset E ⊂ Q,

1/r |E| w(E) (2.7) ≤c . w(Q) |Q| Notice also that the following converse estimate

p |Q| w(Q) ≤ [w]Ap (E ⊂ Q, |E| > 0) (2.8) w(E) |E| holds for all p > 1. Indeed, by H¨older’s inequality,

   p−1 w dx w−1/(p−1) dx , |E|p ≤ E

E

which along with the definition of [w]Ap implies (2.8).

288

ANDREI K. LERNER

3. Maximal operator on associate spaces  p(·)  p (·) Since Lw = Lw−1 , the statement of Theorem 1.1 leads naturally to a question about conditions on a BFS X such that M : X → X ⇒ M : X  → X  . The result below provides a criterion in terms of sparse families and an A∞ -type condition. Its proof is based essentially on the theory of Ap weights. Theorem 3.1. Let X be a BFS such that the Hardy-Littlewood maximal operator M is bounded on X. Let 0 < η < 1. The following conditions are equivalent: (i) M is bounded on X  ; (ii) there exist c, δ > 0 such that for every dyadic grid D and any finite ηsparse family S ⊂ D,

δ     |GQ |      αQ χGQ  ≤ c max αQ χQ  ,   Q∈S |Q| X X Q∈S

Q∈S

where {αQ }Q∈S is an arbitrary sequence of non-negative numbers, and {GQ }Q∈S is any sequence of pairwise disjoint measurable subsets GQ ⊂ Q. Proof. Let us first prove (i) ⇒ (ii). Let g ≥ 0 and gX  = 1. We use the standard Rubio de Francia algorithm [14], namely, set ∞  M kg Rg = , (2M X  )k k=0

where M denotes the k-th iteration of M and M 0 g = g. Then g ≤ Rg and RgX  ≤ 2. Also, M (Rg)(x) ≤ 2M X  Rg(x). Therefore, Rg ∈ A1 . Using the properties of Rg along with (2.7) and H¨ olders inequality (2.1), we obtain that there exist c, δ > 0 such that      αQ χGQ g dx ≤ αQ Rg dx k

Rn

Q∈S

Q∈S

GQ



δ  |GQ | ≤ c αQ Rg dx |Q| Q Q∈S

δ     |GQ | αQ χQ Rg dx ≤ c max Q∈S |Q| Rn Q∈S

δ    |GQ |   ≤ 2c max αQ χQ  .  Q∈S |Q| X 

Q∈S

It remains to take here the supremum over all g ≥ 0 with gX  = 1 and to use (2.2). Turn to the proof of (ii) ⇒ (i). By (2.4), it suffices to prove that the dyadic maximal operator M D is bounded on X  . Let us show that there is c > 0 such that for every f ∈ L1 ∩ X  , (3.1)

M D f X  ≤ cf X  .

Notice that (3.1) implies the boundedness of M D on X  . Indeed, having (3.1) established, for an arbitrary f ∈ X  we apply (3.1) to fN = f χ{|x|≤N } (clearly,

ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

289

fN ∈ L1 ∩ X  ). Letting then N → ∞ and using the Fatou property ((iii) of Definition 2.1), we obtain that (3.1) holds for any f ∈ X  . In order to prove (3.1), by Lemma 2.6, it suffices to show that the operator  MS f = fQ χE(Q) Q∈S

satisfies MS f X  ≤ cf X  for every non-negative f ∈ L1 ∩ X  with c > 0 independent of f and S. Notice that here S = {Qkj }, and Qkj are maximal dyadic cubes forming the set   2n k  . Ωk = x ∈ Rn : M D f (x) > 1−η By duality, it is enough to obtain the uniform boundedness of the adjoint operator !   1  MS f = f χQkj |Qkj | E(Qkj ) j,k on X. Using the Fatou property again, one can assume that S is finite. Take ν ∈ N such that nδ

2 c

∞   1 − η lδ l=ν

2n

≤

1 , 2

where c and δ are the constants from condition (ii). Denote αj,k = ∪j Qkj

1 |Qk j|

 E(Qk j)

f.

\ Ωk+ν = \ Ωk+i+1 , we obtain   αj,k χQkj \Ωk+ν + αj,k χQkj ∩Ωk+ν MS f ≤

Then, using that

∪ν−1 i=0 Ωk+i

j,k

≤ νM f +

j,k ∞  

αj,k χQkj ∩(Ωk+l \Ωk+l+1 ) .

l=ν j,k

Therefore, applying (2.5) along with condition (ii), we obtain MS f X

≤ νM f X +

∞    αj,k χQkj ∩(Ωk+l \Ωk+l+1 ) X l=ν

j,k

≤ νM X f X + 2nδ c

∞   1 − η lδ   αj,k χQkj X 2n l=ν

j,k

1 ≤ νM X f X + MS f X . 2 Since S is finite, by (iv) of Definition 2.1 we obtain that MS f X < ∞. Hence, MS f X ≤ 2νM X f X , and this completes the proof of (ii) ⇒ (i).



290

ANDREI K. LERNER

4. Proof of Theorem 1.1 p(·)

Take X = Lw in Theorem 3.1. All we have to do is to check condition (ii) in this theorem. In order to do that, we need a kind of the reverse H¨older property for the weights (tw(x))p(x) . The following key lemma provides a replacement of such a property which is enough for our purposes. Lemma 4.1. Let 1 < p− ≤ p+ < ∞. Assume that w(·)p(·) ∈ A∞ and that M is p(·) bounded on Lw . Then there exist γ > 1 and c, η > 0, and there is a measure b on n R such that for every cube Q and all t > 0 such that tχQ Lp(·) ≤ 1 one has

|Q|

(4.1)



1 |Q|

≤c

w

1/γ

 (tw(x))γp(x) dx Q

(tw(x))p(x) dx + 2tη b(Q)χ(0,1) (t), Q

and for every finite family of pairwise disjoint cubes π,

 Q∈π

b(Q) ≤ c.

The proof of this lemma is rather technical, and we postpone it until the next Section. Let us see now how the proof of Theorem 1.1 follows. Proof of Theorem 1.1. Let D be a dyadic grid, and let S ⊂ D be a finite 12 sparse family. Let {GQ }Q∈S be a family of pairwise disjoint sets such that GQ ⊂ Q. Take any sequence of non-negative numbers {αQ }Q∈S such that     αQ χQ  p(·) = 1. (4.2)  Lw

Q∈S

By Lemma 2.2 and Theorem 3.1, it suffices to show that there exist absolute constants c, δ > 0 such that

δ  |GQ | (αQ w(x))p(x) dx ≤ c max . (4.3) Q∈S |Q| GQ Q∈S

It follows from (4.2) that αQ χQ Lp(·) ≤ 1 for every Q ∈ S. Therefore, if w αQ ≥ 1, by Lemma 4.1 and H¨older’s inequality along with (4.2) we obtain   (4.4) (αQ w(x))p(x) dx Q∈S:αQ ≥1

GQ



≤

|Q|

Q∈S:αQ ≥1

≤c

 Q∈S:αQ ≥1



|GQ | |Q|

|GQ | |Q|

1/γ 

1 |Q|

1/γ

 (αQ w(x))

dx

Q

1/γ   (αQ w(x)) Q

γp(x)

p(x)

1/γ  |GQ | dx ≤ c max . Q∈S |Q|

The case when αQ < 1 is more complicated because of the additional term on the right-hand side of (4.1). We proceed as follows. Denote Sk = {Q ∈ S : 2−k ≤ αQ < 2−k+1 } (k ∈ N). Let Qki be the maximal cubes from Sk such that every other cube Q ∈ Sk is contained in one of them. Then the cubes Qki are pairwise disjoint (for k fixed).

ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

Set



ψQki (x) =

291

χGQ (x).

Q∈Sk :Q⊆Qk i

Then





(αQ w(x))p(x) dx = GQ

Q∈S:αQ (2n+1 )k αQ } (k ∈ N)

as a union of pairwise disjoint cubes Pjk (Q) satisfying |Ejk (Q)| ≥ 12 |Pjk (Q)|, where Ejk (Q) = Pjk (Q) \ Ωk+1 (Q). From this,    tQ χΩk (Q) ≤ 2M tQ χΩk (Q)\Ωk+1 (Q) , Q∈π

Q∈π

and hence,     tQ χΩk (Q)  

p(·) Lw

Q∈π

Setting tQ =



 Q∈π

    ≤ 2M Lp(·)  tQ χΩk (Q)\Ωk+1 (Q)  w

tQ tQ χΩk (Q) 

, this inequality yields

p(·) Lw

    tQ χΩk (Q)\Ωk+1 (Q)  1 ≤ 2M Lp(·)  w

Q∈π

Since

    tQ χΩk (Q)\Ωk+1 (Q)  

p(·)

Lw

Q∈π

p(·)

.

Lw

    ≤ tQ χΩk (Q) 

p(·)

= 1,

Lw

Q∈π

Lemma 2.2 along with the previous estimate implies,  1 ≤ (tQ w(x))p(x) dx (2M Lp(·) )p+ Ω (Q)\Ω (Q) k k+1 Q∈π w  ≤ 1− (tQ w(x))p(x) dx, Q∈π

Ωk+1 (Q)

which in turn implies (again, by Lemma 2.2)     tQ χΩk+1 (Q)   Q∈π

p(·)

Lw

Q∈π

p(·)

Lw

≤ β,

.

ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

 where β = 1 −

(2M 

1/p+

1

p+ p(·) )

. Hence,

Lw

    tQ χΩk+1 (Q)   and thus, 



p(·) Q∈π tQ χΩk (Q) Lw



(5.1)

    ≤ β tQ χΩk (Q) 

p(·)

Lw

Q∈π

Q∈π

293

Q∈π

p(·)

,

Lw

≤ β k−1 , which by Lemma 2.2 implies (tQ w(x))p(x) dx ≤ β p− (k−1) .

Ωk (Q)

Denote Ω0 (Q) = Q. Then, for ε > 0 we have  ∞   (1+ε)p(x) (tQ w(x)) dx = Q

k=0 ε ≤ αQ

(tQ w(x))(1+ε)p(x) dx

Ωk (Q)\Ωk+1 (Q) ∞ 



2(n+1)(k+1)ε

k=0

∞

(tQ w(x))p(x) dx. Ωk (Q)

Take ε > 0 such that k=0 (2 β ) < ∞. Then, combining the previous estimate with (5.1) and H¨ older’s inequality, we obtain 1

 1+ε   1 (1+ε)p(x) |Q| (tQ w(x)) dx |Q| Q (n+1)ε p− k

Q∈π

≤



vQ (Q)

∞ 

ε 1+ε

Q∈π

≤

1 ! 1+ε

2(n+1)(k+1)ε vQ (Ωk (Q))

k=0

2n+1 ε +

∞ 

1 ! 1+ε

≤ c,

2(n+1)(k+1)ε β p− (k−1)

k=1



and therefore, the proof is complete. p(·)

Lemma 5.2. Assume that M is bounded on Lw . Then there  exist r, k > 1, and a measure b on Rn such that the following properties hold: if Q (tw(x))p(x) dx ≤ 1, then 1/r

  1 (tw(x))rp(x) dx ≤ k (tw(x))p(x) dx + b(Q), (5.2) |Q| |Q| Q Q  and for every finite family of pairwise disjoint cubes π, b(Q) ≤ 2k. Q∈π p+

Proof. Let r and c be the constants from Lemma 5.1. Set k = 2 p− Given a cube Q, denote by A(Q) the set of t > 0 such that  (tw(x))p(x) dx ≤ 1 Q

and (5.3)

|Q|

1 |Q|

1/r

 (tw(x)) Q

rp(x)

dx

 (tw(x))p(x) dx.

>k Q

+1

c.

294

ANDREI K. LERNER

Let tQ = sup A(Q) (if A(Q) = ∅, set tQ = 0). Then  (tQ w(x))p(x) dx < 1. (5.4) Indeed, if

 Q

Q

(tQ w(x))

p(x)

dx = 1, we obtain

|Q|

1 |Q|

1/r

 (tQ w(x))rp(x) dx

≥ k,

Q

and this would contradict Lemma 5.1. Further, we have

1/r   1 (5.5) |Q| (tQ w(x))rp(x) dx = k (tQ w(x))p(x) dx, |Q| Q Q since otherwise (5.3) holds with t = tQ , and by continuity, using also (5.4), we would obtain that tQ + ε ∈ A(Q) for some ε > 0, which contradicts the definition of tQ . Set now 1/r

 1 rp(x) (tQ w(x)) dx . b(Q) = |Q| |Q| Q Then (5.2) holds trivially. Let π be any finite family of pairwise disjoint cubes. Let π  ⊆ π be a maximal  subset such that Q∈π Q (tQ w(x))p(x) dx ≤ 2 (maximal in the sense of the number of elements; this set is not necessarily unique,in general). We claim that π  = π.   Indeed, assume that π = π. Then we have Q∈π Q (tQ w(x)/21/p− )p(x) dx ≤ 1, and by Lemma 5.1,

1/r  p+  1 rp(x) |Q| (tQ w(x)) dx ≤ 2 p− c. |Q| Q  Q∈π

From this and from (5.5),

1/r    1 1  1 p(x) rp(x) (tQ w(x)) dx = |Q| (tQ w(x)) dx ≤ . k |Q| Q 2 Q   Q∈π

Q∈π



Therefore, if P ∈ π \ π , we obtain   Q∈π  ∪{P }

(tQ w(x))p(x) dx ≤

Q

3 , 2

which contradicts the maximality of π  . This proves that π  = π. Hence,   b(Q) = k (tQ w(x))p(x) dx ≤ 2k, Q∈π

Q∈π

Q



which completes the proof. p(·)

Lemma 5.3. Assume that w(·)p(·) ∈ A∞ and that M is bounded on Lw . There exist γ, c > 1 and ε > 0 such that if   0 /  1+ε , (5.6) t ∈ min 1, 1/χQ 1+ε p(·) , max 1, 1/χQ  p(·) Lw

Lw

ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

then



(5.7)

1 |Q|

1/γ

 (tw(x))

γp(x)

1 ≤c |Q|

dx

Q

295

 (tw(x))p(x) dx. Q

Proof. By the definition of A∞ , there is an s > 1 such that w(·)p(·) ∈ As . By (2.6), w(·)p(·) satisfies the reverse H¨ older inequality with an exponent ν > 1. Let r > 1 be the exponent from Lemma 5.1. Take any γ satisfying 1 < γ < min(ν, r). r−γ . Set ε = γ(1+(s−1)r) For every α > 0, 1/γ

1/γ

  1 1 γp(x) γ(p(x)−α) γp(x) (tw(x)) dx = t w(x) dx tα . |Q| Q |Q| Q Next, by (5.6), for all x ∈ Q, γα(1+ε)

tγ(p(x)−α) ≤ 1 + χQ 

p(·)

Lw

and hence,  tγ(p(x)−α) w(x)γp(x) dx

γp(x) (1/χQ 1+ε , p(·) ) Lw

 ≤

Q

w(x)γp(x) dx Q

 +

γα(1+ε)

χQ 

p(·)

Lw

Q

⎛ ⎝

⎞γp(x) w(x) ⎠ χQ 1+ε p(·)

dx.

Lw

Combining this with the previous estimates yields 1/γ

1/γ

  1 1 γp(x) γp(x) (5.8) (tw(x)) dx ≤ w(x) dx tα |Q| Q |Q| Q ⎛ ⎛ ⎞γp(x) ⎞1/γ  ⎟ α(1+ε) ⎜ 1 ⎝ w(x) ⎠ +χQ  p(·) ⎝ dx⎠ tα . 1+ε Lw |Q| Q χQ  p(·) Lw

Let α = mp (Q) be a median value of p over Q, that is, a number satisfying

 |{x ∈ Q : p(x) > mp (Q)}| |{x ∈ Q : p(x) < mp (Q)}| 1 max , ≤ . |Q| |Q| 2 Set E1 = {x ∈ Q : p(x) ≤ mp (Q)} and E2 = {x ∈ Q : p(x) ≥ mp (Q)}. Then |E1 | ≥ 12 |Q| and |E2 | ≥ 12 |Q|. Suppose, for instance, that χQ Lp(·) ≤ 1. Then t ≥ 1. Let us estimate the w first term on the right-hand side of (5.8). Since γ < ν, the reverse H¨older inequality implies 1/γ

  1 1 w(x)γp(x) dx ≤c w(x)p(x) dx. |Q| Q |Q| Q By (2.8) and since |E2 | ≥ 12 |Q|,   p(x) w(x) dx ≤ c Q

E2

w(x)p(x) dx.

296

ANDREI K. LERNER

Using also that t ≥ 1, we obtain 1/γ

 1 γp(x) (5.9) w(x) dx tmp (Q) |Q| Q

≤ ≤

 c mp (Q) t w(x)p(x) dx |Q| E2  c (tw(x))p(x) dx. |Q| Q

Turn to the second term on the right-hand side of (5.8). The boundedness of p(·) M on Lw implies χQ Lp(·) ≤ cχE1 Lp(·) . By Lemma 2.2 (to be more precise, we w w use here a local version of Lemma 2.2; see [3, p. 25] for details),

 1/p+ (E1 )  1/mp (Q) p(x) p(x) w(x) dx ≤ w(x) dx , χE1 Lp(·) ≤ w

E1

E1

where p+ (E1 ) = ess sup p(x). As previously, by (2.8), x∈E1



 w(x)p(x) dx ≤ c

w(x)p(x) dx.

E1

E2

Therefore, combining the previous estimates yields

 1/mp (Q) p(x) w(x) dx . (5.10) χQ Lp(·) ≤ c w

E2

1 Let q = and q  = Then q(1 + ε)γ = r and q  εγ = s−1 . Hence,  H¨ older’s inequality with the exponents q and q along with Lemma 5.1 implies ⎛ ⎞γp(x)  w(x) 1 ⎝ ⎠ dx |Q| Q χQ 1+ε p(·) 1+r(s−1) 1+γ(s−1)

Lw

⎛ 1 ≤⎝ |Q| ≤c

q q−1 .

 Q

dx⎠



w



1 |Q|

⎞1/q

!rp(x)

w(x) χQ Lp(·) w(x)

r−1 q +1

1 − s−1 p(x)

1 |Q|

 w(x)

1 − s−1 p(x)

1/q dx

Q

1/q dx

.

Q

Notice that r 1 r−1 + 1 = +  = (1 + ε)γ + εγ(s − 1) = γ(sε + 1). q q q Therefore, from the previous estimate and from (5.10), ⎛ ⎛ ⎞γp(x) ⎞1/γ  ⎟ mp (Q)ε ⎜ 1 ⎝ w(x) ⎠ χQ  p(·) dx⎠ ⎝ Lw |Q| Q χQ 1+ε p(·) Lw

≤c

|Q|

|Q|sε   r−1 q +1

≤ c[w(·)p(·) ]εAs

1 γ

1 . |Q|

1 |Q|

ε

 w(x)p(x) dx Q

1 |Q|



w(x)− s−1 p(x) dx 1

Q

ε(s−1)

ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

297

From this, using (5.10) again, we obtain ⎛ ⎜ 1 ⎝ |Q|



⎛

⎞1/γ

⎞γp(x)

w(x) ⎠ ⎟ dx⎠ tmp (Q) 1+ε χ  Q Q p(·) Lw   c mp (Q) c t ≤ w(x)p(x) dx ≤ (tw(x))p(x) dx. |Q| |Q| E2 Q mp (Q)(1+ε)

χQ 

p(·)

Lw

⎝

This along with (5.8) and (5.9) proves (5.7). Finally, we note that the proof in the case when χQ Lp(·) ≥ 1 is the same, w  with reversed roles of the sets E1 and E2 . Proof of Lemma 4.1. Assume that tχQ Lp(·) ≤ 1. If t ≥ 1, then the conw clusion of Lemma 4.1 follows immediately from Lemma 5.3. Therefore, it remains to consider the case when t < 1. We may keep all main settings of Lemma 5.3, namely, assume that w(·)p(·) ∈ As , r−γ . and take the same numbers γ and ε = γ(1+(s−1)r) If 1/γ

  1 1 (tw(x))γp(x) dx ≤A (tw(x))p(x) dx, |Q| Q |Q| Q where A > 0 will be determined later, then (4.1) is trivial. Suppose that 1 |Q|

(5.11)

 (tw(x))p(x) dx < Q

1 A



1 |Q|

As in the proof of Lemma 5.3, take q = with the exponents q and q  . We obtain

≤

1 |Q|

1/γ

 (tw(x))γp(x) dx

.

Q

1+r(s−1) 1+γ(s−1)

and apply H¨older’s inequality

1/γ

 (tw(x))γp(x) dx Q

1 |Q|



1

(t 1+ε w(x))rp(x) dx Q

 1+ε

r

1 |Q|



w− s−1 p(x) dx 1

(s−1)ε .

Q

From this, applying H¨ older’s inequality again along with (5.11) yields  1 1 (5.12) (t 1+ε w(x))p(x) dx |Q| Q 1 ε

 1+ε  1+ε

  1 1 p(x) p(x) ≤ (tw(x)) dx w(x) dx |Q| Q |Q| Q 1 ε  γ(1+ε)  1+ε



  1 1 1 γp(x) p(x) ≤ (tw(x)) dx w(x) dx 1 |Q| Q A 1+ε |Q| Q 1/r

 ε 1 1 1 p(·) 1+ε rp(x) 1+ε ≤ ]A s (t w(x)) dx . 1 [w(·) |Q| Q A 1+ε

298

ANDREI K. LERNER 1

Further, from (5.11) and from Lemma 5.3, t 1+ε ≤

1 χQ 

p(·) Lw

(here we assume

that A ≥ c, where c is the constant from Lemma 5.3). Hence, by Lemma 5.2,

1/r   1 1 1 |Q| (t 1+ε w(x))rp(x) dx ≤ k (t 1+ε w(x))p(x) dx + b(Q), |Q| Q Q where b(Q) is defined in the proof of Lemma 5.2. Thus, taking A = max((2k)1+ε [w(·)p(·) ]εAs , c), where c is the constant from Lemma 5.3, and applying (5.12), we obtain

1/r  1 1 |Q| (t 1+ε w(x))rp(x) dx ≤ 2b(Q), |Q| Q which implies

|Q|

1 |Q|

1/r

 (tw(x))

rp(x)

dx

ε

≤ 2t 1+ε p− b(Q).

Q



This along with H¨older’s inequality (since γ < r) proves (4.1). 6. Concluding remarks and open questions

6.1. About the assumption w(·)p(·) ∈ A∞ . We start with the following question. Question 6.1. Is it possible to remove completely the assumption w(·)p(·) ∈ A∞ in Theorem 1.1? Several remarks related to this question are in order. Denote by LH(Rn ) the class of exponents p(·) with p− > 1, p+ < ∞ and such that c c |p(x) − p(y)| ≤ and |p(x) − p∞ | ≤ log(e + 1/|x − y|) log(e + |x|) for all x, y ∈ Rn , where c > 0 and p∞ ≥ 1. Also denote by Ap(·) the class of weights such that sup |Q|−1 χQ Lp(·) χQ Lp (·) < ∞. Q

w

w−1

p(·)

It was shown in [2, 4] that if p(·) ∈ LH(Rn ), then M is bounded on Lw if and only if w ∈ Ap(·) . An important ingredient in the proof in [4] is the fact that if p(·) ∈ LH(Rn ) and w ∈ Ap(·) , then w(·)p(·) ∈ A∞ . Since the boundedness of M p(·) on Lw implies the Ap(·) condition trivially, we see that if p(·) ∈ LH(Rn ), then the assumption that w(·)p(·) ∈ A∞ in Theorem 1.1 is superfluous. However, we do not know whether this assumption can be removed (or at least weakened) in general. It is well known (see, e.g., [3, Th. 3.16]) that if p(·) ∈ LH(Rn ), then M is bounded on Lp(·) . This fact raises the following questions. Question 6.2. Suppose that M is bounded on Lp(·) and w ∈ Ap(·) . Does this imply w(·)p(·) ∈ A∞ ? Question 6.3. Is it possible to replace in Theorem 1.1 the assumption w(·)p(·) ∈ A∞ by the boundedness of M on Lp(·) ?

ON A DUAL PROPERTY OF THE MAXIMAL OPERATOR

299

Question 6.3 is closely related to another open question stated in [7] and [3, p(·) p(·) p. 275]: is it possible to deduce the equivalence M : Lw → Lw ⇔ w ∈ Ap(·) assuming only that M is bounded on Lp(·) ? 6.2. An application. It is a well known principle that if M is bounded on a BFS X and on X  , then some other basic operators in harmonic analysis are also bounded on X. Consider, for instance, a Calder´ on-Zygmund operator T . By this we mean that T is an L2 bounded integral operator represented as  K(x, y)f (y)dy, x ∈ supp f, T f (x) = Rn

with kernel K satisfying |K(x, y)| ≤

c |x−y|n

for all x = y, and for some 0 < δ ≤ 1,

|K(x, y) − K(x , y)| + |K(y, x) − K(y, x )| ≤ c

|x − x |δ , |x − y|n+δ

whenever |x − x | < |x − y|/2. It was shown in [12] that   |T f (x)g(x)|dx ≤ c M f (x)M g(x)dx. Rn

Rn

This estimate along with (2.1) and (2.2) implies that if M is bounded on a BFS X and on X  , then T is bounded on X. Hence, Theorem 1.1 yields the following corollary. Corollary 6.4. Let p : Rn → [1, ∞) be a measurable function such that p− > 1 and p+ < ∞. Let w be a weight such that w(·)p(·) ∈ A∞ . If M is bounded p(·) p(·) on Lw (Rn ), then T is bounded on Lw . As we have mentioned above, it was shown in [4] that if p(·) ∈ LH(Rn ) and p(·) w ∈ Ap(·) , then w(·)p(·) ∈ A∞ and M is bounded on Lw . Therefore, Corollary 6.4 implies the following less general result. p(·)

Corollary 6.5. If p(·) ∈ LH(Rn ) and w ∈ Ap(·) , then T is bounded on Lw . Notice that a closely related result was very recently proved in [11]. Acknowledgement. I am grateful to Alexei Karlovich for valuable remarks on an earlier version of this paper. Also I would like to thank the anonymous referee for detailed comments that improved the presentation. References [1] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR928802 [2] David Cruz-Uribe, Lars Diening, and Peter H¨ ast¨ o, The maximal operator on weighted variable Lebesgue spaces, Fract. Calc. Appl. Anal. 14 (2011), no. 3, 361–374, DOI 10.2478/s13540011-0023-7. MR2837636 [3] David V. Cruz-Uribe and Alberto Fiorenza, Variable Lebesgue spaces, Applied and Numerical Harmonic Analysis, Birkh¨ auser/Springer, Heidelberg, 2013. Foundations and harmonic analysis. MR3026953 [4] D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, Weighted norm inequalities for the maximal operator on variable Lebesgue spaces, J. Math. Anal. Appl. 394 (2012), no. 2, 744–760, DOI 10.1016/j.jmaa.2012.04.044. MR2927495 [5] Lars Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces (English, with English and French summaries), Bull. Sci. Math. 129 (2005), no. 8, 657–700, DOI 10.1016/j.bulsci.2003.10.003. MR2166733

300

ANDREI K. LERNER

[6] Lars Diening, Petteri Harjulehto, Peter H¨ ast¨ o, and Michael R˚ uˇ ziˇ cka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011. MR2790542 [7] L. Diening and P. H¨ ast¨ o, Muckenhoupt weights in variable exponent spaces, preprint. Available at http://www.helsinki.fi/˜hasto/pp/p75 submit.pdf [8] Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR2445437 [9] Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR2463316 [10] Tuomas P. Hyt¨ onen, Michael T. Lacey, and Carlos P´ erez, Sharp weighted bounds for the q-variation of singular integrals, Bull. Lond. Math. Soc. 45 (2013), no. 3, 529–540, DOI 10.1112/blms/bds114. MR3065022 [11] Mitsuo Izuki, Eiichi Nakai, and Yoshihiro Sawano, Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 551–571, DOI 10.5186/aasfm.2015.4032. MR3409692 [12] A. K. Lerner, Weighted norm inequalities for the local sharp maximal function, J. Fourier Anal. Appl. 10 (2004), no. 5, 465–474, DOI 10.1007/s00041-004-0987-3. MR2093912 [13] Wilhelmus Anthonius Josephus Luxemburg, Banach function spaces, Thesis, Technische Hogeschool te Delft, 1955. MR0072440 [14] Jos´ e L. Rubio de Francia, Factorization theory and Ap weights, Amer. J. Math. 106 (1984), no. 3, 533–547, DOI 10.2307/2374284. MR745140 Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13942

Is the Dirichlet space a quotient of DAn ? Richard Rochberg To Bj¨ orn Jawerth, for many good memories. Abstract. We show that the Dirichlet space is not a quotient of the DruryArveson space on the n−ball for any finite n. The proof is based a quantitative comparison of the metrics induced by the Hilbert spaces

1. Statement of the Result We will consider reproducing kernel Hilbert spaces, RKHS’s, on the balls Bn ⊂ Cn , n = 1, 2, ..., ∞. We interpret C∞ as the space 2 (Z+ ) and B∞ as its unit ball. We are interested in D, the Dirichlet space, which consists of holomorphic  2 1 n functions f defined on the unit disk B = D, f (z) = an z , normed using f D =  (n + 1) |an |2 . The space has a reproducing kernel, hz , for evaluating functions at z, given by  1 1 log 1−¯ z w , zw = 0 hz (w) = z¯w 1, zw = 0 +z . We denote the normalized kernels by h The other spaces we consider are the Drury-Arveson spaces, DAn , 1, 2, ..., ∞. The space DAn is the Hilbert space of holomorphic functions on Bn which is defined (n) by the reproducing kernel jz ; for z, w ∈ Bn , jz(n) (w) =

1 . 1 − w, z

Here w, z is the standard inner product on Cn . We will denote the normalized ' (n) kernels by jx and generally omit ”(n)”. We are interested in the following Question 1: Is there, for some finite n, a map Φ : B1 → Bn , and a nonvanishing function λ defined on B1 so that for all z, w ∈ B1 (1.1)

(n)

hz (w) = λ(z)λ(w)jΦ(z) (Φ(w)) ?

The main result in this paper is that this question has a negative answer. 2010 Mathematics Subject Classification. Primary 46E22. Key words and phrases. Reproducing kernel Hilbert space. pseudohyperbolic metric. c 2017 American Mathematical Society

301

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RICHARD ROCHBERG

2. Background The spaces D and DAn have irreducible complete Pick kernels, CPK. An introduction to such spaces is in [AgMc] and we will make free use of the results there. More recent information is in [Sha]. When the general theory of RKHS with CPK is applied to the space D, it insures that the variation of Question 1 in which n is not required to be finite has a positive answer. That result holds in general; if H is an irreducible RKHS of functions on a space X, and if H has a CPK, then there is a map Φ : X → Bn(H) so that the analog of (1.1) holds. Again, n(H) = ∞ may be required. Given such a Φ, denote its range by Φ(X), let Span(jΦ ) be the closed linear span of kernel functions for the points in Φ(X), and let Van(Φ(X)) be the space of functions which vanish on Φ(X). That is, Φ(X) = {Φ(x) : x ∈ X} ⊂ Bn(H) ,   Span(jΦ ) = closed span of jyn(H) : y ∈ Φ(X) ⊂ DAn(H) , . Van(Φ(X))= f ∈ DAn(H) : f (y) = 0 ∀y ∈ Φ(X) ⊂ DAn(H) . n(H)

The map kx → jΦ(x) which takes kernel functions for H to kernel functions in DAn(H) , extends by linearity and continuity to a surjective isometry of H onto ⊥ Span(jΦ ). Also, considering the definitions we see that Span(jΦ ) = Van(Φ(X)). Combining these observations we have that H is the Hilbert space quotient of DAn(H) by Van(Φ(X)) : H ≈ DAn(H) * Van(ΦX (X)) ⊥

≈ Van(ΦX (X)) ≈ Span(jΦ )

This representation of H as a quotient of DAn(H) is the source of the title of this paper. In this situation it is natural to wonder, for each H, what can be said about the optimal value of n(H). The author learned of this question a few years ago in discussions with Ken Davidson and Orr Shalit, and this work began with those conversations. These and similar questions have also been considered by John McCarthy and Orr Shalit [McSh], and by Michael Hartz [Har1]. In particular Theorem 3.1 below is contained in Proposition 11.8 of [Har1]. In that paper the result is obtained through functional analytic considerations. 3. A Reformulation Using the Metric δ We will recast this Hilbert space question as one about isometric mappings between metric spaces. Suppose H is a RKHS of functions  on  X with reproducing kernels {kx : x ∈ X} +x . Define, for all x, w ∈ X, and normalized reproducing kernels k D (3.1)

δ(x, w) = δH (x, w) =

E F  + + 2 1− k x , kw  .

IS THE DIRICHLET SPACE A QUOTIENT OF DAn ?

303

For any x ∈ X let Px be the Hilbert space projection of H onto the span of kx . The functional δ is a metric on X and we have δ(z, w)2 = 1 − Pz Pw 2 = Pz − Pw 2 . The first equality is a straightforward computation. For a proof of the second see [Cob] or [ARSW]. This metric will be our main tool. If H = D the formula for δD does not simplify algebraically. On the other hand, there are informative algebraic rewritings of the formula for δDAn . We begin with the case n = 1. Note that DA1 is the classical Hardy space of the disk. Using the definitions we find that for z, w ∈ B1 = D, G 2 2 (1 − |z| )(1 − |w| ) δDA1 (z, w) = 1 − . |1 − z¯w|2 When we use the fundamental identity, for points x, w ∈ B1 2  (1 − |z|2 )(1 − |w|2 )  z − w  (3.2) 1− =  1 − z¯w  , |1 − z¯w|2 we find that δH 2 = ρ1 , the pseudohyperbolic metric on the disk;    z−w  . ρ1 (z, w) =  1 − z¯w  That metric is characterized by the fact that for z ∈ D, ρ1 (0, z) = |z| , together with the fact that ρ1 is invariant under holomorphic automorphisms of the disk. For general n we have something very similar. From the definitions we see that for z, w ∈ Bn , D E F2   δDAn (z, w) = 1 −  j&z , j+ w  G (1 − z2 )(1 − w2 ) = 1− . 2 |1 − z¯w| Although it is less well known, there is also a pseudohyperbolic metric ρn on Bn . For our purposes a good reference on that metric is [DuWe]. There is an identity for simplifying the expression for δDAn , similar to but more complicated than (3.2). With it one finds that δDAn = ρn . In analogy with n = 1, ρn is characterized by knowing that for any z ∈ Bn (3.3)

ρn (0, z) = z

and that ρn is invariant under holomorphic automorphisms of the ball. Although we will not use this fact, we note in passing that the metric space (Bn , δDAn ) = (Bn , ρn ) is a standard model for complex hyperbolic geometry, [Gol]. Suppose now that Question 1 has a positive answer, and let Φ, λ be the objects guaranteed by that answer. Using (1.1) and the previous discussion of the δ’s and ρ’s, we would have, for all z, w ∈ D, δD (z, w) = δDAn (Φ(z), Φ(w)) = ρn (Φ(z), Φ(w))

304

RICHARD ROCHBERG

(Note that the factors of λ which occur in the definition of Φ cancel completely in this computation. Those factors play no further role.) Hence, to get a negative answer to Question 1 it suffices to get a negative answer to the following question: Question 2: Is there a finite n for which there is an isometric mapping Φ of the metric space (D, δD ) into the metric space (Bn , δDAn )? We will now give a negative answer to that question. Theorem 3.1. Question 2, and hence also Question 1, have negative answers. 4. Preliminary Estimates We want to estimate δD (0, z) for z near the boundary. The situation is clearly rotation invariant so without loss of generality we suppose z is real and positive. For convenience we write z = 1 − σ for small positive σ. We define K by K = − log(2σ − σ 2 )/(1 − σ)2 ∼ − log(2σ), and note that K + ∞ as σ , 0. We then have 1 z2 = , − log(1 − z 2 ) K D 1 (4.1) δD (0, z) = 1 − . K Now we estimate the δD distance that z is moved by a rotation through the small angle σ; that is, we want to estimate δD (z, zeiσ ). From the definitions we have    log 1 − z 2 eiσ  2   2 1 − δD (z, zeiσ ) =    log(1 − z 2 )  2 1 − δD (0, z) =

We have z 2 = 1 − 2σ + σ 2 . Using the Taylor series for log(1 − x) and for exp x we find    log /(2 − i)σ + 2i − 1  σ 2 + O(σ 3 )0 2   2 iσ 2 1 − δD (z, ze ) =     log e−K 2   log(2 − i) + log σ + O(σ)   . =   −K We now use the estimate log σ = −K − log 2 + O(σ) and continue with    −K + log(1 − i/2) + O(σ) 2 2  . (z, zeiσ ) =  1 − δD  −K Write log(1 − i/2) = A + iB with A, B real and A > 0. Hence, with O(1) and O(σ) denoting real quantities, we have 2   O(1)  A + O(σ) 2 iσ  +i (4.2) . 1 − δD (z, ze ) = 1 − K K  2A O(1) + . =1− K K2

IS THE DIRICHLET SPACE A QUOTIENT OF DAn ?

305

The invariant (Poincare-Bergman, hyperbolic) volume of a pseudohyperbolic ball of radius r is a function of the radius only, not the center. If the center is selected to be the origin then one can compute the volume explicitly. We record the formula from [DuWe, (8)], if the radius is r then the volume is (4.3)

V (r) =

r 2n . (1 − r 2 )n

In particular (4.4)

as r → 0

V (r) = r 2n + O(r 2n+1 ) = 2 (1 − r) n

−n

+ O((1 − r)

−n+1

) as r → 1.

5. Proof of the Theorem Proof. Suppose the answer to Question 2 is positive. Select and fix a finite n and a map Φ whose existences are insured by that answer. The ball Bn has a transitive group of biholomorphic automorphisms, each isometric with respect to ρn [DuWe]. Hence, with no loss of generality we assume Φ(0) = 0. Fix a large positive K with the property that, with σ defined as above, N = 2π/σ is an integer. Consider the circle centered at the origin and with radius 1 − σ. On that circle select N equally spaced points {zi } . Now consider the image points {Φ(zi )} . Because Φ is an isometry, and noting (4.1), we have D 1 δDA (0, Φ(zi )) = δD (0, zi ) = 1 − K n Hence all of the " {Φ(zi )} lie on the sphere S in B centered at the origin and with δDAn radius 1 − 1/K. Also, again using the isometry property, and now noting (4.2), we have, for zi = zj , and for some B > 0 " δDA (Φ(zi ), Φ(zj )) = δD (zi , zj ) > B/K. Hence if we pick and fix a small C " > 0, then the δDA balls {Bi } centered at the points {Φ(zi )} and having radius C/K will be disjoint. " These balls have centers on S, the boundary of the ball BδDA (0, 1 − 1/K), and hence certainly do not lie inside that ball. However they do lie inside a slightly large concentric ball whose radius we now estimate. The metric δDAn satisfies a strengthened version of the triangle inequality, [DuWe, Theorem 1. (c)]. Hence if ζ is inside one of the Bi then for large K we can make the following estimates: δDA (0, Φ(zi )) + δDA (Φ(zi ), ζ) 1 + δDA (0, Φ(zi ))δDA (Φ(zi ), ζ) # # 2 1 1− K + CK # ≤ # 1 C2 1+ 1− K K



  1 1 1 2 +C +O =1− 2 K K 3/2 1 . ≤1− 3K The first line is the strengthened triangle inequality for δDA . δDA (0, ζ) ≤

306

RICHARD ROCHBERG

Thus we have N = 2π/σ small balls inside BδDA (0, 1 − 1/ (3K)). By our estimates for their radius and for the distance between their centers we see that the small balls are disjoint. Finally, we have an estimate for the number of them; if K is large then σ < 2πe−K and hence N > eK . However, by comparing volumes, we see that this combination of estimates in impossible. Specifically, from (4.4) we find that, with A and B positive constants that are independent of n, and also independent of K if K is large, we have: N = number of small balls ≥ eK



A VS = volume of each small ball ≥ K n VL = volume of large ball ≤ (BK) .

n

If there were a map Φ as hypothesized then we would have have N VS ≤ VL for all K, no matter how large; but the previous estimates show this is impossible, no matter what the values of A, B.  6. Final Remarks Although the result just proved and various related results are proved more directly in [McSh], this use of δ provides a different insight into what is going on. It is not clear how general the argument is. It does not seem to apply directly to the Hilbert spaces Dα , 0 < α < 1, which are defined by the kernel functions (1 − z¯w)−α . Perhaps this is not surprising. The space D is formally a limiting case of these spaces as α → 0, However it is known that the metric δD is fundamentally different from δDα , 0 < α < 1. That difference is discussed in [Roc]. The theorem is trivially false for α = 1. On the other hand the argument seems to give a similar result for the spaces Hn , the HSRKs of functions on the disk defined by the reproducing kernels n  1 1 log , zw = 0 z¯w 1−¯ zw hz (w) = 1, zw = 0 for n = 2, 3, .... Those spaces are studied in [AMPRS]. It may be that there are local, or even infinitesimal, versions of this argument. Suppose Φ is such a map, normalized by Φ(0) = 0. Perhaps having such a Φ take a small (B1 , δD ) neighborhood of z into (Bn , ρn ), even approximately isometrically, is increasingly difficult, and eventually impossible, as z approaches the boundary. An infinitesimal version might involve a curvature obstacle, either with a curvature defined directly for metric spaces or one involving the Riemannian metrics obtained from the infinitesimal version of the δ  s. The passage to those Riemannian metrics is discussed in [ARSW] and the references mentioned there. On the ball, starting from ρn , that process produces the classical Bergman-Poincare metric, the sectional curvatures of which are always be between two negative constants. If we start from a metric of the form δK on the disk then there are formulas for the curvature of the induced infinitesimal Riemannian metric. If k(z, z¯) is the kernel function then the Riemannian metric is α |dz| with α2 = Δ log k. The curvature is then κ = (−Δ log α) /α2 . When these formulas are used for the Dirichlet space we find the resulting curvature is unbounded below near the boundary.

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Finally, it would be interesting to recast these ideas in Hilbert space terms. The metric δ measures the sine of the angle between reproducing kernels. Hence the analogs of the ”approximate isometries” on the metric spaces would be linear maps between spans of sets of kernel functions, where the maps would be subject to an appropriate rigidity constraint. 7. Afterword Since this was written Michael Hartz has used related ideas to obtain further, related, results [Har2]. In the previous proof we ruled out the existence of an isometric map Φ between two metric spaces. Hartz shows, further, that there cannot even be a bi-Lipshitz map between the spaces. Using this he is able to establish a result for algebraic isomorphisms of the multiplier algebras of the Hilbert spaces which, to put it very roughly, parallels the Hilbert space result above. References Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR1882259 [AMPRS] Arcozzi, N.; Mozolyako, P.; Perfekt, K-M.; Richter, S.; Sarfattisome, G.; Hilbert Spaces Related with the Dirichlet Space arXiv:1512.07532v2 [ARSW] N. Arcozzi, R. Rochberg, E. Sawyer, and B. D. Wick, Distance functions for reproducing kernel Hilbert spaces, Function spaces in modern analysis, Contemp. Math., vol. 547, Amer. Math. Soc., Providence, RI, 2011, pp. 25–53, DOI 10.1090/conm/547/10805. MR2856478 [Cob] L. A. Coburn, Sharp Berezin Lipschitz estimates, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1163–1168 (electronic), DOI 10.1090/S0002-9939-06-08569-8. MR2262921 [DuWe] Peter Duren and Rachel Weir, The pseudohyperbolic metric and Bergman spaces in the ball, Trans. Amer. Math. Soc. 359 (2007), no. 1, 63–76, DOI 10.1090/S0002-9947-0604064-5. MR2247882 [Gol] William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR1695450 [Har1] Hartz, M.; On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces, arXiv:1505.05108. (to appear, Canad. J. Math.) [Har2] Hartz, M.; Embedding Dimension, manuscript, 2015 [McSh] McCarthy, J., Shalit, O.; preliminary manuscript, 2015 [Roc] Richard Rochberg, Structure in the spectra of some multiplier algebras, The corona problem, Fields Inst. Commun., vol. 72, Springer, New York, 2014, pp. 177–200, DOI 10.1007/978-1-4939-1255-1 9. MR3329547 [Sha] Shalit, O.; Operator theory and function theory in Drury-Arveson space and its quotients, arXiv:1308.1081 (to appear in Handbook of Operator Theory). [AgMc]

Department of Mathematics, Campus Box 1146, Washington University, St. Louis, Missouri 63130 E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13928

Characterizations of the Hardy space H 1 (R) and BMO(R) Wael Abu-Shammala, Ji-Liang Shiu, and Alberto Torchinsky Abstract. In this paper we show how to compute the BMO(R) norm in various ways, using the closely related, simpler dyadic and two-sided counterparts. This follows as a consequence of the description of the Hardy space H 1 (R) in terms of dyadic and special atoms. As a result of these characterizations we establish when a bounded linear operator defined on dyadic or two-sided H 1 (R) into a Banach space has a continuous extension to H 1 (R) and when a bounded linear operator that maps a Banach space into dyadic or two-sided BMO(R) maps continuously into BMO(R).

1. Introduction In this paper we seek to elucidate the role that simple atoms, such as the Haar system, play in the theory of the Hardy space H 1 (R). It becomes quickly apparent that the Haar system, or more generally dyadic atoms, do not suffice to span H 1 (R). On the other hand, since arbitrary atoms can be written as the sum of at most three atoms, two dyadic and a special atom, we gain an insight into the structure of H 1 (R), and its dual, BMO(R), by studying these simpler atoms. We describe now the main results. By the Hardy space H 1 (R) we mean the Banach space consisting of those integrable functions which admit an atomic decomposition in terms of L∞ atoms. Recall that a compactly supported function a with vanishing integral is an L∞ atom, or plainly an atom, with defining interval I if  1 a(x) dx = 0 . , and supp(a) ⊆ I, |a(x)| ≤ |I| I H 1 (R) is then ∞ ∞     H 1 (R) = f ∈ L1 (R) : f = λj a j : |λj | < ∞ , 1

1

where the aj are L∞ atoms, the convergence is in the sense of distributions as well as in L1 , and the atomic norm is given by ∞ ∞    f H 1 = inf |λj | : f = λj a j . 1

1

2010 Mathematics Subject Classification. 42B30, 42B35. c 2017 American Mathematical Society

309

310

WAEL ABU-SHAMMALA, JI-LIANG SHIU, AND ALBERTO TORCHINSKY

For these, and all other well–known basic facts used throughout the article, see for instance [12] and [20]. Hd1 (R), or dyadic H 1 (R), is obtained by restricting the defining intervals to be dyadic. Clearly f H 1 ≤ f Hd1 , and the inclusion Hd1 (R) ⊂ H 1 (R) is strict. As for Hd1 (R), the closure of Hd1 (R) in H 1 (R), it turns out to be the space  ∞   f (x) dx = 0 . H A (R) = f ∈ H 1 (R) : 0

More to the point: if {HI } denotes the H 1 normalized Haar system indexed by the dyadic intervals I of R, then sp{HI }, the closed span of the Haar system in H 1 (R), is also H A (R). Now, H A (R) is not a convenient space – for instance, f ∈ H A (R) does not imply that f χ[0,∞) is in H 1 (R) – and we are led to introduce two-sided H 1 , or 1 H2s (R). This is the space of H 1 functions with atomic decompositions in terms of atoms whose defining interval lies on either side of the origin. Endowed with 1 (R) is a proper subspace of H A (R) continuously included in the atomic norm, H2s 1 1 ≤ f Hd1 , and sp 2s {HI }, the closed span of H (R). Also, if f ∈ Hd1 (R), f H2s 1 1 the Haar system in H2s (R), is H2s (R), [19]. As for the atoms themselves, we note that each atom can be written as the sum of at most three atoms, two dyadic and a special atom, [10]. (Special atoms are defined right before Proposition 2 below.) This allows us to identify H 1 (R) as the sum of Hd1 (R) and a space generated by special atoms and to establish the boundedness of linear operators from H 1 (R) taking values in a Banach space X 1 (R). given that they map Hd1 (R) into X. Similar results hold for H A (R) and H2s The space of functions of bounded mean oscillation, BMO(R), consists of those locally integrable functions ϕ with  1 | ϕ(x) − ϕI | dx < ∞ , ϕ∗ = sup I |I| I  1 where ϕI = |I| ϕ(y) dy is the average of ϕ over I, and the sup is taken over all I bounded intervals I. Modulo constants, (BMO,  · ∗ ) is a Banach space. BMOd (R), or dyadic BMO, is defined by restricting the intervals above to be dyadic. The sup is now denoted by  · ∗,d , and modulo two constants (one for each side of the origin), (BMOd ,  · ∗,d ) also becomes a Banach space. Finally, when the above sup is restricted to those I that lie on either side of the origin, the sup is denoted  · 2s , and we have BMO2s (R), or two-sided BMO. Modulo two constants, (BMO2s ,  · 2s ) becomes a Banach space. BMO(R) is the dual of H 1 (R), BMOd (R) is the dual of Hd1 (R), and below we 1 (R) and identify the dual of H A (R). Each note that BMO2s (R) is the dual of H2s 1 of the decompositions of H (R) suggests a characterization of BMO(R) in terms of these dual spaces. These new characterizations of BMO in turn allow us to work in dyadic and dyadic-like settings, and provide us with an effective way to pass from BMOd (R) and BMO2s (R) to BMO(R). We also establish when a bounded linear operator that maps a Banach space X into BMOd (R) actually maps X into BMO(R). The paper is organized as follows. Section 2 is devoted to H 1 (R) and in Section 3 we discuss BMO(R).

CHARACTERIZATIONS OF THE HARDY SPACE H 1 (R) AND BMO(R)

311

2. The Hardy Space H 1 (R) The Haar System. A dyadic interval I is an interval of the form I = In,k = [ k2n , (k + 1)2n ), where k and n are arbitrary integers, positive, negative or 0. Note that I = IL ∪ IR , where the left half IL and the right half IR of I are also dyadic. For each dyadic interval I, the H 1 normalized Haar function HI is given by HI (x) =

1 1 χIL (x) − χI (x) . |I| |I| R

Finally, throughout the paper b will denote the function / 0 b(x) = 12 χ(−1,0) (x) − χ(0,1) (x) . To describe the fit of Hd1 (R) in H 1 (R), let a be an atom. If the origin is not an interior point of the defining interval of a, a is a multiple of a dyadic atom. On the other hand, if the origin is interior to the defining interval of a, let  5  ∞  4  ∞ a(y) dy b(x) − 2 a(y) dy b(x) . a(x) = a(x) + 2 0

0

Since a has vanishing integral the first function above is a linear combination of two dyadic atoms with defining intervals on opposite sides of the origin, and the second is a multiple of (the fixed function) b. Thus Hd1 (R) is of codimension one in H 1 (R), and since Hd1 (R) ⊂ H A (R), actually Hd1 (R) = H A (R). To show that the same is true for the closed span of the Haar system in H 1 (R), we make use of the following observation. Its proof is left to the reader.  Lemma 1. Suppose a locally integrable function ϕ satisfies R HI (x)ϕ(x) dx = 0 for all dyadic intervals I. Then for some constants c, d, ϕ(x) = d χ(−∞,0) (x) + c χ(0,∞) (x) . We then have, Theorem 1. The closed span of the Haar system in H 1 (R) is H A (R). Proof. It suffices to prove that sp d {HI }, the closed span of the Haar system in Hd1 (R), is Hd1 (R). Let L be a bounded linear functional on Hd1 (R) that vanishes on the HI . Then there is ϕ ∈ BMO  d (R) with ϕ∗ ∼ L such that for compactly supported f ∈ Hd1 (R), L(f ) = R f (x)ϕ(x) dx. Now, since L vanishes on the HI , by Lemma 1 above, ϕ(x) = d χ(−∞,0) (x) + c χ(0,∞) (x), and, consequently, for those f,  0  ∞ L(f ) = d f (x) dx + c f (x) dx = 0 . −∞

0

Thus L is the zero functional and we have finished.  The reader will have no difficulty in establishing the quantitative version of Theorem 1 in terms of d(f, H A ), the distance of f in H 1 (R) to H A (R).    ∞  Proposition 1. Let f ∈ H 1 (R). Then d(f, H A ) ∼  0 f (x) dx.

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Structure of Atoms. Since dyadic atoms do not suffice to represent H 1 functions, we consider how far apart dyadic atoms are from arbitrary atoms. The answer is that an arbitrary H 1 atom can be expressed as a sum of at most three atoms, two dyadic and a special atom, [10]. Lemma 2. Let a be an H 1 (R) atom. Then there are at most three atoms aL , aR , bn,k , such that 1. aL and aR are dyadic atoms. 2. For some integers n, k, 0 1 / bn,k (x) = n+1 χ[(k−1)2n , k 2n ] (x) − χ[k 2n ,(k+1)2n ] (x) . 2 3. a = c1 aL + c2 aR + c3 bn,k , where |c1 |, |c2 |, |c3 | ≤ 4 . Proof. Let I be the defining interval for a, and let n be the integer such that 2n−1 ≤ |I| < 2n and k the integer such that I ⊂ [ k 2n , (k + 1)2n ]. Let now  ⎧   ⎨ 1 a(x) − 1 a(y) dy , x ∈ [(k − 1)2n , k 2n ), 2n [(k−1)2n , k 2n ) aL (x) = 4 ⎩ 0, otherwise. Since a is an atom with defining interval I it readily follows that 1  1 1 1 aL ∞ ≤ + ≤ 2−n . ≤ 4 |I| |I| 2 |I| Furthermore, since aL is supported in [(k − 1)2n , k 2n ] and has integral 0, aL is a dyadic atom. Similarly, let  ⎧   ⎨ 1 a(x) − 1 a(y) dy , x ∈ [k 2n , (k + 1)2n ), 2n [k 2n , (k+1)2n ) aR (x) = 4 ⎩ 0, otherwise. aR is supported in [k 2n , (k + 1)2n ], aR ∞ ≤ 2−n , and has integral 0, so aR is also a dyadic atom. Finally put 5 1 4 bn,k (x) = n+1 χ[(k−1)2n , k 2n ] (x) − χ[k 2n , (k+1)2n ] (x) . 2 n n Since  and I ⊂ [(k − 1)2 , (k + 1)2 ] it is also true that  a has vanishing integral − [(k−1)2n , k 2n ] a(y) dy = [k 2n , (k+1)2n ] a(y) dy , and consequently, since a(x) − 4aL (x) − 4aR (x) is equal to ⎧  1 ⎪ n n ⎪ ⎪ ⎨ 2n [(k−1)2n , k 2n ] a(y) dy , x ∈ [(k − 1)2 , k 2 ],  ⎪ 1 ⎪ ⎪ a(y) dy , x ∈ [k 2n , (k + 1)2n ], ⎩ 2n [k 2n , (k+1)2n ]

we have

 a(x) = 4aL (x) + 4aR (x) + 2

 a(y) dy bn,k (x) . [(k−1)2n , k 2n ]

CHARACTERIZATIONS OF THE HARDY SPACE H 1 (R) AND BMO(R)

313

Thus the conclusion follows in this case with c1 = c2 = 4 and  2n ≤ 4.  |a(y)| dy ≤ 2 |c3 | ≤ 2 |I| [(k−1)2n , k 2n ] H 1 (R) as the sum of Banach Spaces. As a first application of the decomposition of atoms we will show that H 1 can be written as the sum of various Banach spaces. We have already seen that H 1 (R) = H A (R) + sp{b} as linear spaces. In fact, the reader will have no difficulty in verifying that H 1 (R) = H A (R) + sp{h}, where h is an arbitrary function in H 1 (R) \ H A (R), and f H 1 ∼ f H A + sp{h} . Furthermore, for integers n, k, let bn,k denote the dyadic dilations and integer translations of b, i.e., the collection of atoms given by / 0 1 χ[(k−1)2n ,k 2n ] (x) − χ[k 2n ,(k+1)2n ] (x) . bn,k (x) = 2n+1 Note that the special atoms bn,k are multiples of dyadic atoms if k is odd, but not if k is even. Also, if k = 0, the support of bn,k lies on one side of the origin. Let Hδ1 (R) denote the space ∞ ∞     λj bnj ,kj : |λj | < ∞ . Hδ1 (R) = ϕ ∈ L1 (R) : ϕ = 1

1

Endowed with the atomic norm ϕHδ1 = inf

∞  1

|λj | : ϕ =

∞ 

 λj bnj ,kj ,

1

(Hδ1 , 

· Hδ1 ) is a Banach space, [1]. Observe that if f ∈ Hδ1 (R), then f ∈ H 1 (R) and f H 1 ≤ f Hδ1 . 1 Similarly, when k = 0 we denote the resulting space Hδ,0 (R). It is clear that if 1 1 1 1 1 . H (R) and H f ∈ Hδ,0 (R), then f ∈ Hδ (R) and f H 1 ≤ f Hδ1 ≤ f Hδ,0 δ δ,0 (R) are the spaces of special atoms alluded to above, [9]. From Lemma 2 it readily follows that Proposition 2. H 1 (R) = Hd1 (R) + Hδ1 (R), and f H 1 ∼ f Hd1 +Hδ1 . The meaning of this decomposition is the following. The Haar system, or more generally the dyadic atoms, divide the line in two regions, (−∞, 0 ] and [ 0, ∞). To allow for the information carried by a dyadic interval to be transmitted to an adjacent dyadic interval, they must be connected. The bn,0 channel information across the origin and the remaining bn,k connect adjacent dyadic intervals that are not subintervals of the same dyadic interval. We also have 1 1 1 +H 1 (R) + Hδ,0 (R), and f H 1 ∼ f H2s . Proposition 3. H 1 (R) = H2s δ,0

This characterization allows us to identify the range of the projection mapping P of H 1 (R) functions f into P f = χ[0,∞) f . It is isomorphic to Ho1 (R), the odd functions of H 1 (R), and consists of those functions in L1 (R+ ) whose Telyakovski˘ı transform also belongs to L1 (R+ ), [11]. The reader will have no difficulty of verifying the following observation, which is useful when considering mappings T : H 1 (R) → X.

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Proposition 4. Let B = B0 + B1 , where B0 , B1 are Banach spaces, and assume T is a linear operator that maps B0 continuously into a Banach space X with norm T 0 . Then T : B → X with norm T  if and only if T : B1 → X is bounded with norm T 1 and T  = max ( T 0 , T 1 ). Proposition 4 applies in particular to Hardy type operators in the setting 1 1 (R) + Hδ,0 (R). Indeed, for 0 ≤ ε ≤ 1, let τε be given by H 1 (R) = H2s  x 1 τε f (x) = f (y) dy , x = 0 . |x|1−ε −x We set 1/p = 1 − ε, 1 ≤ p ≤ ∞ , and consider when τε f is in X = Lp (R). Since 1 (R), the continuity on H 1 (R) is equivalent to that on τε bn,0 = 0 for bn,0 ∈ Hδ,0 1 H2s (R). The case ε = 1 is trivial and merely states τ1 f ∞ ≤ f H 1 . For the remaining cases we begin by observing that for a two-sided atom a with  1/p 1/p ≤  ln |·| ∗,2s . defining interval I, τε a is also supported in I, and R |τε a(x)|p dx  1 Let now f ∈ H2s (R) have the atomic decomposition f (x) = j λj aj . Since the  convergence also takes place in L1 , it readily follows that τε f (x) = j λj τε aj (x), and thus by Minkowski’s inequality, upon taking the infimum over all possible decompositions of f , we get  1/p 1/p 1 . |τε f (x)|p dx ≤  ln | · | ∗,2s f H2s R

1/p

Thus τε maps H 1 (R) into Lp (R) with norm ≤  ln | · | ∗,2s . A similar reasoning applies to the more general operators of Hardy type discussed in [12], which include the Fourier transform. Sublinear operators may be treated in a similar fashion. Consider, for instance, 6ε,d , the maximal operator on H 1 (R) given by M d   1   6 Mε,d f (x) = sup 1−ε  f (y) dy , |I| x∈I I where 0 ≤ ε < 1, and I varies over the collection of dyadic intervals containing x, [5]. If a is a dyadic atom with defining interval I, 6ε,d a(x) ≤ M and, consequently,

  M 6ε,d a ≤ c , p

1 χI (x) , |I|1−ε 0
0 . n2 n=1 Then f ∈ H A (R), and f H 1 ≤ 6 L. On the other hand, since  ∞ fn (x) ln x dx = −n ln 2 + (2 ln 2 − 1) , 0

∞ 1 it readily follows that | 0 f (x) ln x dx| = ∞, and f ∈ / H2s (R). In other words, A f ∈ H (R) and its projection P f is an integrable function with vanishing integral supported in [0, ∞), but P f ∈ / H 1 (R). However, if g ∈ H A (R) vanishes  for x < 0, 1 then P g ∈ H2s (R). For, if g has an atomic decomposition g(x) = j λj aj (x), it also has the two-sided decomposition g(x) =

 j

1 4 λj [aj (x) + aj (−x)]χ[0,∞) (x) . 4

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3. Characterizations of BMO(R) From BMOd (R) to BMO(R). Different characterizations of BMO(R) are suggested when Proposition 4 is restricted to linear functionals. We discuss the dyadic case first. Given a BMO(R) function ϕ, consider the bounded linear functional on H 1 (R) induced by ϕ. When acting on individual atoms two conditions, one for dyadic atoms and the other for special atoms, must be satisfied for this functional to be bounded. The condition on the dyadic atoms suggests that ϕ ∈ BMOd (R), whereas the condition on the bn,k restates that the integral of ϕ is in the Zygmund class. This motivates the following definition. For a locally integrable function ϕ let    1   ϕ(x) dx − ϕ(x) dx , A(ϕ) = sup n+1  n,k 2 [(k−1) 2n ,k 2n ] [k 2n ,(k+1) 2n ] and put Λ = {ϕ ∈ BMOd (R) : A(ϕ) < ∞} ,

  ϕΛ = max ϕ∗,d , A(ϕ) .

Our next result describes how to pass from BMOd (R) to BMO(R). Theorem 3. BMO(R) = Λ. More precisely, if ϕ ∈ BMO(R), then ϕ ∈ Λ and  ϕΛ ≤  ϕ∗ . Also, if ϕ ∈ Λ, then ϕ ∈ BMO(R) and  ϕ∗ ≤ c  ϕΛ . Proof. It is clear that  ϕΛ ≤ c  ϕ∗ . Conversely, suppose that ϕ ∈ Λ and observe that for an atom a = c1 aL + c2 aR + c3 bn,k , we have      a(x)ϕ(x) dx R             ≤ 4  aL (x)ϕ(x) dx + 4  aR (x)ϕ(x) dx + 4  bn,k (x)ϕ(x) dx R

R

R

≤ 4  ϕ ∗,d + 4  ϕ ∗,d + 4 A(ϕ) ≤ 12  ϕΛ .  1 Suppose now that f = ∞ supported and bounded, 1 λn an ∈ H (R) is compactly N  λ a = f in L1 , and that ϕ ∈ Λ is bounded. Since limN →∞ N n n 1 1 λn a n ϕ converges to f ϕ in L1 as N → ∞. Thus   N  lim λn an (x)ϕ(x) dx = f (x)ϕ(x) dx N →∞

and

R

1

R

∞ ∞           |λn | an (x)ϕ(x) dx ≤ c |λn |  ϕΛ .  f (x)ϕ(x) dx ≤ R

R

1  Since the decomposition of f is arbitrary,  R f (x) ϕ(x) dx ≤ c f H 1  ϕΛ . To show that this estimate also holds for arbitrary ϕ ∈ Λ, note that if ϕk denotes the truncation of ϕ at level k, ϕk Λ ≤ c ϕΛ uniformly in k. Thus      f (x)ϕk (x) dx ≤ c f H 1 ϕk Λ ≤ c f H 1 ϕΛ . 1

R

Since f has compact support and ϕ is locally integrable, f ϕ is integrable. Now,   by the dominated convergence theorem, limk→∞ R f (x)ϕk (x) dx = R f (x)ϕ(x) dx, and, consequently,      f (x)ϕ(x) dx ≤ c f H 1 ϕΛ . R

CHARACTERIZATIONS OF THE HARDY SPACE H 1 (R) AND BMO(R)

Finally, since ϕ∗ =

sup f ∈H 1 , f H 1 ≤1

317

     f (x)ϕ(x) dx, R

where f is compactly supported and bounded, ϕ∗ ≤ c ϕΛ .



As an application of the above characterization, the following holds. Proposition 5. Let T be a continuous linear operator defined on a Banach space X which assumes values in BMOd (R) with norm T d . Then T maps X continuously into BMO(R) if and only if for all integers n, k,    1   T f (x) dx − T f (x) dx   ≤ M f X , 2n+1 [(k−1) 2n ,k 2n ] [k 2n ,(k+1) 2n ] and T  = max ( T d , M ). Shifted BMO(R). The process of averaging the translates of functions in BMOd (R) leads to BMO(R), and is an important tool in obtaining results in the latter space once they are known to be true in its dyadic counterpart, BMOd (R), [13]. It is also known that BMO(R) can be obtained as the intersection of BMOd (R) and one of its shifted counterparts, [17]. These results motivate the observations in this section. Given a dyadic interval I = [(k − 1)2n , k 2n ) of length 2n , we call the interval  I = [(k − 1)2n + 2n−1 , k 2n + 2n−1 ) the shifted interval of I by its half-length. Clearly |I  | = |I|, and I  = [(2k − 1)2n−1 , (2k + 1)2n−1 ) is not dyadic. Let J = {Jn,k } be the collection of all dyadic shifted by their half-length, Jn,k = [(k − 1)2n , (k + 1)2n ), all integers n, k, and let BMOds (R) be the space consisting of those locally integrable functions ϕ such that    1 ϕ(x) − ϕJ  dx < ∞ . s ϕ∗, d = sup n,k n,k |Jn,k | Jn,k We then have Theorem 4. BMO(R) = BMOd (R) ∩ BMOds (R). Proof. It is clear that if ϕ ∈ BMO(R), then ϕ∗,d , ϕ∗,ds ≤ ϕ∗ . Conversely, it suffices to show that ϕ ∈ Λ. Since ϕ ∈ BM Od (R) it is enough to verify that A(ϕ) < ∞. For integers n, k, consider    1   ϕ(x) dx − ϕ(x) dx   2n+1 [(k−1) 2n ,k 2n ] n n [k 2 ,(k+1) 2 ]   4 5 4 5  1   ϕ(x) − ϕ dx − ϕ(x) − ϕJn,k dx  Jn,k n+1 2 n n n n [(k−1) 2 ,k 2 ] [k 2 ,(k+1) 2 ]    1   ≤ n+1 ϕ(x) − ϕJn,k  dx ≤ ϕ∗,ds , 2 n n [(k−1)2 ,(k+1) 2 ] =

which implies that A(ϕ) ≤ ϕ∗,ds , and we are done.



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WAEL ABU-SHAMMALA, JI-LIANG SHIU, AND ALBERTO TORCHINSKY

Further Characterizations of BMO(R). We further describe BMO(R) in terms of the duals of the various spaces describing H 1 (R). From H 1(R) = H A (R)+ sp{h} it follows that BMO(R) = B∩(sp{h})∗ , where h ∈ H 1 (R) has [0,∞) h(y) dy = 0. To fix ideas we pick h = −b and introduce the equivalence relation ∼b in BMO(R) as follows. We say that ϕ ∼b ψ if ϕ − ψ = η for some η ∈ BMO with η(y) b(y) dy = 0. We endow these equivalence classes, which we denote by Bb , R with the quotient norm, and observe that the norm in BMO is equivalent to the norm in B ∩ Bb . It is possible, however, to work with a simpler expression. Proposition 6. For a locally integrable function ϕ, let     Ab (ϕ) =  ϕ(y) b(y) dy . R

Let ϕ ∈ BMO(R) be the representative of Φ ∈ B. Then ϕ∗ ∼ max( ΦB , Ab (ϕ) ). Proof. If ϕ ∈ BMO(R), clearly ΦB ≤ ϕ∗ . Also, |Ab (ϕ)| ≤ ϕ∗ bH 1 ≤ ϕ∗ . ˜ B ) . Let A = , Φ b  As for the other inequality, we have ϕ∗ ≤ c max( ΦB∞ ϕ(y) b(y) dy, put ψ(x) = 2A b(x), and observe that ψ ∈ L (R) and ϕ ∼b ψ. R ˜ B ≤ ψ∗ ≤ ψ∞ ≤ |A| = Ab (ϕ), and we have finished.  Then Φ b We also have the following result for BMO2s (R). Proposition 7. A function ϕ ∈ BMO(R) if and only if ϕ ∈ BMO2s (R) and    1    ϕ(y) dy − ϕ(y) dy A0 (ϕ) = sup   < ∞. 2n+1 [−2n ,0] n [0,2n ] Moreover, ϕ∗ ∼ max( ϕ∗,2s , A0 (ϕ) ). We leave the verification of this fact to the reader, and point out an interesting consequence, [3]. Proposition 8. Suppose ϕ ∈ BMO2s (R) is supported in [ 0, ∞). 1. The even extension ϕe of ϕ belongs to BMO(R), and ϕe ∗ = ϕ∗,2s . 2. The odd extension ϕo of ϕ belongs to BMO(R) if and only if   1   ϕ(y) dy  < ∞ , sup n  n 2 [ 0,2n ] and in this case

1  ϕo ∗ ∼ ϕ∗,2s + sup n  n 2

 [ 0,2n ]

  ϕ(y) dy .

As an illustration of the use of the above results we consider the T (1) Theorem, which establishes the continuity in L2 (R) of a standard CZO operator essentially under two kinds of assumptions, the weak boundedness property and the fact that T (1), T ∗ (1) ∈ BMO(R). Indeed, we have [8], Theorem A. Suppose T is a standard CZO that satisfies (WBP) For every interval I, |T χI , χI | ≤ c |I| . (BMO condition) T (1)∗ + T ∗ (1)∗ ≤ c .

CHARACTERIZATIONS OF THE HARDY SPACE H 1 (R) AND BMO(R)

319

Then T is a continuous mapping in L2 (R). In applications it is of interest to state the BMO condition in a form that is easily verified. For instance, in the dyadic setting, the following two conditions may be assumed instead. (1d ) T (1)∗,d + T ∗ (1)∗,d ≤ c . (2d ) For all integers n, k, |T (bn,k ), 1 | + |T ∗ (bn,k ), 1 | ≤ c . Then clearly T (1), T ∗ (1) ∈ BMO(R), and the T (1) Theorem obtains. Similarly, in the two-sided setting, the following two conditions may be used instead of the BMO assumption, (1s ) T (1)∗,2s + T ∗ (1)∗,2s ≤ c . (2s ) For all integers n, |T (bn,0 ), 1 | + |T ∗ (bn,0 ), 1 | ≤ c . Two particular instances of this last observation come to mind. Let T be a CZO with WBP that satisfies (1s ). Also, assume that for any interval I = In,0 , T (χI ) is supported in I, and similarly for T ∗ . Now, since 0 1 / T (bn,0 ), 1 = n+1  T χ[−2n ,0] , χ[−2n ,0] −  T χ[0,2n ] , χ[0,2n ] , 2 by WBP, |T (bn,0 ), 1 | ≤ c. The estimate for T ∗ is obtained in a similar fashion, and therefore (2s ) also holds. Thus T is bounded in L2 (R). Finally, when the kernel of T is even (resp. odd), in x and y, T (1) and T ∗ (1) is even (resp. odd). Now, by Proposition 4, if T (1) is even and T (1) χ[0,∞) ∈ BMO2s (R), then T (1) ∈ BMO(R); similarly for T ∗ (1). On the other hand, if T (1), T ∗ (1) are odd and T (1) χ[0,∞) , T ∗ (1) χ[0,∞) are in BMO2s (R), we also require that     1  1    sup n  T (1)(y) dy , sup n  T ∗ (1)(y) dy  ≤ c . 2 n n n 2 n [0,2 ] [0,2 ] Under these assumptions (1s ) holds and together with (2s ) give the continuity of T in L2 (R). References [1] Wael Abu-Shammala and Alberto Torchinsky, Spaces between H 1 and L1 , Proc. Amer. Math. Soc. 136 (2008), no. 5, 1743–1748, DOI 10.1090/S0002-9939-08-09223-X. MR2373604 [2] Wael Abu-Shammala and Alberto Torchinsky, From dyadic Λα to Λα , Illinois J. Math. 52 (2008), no. 2, 681–689. MR2524660 [3] G´ erard Bourdaud, Remarques sur certains sous-espaces de BMO(Rn ) et de bmo(Rn ) (French, with English and French summaries), Ann. Inst. Fourier (Grenoble) 52 (2002), no. 4, 1187– 1218. MR1927078 [4] Marcin Bownik, Boundedness of operators on Hardy spaces via atomic decompositions, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3535–3542 (electronic), DOI 10.1090/S0002-9939-0507892-5. MR2163588 [5] A. P. Calder´ on, Estimates for singular integral operators in terms of maximal functions, Studia Math. 44 (1972), 563–582. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. MR0348555 [6] A.-P. Calder´ on and A. Torchinsky, Parabolic maximal functions associated with a distribution. II, Advances in Math. 24 (1977), no. 2, 101–171. MR0450888 [7] Ronald R. Coifman, A real variable characterization of H p , Studia Math. 51 (1974), 269–274. MR0358318 [8] Guy David and Jean-Lin Journ´e, A boundedness criterion for generalized Calder´ on-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397, DOI 10.2307/2006946. MR763911 [9] Geraldo Soares De Souza, Spaces formed by special atoms. I, Rocky Mountain J. Math. 14 (1984), no. 2, 423–431, DOI 10.1216/RMJ-1984-14-2-423. MR747289

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[10] S. Fridli, Transition from the dyadic to the real nonperiodic Hardy space, Acta Math. Acad. Paedagog. Nyh´ azi. (N.S.) 16 (2000), 1–8. MR1796256 [11] S. Fridli, Hardy spaces generated by an integrability condition, J. Approx. Theory 113 (2001), no. 1, 91–109, DOI 10.1006/jath.2001.3614. MR1866249 [12] Jos´ e Garc´ıa-Cuerva and Jos´e L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matem´ atica [Mathematical Notes], 104. MR807149 [13] John B. Garnett and Peter W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. MR658065 [14] Alfred Haar, Zur Theorie der orthogonalen Funktionensysteme (German), Math. Ann. 69 (1910), no. 3, 331–371, DOI 10.1007/BF01456326. MR1511592 [15] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR0131498 [16] Robert H. Latter, A characterization of H p (Rn ) in terms of atoms, Studia Math. 62 (1978), no. 1, 93–101. MR0482111 [17] Tao Mei, BMO is the intersection of two translates of dyadic BMO (English, with English and French summaries), C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003–1006, DOI 10.1016/S1631-073X(03)00234-6. MR1993970 [18] Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR1228209 [19] Ji-Liang Shiu, The H1-closure of the Haar system and its dual space, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Indiana University. MR2706032 [20] Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR869816 Department of Mathematics, Indiana University, Bloomington Indiana 47405 E-mail address: [email protected] Department of Mathematics, Indiana University, Bloomington Indiana 47405 E-mail address: [email protected] Department of Mathematics, Indiana University, Bloomington Indiana 47405 E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13930

Four proofs of cocompactness for Sobolev embeddings Cyril Tintarev Abstract. Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some noncompact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the domain space) have subsequences that can be represented as a sum of a well-structured “bubble decomposition” (or defect of compactness) plus a remainder vanishing in the target space. This note is an exposition of different proofs of cocompactness for Sobolev-type embeddings, which employ methods of classical PDE, potential theory, and harmonic analysis.

1. Introduction Cocompactness is a property of continuous embeddings between two Banach spaces, defined as follows. Definition 1.1. Let X be a reflexive Banach space continuously embedded into a Banach space Y . The embedding X → Y is called cocompact relative to a group G of bijective isometries on X if every sequence (xk ), such that gk xk  0 for any choice of (gk ) ⊂ G, converges to zero in Y . In particular, a compact embedding is cocompact relative to the trivial group {I} (and therefore to any other group of isometries), but generally cocompact embeddings are not compact. In this note we study embedding of the Sobolev space H˙ 1,p (RN ), defined as the completion of C0∞ (RN ) with respect to the gradient norm ´ Np 1 ( |∇u|p dx) p , 1 < p < N , into the Lebesgue space L N −p (RN ). This embedding is not compact, but it is cocompact relative to the rescaling group (see (1.1) below). An elementary example of a cocompact embedding that is not compact, due to Jaffard [8], is the embedding of ∞ (Z) into itself, which is cocompact relative to the group of shifts {gn : {xj } → {xj−n }, n ∈ Z}. Indeed, assume that gnk x(k)  0 for any choice of (nk ). Choose a component xkjk of x(k) so that x(k) ∞ ≤ 2|xkjk |. Then, applying the sequence gjk we have x(k) ∞ ≤ 2|gjk x(k) , e0 | → 0. In fact, this type of argument can be used to prove cocompactness of any embedding, where the spaces involved have an unconditional wavelet basis associated with the group, as it is indeed done in [8] for fractional Sobolev spaces, and in [2] for Besov and Triebel-Lizorkin spaces. 2010 Mathematics Subject Classification. Primary 46E35, 46B50, 46B99 Secondary 46E15, 46B20, 47N20. c 2017 American Mathematical Society

321

322

CYRIL TINTAREV

Cocompactness is a tool widely used (often without being referred to by any name, or else as “vanishing lemma” or “inverse embedding”) for proving convergence of functional sequences, in particular in calculus of variations. Cocompactness property implicitly underlies the concentration compactness method of P.-L. Lions, but it is important to stress that the latter was developed for particular function spaces, while cocompactness argument is functional-analytic, and its applications extend beyond the concentration phenomena (see [17]). The earliest cocompactness result known to the author is the proof by E. Lieb [10] of cocompactness of embedding of the inhomogeneous Sobolev space H 1,p into Lq , q ∈ (p, p∗ ), relative to the group of shifts u → u(· − y), y ∈ RN (which is an easy consequence of the cocompactness result discussed here). We refer the reader to a survey [17] of known cocompact embeddings for function spaces. The purpose of this note is to complement that survey of results with four different proofs of cocompactness for Sobolev embeddings. We generalize the second and the fourth proofs found in literature for the case p = 2 to general p, and the fourth proof is shortened in comparison to the original version by means of referring to a known inequality. The third and the fourth proofs can be easily extended to embeddings of Besov and Triebel-Lizorkin spaces, as it is done in, respectively, [2] and (to our best knowledge) in [3]. Cocompactness property plays crucial role in describing the “blow-up” structure of sequences. In presence of a G-cocompact embedding X → Y , any bounded sequence in X, under suitable additional conditions, has a subsequence consisting of asymptotically decoupled “bubbles” of the form gk w with gk ∈ G and w ∈ X, plus a remainder that vanishes in the norm of Y . Existence of such decomposition ∗ was first proved by Solimini [14] for the Sobolev embedding H˙ 1,p (RN ) → Lp (RN ), generalized by several subsequent papers to other spaces of Sobolev type, to more general groups, as well as to Strichartz spaces (see survey [17]), extended to general Hilbert spaces in [13], and to general Banach spaces in [15]. The latter work, however, uses a related, but different, property of Delta-cocompactness, which can be formulated by replacing the weak convergence in the statement of Definition 1.1 with Delta-convergence of T. C. Lim [11]. Definition 1.2. Let X be a metric space. One says a sequence (xn ) ⊂ X has a Delta-limit x if for any y ∈ X there is a sequence of real numbers αn → 0, such that d(xn , y) ≥ d(xn , x) − αn . Delta-convergence and weak convergence coincide in Hilbert spaces, but not in general Banach spaces. Delta-convergence is, however, dependent on a norm, and in separable spaces one can always find an equivalent norm, relative to which Deltaconvergence coincides with weak convergence (van Dulst, [5]). Furthermore, for Besov and Triebel-Lizorkin (including Sobolev) spaces, such equivalent norm exists while it also remains invariant with respect to the rescaling group, defined via the Littlewood-Paley decomposition ([18], Definition 2 in Chapter 5), see Cwikel, [3]. Thus, in the context of spaces of Sobolev type, there is no need to consider Delta-cocompactness as a property distinct from cocompactness. We do not include here the case p = 1, because the natural extension of the notion of cocompactness to a non-reflexive space would invoke the weak-star convergence, which is not defined on H˙ 1,1 . Instead one may regard H˙ 1,1 (RN ) as a closed isometric subspace of the space of functions of bounded variation, and prove ∗ cocompactness of the embedding BV (RN ) → L1 (RN ) for N ≥ 2, as it is done

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in [1]. The proof there is an adaptation of the second proof in this note that takes into account a different chain rule for functions of bounded variation. A larger collection of cocompact embeddings can be produced by restricting homogeneous functional spaces to their subspaces, in particular to inhomogeneous Sobolev spaces. In presence of additional symmetries, such as radial symmetry, cocompactness may even yield compact embedding, such as in the case of subspaces of radial functions. For details on derived cocompact embeddings we refer to [17]. Let λ ∈ R and let (1.1)

Gλ = {gj,y : u(x) → 2jλ u(2j (x − y)), j ∈ Z, y ∈ RN }

∗ with λ = N p−p = N/p∗ . On either of H˙ 1,p (RN ) and Lp (RN ), 1 < p < N , this defines a group of (bijective) linear isometries, often called the rescalings group. In what follows the index λ will be usually omitted. In this note we give four different proofs of the following statement. ∗ Theorem 1.3. The Sobolev embedding H˙ 1,p (RN ) → Lp (RN ), 1 < p < N , is cocompact relative to the rescaling group G N −p . p

It could be possible also to furnish a proof based on atomic decomposition (for ∗ definition see [6]), characterization of the norm in B −N/p ,∞,∞ in terms of the atomic coefficients, and the inequality (4.2), but given that the atomic decomposition is derived from the Littlewood-Paley decomposition, such proof would be too repetitive of the fourth proof here. Throughout this paper we assume that N > p and p ∈ (1, ∞), unless stated otherwise. We will denote an open ball of radius r centered at x as Br (x). We will use the following notations for the conjugate exponent and the critical exponent, p  p,q (RN ) respectively: p∗ = NpN −p , p = p−1 . The quasinorm of the Lorentz space L (for definition see [18]) will be denoted as up,q . The norm of the usual Lebesgue space Lp (RN ), identified with the Lorentz space Lp,p (RN ), will be denoted as up . 2. Capacity-type argument The proof below is given by Sergio Solimini in [14], Section 2. Cocompactness ∗ in H˙ 1,p (RN ) → Lp (RN ) is derived there from cocompactness in the embedding ∗ into the larger space, H˙ 1,p (RN ) → Lp ,∞ (RN ), whose equivalent norm, due to the original characterization of the space by Marcinkiewicz, is ˆ ∗ up∗ ,∞ = sup |E|−1/p |u|. E⊂RN

E

The proof needs the following auxiliary statement. Lemma 2.1. Let r > 0, and let (u(x) − u(z)) dx.

δr (u)(z) = Br (z)

There exist C > 0 such that for every r > 0 every u ∈ H˙ 1,p (RN ), (2.1)

δr (u)p ≤ Cr∇up .

324

CYRIL TINTAREV

Proof. It suffices to prove (2.1) for u ∈ C0∞ (RN ). Writing the function u in polar coordinates ρ, ω centered at a point z ∈ R, we have ˆ ρ ∂t u(t, ω)dt. u(ρ, ω) − u(0) = 0

Integrating with respect to ω over the unit sphere we easily arrive at ˆ  ˆ   |∇u(x)|   ≤ C (2.2) (u(x) − u(z)) dx dx.    Br (z)  |x − z|N −1 Br (z) Inequality (2.1) follows then immediately by applying Young inequality for convolutions, once we note that the L1 -norm of the restriction of 1/|x|N −1 to Br (0) is a multiple of r.  We can now prove Theorem 1.3. Proof. 1. It suffices to prove cocompactness of the embedding H˙ 1,p (RN ) → ∗ Lp ,∞ (RN ). Indeed, there is a continuous embedding H˙ 1,p (RN ) → Lp ,p (RN ) (expressed by the well-known Hardy inequality), and from application of H¨older inequality to the definition of Lorentz quasinorm one has ∗

(2.3)

∗

∗

∗

p −p p upp∗ ≤ Cupp∗ ,p upp∗ −p ,∞ ≤ C∇up up∗ ,∞ . ∗

Thus any sequence bounded in H˙ 1,p (RN ) and vanishing in Lp ,∞ (RN ) will also ∗ vanish in Lp (RN ). 2. Consider a bounded sequence (vk ) ⊂ H˙ 1,p (RN ) which does not converge to zero in the norm of Lp,∞ . Passing to a renamed subsequence, we may assume that vk p,∞ ≥ 4 for some  > 0. Then, by definition of the Marcinkiewicz norm, there exist a sequence of measurable sets Yk ⊂ RN such that ˆ 1 |vk | ≥ 2|Yk | p∗ . Yk

Let us choose integers jk so that the measure of the sets Xk = 2jk Yk falls into the interval [1, 2N ] and let uk = gjk ,0 vk . Note that uk p,∞ = vk p,∞ ≥ 4, and ˆ |uk | ≥ 2. Xk

From Lemma 2.1 it now follows that there exists a r > 0 such that ˆ |δr (uk )| ≤ . Xk

Comparing the last two relations and using the triangle inequality we have  ˆ     uk dx dz ≥ ,    Xk Br (z) and thus there exist points zk ∈ Xk such that       uk dx ≥ 2−1−N .   Br (zk )  In other words,

ˆ uk (· − zk )dx → 0. Br (0)

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325

This implies that uk (x − zk ) does not converge weakly to zero, or, in terms of the N −p original sequence, 2 p jk vk (2jk (· − yk )) does not converge weakly to zero. This ∗ contradiction proves that the embedding of H˙ 1,p (RN ) into Lp ,∞ is cocompact, ∗  and, subsequently, the embedding H˙ 1,p (RN ) → Lp (RN ) is cocompact.

3. Proof by partition of domain and range ∗ Cocompactness of the embedding H˙ 1,2 (RN ) → L2 (RN ) for N > 2 is also proved by Lemma 5.10 in [16], and its argument can be easily modified for general p ∈ (1, N ) giving another proof of Theorem 1.3.

Proof. We may assume without loss of generality that uk ∈ C0∞ (RN ). Let (uk ) ⊂ H˙ 1,p (RN ) and assume that for any (jk ) ⊂ Z and any (yk ) ⊂ RN , gjk ,yk u 0. Let χ ∈ C0∞ (( 21 , 4), [0, 3]), such that |χ | ≤ 2 for all t and χ(t) = t for t ∈ [1, 2]. By ∗ continuity of the embedding H˙ 1,p (RN ) → Lp (RN ), we have for every y ∈ ZN , !p/p∗

ˆ χ(|uk |)

p∗

ˆ ≤C

(|∇uk |p + χ(|uk |)p ),

(0,1)N +y

(0,1)N +y ∗

from which follows, if we take into account that χ(t)p ≤ Ctp for t ≥ 0, ˆ ∗ χ(|uk |)p (0,1)N +y

ˆ

!1−p/p∗

ˆ

≤ C

(|∇uk | + χ(uk ) ) p

χ(|uk |)

p

(0,1)N +y

(0,1)N +y

ˆ

!1−p/p∗

ˆ

≤ C

p∗

(|∇uk |p + χ(|uk |)p )

|uk |p

(0,1)N +y

.

(0,1)N +y

Adding the above inequalities over y ∈ ZN and taking into account that χ(t)p ≤ ∗ Ctp for t ≥ 0, so that

ˆ

ˆ χ(|uk |) ≤ C

p∗ /p |∇uk |

p

RN

RN

≤ C,

p

we get ˆ (3.1) RN

χ(|uk |)

p∗

!1−p/p∗

ˆ ≤ C sup y∈ZN

|uk |

p

.

(0,1)N +y

Let yk ∈ ZN be such that !1−p/p∗

ˆ |uk |

sup y∈ZN

(0,1)N +y

p

!1−p/p∗

ˆ ≤2

|uk | (0,1)N +yk

p

.

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CYRIL TINTAREV

Since uk (· − yk )  0 in H˙ 1,p (RN ) and by the local compactness of subcritical Sobolev embeddings, ˆ ˆ |uk |p = |uk (· − yk )|p → 0. (0,1)N +yk

(0,1)N

Substituting this into (3.1), we get ˆ RN

∗

χ(|uk |)p → 0.

Let χj (t) = 2j χ(2−j t)), j ∈ Z. Note that we may substitute for the original sequence uk a sequence gjk ,0 uk , with arbitrary jk ∈ Z, and so we have ˆ ∗ (3.2) χjk (|uk |)p → 0. RN

Note now that, with j ∈ Z,

ˆ p/p∗ ˆ ∗ χj (|uk |)p ≤C RN

2j−1 ≤|u

which can be rewritten as ˆ ˆ p∗ χj (|uk |) ≤ C (3.3) RN

k

|≤2j+2

|∇uk |p ,

ˆ

2j−1 ≤|uk |≤2j+2

|∇uk |

p RN

χj (|uk |)

p∗

1− pp∗ .

Adding the inequalities (3.3) over j ∈ Z and taking into account that the sets 2j−1 ≤ |uk | ≤ 2j+2 cover RN with a uniformly finite multiplicity, we obtain

ˆ 1−p/p∗ ˆ ˆ p∗ p p∗ |uk | ≤ C |∇uk | sup χj (|uk |) . (3.4) RN

RN

Let jk be such that

ˆ sup j∈Z

RN

χj (|uk |)

p∗

j∈Z

RN

ˆ

1−p/p∗ ≤2

RN

χjk (|uk |)

p∗

1−p/p∗ ,

and note that the right hand side converges to zero due to (3.2). Then from (3.4) ∗  follows that uk → 0 in Lp , which yields the cocompactness. 4. Proof by the wavelet decomposition We give here the proof of cocompactness in Sobolev embeddings by Stephane Jaffard, [8]. Note that Jaffard’s result covers spaces H˙ s,p for general s, and that it has been further extended in [2] to a greater range of embeddings involving Besov and Triebel-Lizorkin spaces, which also admit an unconditional wavelet basis of rescalings of a mother wavelet. This proof is the shortest in this survey, because the hard analytic part of the argument relies on the wavelet analysis. Specifically, we use the following results (see [4, 12] for details). There exists a function ψ : RN → R,

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327

N −p

called mother wavelet, such that the family {2 p j ψ(2j − k); j ∈ Z, k ∈ ZN } ⊂ {gψ, g ∈ G N −p } forms an unconditional basis in H˙ 1,p (RN ) as well as (among others) p ∗ in the Besov space B˙ −N/p ,∞,∞ . Moreover, if cj,k [u] are coefficients in the expansion of u in this basis, one has the following equivalent norm: (4.1)

uB˙ −N/p∗ ,∞,∞ =

sup j∈Z,k∈ZN

|cj,k [u]|.

We can now prove Theorem 1.3. Proof. The starting point of the proof is the following inequality (see [7]): (4.2)

p/p∗

1−p/p∗

up∗ ≤ CuH˙ 1,p (RN ) uB˙ −N/p∗ ,∞,∞ .

Assume that for any (jn ) ⊂ Z and any (yn ) ⊂ RN , gjn ,yn un  0 in H˙ 1,p (RN ). Since cj,k [u] are continuous functionals on H˙ 1,p (RN ) and since for any (j, k), (j  , k ) ∈ Z × ZN there exists a g ∈ G such that ck,j [gu] = ck ,j  [u], it follows that for any (jn , kn ) ∈ Z × ZN , cj,k [un ] → 0 as n → ∞, and, consequently, using (4.1), we have (4.3)

un B˙ −N/p∗ ,∞,∞ =

sup j∈Z,k∈ZN

|cj,k [un ]| → 0.

∗

Then, by (4.2), un → 0 in Lp .



5. Proof via the Littlewood-Paley decomposition The proof below is a modification of the argument given in [9], Chapter 4: we replaced their intermediate step, inequality (4.16) of Proposition 4.8 in [9], with a related inequality (4.2) which we already used above. The proof is generalized from the case p = 2 to the case p ∈ (1, N ). For details concerning the Littlewood-Paley theory we refer the reader to Chapter 5 of [18] or Chapter 5 of [6]. The Littlewood-Paley family of operators {Pj }j∈Z is based on existence of a family of functions {ϕj }j∈Z with the following properties: . (5.1) supp {ϕj ⊂ ξ ∈ RN , 2j−1 ≤ |ξ| ≤ 2j+1 (5.2)



ϕn (ξ) = 1 for all ξ ∈ RN \ {0} .

j∈Z

(5.3)

ϕj (ξ) = ϕ0 (2−j ξ) for all ξ ∈ RN and j ∈ Z.

(5.4)

ϕj−1 (ξ) + ϕj (ξ) + ϕj+1 (ξ) = 1 for all ξ ∈ suppϕj .

Then Pj : H˙ 1,p (RN ) → R, j ∈ Z, are given by Pj u = F −1 ϕ0 (2−j ·)F u, where F is the Fourier transform, say, normalized as a unitary operator in L2 (RN ). ∗ In what follows, we will use the following equivalent norm of B˙ −N/p ,∞,∞ (RN ) (cf (4) in Chapter 5 of [18] or (5.2) in [6]): (5.5)

uB˙ −N/p∗ ,∞,∞ = sup 2− p∗ j Pj u∞ . N

j∈Z

We can now prove Theorem 1.3.

328

CYRIL TINTAREV

Proof. Assume that for any (jn ) ⊂ Z and any (yn ) ⊂ RN , gjn ,yn un  0 in ∗ H˙ 1,p (RN ). By (4.2) it suffices to show that uk → 0 in B˙ −N/p ,∞,∞ (RN ). By (5.5), it suffices then to show that (5.6)

sup 2− p∗ j Pj uk ∞ → 0. N

j∈Z

or, equivalently, that for any sequence (jk ) ⊂ Z, (5.7)

2− p∗ jk Pjk uk ∞ → 0, N

or, equivalently, setting vk = gjk ,0 uk and noting that vk (·−yk )  0 for any sequence (yk ) ⊂ RN , (5.8)

P0 vk ∞ → 0.

Choose any s0 > N/p, so that H˙ s0 ,p (RN ) is compactly embedded into C(RN ). Since P0 vk is bounded in H˙ s0 ,p (RN ), we have vk (· − yk )  0 in H˙ s0 ,p (RN ) for any sequence (yk ) ⊂ RN , and (5.8) follows by compactness of the embedding into ∗  C(RN ). This implies (5.6), and by (5.5) uk → 0 in Lp (RN ).

References [1] Adimurthi and Cyril Tintarev, Defect of compactness in spaces of bounded variation, J. Funct. Anal. 271 (2016), no. 1, 37–48, DOI 10.1016/j.jfa.2016.04.002. MR3494241 [2] Hajer Bahouri, Albert Cohen, and Gabriel Koch, A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Math. 3 (2011), no. 3, 387–411, DOI 10.1142/S1793744211000370. MR2847237 [3] M. Cwikel, Opial’s condition and cocompactness for Besov and Triebel-Lizorkin spaces, in preparation. [4] Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR1162107 [5] D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. (2) 25 (1982), no. 1, 139–144, DOI 10.1112/jlms/s2-25.1.139. MR645871 [6] Michael Frazier, Bj¨ orn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR1107300 [7] Patrick Gerard, Yves Meyer, and Fr´ed´ erique Oru, In´ egalit´ es de Sobolev pr´ ecis´ ees (French), ´ ´ S´ eminaire sur les Equations aux D´eriv´ees Partielles, 1996–1997, Ecole Polytech., Palaiseau, 1997, pp. Exp. No. IV, 11. MR1482810 [8] St´ ephane Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal. 161 (1999), no. 2, 384–396, DOI 10.1006/jfan.1998.3364. MR1674639 [9] Rowan Killip and Monica Vi¸san, Nonlinear Schr¨ odinger equations at critical regularity, Evolution equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 325– 437. MR3098643 [10] Elliott H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), no. 3, 441–448, DOI 10.1007/BF01394245. MR724014 [11] Teck Cheong Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182 (1977). MR0423139 [12] Yves Meyer, Ondelettes et op´ erateurs. I (French), Actualit´ es Math´ ematiques. [Current Mathematical Topics], Hermann, Paris, 1990. Ondelettes. [Wavelets]. MR1085487 [13] I. Schindler and K. Tintarev, An abstract version of the concentration compactness principle, Rev. Mat. Complut. 15 (2002), no. 2, 417–436, DOI 10.5209/rev REMA.2002.v15.n2.16902. MR1951819

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[14] Sergio Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space (English, with English and French summaries), Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 12 (1995), no. 3, 319–337. MR1340267 [15] Sergio Solimini and Cyril Tintarev, Concentration analysis in Banach spaces, Commun. Contemp. Math. 18 (2016), no. 3, 1550038, 33, DOI 10.1142/S0219199715500388. MR3477400 [16] Kyril Tintarev and Karl-Heinz Fieseler, Concentration compactness, Imperial College Press, London, 2007. Functional-analytic grounds and applications. MR2294665 [17] C. Tintarev, Concentration analysis and compactness, in: Adimurthi, K. Sandeep, I. Schindler, C. Tintarev, editors, Concentration Analysis and Applications to PDE ICTS Workshop, Bangalore, January 2012, ISBN 978-3-0348-0372-4, Birkh¨ auser, Trends in Mathematics (2013), 117-141. [18] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkh¨ auser Verlag, Basel, 1983. MR781540 Department of Mathematics, Uppsala University, 75106 Uppsala, Sweden E-mail address: [email protected]

Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13929

Tempered homogeneous function spaces, II Hans Triebel Abstract. This paper deals with homogeneous function spaces of BesovSobolev type within the framework of tempered distributions in Euclidean n-space based on Gauss-Weierstrass semi-groups. Related Fourier-analytical descriptions are incorporated afterwards as so-called domestic norms. This approach avoids the usual ambiguity modulo polynomials when homogeneous function spaces are considered in the context of homogeneous tempered distributions. The motivation to deal with these spaces comes from (nonlinear) heat and Navier-Stokes equations, but also from Keller-Segel sytems and other PDE models of chemotaxis.

Contents 1. Introduction and motivation 2. Definitions and basic assertions 2.1. Preliminaries and inhomogeneous spaces 2.2. Spaces with negative smoothness 2.3. Spaces in the distinguished strip 3. Properties 3.1. Embeddings 3.2. Global inequalities for heat equations 3.3. Hardy inequalities 3.4. Multiplication algebras 3.5. Examples I: Riesz kernels 3.6. Examples II: Mixed Riesz kernels References

1. Introduction and motivation This paper is a complementing survey to [T15]. We try to be selfcontained repeating some definitions and basic assertions. Let Asp,q (Rn ), n ∈ N, A ∈ {B, F } with (1.1)

0 < p, q ≤ ∞

and

s∈R

be the nowadays well-known  function spaces treated in the frame inhomogeneous work of the dual pairing S(Rn ), S  (Rn ) . The theory of these spaces, including 2010 Mathematics Subject Classification. Primary 46E35, 42B35. Key words and phrases. Homogeneous function spaces.

331

c 2017 American Mathematical Society

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their history, references and special cases may be found in [T83, T92, T06]. Here S(Rn ) is the usual Schwartz space of infinitely differentiable rapidly decreasing functions in Rn and S  (Rn ) is its dual, the space of tempered distributions. The related homogeneous function spaces A˙ sp,q (Rn ) again with A ∈ {B, F } and (1.1) are   ˙ n ), S˙  (Rn ) where usually treated in the context of the dual pairing S(R . ˙ n ) = ϕ ∈ S(Rn ) : (Dα ϕ)(0) (1.2) S(R & = 0, α ∈ Nn0 . Recall that ϕ & is the Fourier transform of ϕ. A first brief account of these homogeneous spaces, again with references and special cases, had been given in [T83, Chapter 5]. A more detailed elaborated and updated version of the theory of these homogeneous spaces may be found in [T15, Chapter 2]. It is well known that these homogeneous spaces must be considered modulo polynomials. This causes some disturbing (topological) troubles if one uses homogeneous spaces in some applications, especially in the context of nonlinear PDEs. In [T15] we offered a theory ∗

of tempered homogeneous function spaces Asp,q (Rn ), A ∈ {B, F } in the framework   of the dual pairing S(Rn ), S  (Rn ) where p, q, s are restricted (mostly but not exclusively) to the distinguished strip 1  n (1.3) 0 < p, q ≤ ∞, n −1 −1 + np and subcritical if s < −1 + np . Details may be found in [T15, pp. 1,2]. Of special interest are the critical and supercritical spaces with −1 + np ≤ s < np . This fits in the scheme (1.4). Our own approach to Navier-Stokes equations in [T13, T14] is based on inhomogeneous supercritical spaces (and is local in time). But most of the recent contributions to Navier-Stokes equations in the context of Fourier analysis rely on −1+ n homogeneous critical spaces A˙ p,q p (Rn ). This applies also to other nonlinear PDEs originating from physics. The reader may consult [BCD11] and some more recent papers quoted in [T13, T14]. An additional motivation to deal with tempered homogeneous function spaces comes from chemotaxis, the movement of biological cells or organisms in response to chemical gradients. Detailed descriptions including the biological background and the model equations studied today may be found in the surveys [Hor03, Hor04], [HiP09] and the recent book [Per15]. In addition to the classical approach (smooth functions in bounded domains) a lot of attention has been paid in recent times to study the underlying Keller-Segel systems in the context of the above spaces Asp,q (Rn ), A˙ sp,q (Rn ). The prototype of these equations is given by (1.8)

∂t u − Δu + div (u∇v) = 0,

x ∈ Rn , 0 < t < T ,

(1.9)

∂t v − Δv + αv = u,

x ∈ Rn , 0 < t < T ,

(1.10)

u(·, 0) = u0 ,

x ∈ Rn ,

(1.11)

v(·, 0) = v0 ,

x ∈ Rn ,

where α ≥ 0 is the so-called damping constant. Here u = u(x, t) and v = v(x, t) are scalar functions describing the cell density and the concentration of the chemical signals, respectively. If α = 0 then one is in a similar position as in the case of the Navier-Stokes equations and one can again ask which spaces Asp,q (Rn ), A˙ sp,q (Rn ) should be called critical, supercritical and subcritical for Keller-Segel equations. If u(x, t), v(x, t) is a solution of (1.8), (1.9) then uλ (x, t) = λ2 u(λx, λ2 t), vλ (x, t) = v(λx, λ2 t) is also a solution of (1.8), (1.9) now in Rn × (0, λ−2 T ), where one has to adapt the initial data. Then the spaces ∗ n 0 < p, q ≤ ∞, s = −2 + , (1.12) Asp,q (Rn ), A˙ sp,q (Rn ), Asp,q (Rn ), p are called critical (for Keller-Segel equations), supercritical if s > −2 + np and subcritical if s < −2 + np . Of special interest are the critical and supercritical spaces with −2 + np ≤ s ≤ np . Again this seems to fit in the scheme of (1.3), (1.4). Details may be found in [T16]. But there are some differences compared with Navier-Stokes equations. Homogeneous spaces with (1.3) cover the cases of interest for Navier-Stokes equations in Rn with 2 ≤ n ∈ N. Keller-Segel systems with n = 1 (ordinary nonlinear equations) attracted some attention but they are not covered by our approach in [T16]. According to the literature the most interesting case for Keller-Segel systems is n = 2 (biological cells in so-called Petri dishes in response to chemicals). But then (1.12) (with n = 2) suggests to have a closer ∗

look at (tempered) homogeneous spaces Asp,q (Rn ), 0 < p, q ≤ ∞ in the limiting   cases s = n p1 − 1 and s = np . It is one aim of this paper to complement some corresponding considerations in [T15] in this direction. But on the other hand it is our main aim to continue the study of the tempered homogeneous spaces

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∗

Asp,q (Rn ) mostly with (1.3) asking for further properties. We deal also with some examples based on Riesz kernels demonstrating the decisive differences between the above homogeneous spaces and their (more flexible) inhomogeneous counterparts Asp,q (Rn ). ∗

In Section 2 we introduce the tempered homogeneous spaces Asp,q (Rn ) following closely [T15]. We repeat some basic assertions and prove new ones (mostly in ∗

limiting situations). Section 3 deals with new properties of the spaces Asp,q (Rn ) complementing [T15]. 2. Definitions and basic assertions 2.1. Preliminaries and inhomogeneous spaces. We use standard notation. Let N be the collection of all natural numbers and N0 = N ∪ {0}. Let Rn be Euclidean n-space, where n ∈ N. Put R = R1 , whereas C is the complex plane. Let S(Rn ) be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on Rn and let S  (Rn ) be the space of all tempered distributions on Rn , the dual of S(Rn ). Let D(Rn ) = C0∞ (Rn ) be the collection of all functions f ∈ S(Rn ) with compact support in Rn . As usual D (Rn ) stands for the space of all distributions in Rn . Furthermore, Lp (Rn ) with 0 < p ≤ ∞, is the standard complex quasi-Banach space with respect to the Lebesgue measure, quasi-normed by  1/p n |f (x)|p dx (2.1) f |Lp (R ) = Rn

with the usual modification if p = ∞. Similarly Lp (M ) where M is a Lebesguemeasurable subset of Rn . As usual Z is the collection of all integers; and Zn where n ∈ N denotes the lattice of all points m = (m1 , . . . , mn ) ∈ Rn with mk ∈ Z. Let Qj,m = 2−j m + 2−j (0, 1)n with j ∈ Z and m ∈ Zn be the usual dyadic cubes in Rn , n ∈ N, with sides of length 2−j parallel to the axes of coordinates and 2−j m as n the lower left corner. As usual, Lloc p (R ) collects all locally p-integrable functions f , that is f ∈ Lp (M ) for any bounded Lebesgue measurable set M in Rn . If ϕ ∈ S(Rn ) then  e−ixξ ϕ(x) dx, ξ ∈ Rn , (2.2) ϕ(ξ) & = (F ϕ)(ξ) = (2π)−n/2 Rn

denotes the Fourier transform of ϕ. As usual, F −1 ϕ and ϕ∨ stand for the inverse Fourier transform, given by the right-hand side of (2.2) with i in place of −i. Here xξ stands for the scalar product in Rn . Both F and F −1 are extended to S  (Rn ) in the standard way. Let ϕ0 ∈ S(Rn ) with (2.3)

ϕ0 (x) = 1 if |x| ≤ 1

and ϕ0 (x) = 0 if |x| ≥ 3/2,

and let (2.4)

ϕk (x) = ϕ0 (2−k x) − ϕ0 (2−k+1 x),

x ∈ Rn ,

Since (2.5)

∞  j=0

ϕj (x) = 1

for

x ∈ Rn ,

k ∈ N.

TEMPERED HOMOGENEOUS FUNCTION SPACES, II

335

ϕ = {ϕj }∞ j=0 forms a dyadic resolution of unity. The entire analytic functions ∨ & (ϕj f ) (x) make sense pointwise in Rn for any f ∈ S  (Rn ). s We recall the well-known definitions of the inhomogeneous spaces Bp,q (Rn ) and s n Fp,q (R ). (i) Let 0 < p ≤ ∞,

(2.6) Then (2.7)

0 < q ≤ ∞, 

s ∈ R.

is the collection of all f ∈ S (R ) such that ∞  q 1/q  s (Rn )ϕ = 2jsq (ϕj f&)∨ |Lp (Rn ) 0 and, some N ∈ N. Further explanations and related references may be found in [T14, Section 4.1]. We give a description how the above Fourier-analytical definition of Asp,q (Rn ) with s < 0 can be replaced by corresponding characterizations in terms of heat kernels. Let (2.15)

s0

t−s/2 |Wt f (x)|,

f |C s (Rn ) =

sup x∈Rn ,0 0,  (3.10) Wt f (x) 2 ≤ c t− 2 −1 √ Wτ f (y) t/2

|x−y|≤ t

[T15, (3.16)] with a reference of [Bui83, Lemma 2, p. 172]. One has  t  1/q2  q2   −s/2  −n/2 −sq2 /2    dy dτ Wt f (x) ≤ c t W t τ f (y) τ √ τ |x−y|≤ t 0 (3.11) ∗

≤ c f |F s∞,q2 (Rn ) where we used again (2.26). Then the last embedding in (3.3) with q2 < ∞ follows from (3.2). If q2 = ∞ then one has (2.28), where the indicated short proof with a ∗

reference to [T15, p. 47] is just a modification of (3.10), (3.11). Let f ∈ F s∞,q1 (Rn ). By (2.26), (3.2) and what we already know the last but one inequality in (3.3) follows from (3.12)  ∗ q −q 1/q2  f |F s∞,q (Rn ) ≤ sup τ −s(q2 −q1 )/2 Wτ f (y) 2 1 2

τ >0,y∈Rn

×

sup



t−n/2

 t

x∈Rn ,t>0 ∗

0 q 1− q1 2

≤ c f |C s (Rn ) ∗

≤ c f |F s∞,q1 (Rn ).

1/q2 q1  −sq1 /2   dy dτ τ f (y) W τ √ τ |x−y|≤ t

∗

q1

f |F s∞,q1 (Rn ) q2

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Based on (2.24) and (3.2) one can argue similarly to prove the first embedding in (3.3),  ∞ ∗ ∗ q0  q dτ 1/q1  f |B s∞,q1 (Rn ) ≤ c f |C s (Rn )1− q1 τ −q0 /2 sup Wτ f (z) 0 τ z∈Rn 0 (3.13) ∗

≤ c f |B s∞,q0 (Rn ). This completes the proof of (3.3). In the same way one can prove the inhomogeneous counterpart (3.4) using (2.16)–(2.18) and (3.2). Step 2. We prove (3.5). By (3.3) we may assume 0 < q < p < ∞. Using the domestic equivalent quasi-norms (2.89) one obtains ∗ s+

n

∗ s+

n

B p1 ,qp1 (Rn ) → B p2 ,qp2 (Rn ),

(3.14)

0 < p1 < p2 ≤ ∞,

in the same way as in the inhomogeneous case, [T83, Theorem 2.7.1, p. 129]. This shows that in addition to 0 < q < p < ∞ one may assume −n < s < s + np < 0. ∗ s+ n

older’s We rely again on (2.26), (2.24) and assume f ∈ B p,∞p (Rn ). One has by H¨ inequality  t dτ −n − sq 2 2 |W f (y)|q dy t τ √ τ τ |x−y|≤ t 0   t  pq dτ  sq n p τ − 2 t− 2 |W f (y)| dy ≤c τ √ τ |x−y|≤ t 0 (3.15)  t q nq nq dτ n τ 2p ≤ c sup σ − 2 (s+ p ) Wσ f |Lp (Rn )q t− 2p τ σ>0 0 ∗ s+ n

≤ c f |B p,∞p (Rn )q . This proves the first embedding in (3.5). Similarly one justifies the second embedding based on (2.16), (2.18). Step 3. The sharp embedding (3.8) for inhomogeneous spaces goes back to [SiT95], based on [Jaw77, Fra86]. One may also consult [ET96, p. 44] and the additional references within. If 0 < p < ∞, 0 < q ≤ ∞ and σp < σ < n/p then the spaces ∗

Aσp,q (Rn ) and Aσp,q (Rn ) coincide locally, (3.16)

∗

g |Aσp,q (Rn ) ∼ g |Aσp,q (Rn ),

g ∈ S  (Rn ),

supp g ⊂ Ω,

with, say, Ω = {y : |y| < 1}. This is covered by [T15, Proposition 3.52, p. 107]. Then one has for some c1 > 0, c2 > 0 (3.17)

∗

∗

∗

c1 f |B sp11 ,v (Rn ) ≤ f |F sp,q (Rn ) ≤ c2 f |B sp00 ,u (Rn )

if, in addition s1 > 0 (then also s > 0, s0 > 0), and supp f ⊂ Ω. All spaces in (3.17) have the same homogeneity s − np according to (2.85). Then one can extend (3.17) by homogeneity to all compactly supported functions f . The rest is now a matter of the Fatou property of the underlying spaces as explained in Remark 2.11 resulting in (3.7) so far for s1 > 0. By [T15, Proposition 3.41, p. 97] ∗ ∗  ∨ n I˙σ f = |ξ|σ f& , (3.18) I˙σ Asp,q (Rn ) = As−σ p,q (R ),

TEMPERED HOMOGENEOUS FUNCTION SPACES, II

351

is a lift within the distinguished strip. This extends (3.7) to all spaces which can be reached in this way, based on what we already know. This covers all spaces with p1 < ∞. If p1 = ∞ and p < p2 < ∞, s2 − pn2 = s1 , then one has by (3.7) with ∗

B sp22 ,v (Rn ) on the right-hand side and (3.14) (3.19)

∗

∗

∗

p ≤ v.

1 F sp,q (Rn ) → B sp22 ,v (Rn ) → B s∞,v (Rn ),

As for the sharpness of (3.20)

∗

∗

p≤v

1 F sp,q (Rn ) → B s∞,v (Rn ),

we may assume by lifting s > 0. Recall s1 < 0. Then one obtains by (2.34) and (3.16) (3.21)

∗

s1 s 1 (Rn ) → B s∞,v (Rn ) → B∞,v (Rn ) Fp,q

if supp f ⊂ Ω. But the embedding (3.8) and its sharpness is a local matter. This means that (3.21) requires p ≤ v. This proves part (iii).  Remark 3.2. The embedding (3.5) for the inhomogeneous spaces is known and goes back to [Mar95, Lemma 16, p. 253] (with a short Fourier-analytical proof). 3.2. Global inequalities for heat equations. Let Asp,q (Rn ) be the inhomogeneous spaces as introduced in Section 2.1 and let Wt w be given by (2.12), (2.13). Let 1 ≤ p, q ≤ ∞ (p < ∞ for F -spaces), s ∈ R and d ≥ 0. According to [T14, Theorem 4.1, p. 114] there is a constant c > 0 such that for all t with 0 < t ≤ 1 and all w ∈ Asp,q (Rn ), (3.22)

n s n td/2 Wt w |As+d p,q (R ) ≤ c w |Ap,q (R ). ∗

We ask for a counterpart in terms of the tempered homogeneous spaces Asp,q (Rn ) according to Definition 2.8(i) in the distinguished strip (2.69). Theorem 3.3. Let n ∈ N. Let 1 < p < ∞, 1 ≤ q ≤ ∞ and  1 n (3.23) n −1 0,

λ > 0,

is an immediate consequence of (2.12), (2.13). We replace w in (3.25) by w(λ·), λ > 0. Using the homogeneity (2.85) one obtains (3.27)

∗

n

n

∗

s− p n w |Asp,q (Rn ). td/2 λs+d− p Wtλ2 w |As+d p,q (R ) ≤ c λ ∗

This proves (3.24) for all w ∈ Asp,q (Rn ) and all t > 0.



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Remark 3.4. One can extend the above theorem to the limiting cases covered by Theorem 2.10. Of special interest may be the spaces with s + d = n/p. In particular, if  1 n (3.28) 1 < p < ∞, n − 1 < s < 0 and s + d = , p p then ∗

n

n

s

∗

p t 2p Wt w |B p,1 (Rn ) ≤ c t 2 w |B sp,1 (Rn )

(3.29) ∗

for all w ∈ B sp,1 (Rn ) and all t > 0. The special case n < p < ∞ and s = −1 + may be of special interest in the context of Navier-Stokes equations, that is ∗n ∗ n −1 √ p p (3.30) t Wt w |B p,1 (Rn ) ≤ c w |B p,1 (Rn ) ∗

n

n p

−1

p (Rn ) and all t > 0. for all w ∈ B p,1

∗

3.3. Hardy inequalities. We have a closer look at some spaces Asp,q (Rn ) according to Definition 2.8(i) consisting entirely of regular tempered distributions. Let 0 < p < ∞, 0 < q ≤ ∞ and   1 n n n − =s− . (3.31) σp = max 0, n − 1 < s < , p p r p ∗

Then 1 < r < ∞. Let s/2 < m ∈ N. Then F sp,q (Rn ) is the collection of all regular n tempered distributions f ∈ S  (Rn ) ∩ Lloc 1 (R ) such that    ∞ q dt 1/q  s   t(m− 2 )q ∂tm Wt f (·) |Lp (Rn ) + f |Lr (Rn ) (3.32)  t 0 ∗

is finite. This may be interpreted as an equivalent domestic quasi-norm in F sp,q (Rn ) as introduced in Definition 2.8(i) if s > 0. Furthermore,   ∞  q dt 1/q  s   n t(m− 2 )q ∂tm Wt f (·) |Lp (Rn ), (3.33) f |Lr (R ) ≤ c  t 0 ∗

∗

f ∈ F sp,q (Rn ). Similarly for B sp,q (Rn ) under the additional restriction 0 < q ≤ r. Then  ∞ q dt 1/q  s n (3.34) f |Lr (R ) ≤ c t(m− 2 )q ∂tm Wt f |Lp (Rn ) , t 0 ∗

f ∈ B sp,q (Rn ). Details may be found in [T15, Section 3.3, pp. 57–62]. But instead ∗

of Lr , having the same homogeneity − nr = s − np as Asp,q (Rn ) one can use weighted Lp -spaces and weighted Lq -spaces with the same homogeneity. For this purpose we recall first the Hardy inequalities for the corresponding inhomogeneous spaces Asp,q (Rn ). Let p, q, s and r be as as above. Then there is a constant c > 0 such that    n s  |x| r f (x)p dx ≤ c f |Fp,q (3.35) (Rn )p |x|n |x|≤1

TEMPERED HOMOGENEOUS FUNCTION SPACES, II

353

s for all f ∈ Fp,q (Rn ). Let, in addition, 0 < q ≤ r. Then there is a constant c > 0 such that    n s  |x| r f (x)q dx ≤ c f |Bp,q (3.36) (Rn )q n |x| |x|≤1 s for all f ∈ Bp,q (Rn ). Details and further explanations may be found in [T01, Theorem 16.3, p. 238]. We extend these assertions to the related tempered homogeneous spaces. Let σp be as in (3.31).

Theorem 3.5. Let 0 < p < ∞, (3.37)

σp < s
0 such that ∗   n  |x| r f (x)p dx ≤ c f |F sp,q (Rn )p n |x| Rn

∗

for all f ∈ F sp,q (Rn ). (ii) Let 0 < q ≤ r. Then there is a constant c > 0 such that  ∗   n  |x| r f (x)q dx ≤ c f |B sp,q (Rn )q (3.39) n |x| Rn ∗

for all f ∈ B sp,q (Rn ). Proof. Using (3.35) and (3.16) one has  ∗   n  |x| r f (x)p dx ≤ c f |F sp,q (Rn )p (3.40) n |x| Rn ∗

∗

for all f ∈ F sp,q (Rn ) with supp f ⊂ {y : |y| ≤ 1}. Let λ > 1 and f ∈ F sp,q (Rn ) with supp f ⊂ {y : |y| ≤ λ}. We insert f (λ·) in (3.40). By the homogeneity (2.85) and an obvious counterpart of the left-hand side of (3.40) with the same homogeneity ∗

one can extend (3.40) to all compactly supported f ∈ F sp,q (Rn ). The rest is now a matter of the Fatou property as explained in Remark 2.11. This proves (3.38). The proof of (3.39) is similar.  Remark 3.6. Let p, s be as in (3.37) and 0 < q ≤ ∞. Let s/2 < m ∈ N. Then both (3.32) and    n   ∞ 1/p q dt 1/q   s    |x| r f (x)p dx t(m− 2 )q ∂tm Wt f (·) |Lp (Rn ) + (3.41)  n t |x| Rn 0 ∗

are equivalent domestic quasi-norms in F sp,q (Rn ). Furthermore (3.42)   ∞   1/p  q dt 1/q  n   (m− s2 )q  m n    |x| r f (x)p dx ∂ ≤ c t W f (·) |L (R )   t p t |x|n t Rn 0 and (3.43)



∞    1/p   n  j ∨ q 1/q  jsq  n  &) (·)  |x| r f (x)p dx (ϕ ≤ c 2 |L (R ) f   p |x|n Rn j=−∞

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HANS TRIEBEL ∗

for some c > 0 and all f ∈ F sp,q (Rn ). Here {ϕj } is the usual dyadic resolution of unity in Rn \ {0}. This is covered by Theorem 2.10. If, in addition, 0 < q ≤ r then  1/q  ∞ q dt 1/q     n s  |x| r f (x)q dx t(m− 2 )q ∂tm Wt f |Lp (Rn ) + (3.44) t |x|n Rn 0 ∗

is an equivalent domestic quasi-norm in B sp,q (Rn ). There are obvious counterparts of (3.42), (3.43). 3.4. Multiplication algebras. Let Asp,q (Rn ) be the inhomogeneous spaces as introduced in (2.6)–(2.9). Recall that Asp,q (Rn ) is called a multiplication algebra if f1 f2 ∈ Asp,q (Rn ) for any f1 ∈ Asp,q (Rn ), f2 ∈ Asp,q (Rn ) and if there is a constant c > 0 such that f1 f2 |Asp,q (Rn ) ≤ c f1 |Asp,q (Rn ) · f2 |Asp,q (Rn )

(3.45)

for all f1 ∈ Asp,q (Rn ), f2 ∈ Asp,q (Rn ). As for (technical) details we refer the reader to [T13, Section 1.2.5], [T14, Section 3.2.4] and the literature within. Let 0 < p, q ≤ ∞ (p < ∞ for F -spaces) and s ∈ R. Then the following assertions are pairwise equivalent: (a) Asp,q (Rn ) is a multiplication algebra. (b) s > 0 and Asp,q (Rn ) → L∞ (Rn ). (c) Either (3.46)

, s Asp,q (Rn ) = Bp,q (Rn ) with

or (3.47)

, s (Rn ) with Asp,q (Rn ) = Fp,q

s > n/p where 0 < p, q ≤ ∞, s = n/p where 0 < p < ∞, 0 < q ≤ 1,

s > n/p where 0 < p < ∞, 0 < q ≤ ∞, s = n/p where 0 < p ≤ 1, 0 < q ≤ ∞.

These assertions have a long history, related references may be found in [T13, Sec∗

tion 1.2.3, pp. 12–13]. Let Asp,q (Rn ) be the tempered homogeneous spaces as introduced in Definition 2.8. One may again ask which of these spaces is a multiplication algebra with ∗

∗

∗

f1 f2 |Asp,q (Rn ) ≤ c f1 |Asp,q (Rn ) · f2 |Asp,q (Rn )

(3.48)

as the obvious counterpart of (3.45). If one replaces f1 by f1 (λ·) and f2 by f2 (λ·), λ > 0, then it follows from (2.85) that (3.49)

∗

n

∗

∗

f1 f2 |Asp,q (Rn ) ≤ c λs− p f1 |Asp,q (Rn ) · f2 |Asp,q (Rn ).

In other words, only spaces with s = n/p have a chance to be multiplication algebras. This applies to the spaces according to Definition 2.8(iii),(iv) with the following outcome. ∗

Theorem 3.7. A space Asp,q (Rn ) as introduced in Definition 2.8 is a multiplication algebra if, and only if, either (3.50)

∗

∗

Asp,q (Rn ) = F sp,q (Rn ),

0 < p ≤ 1,

0 < q ≤ ∞,

s = n/p

TEMPERED HOMOGENEOUS FUNCTION SPACES, II

355

or (3.51)

∗

∗

0 < p < ∞,

Asp,q (Rn ) = B sp,q (Rn ), ∗

0 < q ≤ 1,

s = n/p.

∗

Proof. Let f1 , f2 ∈ Asp,q (Rn ) with Asp,q (Rn ) as in (3.50), (3.51). Let, in addition, supp f1 ⊂ Ω, supp f2 ⊂ Ω with Ω = {y : |y| < 1}. Then (3.16) can be extended to the spaces in (3.50), (3.51), again with a reference to [T15, Proposition 3.52, p. 107], that is (3.52)

∗

n n/p n fk |An/p p,q (R ) ∼ fk |Ap,q (R ),

k = 1, 2.

n/p

According to (3.46), (3.47) the spaces Ap,q (Rn ) in question are multiplication algebras. Using (3.49) with s = n/p it follows (3.53)

∗

∗

∗

f1 f2 |Asp,q (Rn ) ≤ c f1 |Asp,q (Rn ) · f2 |Asp,q (Rn ) ∗

for all compactly supported f1 , f2 ∈ Asp,q (Rn ). The rest is now a matter of the Fatou property as explained in Remark 2.11. As mentioned after (3.49) spaces ∗



Asp,q (Rn ) with s < n/p cannot be multiplication algebras.

Remark 3.8. In connection with possible applications to Navier-Stokes equations and Keller-Segel systems as described in Section 1 those multiplication algebras might be of interest which are also Banach spaces (and not only quasi-Banach ∗

∗

n/p

spaces). This applies to F n1,q (Rn ), 1 ≤ q ≤ ∞, and B p,1 (Rn ), 1 ≤ p < ∞. ∗

n/p

The assertion that B p,1 (Rn ) is a multiplication algebra has already been obn/p served in [Pee76, p. 148], denoted there as B˙ p,1 (Rn ), considered in the context   ˙ n ), S˙  (Rn ) . of S(R 3.5. Examples I: Riesz kernels. We discuss some examples which illumi∗

nate the different nature of the tempered homogeneous spaces Asp,q (Rn ) in the   framework of the dual pairing S(Rn ), S  (Rn ) and their inhomogeneous counterparts Asp,q (Rn ). We begin with a simple observation. Let f (x) = 1, x ∈ Rn . By (2.12) or (2.13) ∗

one has Wt f (x) = 1, t > 0, x ∈ Rn . Then it follows from (2.29) with B s∞,∞ (Rn ) = ∗

∗

s C s (Rn ) that f ∈ C s (Rn ) for any s < 0. On the other hand f ∈ C s (Rn ) = B∞,∞ (Rn ) for all s ∈ R. But we are mainly interested to have a closer look at the kernels of the Riesz transform

(3.54)

hσ (x) = |x|−σ ,

0 < σ < n,

x ∈ Rn .

The interest in these kernels comes from (3.55)

+σ (ξ) = cσ |ξ|σ−n , h

0 < σ < n,

cσ > 0,

0 = ξ ∈ Rn ,

[Ste70, pp. 117, 73], [LiL97, Theorem 5.9, p. 122], and the related Riesz potentials   ∨ f (y) (3.56) hσ f& (x) = c dy, 0 < σ < n, x ∈ Rn . n−σ Rn |x − y|

356

HANS TRIEBEL ∗

Let again Asp,q (Rn ) be the tempered homogeneous spaces in the distinguished strip ∗

as introduced in Definition 2.8. If hσ ∈ Asp,q (Rn ) then one has by (2.85) for λ > 0, ∗ ∗ ∗      n (3.57) λ−σ  |x|−σ |Asp,q (Rn ) =  |λx|−σ |Asp,q (Rn ) = λs− p  |x|−σ |Asp,q (Rn ). One obtains s =

n p

∗

− σ as a necessary condition for hσ ∈ Asp,q (Rn ). (Recall that for ∗

fixed p, q and A ∈ {B, F } there is no monotonicity of Asp,q (Rn ) with respect to s in contrast to the related inhomogeneous spaces). There is a temptation to use the ∗

equivalent domestic quasi-norms (2.87), (2.89) to clarify whether hσ ∈ Asp,q (Rn ) where s − np = −σ = − nr is covered by (2.69). But this requires that one first ∗

∗

ensures hσ ∈ C −σ (Rn ) as the anchor space of Asp,q (Rn ) with s = ∗

n p

− σ which are

continuously embedded in C −σ (Rn ). Furthermore we wish to compare the outcome with corresponding assertions for the inhomogeneous spaces Asp,q (Rn ). Theorem 3.9. Let n ∈ N and 0 < σ < n. (i) Let 0 < p ≤ ∞,

(3.58)

0 < q ≤ ∞,

n

 1 n −1 0

∗

where f (t) is the decreasing rearrangement of the Lebesgue-measurable function f in Rn . There are two positive constants c, c such that for any R > 0, . . (3.63)  x ∈ Rn : hσ (x) ≥ R−σ  = c Rn = t =  τ > 0 : h∗σ (τ ) ≥ c t−σ/n . Then h∗σ (t) = c t−σ/n and hσ ∈ Lr,∞ (Rn ). Let 1 < r1 < r < r2 < ∞. By (3.7) one has (3.64) j = 1, 2. With

∗

∗

Lrj (Rn ) = F 0rj ,2 (Rn ) → C −σj (Rn ), 1 r

=

1−θ r1

+ rθ2 , σ = (1 − θ)σ1 + θσ2 =

σj = n/rj , n r,

one has by real interpolation

(3.65) Lr,∞ (Rn ) = ∗ ∗   ∗  Lr1 (Rn ), Lr2 (Rn ) θ,∞ → C −σ1 (Rn ), C −σ2 (Rn ) θ,∞ → C −σ (Rn )

TEMPERED HOMOGENEOUS FUNCTION SPACES, II

357

As far as Lorentz spaces and the above real interpolations are concerned one may consult [T78, Theorem 1.18.6/1, p. 133, Theorem 2.4.1, p. 182] (and their proofs). In particular, ∗

hσ ∈ L nσ ,∞ (Rn ) → C −σ (Rn ),

(3.66)

justifies to deal with hσ in the context of tempered homogeneous spaces. Step 2. We prove part (i). Let ϕj (ξ) = ϕ0 (2−j ξ), j ∈ Z, be as in (2.32). Then one has by (3.55) (3.67) (3.68) and (3.69)

+σ (ξ) = c 2j(σ−n) |2−j ξ|σ−n ϕ0 (2−j ξ), ϕj (ξ) h 

+σ (ξ) ϕj (ξ) h

∨

  +σ ∨ (2j x), (x) = c 2jσ ϕ0 h

j ∈ Z, j ∈ Z,

 j  +σ )∨ |Lp (Rn ) = c 2j(σ− np ) (ϕ0 h +σ )∨ |Lp (Rn ), (ϕ h

j ∈ Z.

Inserted in (2.89) (what is now justified by Step 1) one obtains (3.70)

∗

hσ ∈ B sp,q (Rn ) if, and only if, s = ∗

n

n − σ, p

q = ∞.

−σ

p By (3.7) with v = p < ∞ one has hσ ∈ F p,q (Rn ). Part (i) follows now from our comments after (3.57). Step 3. We prove part (ii). In the starting terms in (2.7), (2.9) one can replace ϕ0 according to (2.3) with ϕ0 (x) = ϕ0 (−x) ≥ 0, x ∈ Rn , by the convolution 2 & = cϕ +0 (x) ≥ 0 for some c > 0. One has ϕ = ϕ0 ∗ ϕ0 . Then ϕ(x)  ∨  1  + (3.71) ϕhσ (x) = c ϕ∨ (y) dy ≥ c |x|−σ , |x| ≥ 1, |x − y|σ n R

+σ )∨ ∈ Lp (Rn ) requires σp > n (in particular c > 0, c > 0. This shows that (ϕh s (Rn ) requires p > 1). By (3.69) with j ∈ N it follows that hσ ∈ Bp,q (3.72)

either

s
0,

x ∈ Rn ,

c > 0, and ∗

n δ ∈ B −n ∞,q (R )

(3.77)

if, and only if, q = ∞.

∗

Hence δ ∈ C −n (Rn ). Inserting (3.76) in (2.26) with q < ∞ one obtains for t > 0 and some c > 0  t  ∗ −n n q dτ nq/2 −nq/2 δ |F ∞,q (R ) ≥ c t−n/2 (3.78) = ∞. τ dy √ τ τ |y|≤ t 0 Hence ∗

n δ ∈ F −n ∞,q (R ),

(3.79)

0 < q < ∞.

We extend now the lift I˙σ according to (3.18) to the usual homogeneous spaces as described in [T15, Proposition 2.18, p. 23]. In particular, s−σ s (Rn ) = F˙ 1,q (Rn ). I˙σ F˙ 1,q

(3.80) Recall the duality (3.81)

−s s n F˙ 1,q (Rn ) = F˙ ∞,q  (R ),

s ∈ R,

1 < q < ∞,

1 1 +  = 1, q q

covered by [FrJ90, (5.2), p. 70]. This gives the possibility to shift the lifting (3.80) by duality to (3.82)

∗

∗

n I˙σ F s∞,q (Rn ) = F s−σ ∞,q (R ),

s ∈ R,

1 < q < ∞.

Then one has by (3.55) and δ& = c = 0, (3.83)

   & ∨ (x) = c I˙−σ δ(x). hn−σ (x) = c h∨ σ (x) = c hσ δ

By (3.79) one obtains (3.84)

∗

n hn−σ ∈ F σ−n ∞,q (R ),

1 0; • t = 0, 0 < p < ∞ and 0 < q ≤ min(p, 2); • t = 0, p = ∞ and q ≤ 1. Remark 3.2. Sufficient conditions for the embedding in (3.1) have been considered by Schmeisser [24] and Hansen [8]. Both used different methods than we do. Schmeisser used characterizations by differences and concentrated on the Banach t+ε t B(Rd ) → Bp,q (Rd ) with ε > 0 and q0 , q arbitrary space case. Hansen showed Sp,q 0 by applying wavelet characterizations. t t (Rd ) and Sp,q B(Rd ) in We summarize what is known about the relation of Bp,q the remaining cases. To get a full picture one has to take into account also Theorem 3.6 in case t = 0. Two subsets X, Y of S  (Rd ) we shall call not comparable if

X \ Y = ∅

and

Y \ X = ∅ .

Proposition 3.3. Let d ≥ 2. (i) Let 1 ≤ p ≤ ∞, 0 < q ≤ ∞ and t < 0. Then we have t t Bp,q (Rd ) → Sp,q B(Rd ) .

(ii) Let t = 0, p = 2, 1 < p < ∞ and min(2, p) < q < max(2, p). Then 0 0 Bp,q (Rd ) and Sp,q B(Rd ) are not comparable. 0 0 (Rd ) and S1,q B(Rd ) are not (iii) Let t = 0, p = 1 and 1 < q < ∞. Then B1,q comparable. 0 0 (Rd ) and S∞,q B(Rd ) are (iv) Let t = 0, p = ∞ and 1 < q < ∞. Then B∞,q not comparable. t t (Rd ) and Sp,q B(Rd ) are (v) Let t < 0, 0 < p < 1 and 0 < q ≤ ∞. Then Bp,q not comparable.

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VAN KIEN NGUYEN AND WINFRIED SICKEL

t

critical line t t B(Rd ) → Bp,q (Rd ) Sp,q

0

1 p

1 not comparable t t Bp,q (Rd ) → Sp,q B(Rd )

t t Figure 1. Comparison of Sp,q B(Rd ) and Bp,q (Rd )

The embedding (3.1) is optimal in the following sense. Theorem 3.4. Let d ≥ 2, 0 < p0 , p, q0 , q ≤ ∞ and t0 , t ∈ R. Let p, q and t be fixed. Within all spaces Spt00 ,q0 B(Rd ) satisfying t Spt00 ,q0 B(Rd ) → Bp,q (Rd ) t the class Sp,q B(Rd ) is the largest one.

Remark 3.5. Comparing Theorems 3.1 and 3.4 it is natural to ask also for the t t t B(Rd ) → Bp,q (Rd ) in the other direction, i.e., we fix Sp,q B(Rd ) optimality of Sp,q t0 d and look for spaces Bp0 ,q0 (R ) such that (3.1) is true. For this we consider a 2 2 B(Rd ) → B1,2 (Rd ). On the other hand, special situation. Theorem 3.1 yields S1,2 a Sobolev-type embedding and Theorem 3.1 imply 3/2

3/2

2 B(Rd ) → S2,2 B(Rd ) → B2,2 (Rd ) , S1,2

see the comments at the beginning of Subsection 4.10. But for d ≥ 2 these isotropic 3/2 2 Besov spaces B1,2 (Rd ) and B2,2 (Rd ) are not comparable. Hence, an optimality in such a wide sense is not true. 3.2. The embedding of isotropic spaces into dominating mixed spaces. Theorem 3.6. Let d ≥ 2, 0 < p, q ≤ ∞ and t ∈ R. Then we have (3.2)

td t Bp,q (Rd ) → Sp,q B(Rd )

if and only if one of the following conditions is satisfied • t > max(0, p1 − 1); • t = 0, 1 < p ≤ ∞ and max(2, p) ≤ q ≤ ∞; • 0 < p ≤ 1, t = p1 − 1 and q = ∞. Remark 3.7. Again we have to refer to Hansen [8] for an earlier result in this td+ε t (Rd ) → Sp,q B(Rd ) with ε > 0 and q0 , q arbitrary. direction. He proved Bp,q 0 Also in this situation we summarize what is known about the relation of td t (Rd ) and Sp,q B(Rd ) in the remaining cases. Bp,q Proposition 3.8. Let d ≥ 2. (i) Let 0 < p, q ≤ ∞ and t < 0. Then we have t td Sp,q B(Rd ) → Bp,q (Rd ) .

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

369

td (ii) Let 0 < p < 1, 0 < t < p1 − 1 and 0 < q ≤ ∞. Then Bp,q (Rd ) and t B(Rd ) are not comparable. Sp,q 0 0 B(Rd ) → Bp,q (Rd ) follows. (iii) Let 0 < p < 1, t = 0 and 0 < q ≤ p. Then Sp,q 0 0 (Rd ) and Sp,q B(Rd ) are (iv) Let 0 < p < 1, t = 0 and p < q ≤ ∞. Then Bp,q not comparable.

Remark 3.9. Obviously the case t = 0 is covered by Proposition 3.3. The set {(p, 0) : 1 ≤ p ≤ ∞} is part of the critical line of (3.2). Also for us it was surprising that the critical line for 0 < p < 1 is given by p1 − 1. t=

1 p

−1

critical line

t not comparable

td t (Rd ) → Sp,q B(Rd ) Bp,q

1 t+(d−1)( p −1) Bp,q (Rd )

0

t → Sp,q B(Rd )

1

1 p

t td Sp,q B(Rd ) → Bp,q (Rd )

td t Figure 2. Comparison of Bp,q (Rd ) and Sp,q B(Rd )

These embeddings are optimal in the following sense. Theorem 3.10. Let d ≥ 2, 0 < p0 , p, q0 , q ≤ ∞ and t0 , t ∈ R. Let p, q and t be fixed. Within all spaces Bpt00 ,q0 (Rd ) satisfying t B(Rd ) Bpt00 ,q0 (Rd ) → Sp,q td (Rd ) is the largest one. the class Bp,q

Theorem 3.11. Let d ≥ 2, 0 < p0 , p, q0 , q ≤ ∞ and t0 , t ∈ R. Let p, q and t be fixed. Within all spaces Spt00 ,q0 B(Rd ) satisfying td Bp,q (Rd ) → Spt00 ,q0 B(Rd ) t B(Rd ) is the smallest one. the class Sp,q

Remark 3.12. Let us come back to the chain of embeddings td t t (Rd ) → S2t W (Rd ) = S2,2 B(Rd ) → W2t (Rd ) = B2,2 (Rd ) W2td (Rd ) = B2,2

discussed in the Introduction. Employing Theorems 3.4, 3.10 and 3.11 we obtain in case d ≥ 2 the following optimality assertions. • Within all spaces Spt00 ,q0 B(Rd ) satisfying Spt00 ,q0 B(Rd ) → W2t (Rd ) the class t B(Rd ) = S2t W (Rd ) is the largest one. S2,2 • Within all spaces Bpt00 ,q0 (Rd ) satisfying Bpt00 ,q0 (Rd ) → S2t W (Rd ) the class td B2,2 (Rd ) = W2td (Rd ) is the largest one. • Within all spaces Spt00 ,q0 B(Rd ) satisfying W2td (Rd ) → Spt00 ,q0 B(Rd ) the class S2t W (Rd ) is the smallest one.

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As already indicated in Figure 2, there is a bridge between Theorem 3.6 (first bullet) and Proposition 3.3(i). Lemma 3.13. Let 0 < p < 1 and 0 < t < 1 t+(d−1)( p −1)

Bp,q

1 p

− 1. Then

t (Rd ) → Sp,q B(Rd )

holds. 4. Proofs To prove our main results we will apply essentially four different tools: Fourier multipliers; complex interpolation; assertions on dual spaces; and some test functions. In what follows we collect what is needed. 4.1. Fourier multipliers. Let us recall some Fourier multiplier assertions. For a compact subset Ω ∈ Rd we introduce the notation . d  d d LΩ p (R ) = f ∈ S (R ) : supp Ff ⊂ Ω , f ∈ Lp (R ) . For those spaces improved convolution inequalities hold, see [36, Proposition 1.5.1]. Lemma 4.1. Let Ω and Γ be compact subsets of Rd . Let 0 < p ≤ ∞ and put u := min(p, 1). Then there exists a positive constant c such that F −1 M Ff |Lp (Rd ) ≤ c F −1 M |Lu (Rd ) · f |Lp (Rd ) d −1 M ∈ LΓu (Rd ). holds for all f ∈ LΩ p (R ) and all F

Lemma 4.1 and the homogeneity properties of the Fourier transform yield the following. d be the two decompositions of unity deLemma 4.2. Let (ψj )∞ ¯ )k∈N ¯ j=0 and (ϕk 0 fined in (2.1) and (2.2),(2.3), respectively. Let u = min(1, p). Then there exists a positive constant C such that

(4.1)

 F −1 (ψj ϕk¯ Ff ) |Lp (Rd ) ≤ C F −1 ϕk¯ Ff |Lp (Rd )

and (4.2)

¯

F −1 (ϕk¯ ψj Ff )|Lp (Rd ) ≤ C 2(jd−|k|)( u −1) F −1 ψj Ff |Lp (Rd ) 1

hold for all j ∈ N0 , all k ∈ Nd0 and all f ∈ S  (Rd ) with finite right-hand sides. Proof. For k¯ ∈ Nd and j ∈ N0 we put 0

Ωk¯

= {x ∈ Rd : |xi | ≤ 2ki +1 , i = 1, . . . , d},

Γj

= {x ∈ Rd : sup |xi | ≤ 2j+1 }. i=1,...,d

Step 1. Proof of (4.1). For f ∈ S  (Rd ) we put g := F −1 ϕk¯ Ff . Then we have Ω g ∈ Lp k¯ (Rd ) and g(2−j ·) ∈ LΓp 0 (Rd ). Observe that jd

(4.3)

F −1 ψj Fg|Lp (Rd ) = 2− p (F −1 ψj Fg)(2−j ·)|Lp (Rd )   jd = 2− p F −1 ψj (2j ·)F[g(2−j ·)] |Lp (Rd ).

Let j ∈ N. Lemma 4.1 together with supp ψj (2j ·) ⊂ Γ0 yield jd

(4.4)

F −1 ψj Fg|Lp (Rd ) ≤ c 2− p F −1 (ψ1 (2 · )) |Lu (Rd ) · g(2−j ·)|Lp (Rd ) ≤ CF −1 ψ0 |Lu (Rd ) · g|Lp (Rd ).

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

371

A similar argument yields the estimate of F −1 ψ0 Fg. This proves (4.1). Γ Step 2. To prove (4.2), we put h := F −1 ψj Ff . Then we have h ∈ Lp j (Rd ), hence h(2−j ·) ∈ LΓp 0 (Rd ). In addition we know that supp ϕk¯ (2j ·) ⊂ Γ1 if ψj ·ϕk¯ = 0. Observe, that the condition supp ψj ∩ supp ϕk¯ = ∅ implies max ki − 1 ≤ j ≤ max ki + 1.

(4.5)

i=1,...,d

i=1,... ,d

Let k¯ = (k1 , . . . , kd ) ∈ Nd0 such that k1 , . . . , kd = 0. Using Lemma 4.1 we obtain jd

(4.6)

F −1 ϕk¯ Fh|Lp (Rd ) = 2− p (F −1 ϕk¯ Fh)(2−j ·)|Lp (Rd ) / 0 jd = 2− p F −1 ϕk¯ (2j ·)F[h(2−j ·)] |Lp (Rd ) jd

≤ c 2− p F −1 [ϕk¯ (2j ·)]|Lu (Rd ) · h(2−j ·)|Lp (Rd ) ≤ c F −1 [ϕk¯ (2j ·)]|Lu (Rd ) · h|Lp (Rd ).

We put ¯j := (j, . . . , j). The homogeneity properties of the Fourier transform lead to ¯ ¯ ¯

(4.7)

F −1 [ϕk¯ (2j ·)]|Lu (Rd ) = F −1 [ϕ¯1 (2−k+j+1 ·)]|Lu (Rd ) ¯

≤ c1 2(jd−|k|)( u −1) F −1 ϕ¯1 |Lu (Rd ). 1

¯ An obvious modification yields Inserting this into (4.6) we get (4.2) for those k. ¯ the estimate for the remaining k. The proof is complete.  4.2. Complex interpolation. For the basics of the classical complex interpolation method of Calder´on we refer to the original paper [4] and the monographs [2, 3, 17, 35]. In the meanwhile it is well-known that the complex interpolation method extends to specific quasi-Banach spaces, namely those, which are analytically convex, see [14]. Note that any Banach space is analytically convex. The following Proposition, well-known in case of Banach spaces, see [3, Theorem 4.1.2], [17, Theorem 2.1.6] or [35, Theorem 1.10.3.1], can also be extended to the quasiBanach case, see [14]. Proposition 4.3. Let 0 < Θ < 1. Let (X1 , Y1 ) and (X2 , Y2 ) be two compatible couples of quasi-Banach spaces. In addition, let X1 +Y1 , X2 +Y2 be analytically convex. If T is in L (X1 , X2 ) and in L (Y1 , Y2 ), then the restriction of T to [X1 , Y1 ]Θ is in L ([X1 , Y1 ]Θ , [X2 , Y2 ]Θ ) for every Θ. Moreover, T : [X1 , Y1 ]Θ → [X2 , Y2 ]Θ  ≤ T : X1 → X2 1−Θ T : Y1 → Y2 Θ . t (Rd ) are analytically convex, see It is not difficult to see that all spaces Bp,q [18] or [14]. By means of Theorem 7.8 in [14] and the wavelet characterization of t t B(Rd ), see [39], one can derive that also the spaces Sp,q B(Rd ) are analytically Sp,q convex.

Proposition 4.4. Let ti ∈ R, 0 < pi , qi ≤ ∞, i = 1, 2, and   min max(p1 , q1 ), max(p2 , q2 ) < ∞ . Let t0 , p0 and q0 be given by 1 1−Θ Θ = + , p0 p1 p2

1 1−Θ Θ = + , q0 q1 q2

t0 = (1 − Θ)t1 + Θt2 .

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VAN KIEN NGUYEN AND WINFRIED SICKEL

Then Bpt00 ,q0 (Rd ) = [Bpt11 ,q1 (Rd ), Bpt22 ,q2 (Rd )]Θ and Spt00 ,q0 B(Rd ) = [Spt11 ,q1 B(Rd ), Spt22 ,q2 B(Rd )]Θ . Remark 4.5. Complex interpolation of isotropic Besov spaces has been studied at various places, we refer to [3, Theorem 6.4.5] and [35, 2.4.1] as well as to the references given there. The extension to the quasi-Banach case has been done by Mendez and Mitrea [18], see also [14]. Vybiral [39, Theorem 4.6] has proved a corresponding result for sequence spaces associated to Besov spaces of dominating mixed smoothness. However, these results can be shifted to the level of function spaces by suitable wavelet isomorphisms, see [39, Theorem 2.12]. Here we would like to mention that all these extensions of the complex method to the quasi-Banach case in the particular situation of Besov spaces are based on investigations about corresponding Calder´on products, an idea, which goes back to the fundamental paper of Frazier and Jawerth [7] (where Triebel-Lizorkin spaces are treated). 4.3. Dual spaces. Next we will recall some results about the dual spaces of t t (Rd ) and Sp,q B(Rd ). For 1 < p < ∞ the conjugate exponent p is determined Bp,q by p1 + p1 = 1. If 0 < p ≤ 1 we put p = ∞ and if p = ∞ we put p = 1. For us it will be convenient to switch to the closure of S(Rd ) in these spaces. t t ˚p,q Definition 4.6. (i) By B (Rd ) we denote the closure of S(Rd ) in Bp,q (Rd ). t t ˚p,q (ii) By S B(Rd ) we denote the closure of S(Rd ) in Sp,q B(Rd ).

Recall that t t ˚p,q B (Rd ) = Bp,q (Rd )

⇐⇒

max(p, q) < ∞

and t t ˚p,q S B(Rd ) = Sp,q B(Rd )

⇐⇒

max(p, q) < ∞ .

Because of the density of S(R ) in these spaces any element of the dual space can be interpreted as an element of S  (Rd ). Hence, a distribution f ∈ S  (Rd ) belongs t ˚p,q to the dual space (B (Rd )) if and only if there exists a positive constant c such that d

holds for all ϕ ∈ S(Rd ).

t (Rd ) |f (ϕ)| ≤ c ϕ|Bp,q t ˚p,q B(Rd ). Similarly for S

Proposition 4.7. Let t ∈ R. (i) If 1 ≤ p < ∞ and 0 < q ≤ ∞, then it holds t d t d ˚p,q ˚p,q [B (Rd )] = Bp−t and [S B(Rd )] = Sp−t  ,q  (R )  ,q  B(R ). (ii) If 0 < p < 1 and 0 < q ≤ ∞, then 1 −t+d( p −1)

t ˚p,q [B (Rd )] = B∞,q

(Rd )

and

−t+ 1 −1

t ˚p,q [S B(Rd )] = S∞,q p

B(Rd )

Proof. The proof in the isotropic case can be found in [36, Section 2.11], see in particular Remark 2.11.2/2. For the dominating mixed smoothness we refer to [8, Subsection 2.3.8], at least if 0 < p, q < ∞. Here we only give a proof in case q = ∞ for the Besov spaces of dominating mixed smoothness following essentially the arguments given in [34, 2.5.1] for the isotropic case.

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

373

t d ˚p,∞ Step 1. We shall prove [S B(Rd )] = Sp−t  ,1 B(R ) for 1 ≤ p < ∞. Substep 1.1. Let (ϕj )j∈N0 be the univariate smooth dyadic decomposition of unity used in the definition of the spaces. We put

ϕ˜j := ϕj−1 + ϕj + ϕj+1 , j = 0, 1, . . . , d d ¯ with ϕ−1 ≡ 0. For k ∈ N0 we define ϕ˜k¯ := ϕ˜k1 ⊗ . . . ⊗ ϕ˜kd . With f ∈ Sp−t  ,1 B(R ) and ψ ∈ S(Rd ) we have          |f (ψ)| =  (F −1 ϕk¯ Ff )(ψ) =  (F −1 ϕ˜k¯ FF −1 ϕk¯ Ff )(ψ) d ¯ k∈N

(4.8)

d ¯ k∈N

0  0    −1 −1 =  (F ϕk¯ Ff )(F ϕ˜k¯ Fψ) d ¯ k∈N 0

 ¯   ¯      ≤  2−|k|t F −1 ϕk¯ Ff k¯ 1 (Lp ) ·  2|k|t F −1 ϕ˜k¯ Fψ k¯ ∞ (Lp ). Observe that  |k|t     2 ¯ F −1 ϕ˜k¯ Fψ ¯ ∞ (Lp ) ≤ k

  ¯    −1  2|k|t Fϕk+ ψ k¯ ∞ (Lp ) ¯ ¯F ¯ ∞ ≤1 

(4.9)

t ≤ c ψ|Sp,∞ B(Rd )

for some c > 0 independent of ψ and f . Consequently d t d |f (ψ)| ≤ c f |Sp−t  ,1 B(R ) · ψ|Sp,∞ B(R ) t ˚p,∞ B(Rd )] . which means f ∈ [S Substep 1.2. Next we prove the reverse direction. We assume that the generator of our smooth dyadic decomposition of unity is an even function. Then ϕk¯ (−x) = ϕk¯ (x) follows for all x ∈ Rd an all k¯ ∈ Nd0 . Let c0 (Lp ) denote the space of all sequences (ψk¯ )k¯ of measurable functions such that lim  ψk¯ |Lp (Rd ) = 0 ¯ |k|→∞

equipped with the norm  (ψk¯ )k¯ |c0 (Lp ) := sup  ψk¯ |Lp (Rd ) . d ¯ k∈N 0

Observe that

¯

J : g → (2|k|t F −1 ϕk¯ Fg)k∈N d ¯ 0

t ˚p,∞ is isometric and bijective if J is considered as a mapping from S B(Rd ) onto a closed subspace Y of c0 (Lp ). Here we use the fact that ¯

lim  2|k|t F −1 ϕk¯ Fg |Lp (Rd ) = 0

¯ |k|→∞

t ˚p,∞ B(Rd ). holds for all g ∈ S t ˚p,∞ B(Rd )] . Hence, by defining Let f ∈ [S     ¯ 2−|k|t ψk¯ , f˜ (ψk¯ )k¯ := f

(ψk¯ )k¯ ∈ Y ,

d ¯ k∈N 0

f˜ becomes a linear and continuous functional on Y satisfying  f˜ |Y → C = t ˚p,∞  f | [S B(Rd )] . Now, by the Hahn-Banach theorem, there exists a linear and

374

VAN KIEN NGUYEN AND WINFRIED SICKEL

continuous extension of f˜ to a continuous linear functional on the space c0 (Lp ). It is known that [c0 (Lp )] = 1 (Lp ) and any g ∈ [c0 (Lp )] can be represented in the form  gk¯ (x) ψk¯ (x) dx , (ψk¯ )k¯ ∈ c0 (Lp ) , (4.10) g((ψk¯ )k¯ ) = d ¯ k∈N 0

Rd

where the functions gk¯ satisfy  g |1 (Lp ) =



gk¯ |Lp (Rd ) < ∞ ,

d ¯ k∈N 0

see [35, Lemma 1.11.1]. Applying this with g = f˜ we find t ˚p,∞ (4.11)  (fk¯ )k¯ |1 (Lp ) =  f˜ |[c0 (Lp )]  =  f | [S B(Rd )]  for an appropriate sequence (fk¯ )k¯ . In view of (4.10), the definition of f˜, the Plancherel identity and the symmetry condition with respect to (ϕk¯ ) we obtain      ¯ f (ψ) = f F −1 ϕk¯ Fψ = f˜ (2|k|t F −1 ϕk¯ Fψ)k¯ d ¯ k∈N 0

(4.12)

=



¯ |k|t

2

d ¯ k∈N 0

=





¯

2|k|t



Rd

Rd

d ¯ k∈N 0

fk¯ (x) (F −1 ϕk¯ Fψ)(x) dx ψ(x) (F −1 ϕk¯ Ffk¯ )(x) dx =



  ¯ 2|k|t F −1 ϕk¯ Ffk¯ (ψ)

d ¯ k∈N 0

for any ψ ∈ S(Rd ). This leads to the identity  ¯ ¯ 2|k+|t ϕk¯ ϕ+ ϕk¯ Ff = ¯ Ffk+ ¯ k ¯ ¯ , ¯ ∞ ≤1  j +kj ≥0, j=1,...,d

valid in S  (Rd ). Consequently, by using a standard convolution inequality and a homogeneity argument, we have  F −1 ϕk¯ Ff |Lp (Rd )  ≤

¯ ¯

d 2|k+|t  F −1 ϕk¯ ϕ+ ¯ Ffk+ ¯ k ¯ ¯ |Lp (R )

¯ ∞ ≤1  j +kj ≥0, j=1,...,d

(4.13)



≤

¯ ¯

d d 2|k+|t  F −1 [ϕk¯ ϕ+ ¯ k ¯ ] |L1 (R ) · fk+ ¯ ¯|Lp (R )

¯ ∞ ≤1  j +kj ≥0, j=1,...,d

≤ c1



¯ ¯

d 2|k+|t fk+ ¯ ¯|Lp (R )

¯ ∞ ≤1  j +kj ≥0, j=1,...,d

¯ Therefore with c1 independent of f and k. d f |Sp−t ¯ )k ¯ |1 (Lp )  ,1 B(R ) ≤ c2  (fk

follows. This together with (4.11) proves that d d  ˚t f |S −t  B(R ) ≤ c2 f |[Sp,∞ B(R )]  p ,1

holds with a constant c2 independent of f . −t+ 1 −1 t ˚p,∞ B(Rd )] = S∞,1 p B(Rd ) for 0 < p < 1. Step 2. We shall prove [S

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON t− 1 +1

t Substep 2.1. The embedding Sp,∞ B(Rd ) → S1,∞p

375

B(Rd ) implies

1

t ˚p,∞ ˚t− p +1 B(Rd ). S B(Rd ) → S 1,∞

Duality and Step 1 yield −t+ 1 −1

S∞,1 p

t ˚p,∞ B(Rd ) → [S B(Rd )] .

t ˚p,∞ Substep 2.2. Let f ∈ [S B(Rd )] . Following Hansen [8, page 75] we choose a d d ¯ point xk¯ ∈ R for any k ∈ N0 such that

1 −1 F ϕk¯ Ff |L∞ (Rd ) ≤ |(F −1 ϕk¯ Ff )(xk¯ )| ≤ F −1 ϕk¯ Ff |L∞ (Rd ) . 2 Then we define the function  1 ¯ a¯(F −1 ϕ¯)(x¯ − x) 2||(−t+ p −1) , x ∈ Rd . ψ(x) := (4.14)

¯ ||≤n

Obviously ψ ∈ S(Rd ). An easy calculation yields t  ψ |Sp,∞ B(Rd )p   ¯  −1 F = sup 2|k|tp   d ¯ k∈N0

 ¯ ∞ ≤1  ¯ |≤n ¯ |k+

×

(4.15) ¯

≤ sup 2|k|tp d ¯ k∈N 0



¯

d ¯ k∈N 0

1

p     −ix(k+ ¯ ) ¯ ξ ak+ ( · ) Lp (Rd ) ¯ ¯ ϕk ¯ (ξ) ϕk+ ¯ ¯(−ξ) e 

   |k+ 1 ¯ −1 d p  2 ¯ |(−t+ p −1) a [ϕk¯ ϕk+ ¯ ¯ F ¯ ¯(−·)]( · ) Lp (R ) k+

¯ ∞ ≤1  ¯ |≤n ¯ |k+

≤ c1 sup 2|k|tp

¯ ¯

2|k+|(−t+ p −1)



   |k+ 1 ¯ −1 d p  2 ¯ |(−t+ p −1) a [ϕk+ ¯ ¯ F ¯ ¯(−·)]( · ) Lp (R ) k+

¯ ∞ ≤1  ¯ |≤n ¯ |k+

where the last inequality is a consequence of Lemma 4.1 and a homogeneity argument. Observe that ¯ ¯

|k+|(1− p ) d d F −1 [ϕk+  Fϕ¯1 |Lp (Rd ) ¯ ¯(−·)]( · ) |Lp (R ) =  Fϕk+ ¯ ¯ |Lp (R ) = 2 1

if ki + i ≥ 1 for all i = 1, . . . , d. If min(ki + i ) = 0 one has to modify this in an obvious way. Altogether we have found   1/p t p B(Rd ) ≤ c2 sup |ak+ ≤ c3 sup |ak¯ |  ψ |Sp,∞ ¯ ¯| d ¯ k∈N 0

¯ |k|≤n

¯ ∞ ≤1  ¯ |≤n ¯ |k+

with c3 independent of n. This estimate can be used to derive           1 1 ¯ ¯ |k|(−t+ −1) −1 |k|(−t+ −1) −1    d d ak¯ 2 (F ϕk¯ Ff )(xk¯ ) =  ak¯ 2 (f ∗ F ϕk¯ )(xk¯ )  ¯ |k|≤n

(4.16)

¯ |k|≤n

= |f (ψ)| t t ˚p,∞ B(Rd )]  · ψ|Sp,∞ B(Rd ) ≤ f | [S t ˚p,∞ B(Rd )]  · sup |ak¯ | . ≤ c3 f | [S ¯ |k|≤n

376

VAN KIEN NGUYEN AND WINFRIED SICKEL

Employing (4.14) and the fact that the ak¯ can be chosen as we want, for instance such that ak¯ (F −1 ϕk¯ Ff )(xk¯ ) = |(F −1 ϕk¯ Ff )(xk¯ )| , we find 

1 ¯ t ˚p,∞ 2|k|(−t+ p −1)  F −1 ϕk¯ Ff |L∞ (Rd ) ≤ c3 f | [S B(Rd )]  .

¯ |k|≤n

Here c3 is independent of f and n. For n → ∞ we obtain −t+ 1 −1

 f |S∞,1 p

t ˚p,∞ B(Rd ) ≤ c3 f | [S B(Rd )]  .



The proof is complete.

4.4. Test functions. Let d ≥ 2. Before we are going to define some test functions we mention a few more properties of our smooth decompositions of unity. As a consequence of the definitions we obtain ϕk¯ (x) = 1

3 kj 2 ≤ xj ≤ 2kj , 4

if

j = 1, . . . , d ,

and minj=1,...,d kj > 0. In case minj=1,...,d kj = 0 the following statement is true: ϕk¯ (x) = 1 if 34 2kj ≤ xj ≤ 2kj holds for all j such that kj > 0 and 0 ≤ xj ≤ 1 for the remaining components. Now we switch to (ψ ) . For  ∈ N it follows  ψ (x) = 1 on the set

x:

  sup |xj | ≤ 2 \ x : j=1,...,d

sup |xj | ≤ j=1,...,d

3  2 . 4

Example 1. Let g ∈ C0∞ (Rd ) be a function such that supp g ∈ B(0, ) for  > 0 small enough (0 <  < 18 ) and |F −1 g(ξ)| > 0 on [−π, π]d . For  ∈ N we define the family of functions f by Ff (ξ) :=

(4.17)

 

7 7 aj g(ξ1 − 2 , ξ2 − 2j , ξ3 , . . . , ξd ) , 8 8 j=1

ξ ∈ Rd ,

where the sequence (aj )j=1 of complex numbers will be chosen later on. Note that (4.18)

7 7 supp g(· − 2 , · − 2j , ·, . . . , ·) ⊂ {x : ϕk¯ (x) = 1, k¯ = (, j, 0, . . . , 0)} 8 8 ⊂ {x : ψ (x) = 1} , j = 1, . . . ,  .

It follows F −1 [ψm Ff ] = δm, f and / 0 7 7 aj F −1 g(ξ1 − 2 , ξ2 − 2j , ξ3 , . . . , ξd ) . F −1 [ϕk¯ Ff ] = δk,(,j,0,...,0) ¯ 8 8

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

377

Hence

   t  f |Bp,q (Rd ) = 2t F −1 [ψ Ff ](·)Lp (Rd )     7 7    = 2t  aj F −1 [g(ξ1 − 2 , ξ2 − 2j , ξ3 , . . . , ξd )] Lp (Rd ) 8 8 j=1      j 7    = 2t F −1 g(x) aj e 8 i(2 x1 +2 x2 ) Lp (Rd )

(4.19)

j=1     j 7    2t  aj e 8 i(2 x1 +2 x2 ) Lp ([−π, π]2 ) . j=1

For the last step we used that F −1 g is rapidly decreasing, |F −1 g(ξ)| > 0 on [−π, π]d and that Rd can be written as 3 [2mπ, 2(m + 1)π) . Rd = m∈Zd

In case 1 < p < ∞ a Littlewood-Paley characterization of Lp ([−π, π]2 ) yields t  f |Bp,q (Rd ) 2t

(4.20)

 

|aj |2

1/2 .

j=1

Similarly t B(Rd )  f |Sp,q

=

 

 q 1/q  2(j+)tq F −1 [ϕ(,j,0,...,0) Ff ](·) Lp (Rd )

j=1

(4.21)

  q 1/q  / 0 7 7 = 2(j+)tq |aj |q F −1 g(ξ1 − 2 , ξ2 − 2j , ξ3 , . . . , ξd ) Lp (Rd ) 8 8 j=1  1/q    = 2t  F −1 g Lp (Rd ) 2jtq |aj |q . j=1

Example 2. In case p = ∞ nontrivial periodic functions are contained in t t (Rd ) and S∞,q B(Rd ). So we can work directly with lacunary series. Let B∞,q (4.22)

f (x) :=

 



aj ei(2

x1 +2j x2 )

,

x = (x1 , . . . , xd ) ∈ Rd .

j=1

Then

F −1 [ψm Ff ] = δm, f

and

F −1 [ϕk¯ Ff ] = δk,(,j,0,...,0) aj ei(2 x1 +2 x2 ) ¯ follow. For aj ≥ 0 for all j this will allow us to calculate the quasi-norms in t t B∞,q (Rd ) and S∞,q B(Rd ). We obtain in the first case    t (Rd ) = 2t F −1 [ψ Ff ](·)L∞ (Rd )  f |B∞,q (4.23)



j

     j   aj ei(2 x1 +2 x2 )  = 2t aj . = 2t sup  x∈Rd

j=1

j=1

378

VAN KIEN NGUYEN AND WINFRIED SICKEL

Concerning the dominating mixed smoothness we conclude t  f |S∞,q B(Rd ) =

 

 q 1/q  2(j+)tq F −1 [ϕ(,j,0,...,0) Ff ](·) L∞ (Rd )

j=1

(4.24)

= 2t

 

2jtq |aj |q

1/q .

j=1

Example 3. Let us consider a function g ∈ C0∞ (R) such that supp g ⊂ {x ∈ R : 3/2 ≤ |x| ≤ 2}. For j ∈ N, k¯ ∈ Nd we define gj (t) = g(2−j+1 t)

and gk¯ (x) = gk1 (x1 ) . . . gkd (xd ) ,

Let ∇ :=

-

¯ ∞= k¯ ∈ Nd , k

.

t ∈ R, x ∈ Rd .

 ∈ N.

,

Then, if k¯ ∈ ∇ , we have supp gk¯ ⊂ {x : ϕk¯ (x) = 1} ⊂ {x : ψ (x) = 1}. We define the family of test functions  f = ak¯ F −1 gk¯ ,

 ∈ Nd .

¯ k∈∇

The coefficients (ak¯ )k¯ will be chosen later on. By construction we have

    1/q  −1 0 d d q   f |Sp,q B(R ) = F [ϕk¯ Ff ](·) Lp (R ) d ¯ k∈N 0

(4.25) =



q   |ak¯ |q F −1 gk¯ Lp (Rd )

1/q .

¯ k∈∇

Observe that  −1     1  1 ¯ ¯ F gk¯ Lp (Rd ) = 2(|k|−d)(1− p )  F −1 g L (Rd ) = C 2|k|(1− p ) ¯ p 1 for an appropriate C > 0 (independent of ). Consequently we obtain

 1/q 1 ¯ 0 f |Sp,q B(Rd ) = C |ak¯ |q 2|k|(1− p )q ,  ∈ N. ¯ k∈∇

Next we compute 0 f |Bp,q (Rd )

=

 ∞

   −1 F [ψj Ff ](·)Lp (Rd )q

j=0

       =  ak¯ F −1 gk¯ Lp (Rd ).

(4.26)

¯ k∈∇

Recall, for 0 < p0 < p < p1 < ∞ we have 1

−1

1

−1

0 Spp00,p p B(Rd ) → Sp,2 F (Rd ) → Spp11,p p B(Rd ) ,

1/q

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

379

0 see [11], and Sp,2 F (Rd ) = Lp (Rd ), 1 < p < ∞, see [16]. These arguments lead to       ¯ 1 1  p1    ak¯ F −1 gk¯ Lp (Rd ) ≤ C1 2|k|( p0 − p )p |ak¯ |p F −1 gk¯ |Lp0 (Rd )p  ¯ k∈∇

¯ k∈∇

= C2

(4.27)

 

¯

¯

2|k|( p0 − p )p |ak¯ |p 2|k|(1− p0 )p 1

1

¯ k∈∇

= C2

 

¯

|ak¯ |p 2|k|(1− p )p 1

 p1

1

 p1

.

¯ k∈∇

Similarly we have   ¯ 1 1  p1    p1 1 ¯ 2|k|( p1 − p )p |ak¯ |p F −1 gk¯ |Lp1 (Rd )p = C3 |ak¯ |p 2|k|(1− p )p ¯ k∈∇

¯ k∈∇

       ≤ C4  ak¯ F −1 gk¯ Lp (Rd ) .

(4.28)

¯ k∈∇

Altogether we have proved in case 1 < p < ∞  p1   1 ¯ 0 (Rd ) |ak¯ |p 2|k|(1− p )p , f |Bp,q ¯ k∈∇

where the positive constants behind do not depend on  ∈ N. Example 4. We consider the same basic functions gk¯ as in Example 3. This time we define f :=

 

aj F −1 g¯j ,

¯j := (j, 1, . . . , 1) .

j=1

As above we conclude t f |Sp,q B(Rd )

 

q   2jtq |aj |q F −1 g¯j Lp (Rd )

j=1

(4.29)

 

1

1/q

1/q

2j(t+1− p )q |aj |q

j=1

and t f |Bp,q (Rd )



 

1 j(t+1− p )q

2

1/q |aj |

q

,

j=1

where the constants behind do not depend on . Notice that we do not need the restriction 1 < p < ∞ here. It is true for all p. Example 5. This will be one more modification of Example 3. Let gk¯ be defined as there. We put f :=

  j=1

aj F −1 g¯j

¯j = (j, . . . , j) .

380

VAN KIEN NGUYEN AND WINFRIED SICKEL

Then we have t f |Sp,q B(Rd )

=

 

tdjq

2



q

|aj | F

−1

q  g¯j Lp (Rd )

j=1

(4.30)

 

1 jd(t+1− p )q

2

1/q

1/q |aj |

q

j=1

and t f |Bp,q (Rd )



 

1 jd( dt +1− p )q

2

1/q |aj |

q

,

j=1

where the constants behind do not depend on . Also here the restriction 1 < p < ∞ is not needed. Example 6. This example is taken from [36, 2.3.9]. Let  ∈ S(Rd ) be a function such that supp F ⊂ {ξ : |ξ| ≤ 1}. We define hj (x) := (2−j x) ,

x ∈ Rd ,

j ∈ N.

For all p, q, t we conclude t (Rd ) =  hj |Lp (Rd ) = 2jd/p   |Lp (Rd ) ,  hj |Bp,q

j ∈ N.

Similarly, also for all p, q, t, we obtain t B(Rd ) =  hj |Lp (Rd ) = 2jd/p   |Lp (Rd ) ,  hj |Sp,q

j ∈ N.

As an immediate consequence of these two identities we get the following. Lemma 4.8. Let 0 < p0 , p1 , q0 , q1 ≤ ∞ and t0 , t1 ∈ R. (i) An embedding Spt00 ,q0 B(Rd ) → Bpt11 ,q1 (Rd ) implies p0 ≤ p1 . (ii) An embedding Bpt00 ,q0 (Rd ) → Spt11 ,q1 B(Rd ) implies p0 ≤ p1 . 4.5. Proof of Theorem 3.1 – sufficiency. Step 1. Preparations. For k¯ ∈ Nd0 we define k¯ := {j ∈ N0 :

supp ψj ∩ supp ϕk¯ = ∅}

Δj := {k¯ ∈ Nd0 :

supp ψj ∩ supp ϕk¯ = ∅}.

and j ∈ N0 Recall that the condition supp ψj ∩ supp ϕk¯ = ∅ implies max ki − 1 ≤ j ≤ max ki + 1,

i=1,...,d

i=1,... ,d

see (4.5). Consequently we obtain (4.31)

|k¯ | 1 , k¯ ∈ Nd0

By definition we have (4.32)

ψj (x) =

 ¯ k∈Δ j

and

|Δj | (1 + j)d−1 , j ∈ N0 .

ϕk¯ (x)ψj (x) ,

x ∈ Rd .

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

381

Step 2. Let t > 0 and let u = min(1, p). Employing (4.32) we find ∞    q     t (Rd )q = 2jtq  F −1 ϕk¯ ψj Ff Lp (Rd ) f |Bp,q ¯ k∈Δ j

j=0

(4.33) ≤

∞ 

2jtq

 

F −1 ϕk¯ ψj Ff |Lp (Rd )u

q/u .

¯ k∈Δ j

j=0

Using (4.1) it follows (4.34)

t f |Bp,q (Rd )q

≤C

∞  4 

q u

2

2

F

−1

5u q/u ϕk¯ Ff |Lp (R ) . d

¯ k∈Δ j

j=0

If

¯ ¯ (j−|k|)t |k|t

≤ 1 then we have ∞  

t f |Bp,q (Rd )q ≤ C

¯

¯

2(j−|k|)tq 2|k|tq F −1 ϕk¯ Ff |Lp (Rd )q

¯ j=0 k∈Δ j

(4.35) ≤ C1

 

¯

2|k|tq F −1 ϕk¯ Ff |Lp (Rd )q .

d j∈ ¯ ¯ k∈N k 0

¯

The last inequality is due to 2(j−|k|)tq ≤ c2 since t > 0 and j − 1 ≤ maxi=1, ... ,d ki ≤ j + 1, see (4.5). In the case uq > 1 we use H¨older’s inequality with 1 = uq + (1 − uq ). (4.34) implies ∞    /  q−u  u ¯ ¯ 0 qu t 2(j−|k|)t q−u (Rd )q ≤ C 2|k|tq F −1 ϕk¯ Ff |Lp (Rd )q . f |Bp,q ¯ j=0 k∈Δ j

¯ k∈Δ j

Observe, for t > 0 we have sup j∈N0

  /

q−u qu  0 q−u u

< ∞,

¯ k∈Δ j

see (4.5). Hence (4.36)

¯

2(j−|k|)t

t f |Bp,q (Rd )q ≤ C2

 

¯

2|k|tq F −1 ϕk¯ Ff |Lp (Rd )q .

d j∈ ¯ ¯ k∈N k 0

Finally, from (4.35), (4.36) together with k¯ 1 we conclude  ¯ t f |Bp,q (Rd )q ≤ C3 2|k|tq F −1 ϕk¯ Ff |Lp (Rd )q . d ¯ k∈N 0

This proves (3.1). Step 3. Let t = 0. Substep 3.1. First we assume that q ≤ min(p, 1). From (4.34) with t = 0 we have ∞   0 f |Bp,q (Rd )q ≤ C1 F −1 ϕk¯ Ff |Lp (Rd )q ¯ j=0 k∈Δ j

(4.37) = C1

 

d j∈ ¯ ¯ k∈N k 0

F −1 ϕk¯ Ff |Lp (Rd )q .

382

VAN KIEN NGUYEN AND WINFRIED SICKEL

Since k¯ 1 we obtain 0 0 f |Bp,q (Rd )q ≤ C2 f |Sp,q B(Rd )q .

Substep 3.2. Let 1 < p < ∞ and 0 < q ≤ min(2, p). Our main tool will be the following Littlewood-Paley assertion. With 1 < p < ∞ there exist positive constants A, B such that    1/2     (4.38) A  f |Lp (Rd ) ≤  |F −1 ϕk¯ Ff |2 Lp (Rd ) ≤ B  f |Lp (Rd ) k∈Nd 0

holds for all f ∈ Lp (Rd ), see Lizorkin [15, 16] or Nikol’skij [19, 1.5.6]. This will be applied to f replaced by F −1 ψj Ff . We proceed as in Step 1. Employing (4.32) and (4.38) we find 0 (Rd )q = f |Bp,q

∞     −1 F ψj Ff Lp (Rd )q j=0

(4.39)

∞   q 1/2  1    −1 2 d  |F ϕ ψ Ff | (R )  L  . ¯ j p k Aq j=0 ¯

≤

k∈Δj

Because of  · |Lp (2 ) ≤  · |min(2,p) (Lp ) ≤  · |q (Lp ) we deduce 0 (Rd )q ≤ f |Bp,q

∞ 1   F −1 ϕk¯ ψj Ff |Lp (Rd )q Aq j=0 ¯ k∈Δj

(4.40) ≤ c

∞  

F −1 ϕk¯ Ff |Lp (Rd )q

¯ j=0 k∈Δ j

where we used in the last step (4.1). As in Step 1 we can continue the estimate by changing the order of summation and using |k¯ | 1.  4.6. Proof of Theorem 3.6 – sufficiency. Step 1. Let us prove (3.2) in case t > max(0, p1 − 1). We put u := min(1, p). From the previous proof we know  ψj (x)ϕk¯ (x) , x ∈ Rd . (4.41) ϕk¯ (x) = j∈ k ¯

This identity yields t f |Sp,q B(Rd )q =



   q ¯    2|k|tq  F −1 ψj ϕk¯ Ff Lp (Rd ) .

d ¯ k∈N 0

j∈ k ¯

Applying |a + b| ≤ a + b and (4.2) we find  ¯      q/u t F −1 ψj ϕk¯ Ff Lp (Rd )u f |Sp,q B(Rd )q ≤ 2|k|tq u

u

u

d ¯ k∈N 0

(4.42)

≤C



j∈ k ¯ ¯

2|k|tq

   u q/u ¯ 1 2(jd−|k|)( u −1) F −1 ψj Ff |Lp (Rd ) .

d ¯ k∈N 0

j∈ k ¯

Because of (4.31) this implies   ¯ ¯ 1 t f |Sp,q B(Rd )q ≤ c 2|k|tq 2(jd−|k|)( u −1)q F −1 ψj Ff |Lp (Rd )q . d j∈ ¯ ¯ k∈N k 0

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

383

Consequently t (4.43) f |Sp,q B(Rd )q ≤ c

∞ 

2jdtq F −1 ψj Ff |Lp (Rd )q



¯

2(jd−|k|)( u −1−t)q . 1

¯ k∈Δ j

j=0

It is easily derived from (4.5) and the restriction t > u1 − 1 that  ¯ 1 sup 2(jd−|k|)( u −1−t)q < ∞ . j∈N0 ¯ k∈Δj

Hence t f |Sp,q B(Rd )q ≤ c1

∞ 

2jdtq F −1 ψj Ff |Lp (Rd )q

j=0

follows. Step 2. Let t = 0, 1 < p ≤ ∞ and max(2, p) ≤ q ≤ ∞. We shall argue by duality. We have Sp0 ,q B(Rd ) → Bp0 ,q (Rd ) , see Theorem 3.1. 4.7(i) can be used to prove the claim. Step 3. Let 0 < p ≤ 1, t = p1 − 1 and q = ∞. Applying (4.41) we find 1

−1

¯

p B(Rd ) = sup 2|k|( p −1) F −1 ϕk¯ Ff |Lp (Rd ) f |Sp,∞ 1

d ¯ k∈N 0

    ¯ 1    = sup 2|k|( p −1)  F −1 ψj ϕk¯ Ff Lp (Rd ).

(4.44)

d ¯ k∈N 0

Making use of (4.2), this implies 1

−1

¯

p f |Sp,∞ B(Rd )p ≤ sup 2|k|( p −1)p 1

d ¯ k∈N 0

j∈ k ¯

      F −1 ψj ϕk¯ Ff Lp (Rd )p j∈ k ¯

¯ 1 −1)p |k|( p

≤ c1 sup 2

(4.45)

d ¯ k∈N 0

= c1 sup

d ¯ k∈N 0

 

 

¯

2(jd−|k|)( p −1)p F −1 ψj Ff |Lp (Rd )p

j∈ k ¯ 1 jd( p −1)p

2

1



 F −1 ψj Ff |Lp (Rd )p .

j∈ k ¯

Taking into account (4.31), we obtain 1

−1

p f |Sp,∞ B(Rd ) ≤ c2 sup 2jd( p −1) F −1 ψj Ff |Lp (Rd ) . 1

j∈N0



The proof is complete.

Proof of Lemma 3.13. We follows the arguments in proof of Theorem 3.6 (sufficiency). As in Step 1 we conclude t f |Sp,q B(Rd )q ≤ c

∞ 

2jd( p −1)q F −1 ψj Ff |Lp (Rd )q 1



¯

2|k|(t− p +1)q . 1

¯ k∈Δ j

j=0

see (4.43). By means of (4.5) we find  ¯ k∈Δ j

¯

2|k|(t− p +1)q ≤ C1 2j(t− p +1)q 1

1

j+1  =0

1

2(t− p +1)q

d−1

1

≤ C2 2j(t− p +1)q

384

VAN KIEN NGUYEN AND WINFRIED SICKEL

since t −

1 p

+ 1 < 0. Hence

t f |Sp,q B(Rd )q ≤ c C2

∞ 

2jd( p −1)q 2j(t− p +1)q F −1 ψj Ff |Lp (Rd )q . 1

1

j=0



This proves the claim. 4.7. Proof of Theorem 3.1 – necessity. Lemma 4.9. (i) Let 0 < p < ∞ and 0 < q ≤ ∞. Then the embedding 0 0 Sp,q B(Rd ) → Bp,q (Rd )

implies

q ≤ min(2, p).

(ii) Let 0 < q ≤ ∞. Then the embedding 0 0 B(Rd ) → B∞,q (Rd ) S∞,q

implies

q ≤ 1.

Proof. Step 1. We prove (i). Substep 1.1. We show necessity of q ≤ 2. Temporarily we assume 1 < p < ∞. 0 We use our test functions from Example 1, see (4.17). The embedding Sp,q B(Rd ) → 0 d Bp,q (R ) implies the existence of a constant c such that  

|aj |2

1/2

0 t  f |Bp,q (Rd ) ≤ c  f |Sp,q B(Rd )

j=1

 

|aj |q

1/q

j=1

where c does not depend on  and (aj )j , see (4.20) and (4.21). This requires q ≤ 2. Now we turn to 0 < p ≤ 1. Again we shall work with Example 1. For any such p there exists some real number Θ ∈ (0, 1) such that 1−Θ Θ 2 = + . 3 p 2 Lyapunov’s inequality  h |L3/2 ([−π, π]2 ) ≤  h |Lp ([−π, π]2 )1−Θ  h |L2 ([−π, π]2 )Θ , valid for all h ∈ Lp ([−π, π]2 ) ∩ L2 ([−π, π]2 ), in combination with the LittlewoodPaley characterization of L3/2 and L2 , leads us to  

|aj |2

1/2

    1−Θ   Θ/2 j 7    ≤ c aj e 8 i(2 x1 +2 x2 ) Lp ([−π, π]2 ) |aj |2

j=1

j=1

j=1

with c independent of  and (aj )j . Hence   j=1

|aj |2

1/2

    j 7    ≤ c aj e 8 i(2 x1 +2 x2 ) Lp ([−π, π]2 ) . j=1

Taking into account (4.19) we can argue as in case 1 < p < ∞. Substep 1.2. We show necessity of q ≤ p. Therefore we use Example 3. In case ¯ 1 1 < p < ∞ we choose ak¯ = 2|k|( p −1) . Then almost immediately we can conclude q ≤ p. Now we turn to the remaining cases. Assume that there exist 0 < p ≤ 1 and p < q ≤ 2 such that 0 0 B(Rd ) → Bp,q (Rd ) . Sp,q

ISOTROPIC AND DOMINATING MIXED BESOV SPACES: A COMPARISON

385

In this situation we may choose a triple (p1 , q1 , Θ) such that 1 1 Θ 1−Θ Θ 1−Θ + and + . = = 1 < p1 < q1 ≤ 2, Θ ∈ (0, 1), p1 p 2 q1 q 2 Then it follows from Proposition 4.4 that 0 0 Sp01 ,q1 B(Rd ) = [Sp,q B(Rd ), S2,2 B(Rd )]Θ

and 0 0 Bp01 ,q1 (Rd ) = [Bp,q (Rd ), B2,2 (Rd )]Θ .

Proposition 4.3 yields Sp01 ,q1 B(Rd ) → Bp01 ,q1 (Rd ). But this is a contradiction to Example 3. 0 B(Rd ) → Step 2. To prove (ii) we use Example 2, see (4.22). The embedding S∞,q 0 d B∞,q (R ) implies the existence of a constant c such that  

0 0 |aj | =  f |B∞,q (Rd ) ≤ c  f |S∞,q B(Rd )

j=1

 

|aj |q

1/q

j=1

where c does not depend on  and (aj )j , see (4.23) and (4.24). Choosing aj = 1 it is obvious that this can happen only if q ≤ 1.  Proof of Theorem 3.1 - necessity. Step 1. Let t = 0. Then the necessity of q ≤ min(p, 2) if p < ∞ and of q ≤ 1 if p = ∞ follows from Lemma 4.9. Step 2. It remains to deal with t < 0. We employ the test functions from 1 Example 5. Choosing aj = δj, 2−jd(t+1− p ) , we find t  f |Sp,q B(Rd ) 1

and

t  f |Bp,q (Rd ) 2−t(d−1)

t with equivalence constants independent of . Hence, the embedding Sp,q B(Rd ) → t (Rd ) can not hold. The proof is complete.  Bp,q

4.8. Proof of Theorem 3.6 – necessity. By means of the same arguments as used in proof of Lemma 4.9 the following dual assertion can be proved. Lemma 4.10. Let 1 < p ≤ ∞ and 0 < q ≤ ∞. Then the embedding 0 0 Bp,q (Rd ) → Sp,q B(Rd )

implies

q ≥ max(p, 2).

Proof of Theorem 3.6 - necessity. Step 1. Let 0 < p ≤ 1 and t = p1 − 1. td t Assume that there is some q < ∞ such that Bp,q (Rd ) → Sp,q B(Rd ) holds. Then 0 d 0 d Proposition 4.7 yields S∞,q B(R ) → B∞,q (R ). In view of Lemma 4.9 this implies q  ≤ 1, hence q = ∞. Step 2. The necessity of the restrictions in case t = 0 follows by Lemma 4.10. Step 3. It remains to deal with t < max(0, p1 − 1). Step 3.1. Let 0 < p, q ≤ ∞ and t < 0. We employ the test functions from 1 Example 4. Choosing aj = δj, 2−j(t+1− p ) , we find td  f |Bp,q (Rd ) 2t(d−1)

and

t  f |Sp,q B(Rd ) 1

with equivalence constants independent of . With  → ∞ it becomes clear that td t (Rd ) → Sp,q B(Rd ) Bp,q

can not hold.

386

VAN KIEN NGUYEN AND WINFRIED SICKEL

Step 3.2. Let 0 < p < 1 and 0 ≤ t < Proposition 4.7 yields −t+ 1 −1

S∞,q p

1 p

td t − 1. We assume Bp,q (Rd ) → Sp,q B(Rd ). 1 d(−t+ p −1)

B(Rd ) → B∞,q

(Rd ) .

¯ Since d(−t + p1 − 1) > −t + p1 − 1 it is enough to use gk¯ (x) := eikx , k¯ ∈ Zd , as test functions to disprove this embedding. 

4.9. Proof of Propositions 3.3, 3.8. Proof of Proposition 3.3. Part (i) in case 1 ≤ q ≤ ∞ follows by duality from Theorem 3.1, see Proposition 4.7. To cover also the cases 0 < q < 1 we argue as in proof of Theorem 3.6 td t (Rd ) by Bp,q . Observe in this connection that (sufficiency) replacing Bp,q  ¯ sup 2(|k|−j)tq < ∞ . j∈N0 ¯ k∈Δj

Parts (ii)-(iv) are immediate consequences of Theorems 3.1, 3.6. t t B(Rd ) → Bp,q (Rd ). It Now we turn to the proof of (v). Theorem 3.1 yields Sp,q t t (Rd ) → Sp,q B(Rd ). We will argue by contradiction. Therefore remains to prove Bp,q t d t d we assume Bp,q (R ) → Sp,q B(R ). In case 0 < q < ∞ Proposition 4.7 yields −t+ 1 −1

S∞,q p

1 −t+d( p −1)

B(Rd ) → B∞,q

(Rd ) .

¯ Since d( p1 − 1) > p1 − 1 it is enough to use gk¯ (x) := eikx , k¯ ∈ Zd , as test functions to disprove this embedding. In case q = ∞ we make use of the implication t t t t ˚p,q ˚p,q (Rd ) → Sp,q B(Rd ) =⇒ B (Rd ) → S B(Rd ) Bp,q

and argue as before.  Proof of Proposition 3.8. To prove (i) we follow the arguments used in t td (Rd ) by Bp,q (Rd ). proof of Theorem 3.1 (sufficiency) by replacing Bp,q td t (Rd ) → Sp,q B(Rd ) follows from Theorem Concerning (ii)-(iv), observe that Bp,q 3.6. Now we split our investigations into two cases: 0 < t < p1 − 1, t = 0. t td Step 1. Let 0 < p < 1 and 0 < t < p1 − 1. Sp,q B(Rd ) → Bp,q (Rd ) follows from Example 4 with aj := δj, . Step 2. Let 0 < p < 1 and t = 0. Theorem 3.1 yields 0 0 Sp,q B(Rd ) → Bp,q (Rd )

⇐⇒

0 < q ≤ p.

0 0 Hence, in case q > p the spaces Sp,q B(Rd ) and Bp,q (Rd ) are not comparable.



4.10. Proofs of the optimality assertions. First we recall some well-known t results about embeddings of Besov spaces. Let 0 < p ≤ p0 ≤ ∞. Then Bp,q (Rd ) → t0 d Bp0 ,q0 (R ) holds if and only if either t0 −

d d 0 independent of . The embedding Bpt00 ,q0 (Rd ) → Sp,q B(Rd ) yields  t 1 1 d d 0 ≥d t+1− +1− ⇐⇒ t0 − d ≥ dt − . d p0 p p0 p t0

Now, if t0 − pd0 = dt − dp , we apply Example 5, this time with aj := 2−jd( d +1− p0 ) , to obtain q0 ≤ q. All together we conclude 1

td Bpt00 ,q0 (Rd ) → Bp,q (Rd ) ,



see the above comments on embeddings.

td Proof of Theorem 3.11. Assuming Bp,q (Rd ) → Spt00 ,q0 B(Rd ) Lemma 4.8 implies p ≤ p0 . Example 5 with aj := δj, , j = 1, . . . , , yields 1

 f |Spt00 ,q0 B(Rd ) = C 2d(t0 +1− p0 )

and

1

td  f |Bp,q (Rd ) 2d(t+1− p )

td with C > 0 independent of . The embedding Bp,q (Rd ) → Spt00 ,q0 B(Rd ) implies   1 1 1 1 ≤d t+1− ⇐⇒ t0 − d t0 + 1 − ≤t− . p0 p p0 p

Working with Example 5 in the case t0 − p10 = t − p1 , choose aj := 2−jd(t+1−1/p) , we obtain q ≤ q0 . Taking into account the above comments on embeddings we arrive t B(Rd ) → Spt00 ,q0 B(Rd ).  at Sp,q

388

VAN KIEN NGUYEN AND WINFRIED SICKEL

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Contemporary Mathematics Volume 693, 2017 http://dx.doi.org/10.1090/conm/693/13941

An iteratively reweighted least squares algorithm for sparse regularization Sergey Voronin and Ingrid Daubechies Abstract. We present a new algorithm and the corresponding convergence analysis for the regularization of linear inverse problems with sparsity constraints, applied to a new generalized sparsity promoting functional. The algorithm is based on the idea of iteratively reweighted least squares, reducing the minimization at every iteration step to that of a functional including only 2 -norms. This amounts to smoothing of the absolute value function that appears in the generalized sparsity promoting penalty we consider, with the smoothing becoming iteratively less pronounced. We demonstrate that the sequence of iterates of our algorithm converges to a limit that minimizes the original functional.

1. Introduction Over the last several years, an abundant number of algorithms (e.g. [2, 4, 16, 17]) have been proposed for the minimization of the 1 -penalized functional Fτ (x) = Ax − b22 + 2τ x1 , where the matrix A, the vector x and the constant τ are, respectively, in RM ×N , RN and R+ . This functional has a number of interesting applications, such as image restoration [6], face recognition [15], and in inverse problems from geophysics [10]; one can also view the recovery of corrupted low rank matrices as a generalization (since it typically penalizes the 1 norm of the N singular values, i.e. the nuclear norm of the matrix) [14]. The x1 = k=1 |xk | penalty is the closest norm to the 0 -penalty (the count of non-zeros in a signal), and the relationship between the two has been brought into focus by compressive sensing [3]. Since x1 is not differentiable due to the absolute value function | · |, standard gradient based techniques cannot be directly applied for the minimization of Fτ . In this paper, we consider a more general functional of which Fτ is a particular case. The new functional introduced in [11] which the algorithm in this paper can minimize is Fq,λλ (x): Fq,λλ (x) = Ax − b22 + 2

N 

λk |xk |qk

k=1

where the coefficients qk and λk may be different for each 1 ≤ k ≤ N , with 1 ≤ qk ≤ 2 for each k. The more general functional makes it possible to treat different components of x differently, corresponding to their different roles. A simple example 2010 Mathematics Subject Classification. Primary 65F10. c 2017 by the authors under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License (CC BY NC SA 4.0).

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SERGEY VORONIN AND INGRID DAUBECHIES

with a half sparse, half dense signal is illustrated in the Numerics section; in that case, imposing a sparsity inducing penalty on all coefficients is not ideal for proper recovery. Another important instance is the case when the penalization contains a multiscale representation (e.g. the wavelet decomposition) of an object to be reconstructed/approximated. In this case, one has an extra matrix W , representing the transform to wavelet coefficients, and the minimization problem for w = W x takes the form: , 8 N  λk |wk |qk w ¯ = arg min AW −1 w − b22 + 2 w

k=1

If W is a wavelet transform, then the different entries of the vector w fulfill distinctly different roles, with some being responsible for coarse scales and others for fine details. In this case, the total number of possible coefficients corresponding to coarse scales is typically quite limited, with each of them crucial to the overall model (e.g. [10]). Thus, we do not necessarily want to impose a sparsity-promoting penalty on these coefficients, which means we would choose qk > 1 for them in the penalty function. On the other hand, the coefficients corresponding to fine scales are typically quite sparse in the object to be reconstructed, and the inversion procedure might, without appropriate regularization, be prone to populate them with noisy features; in this case, a sparsity promoting choice qk = 1 would be indicated for those k. Two approaches are commonly used by various algorithms for the minimization of functionals that, like Fq,λλ , involve a non-smooth absolute value term. The first approach handles the non-smooth minimization problem directly. For instance, for our original Fτ , one uses the soft-thresholding operation [4] on R, defined by: ⎧ ⎨ x − τ, x ≥ τ ; 0, −τ ≤ x ≤ τ ; Sτ (x) = ⎩ x + τ, x ≤ −τ . For a vector of N elements, soft-thresholding is then defined component-wise by setting, ∀ k = 1, . . . , N , (Sτ (x))k = Sτ (xk ). The .use of soft-thresholding relies on the identity Sτ (β) = arg mina (a − β)2 + 2τ |a| for scalars a and β, which for vectors x and b translates to: . (1.1) Sτ (b) = arg min x − b22 + 2τ x1 x

The simplest example is the Iterative Soft Thresholding Algorithm (ISTA) [4]: (1.2)

xn+1 = Sτ (xn + AT b − AT Axn )

which for an initial x0 and with A2 < 1 (easily accomplished by rescaling; A2 is the operator norm of A from 2 to 2 , also called the spectral norm of A), converges slowly but surely to the 1 -minimizer. A faster variation on this scheme, known as FISTA [1], is frequently employed; the thresholding function can also be adjusted to correspond to more general penalties [12]. Along the same line of thinking, algorithms based on the dual space of the 1 -norm have been proposed [16], with the dual being the ∞ -norm.

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The second approach to algorithms minimizing the 1 -based functional involves some kind of smoothing. One idea is to replace the entire functional by a smooth approximation. This can be done, for instance, by convolving the absolute value function with narrow Gaussians [13]. This approach then allows for the use of standard gradient based methods (such as Conjugate Gradients) for the minimization of the approximate smooth functional. The main problem with this approach is that we are then minimizing a slightly different functional from the original that does not necessarily have the same properties that the original penalty possesses. The algorithm described in this paper replaces the |xk |qk term in Fq,λλ with a smoothened version that tends to the original as the iterates progress towards the limit. This algorithm builds upon the original iteratively reweighted least squares (or IRLS) method proposed in [5] (as well as earlier work in [6, 8]), extending it to the unconstrained case and to a more general penalty. The idea can be illustrated simplest for the qk = 1 case. Consider the approximation: |xk | =

x2 x2k x2 = "k2 ≈ " 2 k |xk | xk xk + 2

where in the rightmost term, a small  = 0 is used, to insure the denominator is finite, regardless of the value of xk . Thus, at the n-th iteration, a reweighted 2 -approximation to the 1 -norm of x is of the form: x1 ≈

N 

 x2 " nk = w ˜kn x2k 2 2 (x ) +  n k k=1 k=1 N

where the right hand side is a reweighted two-norm with weights: (1.3)

1 . w ˜kn = " n (xk )2 + 2n

 n n 2 ˜k (xk ) is a close approximation to xn 1 . In the same Clearly, it follows that k w way, we can use the slightly more general weights: (1.4)

wkn =

1 [(xnk )2 + 2n ]

2−qk 2

.

for the approximation |xnk |qk ≈ wkn (xnk )2 to hold; these can then deal with the case 1 ≤ qk ≤ 2. We shall use a sequence {n } such that n → 0 as n → ∞. We note that it is important that n > 0 for all n for a rigorous convergence proof. In an approach ˜kn and hence to where n = 0, entries k for which xnk = 0 would lead to diverging w n  xk = 0 for all subsequent n > n. This is OK if the k-th entry of the minimizer is indeed zero; if (as is typically the case) this cannot be guaranteed, convergence would fail. A precise choice of the sequence {n } is important for convergence analysis. Although in practice, different approaches can work, the rate at which the -sequence converges needs to match that of the iterates xn . In our analysis, we will use the following definition:  1   (1.5) n = min n−1 , xn − xn−1 2 + αn 2 ,

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where α ∈ (0, 1) is some fixed number. The resulting algorithm we present and analyze is very similar in form to (1.2): (1.6)

= xn+1 k

 n  1 x + (AT b)k − (AT Axn )k 1 + λk qk wkn k

for

k = 1, . . . , N,

with the thresholding replaced by an iteration-dependent scaling operation using the weights (1.4). The algorithm is found to be numerically competitive with the thresholding based schemes for the 1 -case but has the advantage that it can handle the minimization of more general functionals of the form Fq,λλ . The main contribution of this paper is a detailed proof of convergence, the methodology of which can be readily applied to analyze similar schemes. An added advantage of a scheme in which all terms are quadratic in the unknown x is that it can be combined with a conjugate gradient approach to speed up the algorithm. In [11] such an algorithm was proposed, and convergence proved if at each reweighted step, the conjugate gradient scheme was pursued to convergence. In [7], the more general and more realistic situation is considered, where only some conjugate gradient steps are taken at each iteration. In both cases, the choice of {n } (e.g. (1.5)), remains crucial for the convergence analysis. 2. Constructions 2.1. Analysis of the generalized sparsity inducing functional. Here, we derive and comment on the optimality conditions of the functional: (2.1)

F (x) = Ax − b22 + 2

N 

λk |xk |qk ,

k=1

for the range 1 ≤ qk ≤ 2, where in (2.1), we drop the subscripts q and λ for convenience. Notice that since (2.1) is convex for the range of qk specified, every local minimizer is a global minimizer of the functional. The optimality conditions for a general vector x with components xk for k ∈ (1, . . . , N ) can be written down in component-wise form, as derived in Lemma 2.1 below. Note that as F1,λλ is a special case of Fq,λλ , the component-wise conditions below reduce to the well known optimality conditions of the 1 penalized functional when qk = 1 for all k. Lemma 2.1. The conditions for the minimizer of the functional F (x) as defined in (2.1) are: (2.2)

{AT (b − Ax)}k T {A  T (b − Ax)}k {A (b − Ax)}k 

= λk sgn(xk )qk |xk |qk −1 , xk = 0 = 0, xk = 0 ≤ λk , xk = 0

(1 ≤ qk ≤ 2) (qk > 1) (qk = 1)

Proof. Since for the case 1 ≤ qk ≤ 2, F (x) is convex, any local minimizer is necessarily global. Thus, to characterize the minimizer, it is necessary only to work out the conditions corresponding to F (x) ≤ F (x + tz) for all sufficiently small t ∈ R and all z ∈ RN . F (x) ≤ F (x + tz) implies that: (2.3)

t2 ||Az||2 + 2tz, AT (Ax − b) + 2

N  k=1

λk (|xk + tzk |qk − |xk |qk ) ≥ 0.

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We derive N conditions, one for each index k ∈ {1, . . . , N }; for the k-th condition, we consider z of the special form z = zk ek (i.e. only the k-th entry of z differs from 0). We separately analyze the cases xk = 0 and xk = 0, starting with the former. When xk = 0, the function f (t) = |xk + tzk |qk is C ∞ at t = 0. Using a Taylor series expansion around 0, we then get f (t) = f (0) + tf  (0) + O(t2 ). In addition, sgn(xk + tzk ) = sgn(xk ) for sufficiently small t. Keeping t fixed we analyze both signs of xk . For xk > 0, we have sgn(xk ) = 1 and |xk + tzk | = xk + tzk , so that: f (t) = (xk + tzk )qk =⇒ f  (t) = sgn(xk )qk zk (xk + tzk )qk −1 = sgn(xk )qk zk |xk + tzk |qk −1 . When xk < 0, we have sgn(xk ) = −1 and |xk + tzk | = −(xk + tzk ), so that: f (t) = (−xk − tzk )qk =⇒ f  (t) = −qk zk (−xk − tzk )qk −1 = sgn(xk )qk zk |xk + tzk |qk −1 . Thus, f  (0) = sgn(xk )qk zk |xk |qk −1 for all xk = 0. Thus, there exists a constant C > 0 such that the Taylor expansion of f becomes: f (t)

= |xk + tzk |qk = |xk |qk + t sgn(xk )qk zk |xk |qk −1 + O(t2 ) ≤ |xk |qk + t sgn(xk )qk zk |xk |qk −1 + Ct2 ,

This implies in particular that |xk + tzk |qk − |xk |qk ≤ t sgn(xk )qk zk |xk |qk −1 + Ct2 . Using this and z = zk ek in (2.3) gives:   t2 A(zk ek )2 +2tzk ek , AT (Ax−b) +2λk t sgn(xk )qk zk |xk |qk −1 + Ct2 ≥ 0     =⇒ t2 A(zk ek )2 + 2Cλk +2t zk {AT (Ax − b)}k + λk sgn(xk )qk zk |xk |qk −1 ≥ 0. The first term can be made arbitrary small with respect to the second; the inequality will thus hold for both t > 0 and t < 0 iff: zk {AT (Ax − b)}k + λk sgn(xk )qk zk |xk |qk −1 = 0, which leads to: {AT (b − Ax)}k = λk sgn(xk )qk |xk |qk −1 , xk = 0. Note that when qk = 1 we recover the familiar condition for minimization of the 1 -functional: {AT (b − Ax)}k = λk sgn(xk ) , xk = 0. When xk = 0, recalling that z = zk ek , (2.3) gives: (2.4)

t2 ||A(zk ek )||2 + 2tzk ek , AT (Ax − b) + 2λk |t|qk |zk |qk ≥ 0.

Making the substitutions t2 = |t|2 , t = |t| sgn(t), we obtain   (2.5) |t|2 ||A(zk ek )||2 + |t| 2 sgn(t)zk {AT (Ax − b)}k + 2λk |t|qk −1 |zk |qk ≥ 0. In this case, we have to consider the case qk = 1 and qk > 1 separately. When qk > 1 we have that: |t|2 ||Az||2 + 2λk |t|qk |zk |qk + 2|t| sgn(t)zk {AT (Ax − b)}k ≥ 0.

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Since qk > 1, the first two terms on the left have greater powers of |t| than the last term and can be made arbitrarily smaller by picking t small enough. This means we must have: 2 sgn(t)zk {AT (Ax − b)}k ≥ 0 for all t, which can be true only if {AT (Ax − b)}k = 0. Thus, we conclude that the condition is: {AT (b − Ax)}k = 0 , xk = 0 (qk > 1). For qk = 1, applying a similar argument to (2.5) leads to:   |t|2 ||Az||2 + |t| 2 sgn(t)zk {AT (Ax − b)}k + 2λk |zk | ≥ 0 =⇒ sgn(t)zk {AT (Ax − b)}k + λk |zk | ≥ 0. Now consider the two cases: where t has the same sign as zk , sgn(t) = sgn(zk ) or the opposite sign, sgn(t) = − sgn(zk ). They lead to, respectively: {AT (Ax − b)}k + λk ≥ 0 and

− {AT (Ax − b)}k + λk ≥ 0,

so we obtain the condition:   T {A (b − Ax)}k  ≤ λk , xk = 0 (qk = 1). Thus, we can summarize the component-wise conditions for the minimizer of F (x) 

as in (2.2).

The conditions derived in Lemma 2.1 allow us to pick a strategy for selecting {λk }. As an example, for the case qk = 1 and λk = λ for all k we have that for λ > AT b∞ , the optimal solution is the zero vector. Hence, we typically would start at some value of λ just below AT b∞ where the zero vector is a good initial guess. We can then iteratively decrease λ and use the previous solution as the initial guess at the next lower λ while we go down to some target residual. Well-known techniques such as the L-curve method [9] apply here. 2.2. Derivation of the algorithm. The iteratively reweighted least squares (IRLS) algorithm given by scheme (1.6) with weights (1.4) follows from the construction of a surrogate functional (2.6) which we will use in our analysis, as presented in Lemma 2.2 below. In our constructions, we split the index set 1 ≤ k ≤ N into two parts: Q1 = {k : 1 ≤ qk < 2} and Q2 = {k : qk = 2}. Lemma 2.2. Define the surrogate functional: G(x, a, w, ) = (2.6)

+

Ax − b22 − A(x − a)22 + x − a22   qk    λk qk wk (xk )2 + 2 + (2 − qk )(wk ) qk −2 k∈Q1

+

 /   0 2λk (xk )2 + 2 (wk2 − 2wk + 2) .

k∈Q2

Then the minimization procedure wn = arg minw G(xn , a, w, n ) defines the iteration dependent weights: (2.7)

wkn =

1 [(xnk )2 + (n )2 ]

2−qk 2

.

AN ITERATIVELY REWEIGHTED LEAST SQUARES ALGORITHM

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In addition, the minimization procedure xn+1 = arg minx G(x, xn , wn , n ) produces the iterative scheme:  n  1 (x )k − (AT Axn )k + (AT b)k . = (2.8) xn+1 k 1 + λk qk wkn Proof. For the derivation of the weights from wn = arg minw G(xn , a, w, n ), we take only the terms of G that depend on w. We derive separately the weights for k ∈ Q1 and k ∈ Q2 . First, for k ∈ Q1 : qk 5 ∂ 4 qk wk ((xnk )2 + (n )2 ) + (2 − qk )(wk ) qk −2 = 0 ∂wk qk  qk −1 =⇒ qk (xnk )2 + (n )2 + (2 − qk ) (wk ) qk −2 = 0 qk − 2 1 =⇒ wkn = 2−qk . [(xnk )2 + (n )2 ] 2 Next, for k ∈ Q2 , we have:     0 ∂ / 2λk (xnk )2 + (n )2 (wk2 − 2wk + 2) = 2λk (xnk )2 + (n )2 (2wk − 2) = 0 ∂wk =⇒ wkn = 1 Notice that this implies that (2.7) is valid for k in both sets Q1 and Q2 since for k ∈ Q2 , qk = 2 and (2.7) gives wkn = 1 as required. Next, we verify that the definition:   n n = arg min G(x, x , w ,  ) xn+1 n k x

recovers the iterative scheme (2.8). Using that we have: G(x, xn , wn , n ) (2.9)

wkn

k

= 1 for k ∈ Q2 , as just derived,

= Ax − b22 − A(x − xn )22 + x − xn 22   qk  + λk qk wkn ((xk )2 + (n )2 ) + (2 − qk )(wkn ) qk −2 k∈Q1

+

 /  0 2λk (xk )2 + (n )2 .

k∈Q2

To prove (2.8), we again separately analyze the cases k ∈ Q1 and k ∈ Q2 . We take the gradient of (2.9) with respect to x, then take the k-th component and set to zero. For k ∈ Q1 , removing terms of (2.9) that do not depend on x, we get: ⎛ ⎞  ∂ ⎝ Ax − b22 − A(x − xn )22 + x − xn 22 + λl ql wln x2l ⎠ ∂xk l∈Q1 ⎛ ⎞    ∂ ⎝ = x22 − 2 x, xn + AT b − AT Axn + λl ql wln x2l ⎠ = 0 ∂xk l∈Q1

and the result is: −2{AT b}k + 2{AT Axn }k + 2xk − 2xnk + 2λk qk wkn xk = 0.

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Then we solve for xk and define xn+1 to be the result: k xk (1 + λk qk wkn ) xn+1 k

=⇒

= xnk + {AT b}k − {AT Axn }k - n . 1 x + AT b − AT Axn k . = 1 + λk qk wkn

For k ∈ Q2 , wkn = 1 and we obtain: ⎛ ⎞    ∂ ⎝ x22 − 2 x, xn + AT b − AT Axn + 2λl x2l ⎠ ∂xk l∈Q2

= −2{A b}k + 2{A Ax }k + 2xk − T

T

n

2xnk

+ 4λk xk = 0.

which, upon solving for xk , yields the scheme: - n . 1 x + AT b − AT Axn k . = xn+1 k 1 + 2λk Thus, it follows that (2.8) holds for all 1 ≤ k ≤ N .



Remark 2.3. Assume that as n → ∞, x → x and n → 0. Notice that with the weights in (2.7), we have that: n

wkn (xnk )2 =

(xnk )2 ((xnk )2

+ (n

)2 )

2−qk 2

→

x2k (x2k

+ 0)

2−qk 2

= |xk |qk as n → ∞, if xk = 0.

Next, observe the result of the computation: qk   qk wkn (xk )2 + (n )2 + (2 − qk )(wkn ) qk −2     k  n 2  qk2−2 + 22  qk2−2 q q−2  n 2 2 2 k (2.10) = qk (xk ) + (n ) + (2 − qk ) (xk ) + (n ) qk   n 2 = 2 (xk ) + (n )2 2 . It follows from (2.9) and qk = 2, wkn = 1 for k ∈ Q2 that: G(xn , xn , wn , n ) = Axn − b22   qk  + λk qk wkn ((xk )2 + (n )2 ) + (2 − qk )(wkn ) qk −2 k∈Q1

+

 /  0 2λk (xk )2 + (n )2 ,

k∈Q2

which using (2.10), reduces to:     qk  2 (2.11) Axn − b22 + 2 λk (xnk )2 + (n )2 2 + 2 λk (xnk )2 + (n )2 2 . k∈Q1

k∈Q2

Thus, we recover: (2.12)

G(xn , xn , wn , n ) = Axn − b22 + 2

N 

  qk λk (xnk )2 + (n )2 2 ,

k=1

As n → ∞, assuming xn → x and n → 0, we have that: lim G(xn , xn , wn , n ) = Ax − b22 + 2

n→∞

N 

λk |xk |qk ,

k=1

so we recover the functional (2.1) we would like to minimize.

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2.3. Summary of argument flow. Notation: With some abuse of notation, we will denote by {an } the sequence (an )n∈N , and write {anl }, {anlr } for subsequences (anl )l∈N , (anlr )r∈N , respectively. By F we will refer to the functional Fq,λλ (x) in (2.1). We demonstrate that for our set of iterates {xn } from (1.6), we x), where x ¯ is have convergence to the minimizing value, i.e. limn→∞ F (xn ) = F (¯ such that F (¯ x) ≤ F (x) for all x. Under some conditions on F , the minimizer will ¯. These statements will all follow from be unique. In that case, we have that xn → x a few properties of F and G (from (2.1) and (2.6)) and the sequence of iterates xn from (1.6), which we now state, and which will be proved in Section 3: (1) 0 ≤ F (xn ) ≤ G(xn , xn , wn , n ), ∀ n. (2) G(xn , xn , wn , n ) ≤ G(xn−1 , xn−1 , wn−1 , n−1 ), ∀ n. (3) ∃ subsequence {xnl } of {xn } for which liml→∞ [G(xnl , xnl , wnl , nl ) − F (xnl )] = 0. (4) xn  is bounded, which implies that any subsequence of {xn } has a weakly convergent subsequence; in particular {xnl } has a convergent subsequence {xnlr }. (5) The limit x ¯ of the particular convergent subsequence {xnlr } satisfies the optimality conditions of F (i.e. F (¯ x) ≤ F (x) for all x). We now show that these statements suffice to conclude that limn→∞ F (xn ) = F (¯ x), an important result, as it states that the iterates converge to the minimizing value of the functional. First, let us define the sequence {gn } := G(xn , xn , wn , n ). Note from (1) and (2) that {gn } is bounded from below and monotonically decreasing; it follows that this sequence converges as n → ∞, say to some g¯. Consequently, {G(xnl , xnl , wnl , nl )} = {gnl } converges to g¯ as l → ∞. By (3) it then follows that ¯, it follows {F (xnl )} also converges to g¯ as l → ∞. Since we know that xnlr → x x)}; consequently g¯ = F (¯ x) and from the continuity of F that {F (xnlr )} → {F (¯ hence F (xnl ) → F (¯ x) as l → ∞, where F (¯ x) ≤ F (x) for all x. x). Note that for any σ > 0, ∃L such Finally, we like to show that F (xn ) → F (¯ x)| < σ. Next, for every n ≥ nl ≥ l, we have that ∀l ≥ L we have that |F (xnl ) − F (¯ that: F (xnl ) = gnl ≥ gn = G(xn , xn , wn , n ) ≥ F (xn ) where gnl ≥ gn since nl ≤ n. So this means that F (xnl ) ≥ F (xn ) and we know from x)| = F (xnl )−F (¯ x) < σ, which implies that F (xn )−F (¯ x) < before that |F (xnl )−F (¯ x) ≤ F (x) for all x. It follows that σ for n ≥ nl , where we have used that F (¯ x). This implies, in particular, that for any accumulation point x ˆ F (xn ) → F (¯ of {xn }, we have F (ˆ x) = F (¯ x) (since x ˆ is the limit of a subsequence of {xn } and F is continuous). In the case that the minimizer of F is unique and equal to x ¯, it follows that x ¯ is the only possible accumulation point of {xn }, i.e. that n ¯. The majority of the work in the convergence argument which follows goes x →x into introducing a proper construction for the {n } sequence and showing that the properties (1) - (5) hold for this choice. 3. Analysis of the IRLS algorithm Having set out the fundamentals (derivation of the scheme and outline of the convergence proof), we now analyze the IRLS scheme in (1.6), with weights wkn defined by (1.4) and {n } as defined by (1.5); we establish convergence by proving properties (1) to (5) from Section 2.3. We will assume that A2 < 1. (I.e., A has spectral or operator norm, or equivalently largest singular value, less than 1, which

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can be accomplished by simple rescaling. The largest singular value can typically be estimated accurately using a few iterations of the power scheme.) Lemma 3.1. Let the surrogate functional G be given by (2.6) of Lemma 2.2 and F be the functional in (2.1). Then property (1) above holds. Proof. The proof follows by direct verification using the result of Remark 2.3. G(xn , xn , wn , n ) = Axn − b22 + 2

N 

  qk λk (xnk )2 + (n )2 2

k=1

≥ F (x ) = Ax − n

n

b22

+2

N 

λk |xnk |qk ≥ 0.

k=1

 Lemma 3.2. Assume that the spectral norm of A is bounded by 1, i.e. A2 < 1. Then the sequence of iterates {xn } generated by (1.6) satisfies xn − xn−1 2 → 0 and the xn are bounded in 1 -norm (xn 1 ≤ K for some K ∈ R). Proof. Using the results from Lemma 2.2, we write down a sequence of inequalities: G(xn+1 , xn+1 , wn+1 , n+1 ) ≤ ≤ ≤ ≤

G(xn+1 , xn+1 , wn , n+1 ) [A] G(xn+1 , xn , wn , n+1 ) n+1

n

[B]

n

, x , w , n ) [C] G(x n G(x , xn , wn , n ). [D]

We now offer explanations for [A − D]. First, [A] follows from wn+1 = arg min G(xn+1 , a, w, n+1 ). w

Next for [B], we have: (3.1) G(xn+1 , xn , wn , n+1 ) − G(xn+1 , xn+1 , wn , n+1 ) = xn − xn+1 22 − A(xn − xn+1 )22 , Now A(x − xn )2 ≤ A2 x − xn 2 < x − xn 2 for A2 < 1, so that x − xn 22 − A(x − xn )22 > 0. Next, [C] follows from n+1 ≤ n (directly from (1.5)). Finally, [D] follows from xn+1 = arg min G(x, xn , wn , n ). x We now set up a telescoping sum of non-negative terms, using the inequalities [A − D] above: P  

G(xn+1 , xn , wn , n+1 ) − G(xn+1 , xn+1 , wn , n+1 )



n=1

≤

P    G(xn , xn , wn , n ) − G(xn+1 , xn+1 , wn+1 , n+1 ) n=1

= G(x1 , x1 , w1 , 1 ) − G(xP +1 , xP +1 , wP +1 , P +1 ) ≤ G(x1 , x1 , w1 , 1 ) =: C ∈ R , where we have used that G(xn , xn , wn , n ) is always ≥ 0. Using (3.1), it follows that: P   n  x − xn+1 22 − A(xn − xn+1 )22 ≤ C. n=1

AN ITERATIVELY REWEIGHTED LEAST SQUARES ALGORITHM

401

Since A(xn − xn+1 )22 ≤ A22 xn − xn+1 22 and A2 < 1: xn − xn+1 22 − A(xn − xn+1 )22

≥ xn − xn+1 22 − A22 xn − xn+1 22 = γxn − xn+1 22 ,

where γ := (1 − A22 ) > 0. Consequently, we have: γ

P 

xn − xn+1 22

n=1

≤

P   n  x − xn+1 22 − A(xn − xn+1 )22 ≤ C n=1

=⇒

∞ 

xn − xn+1 22 < ∞

n=1

=⇒

xn − xn+1 2 → 0.

To prove that the {xn } are bounded, we use the result from Remark 2.3: G(xn , xn , wn , n ) = Axn − b22 + 2

N 

  qk λk (xnk )2 + (n )2 2 ≥ λk |xnk |qk ,

k=1

It follows that:

 q1

 q1 k k 1 1 |xnk | ≤ G(xn , xn , wn , n ) ≤ G(x1 , x1 , w1 , 1 ) λk λk  q1

k 1 ≤ max G(x1 , x1 , w1 , 1 ) =: C1 k∈{1,...,N } λk This implies the boundedness of {xn }, since xn 1 =

N 

|xnk | ≤ N C1 .



k=1

By Lemma 3.2 we have that property (2) holds; moreover (4) (the boundedness of the xn 1 ) is established as well. The next lemma demonstrates property (3) and the existence of a convergent subsequence xnlr . Lemma 3.3. There exists a subsequence {nl } of {n } such that every member of the subsequence is defined by: 1  nl = xnl − xnl −1 2 + αnl 2 < nl −1 . Additionally, there is a subsequence {nlr } of this subsequence such that {xnlr } is convergent. Proof. By the definition of the n ’s in (1.5) and by Lemma 3.2, we know that n → 0, since xn − xn−1  → 0 and αn → 0. It follows that a subsequence {nl } must exist such that nl < nl −1 , for otherwise, the monotonicity n+1 ≤ n combined with n > 0 for all n would imply the existence of N0 such that for n ≥ N0 , n+1 = n , implying that the sequence of n ’s would not converge to zero. The fact that nlr exists is a consequence of the boundedness of the iterates {xn } and hence that of {xnl }, Lemma 3.2, and the standard fact that any bounded sequence  in RN has at least one accumulation point.

402

SERGEY VORONIN AND INGRID DAUBECHIES

By Lemma 3.3 and Lemma 3.2, we have that nl → 0 as l → ∞. Thus, together with (2.12), it follows that (3) holds. Lemma 3.4. The limit x ¯ of the converging subsequence {xnlr } satisfies the optimality conditions (2.2) of the convex functional (2.1): = λk sgn(xk )qk |xk |qk −1 , xk = 0 xk = 0 ≤ λk , = 0, xk = 0

T  {AT (b − Ax)}k {A (b − Ax)}k  {AT (b − Ax)}k

(3.2)

(1 ≤ qk ≤ 2) (qk = 1) (qk > 1)

Proof. For each k, we consider three separate cases, depending on the limit x ¯k . (1) xk = 0 and 1 ≤ qk ≤ 2, (2) xk = 0 and qk = 1, (3) xk = 0 and qk > 1. ¯, and since xn − xn+1  → 0 (by Lemma 3.2), we have that: Since xnlr → x →x ¯k . We can rewrite the iterative scheme (1.6) as:

n +1 xk lr

xn+1 (1 + λk qk wkn ) = xnk + {AT (b − Axn )}k k Specializing this to {xnlr } and reordering terms, we have: n

n +1

λk qk wk lr xk lr

n

n +1

= x k lr − x k lr

+ {AT (b − Axnlr )}k .

Since the right hand side converges to a limit as r → ∞, so must the left hand side; we obtain: n

n +1

lim wk lr xk lr

(3.3)

r→∞

=

1 {AT (b − Ax)}k . λk qk

We will use this to compute {AT (b − A¯ x)}k and to verify that (2.2) is satisfied. We n n +1 are thus interested in the value of limr→∞ wk lr xk lr . nlr In case (1), limr→∞ xk = x¯k = 0, we obtain n +1

n

n +1

lim wk lr xk lr

r→∞

n

n

= lim wk lr xk lr l→∞

n

where we have used that limr→∞ it follows that:

xk lr

xk lr n xk lr

n

n

= lim xk lr wk lr , l→∞

+1

n xk lr

= 1, since xn+1 − xn  → 0. Using (1.4), n

n

n +1

lim wk lr xk lr

r→∞

=

=

xk lr xk lim / = 2−qk k 0 2−q r→∞ nlr 2 2 2 ((xk ) + 0) 2 (xk ) + (nlr )2 sgn(xk ) |xk | |xk |2−qk

= sgn(xk )|xk |qk −1 .

Thus, from (3.3), we obtain that: {AT (b − Ax)}k = λk qk sgn(xk )|xk |qk −1 , in accordance with (2.2).

AN ITERATIVELY REWEIGHTED LEAST SQUARES ALGORITHM

403

In case (2) and (3), limr→∞ xnlr = xk = 0, and we still have that (3.3) holds. Writing out (1.6) for xnklr in terms of xnklr −1 , we obtain: n −1 nlr xk

λk qk wk lr

n −1

= xk lr

− xk lr + {AT (b − Axnlr −1 )}k . n

which gives the limit: n −1 nlr xk

lim wk lr

(3.4)

r→∞

=

1 {AT (b − Ax)}k . λk qk

We define βk to be: βk :=

1 {AT (b − Ax)}k λk qk

To prove that (2.2) is satisfied, we must show that |βk | ≤ 1 for case (2) and that βk = 0 for case (3). We first write down some relations involving βk which we will use. Note that n −1 n by (3.4), limr→∞ wk lr xk lr = βk . If βk = 0, it follows that for every σ ∈ (0, 1), ∃r0 such that for every r ≥ r0 :  2  −2  n 2 n −1 n n −1 wk lr xk lr > (1 − σ)βk2 =⇒ xk lr > (1 − σ)βk2 wk lr =  2−qk n −1 (1 − σ)βk2 (xk lr )2 + (nlr −1 )2 Since nlr < nlr −1 , it follows that for r sufficiently large::  2−qk  nlr 2 n −1 xk > (1 − σ)βk2 (xk lr )2 + (nlr )2  2−qk n −1 = (1 − σ)βk2 (xk lr )2 + xnlr − xnlr −1 2 + αnlr  2−qk n −1 n n −1 ≥ (1 − σ)βk2 (xk lr )2 + |xk lr − xk lr | + αnlr  2−qk n −1 n n −1 > (1 − σ)βk2 (xk lr )2 + |xk lr − xk lr | where we have used in the last part that αnlr → 0. To simplify notation, let us set n −1 n n −1 u = x k lr and v = xk lr − xk lr . Then in terms of u and v, we have:  2−qk (3.5) (u + v)2 > (1 − σ)βk2 u2 + |v| Notice that for any K > 0:

2 √ 1 1 0≤ Ku − √ v = Ku2 + v 2 − 2uv K K It follows that:

 1 2 1 2 2 (3.6) (u + v) = u + 2uv + v ≤ u + Ku + v + v = (1 + K)u + 1 + v2 K K 2

2

2

2

2

Using (3.6) in (3.5), we get: (3.7)

 2−qk 1 (1 − σ)βk2 u2 + |v| < (1 + K)u2 + (1 + )v 2 . K

404

SERGEY VORONIN AND INGRID DAUBECHIES

Let us now consider case (2) where qk = 1. We assume that βk > 1 and derive a contradiction. Rearranging terms in (3.7) yields: (3.8)

  1 (1 − σ)βk2 − (1 + K) u2 < (1 + )v 2 − (1 − σ)βk2 |v| = K

 1 2 (1 + )|v| − (1 − σ)βk |v| K

Since we assume that βk2 > 1, we can choose our σ < 1 small enough such that (1 − σ)βk2 > 1; once σ is fixed, we can choose K > 0 small enough such that (1 − σ)βk2 ≥ (1 + K); with these choices of σ and K, the left hand side of (3.8) ≥ 0. With this fixed choice of σ and K we analyze the right hand side of (3.8). Note that by Lemma 3.2, we have that xnlr − xnlr −1 2 → 0 as r → ∞. This means   1 −1 that |v| → 0 as r → ∞. For sufficiently large r, we will have |v| < 1 + K (1 − σ)βk2 , implying that the right hand side of (3.8) would then be ≤ 0. This is in contradiction with the left hand side of this strict inequality (3.8) being ≥ 0. It follows that the assumption |βk | > 1 is not correct. Hence, we have |βk | ≤ 1 which implies that |{AT (b − Ax)}k | ≤ λk , consistent with (2.2). Finally, consider case (3) with qk > 1. We assume that |βk | > 0 and derive a contradiction. In this case, (3.7) does not simplify further:

  2 2−qk 1 2 2 < (1 + K)u + 1 + (3.9) βk (1 − σ) u + |v| v 2 for all K > 0. K This means in particular that:

 1 βk2 (1 − σ)u2(2−qk ) < (1 + K)u2 + 1 + v 2 and K

 1 βk2 (1 − σ)|v|(2−qk ) < (1 + K)u2 + 1 + v2 . K Then the average of the terms is also smaller than this quantity:

   1 2 1 βk (1 − σ) u2(2−qk ) + |v|(2−qk ) < (1 + K)u2 + 1 + v2 . 2 K Rearranging terms again, we have:



 1 2 1 1 βk (1 − σ) − (1 + K)u2(qk −1) < 1 + u2(2−qk ) v 2 − βk2 (1 − σ)|v|(2−qk ) . 2 K 2 Since qk > 1 and thus 2 − qk < 1, we have that for v sufficiently small (obtained by taking r sufficiently large), the right hand side is negative, by the same logic as in the previous case (because |βk | > 0 by assumption, the first term will go to zero faster than the second when v → 0 as r → ∞). Thus, by the above inequality, for r sufficiently large, the left hand side, bounded above by the negative right hand side, must be negative as well. Since u2(2−qk ) is non-negative, that is possible only when: 1 2 β (1 − σ) − (1 + K)u2(qk −1) < 0 2 k n −1 for r sufficiently large. However, since limr→∞ u2(qk −1) = limr→∞ (xk lr )2(qk −1) = 0, this condition cannot be satisfied for large r. This contradicts our original assumption that |βk | > 0. Hence, we conclude that βk = 0. It follows that  {AT (b − Ax)}k = 0, which is the right optimality condition.

AN ITERATIVELY REWEIGHTED LEAST SQUARES ALGORITHM

405

Lemma 3.4, together with the proceeding Lemmas in this section, show that properties (1) to (5) of Section 2.3 hold. It thus follows from the argument in x). Section 2.3 that we have F (xn ) → F (¯

4. Numerics We now discuss some aspects of the numerical implementation and performance of the IRLS algorithm. We first illustrate performance for the case qk = 1 for all k, where it’s easiest to compare with existing algorithms. Then we discuss a simple example concerning a case where different values of qk can be used. An implementation of the scheme as given by (1.6) has the same computational complexity as ISTA in (1.2). Not surprisingly, the performance of the two schemes is also similar. However, our numerical experiments indicate that the speed-up idea behind FISTA as described in [1] is also effective for the IRLS algorithm. FISTA was designed to minimize the function f (x) + g(x), where f is a continuously differentiable convex function with Lipschitz continuous gradient (i.e., ∇f (x) − ∇f (y)2 ≤ Lx − y2 for some constant L > 0), and g is a continuous convex function such as 2τ x1 in the 1 -penalized functional. FISTA uses the proximal mapping function: ,  2 8  1 L  pL (y) = arg min g(x) + x − (y − ∇f (y))  x 2 L 2

to define the following algorithm: y1 xn+1 (4.1) tn+1 y n+1

= x 0 ∈ RN

, t1 =,1 , and for n = 1, 2, . . . ,  2 8  1 L n n n   = pL (y ) = arg min g(x) + x − (y − ∇f (y )) x 2 L " 1 + 1 + 4t2n = 2 tn − 1 n+1 = xn+1 + (x − xn ). tn+1

In the case that f (x) = Ax − b22 and g(x) = 2τ x1 , we obtain: ∇f (x) − ∇f (y)2

= 2AT Ax − 2AT Ay2 = 2AT A(x − y)2 ≤ 2AT A2 x − y2 ,

which implies that when A is scaled such that A2 ≈ 1, the Lipschitz constant can be taken to be L = 2. It follows that:  2    L 1  = 2τ x1 + x − (y − AT (Ay − b))2 . ∇f (y)) (4.2) g(x) +  x − (y −   2 2 L 2 Using (4.2) in (4.1), we obtain:   2  (4.3) xn+1 = arg min 2τ x1 + x − (y n − AT (Ay n − b))2 = x   Sτ y n − AT Ay n + AT b ,

406

SERGEY VORONIN AND INGRID DAUBECHIES

where we have used (1.1). We note that (4.3) is very similar to the ISTA scheme in (1.2), except the thresholding is applied to {y n }. In the same way, we can coin the FIRLS algorithm by performing the steps in (4.1), using

-

1

(4.4) xn+1 = 1+τ

[(ykn )2

+ (n )2 ]

1 2

. y n − AT Ay n + AT b k

for k = 1, . . . , N

in place of (4.3). With the more general weights given by (1.4), we can specialize this algorithm to our functional (2.1). We now demonstrate some results of simple numerical experiments. We begin with the qk = 1 case for all k. We also let the regularization parameter be the same for all k, setting λk = τ . For the first test, we use two differently conditioned random matrices (built up via a reverse SVD procedure with orthogonal random matrices U and V , obtained by performing a QR factorization on Gaussian random matrices, and a custom diagonal matrix of singular values S, to form a 1000 × 1000 matrix A = U SV T ), and a sparse signal x with 5% non-zeros. We form b = Ax and use the different algorithms to recover x ˜ using a single run of 300 iterations max |AT b| . In Figure 1, we plot the decrease of 1 -functional values F1 (xn ) with τ = 105 n −x and recovery percent errors 100 xx versus the iterate number n, using four algorithms: IRLS, FIRLS, ISTA, and FISTA for two matrix types: A1 , with singular values logspaced between 1 and 0.1 and A2 , with singular values logspaced between 1 and 10−4 . We see that the performance of ISTA/IRLS and FISTA/FIRLS are mostly similar, with better recovery using FISTA in the well-conditioned case, but almost identical performance in the worst-conditioned case. In Figure 2, we run a compressive sensing experiment. We again take the 1000 × 1000 matrix of type A2 . Now we use a staircase-like sparse vector x with about 12% non-zeros. After we form b = Ax, we zero out all but the first 13 of the rows of A and b forming Ap and bp (i.e. we only keep a portion of the measurements). We then recover solution x ˜ using Ap and bp while employing a continuation scheme across 20 different values of τ , starting with a zero initial guess at τ = max |AT b| |AT b| and proceeding down to τ = max 50000 , while reusing the previous solutions as the initial guess at each new value of τ . From Figure 2, we can see that the recovered solutions with FIRLS and FISTA are very similar. We illustrate the use of the more general functional in (2.1) in Figure 3. We use the same setup as before, with the different algorithms running across multiple values of the regularization parameter τ , which is fixed for all k. However, we use a more complicated input signal, whose first half is sparse and whose second half is entirely dense. For this reason, in the IRLS schemes, we take qk = 1 for the first half of the weights (for indices k from 1 to n2 ) and qk = 1.9 for the second half (for indices k from n2 + 1 to n). We observe that the recovered signal with the IRLS algorithms is superior to that of the ISTA/FISTA schemes which utilize qk = 1 for all entries. Of course, setting the values of qk for individual coefficients maybe difficult in practice unless one knows the distribution of the sparser and denser parts in advance, although in applications, some information of this nature may be available from the setup of the problem.

AN ITERATIVELY REWEIGHTED LEAST SQUARES ALGORITHM

407

Finally, in Figure 4, we show the result of an image reconstruction experiment. We use two images, blurred with a Gaussian source and corrupted by Gaussian noise. The first image is 170 × 120 and the second is 125 × 125. In both cases, the blurring source is a 2D Gaussian function with support on a 9 × 9 grid with σ = 2.5 and max amplitude of 2.9. We then take the blurred image (obtained via convolution with the blurring source) and add white Gaussian noise, so that the signal to noise ratio is 25. We then recover a corrected image using an application of wavelet denoising followed by IRLS, from the blurred and noisy image. The IRLS algorithms is run over 30 parameters τ with 40 iterations each, in a setup similar to that used for Figure 2. The matrices we use in the inversion are derived from the blur source itself, so this is a non-blind deconvolution. The problem, however, is still challenging and the resulting images are much improved from their blurred and noisy counterparts. We have noticed that the use of qk < 1 can yield, in some instances, slighter sharper reconstructions in the same number of iterations.

irls firls ista fista

10

-1

10

-2

10

-3

10

-4

% error

0

10

F(x n )

F(x n ) vs n

1

10

50

100

150

200

250

300

10

2

10

1

10

0

10

-1

10

-2

10

-3

ERROR vs n irls firls ista fista

50

iteration n F(x ) vs n

0

10

-1

10

-2

10

-3

10

-4

10

150

200

250

300

2

irls firls ista fista

irls firls ista fista

% error

F(x n )

10

100

iteration n ERROR vs n

n

50

100

150

iteration n

200

250

300

10

1

50

100

150

200

250

iteration n

Figure 1. Functional values F (xn ) and recovery percent errors n −x versus the iterate number n for better and worse con100 xx ditioned matrices (medians over 10 trials).

300

408

SERGEY VORONIN AND INGRID DAUBECHIES

SIGNAL TO RECOVER

x(i)

50

x

0

-50 0

200

400

600

800

1000

i

50

50

x irls

0

x firls

0

-50

-50 0

200

400

600

800

50

1000 x ista

0

200

400

600

800

50

1000 x fista

0

0

-50

-50 0

200

400

600

800

1000

0

200

400

600

800

1000

Figure 2. Row 1: sparse model x and the recovery percent errors vs τ . Rows 2 and 3: final recovered solution with algorithms IRLS, FIRLS, ISTA, FISTA vs x.

AN ITERATIVELY REWEIGHTED LEAST SQUARES ALGORITHM

409

SIGNAL TO RECOVER

2

x

1.5 1

x(i)

0.5 0 -0.5 -1 -1.5 -2

0

200

400

600

800

1000

i

2

2 x irls

1.5 1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

0

200

400

600

800

1000

2

-2

0

200

400

600

800

1000

2 x ista

1.5

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5 0

200

400

600

800

1000

x fista

1.5

1

-2

x firls

1.5

-2

0

200

400

600

800

1000

Figure 3. Row 1: half sparse / half dense model x and the recovery percent errors vs τ . Rows 2 and 3: final recovered solution with algorithms IRLS, FIRLS, ISTA, FISTA vs x.

410

SERGEY VORONIN AND INGRID DAUBECHIES

Figure 4. Two examples of blurred and noisy image reconstruction with IRLS. In each row from left to right: original image, blurred and noisy image, and two reconstructions with IRLS using q = 1 and q = 0.85.

5. Conclusions This manuscript presents a new iterative algorithm for obtaining regularized solutions to least squares systems of equations with sparsity constraints. The proposed iteratively reweighted least squares algorithm extends the work of [5] and is similar in form to the popular ISTA and FISTA algorithms [1, 4]; it has the added benefit of being able to minimize a more general sparsity promoting functional. The main contribution of this work is the analysis of the algorithm, relying on matching the approximation rate to the original functional of a smoothened surrogate functional to the speed of convergence of the iterates; this methodology can likely also be applied to other situations. The presented IRLS algorithm (1.6) is very simple to implement and use; it offers performance similar to popular thresholding schemes, including the speedup benefit from the FISTA formulation. Because the surrogate functionals are all quadratic in the xk , they lend themselves naturally to the use of a conjugate gradient approach, which enables further speed-up, as shown elsewhere [7, 11]. References [1] Amir Beck and Marc Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci. 2 (2009), no. 1, 183–202, DOI 10.1137/080716542. MR2486527 [2] Jian-Feng Cai, Stanley Osher, and Zuowei Shen, Linearized Bregman iterations for compressed sensing, Math. Comp. 78 (2009), no. 267, 1515–1536, DOI 10.1090/S0025-5718-0802189-3. MR2501061 [3] Emmanuel J Cand` es and Michael B Wakin. An introduction to compressive sampling. Signal Processing Magazine, IEEE, 25(2):21–30, 2008. [4] Ingrid Daubechies, Michel Defrise, and Christine De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57 (2004), no. 11, 1413–1457, DOI 10.1002/cpa.20042. MR2077704 [5] Ingrid Daubechies, Ronald DeVore, Massimo Fornasier, and C. Sinan G¨ unt¨ urk, Iteratively reweighted least squares minimization for sparse recovery, Comm. Pure Appl. Math. 63 (2010), no. 1, 1–38, DOI 10.1002/cpa.20303. MR2588385

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[6] M´ ario A. T. Figueiredo, Jos´ e M. Bioucas-Dias, and Robert D. Nowak, Majorizationminimization algorithms for wavelet-based image restoration, IEEE Trans. Image Process. 16 (2007), no. 12, 2980–2991, DOI 10.1109/TIP.2007.909318. MR2472805 [7] Massimo Fornasier, Steffen Peter, Holger Rauhut, and Stephan Worm, Conjugate gradient acceleration of iteratively re-weighted least squares methods, Comput. Optim. Appl. 65 (2016), no. 1, 205–259, DOI 10.1007/s10589-016-9839-8. MR3529140 [8] Jean Jacques Fuchs. Convergence of a sparse representations algorithm applicable to real or complex data. IEEE Journal of Selected Topics in Signal Processing, 1(4):598–605, 2007. [9] Per Christian Hansen. The L-curve and its use in the numerical treatment of inverse problems. IMM, Department of Mathematical Modelling, Technical Universityof Denmark, 1999. [10] Frederik J Simons, Ignace Loris, Guust Nolet, Ingrid C Daubechies, S Voronin, JS Judd, Ph A Vetter, J Charl´ety, and C Vonesch. Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity. Geophysical journal international, 187(2):969–988, 2011. [11] Sergey Voronin, Regularization of linear systems with sparsity constraints with applications to large scale inverse problems, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Princeton University. MR3122037 [12] S. Voronin and R. Chartrand. A new generalized thresholding algorithm for inverse problems with sparsity constraints. ICASSP, 2013. [13] S. Voronin, G. Ozkaya, and D. Yoshida. Convolution based smooth approximations to the absolute value function with application to non-smooth regularization. ArXiv e-prints, August 2014. [14] John Wright, Arvind Ganesh, Shankar Rao, Yigang Peng, and Yi Ma. Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization. In Advances in neural information processing systems, pages 2080–2088, 2009. [15] Allen Y Yang, S Shankar Sastry, Arvind Ganesh, and Yi Ma. Fast l1-minimization algorithms and an application in robust face recognition: A review. In Image Processing (ICIP), 2010 17th IEEE International Conference on, pages 1849–1852. IEEE, 2010. [16] Junfeng Yang and Yin Zhang, Alternating direction algorithms for 1 -problems in compressive sensing, SIAM J. Sci. Comput. 33 (2011), no. 1, 250–278, DOI 10.1137/090777761. MR2783194 [17] Wotao Yin, Stanley Osher, Donald Goldfarb, and Jerome Darbon, Bregman iterative algorithms for l1 -minimization with applications to compressed sensing, SIAM J. Imaging Sci. 1 (2008), no. 1, 143–168, DOI 10.1137/070703983. MR2475828 Mathematics, Tufts University, Medford, Massachusetts 02155 E-mail address: [email protected] Mathematics, Duke University, Durham, North Carolina 27708 E-mail address: [email protected]

SELECTED PUBLISHED TITLES IN THIS SERIES

693 Michael Cwikel and Mario Milman, Editors, Functional Analysis, Harmonic Analysis, and Image Processing, 2017 690 Andr´ es Eduardo Caicedo, James Cummings, Peter Koellner, and Paul B. Larson, Editors, Foundations of Mathematics, 2017 689 Erica Flapan, Allison Henrich, Aaron Kaestner, and Sam Nelson, Editors, Knots, Links, Spatial Graphs, and Algebraic Invariants, 2017 688 Jeffrey Bergen, Stefan Catoiu, and William Chin, Editors, Groups, Rings, Group Rings, and Hopf Algebras, 2017 687 Fernanda Botelho, Raena King, and T. S. S. R. K. Rao, Editors, Problems and Recent Methods in Operator Theory, 2017 686 Alp Bassa, Alain Couvreur, and David Kohel, Editors, Arithmetic, Geometry, Cryptography and Coding Theory, 2017 685 Heather A. Harrington, Mohamed Omar, and Matthew Wright, Editors, Algebraic and Geometric Methods in Discrete Mathematics, 2017 684 Anna Beliakova and Aaron D. Lauda, Editors, Categorification in Geometry, Topology, and Physics, 2017 683 Anna Beliakova and Aaron D. Lauda, Editors, Categorification and Higher Representation Theory, 2017 682 Gregory Arone, Brenda Johnson, Pascal Lambrechts, Brian A. Munson, and Ismar Voli´ c, Editors, Manifolds and K-Theory, 2017 681 Shiferaw Berhanu, Nordine Mir, and Emil J. Straube, Editors, Analysis and Geometry in Several Complex Variables, 2017 680 Sergei Gukov, Mikhail Khovanov, and Johannes Walcher, Editors, Physics and Mathematics of Link Homology, 2016 679 Catherine B´ en´ eteau, Alberto A. Condori, Constanze Liaw, William T. Ross, and Alan A. Sola, Editors, Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions, 2016 678 Joseph Auslander, Aimee Johnson, and Cesar E. Silva, Editors, Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, 2016 677 Delaram Kahrobaei, Bren Cavallo, and David Garber, Editors, Algebra and Computer Science, 2016 676 Pierre Martinetti and Jean-Christophe Wallet, Editors, Noncommutative Geometry and Optimal Transport, 2016 675 Ana Claudia Nabarro, Juan J. Nu˜ no-Ballesteros, Ra´ ul Oset Sinha, and Maria Aparecida Soares Ruas, Editors, Real and Complex Singularities, 2016 674 Bogdan D. Suceav˘ a, Alfonso Carriazo, Yun Myung Oh, and Joeri Van der Veken, Editors, Recent Advances in the Geometry of Submanifolds, 2016 673 Alex Martsinkovsky, Gordana Todorov, and Kiyoshi Igusa, Editors, Recent Developments in Representation Theory, 2016 672 Bernard Russo, Asuman G¨ uven Aksoy, Ravshan Ashurov, and Shavkat Ayupov, Editors, Topics in Functional Analysis and Algebra, 2016 671 Robert S. Doran and Efton Park, Editors, Operator Algebras and Their Applications, 2016 670 Krishnendu Gongopadhyay and Rama Mishra, Editors, Knot Theory and Its Applications, 2016 oller, Pieter Moree, and Thomas Ward, Editors, 669 Sergiˇı Kolyada, Martin M¨ Dynamics and Numbers, 2016 668 Gregory Budzban, Harry Randolph Hughes, and Henri Schurz, Editors, Probability on Algebraic and Geometric Structures, 2016

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/conmseries/.

CONM

693

ISBN 978-1-4704-2836-5

AMS

9 781470 428365 CONM/693

Functional Analysis, Harmonic Analysis, Image Processing • Cwikel and Milman, Editors

This volume is dedicated to the memory of Bj¨orn Jawerth. It contains original research contributions and surveys in several of the areas of mathematics to which Bj¨orn made important contributions. Those areas include harmonic analysis, image processing, and functional analysis, which are of course interrelated in many significant and productive ways. Among the contributors are some of the world’s leading experts in these areas. With its combination of research papers and surveys, this book may become an important reference and research tool. This book should be of interest to advanced graduate students and professional researchers in the areas of functional analysis, harmonic analysis, image processing, and approximation theory. It combines articles presenting new research with insightful surveys written by foremost experts.