Function Classes on the Unit Disc: An Introduction [2nd rev. and ext. ed.] 9783110630855, 9783110628449

Citation preview

Miroslav Pavlović Function Classes on the Unit Disc

De Gruyter Studies in Mathematics

|

Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Karl-Hermann Neeb, Erlangen, Germany

Volume 52

Miroslav Pavlović

Function Classes on the Unit Disc |

An Introduction 2nd edition

Mathematics Subject Classification 2010 46E10, 46E15, 46E30, 30H10, 30H20, 30H25, 30H30, 30H35, 31A05,31C45,30J05, 30J15, 30C62, 30C55, 46A16, 47B33 Author Miroslav Pavlović Belgrade Serbia [email protected]

ISBN 978-3-11-062844-9 e-ISBN (PDF) 978-3-11-063085-5 e-ISBN (EPUB) 978-3-11-062865-4 ISSN 0179-0986 Library of Congress Control Number: 2019946053 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

|

Dedicated to the memory of my parents

Preface This edition of Function classes on the unit disc differs from the first edition in that I corrected several mathematical and enormously many typographical errors. I hope that now the formulations of statements are not confusing. Besides, I updated the bibliography with a choice of new papers that appeared (not only) in the period July 2013–November 2018; most of them are commented in Further notes and results, but some new results are included into the text. This forced me to remove some of the former text; for example, the proof of my theorem on Lp integrability of lacunary series with values in quasi-Banach spaces, a section devoted to composition operators between analytic Lipschitz spaces, generalized Carleman’s inequality, and a few theorems that were rather isolated from other material of the book. On the other hand, I shortened some of my proofs and added some proofs that I found after publishing the first edition: the proofs of Spencer’s area theorem, the equivalence of three definitions of weak Hardy spaces, the celebrated Hardy–Littlewood inequality, etc. The reader will also find sections devoted to the integration operator between H p and H q , on Triebel–Lizorkin spaces on the circle and disc, on the generalized area theorem, on a generalized Calderon’s theorem (with slightly modified original proof of Calderon), and a conjecture concerning the (C, α)-convergence of Fourier series for α < 0. The order of chapters is changed so that Littlewood–Paley theory comes after two chapters on Lipschitz spaces. The former chapter Coefficient multipliers is divided into two chapters: Multipliers on H p , BMOA, and Besov spaces and – Decomposition of spaces with subnormal weights and applications. The former chapter Subharmonic behavior and mixed norm spaces is also divided into two chapters: Subharmonic behavior and Bergman-type spaces and Mixed norm spaces with nonstandard weights. The chapter Vector-valued analytic functions is now Appendix B. The reader should not be surprised by the fact that the intersection of this book with other books from the area is almost empty. “It is desirable to diminish intersections with the above-mentioned books by Zygmund, Garnett, and Zhu” – this is a sentence from the response of a reviewer to the publisher’s request to evaluate my proposal. So the reader will not find, e. g., the proof of the Carleson measure theorem, which can be found in hundreds of books. Instead, she/he will find new information about connection of a Carleson measure and the Nevanlinna counting function. On the other hand, the methods used in considering BMOA, Besov–Lipschitz spaces, Bergman spaces with mixed norms, and Hardy spaces differ very much from the methods used in other books.

https://doi.org/10.1515/9783110630855-201

VIII | Preface Acknowledgment I am grateful to de Gruyter and Apostolos Damialis for giving me a chance to revise and update the first version of the book. Miroslav Pavlović Belgrade January 2019

Preface to the first edition This is an attempt to write a book that differs as much as possible from the existing1 books in this area. Although the main protagonists of the story, Hardy, Bergman, Besov, Lipschitz, Bloch, Hardy–Sobolev, BMO, etc., are well-known from many books, some new properties of them are described, whereas verifications of known properties are in many cases new. The reader is assumed to be well acquainted with complex analysis and the theory of Lebesgue integration, which includes the fundamental facts of the harmonic functions theory – Fatou’s theorem on radial limits of the Poisson integral of a complex Borel measure, along with the canonical isometry between the harmonic Hardy space hp and the Lebesgue space Lp (p > 1). The knowledge of a minimum of the theory of Fourier series and Banach space techniques is also desirable (although Appendix A contains necessary facts in the wider framework of quasi-Banach spaces). All this, and much more, can be found in Rudin’s Real and Complex Analysis. Some deep facts on Lebesgue spaces and maximal functions stated without proofs in Appendix C, e. g., the Fefferman–Stein vector maximal theorem and a theorem of Nikishin, only should be understood and taken as granted. One more fact of such deepness is used in Chapter 6 and concerns the real interpolation between Hardy spaces, but it arises because of the athors’s ineffectiveness to find a simple proof, which certainly exists, of a theorem on radial limits of “Hardy–Bloch” functions. The author hopes that applications of these theorems in this text show their strength and that this can motivate the reader to learn the corresponding theories. The exposition is not linear, but the reader can be sure that there are no circular arguments in the text. Acknowledgment I am grateful to the following people: Greg Knese (Washington University in St. Louis), Oscar Blasco (University of Valencia), Jie Xiao (Memorial University of Newfoundland), Dragan Vukotić (Universidad Auto noma de Madrid), Wolfgang Lusky (University of Paderborn), Jorge Hounie (Universidade Federal de Sao Carlos), John Garnett (University of California, LA), José Ángel Peláez (University of Malaga), Aimo Hinkkanen (University of Illinois at Urbana-Champaign), and Ern Gun Kwon (Andong National University), who sent me texts that were unavailable in Belgrade. Miroslav Pavlović Belgrade, July 2013

1 in the author’s head https://doi.org/10.1515/9783110630855-202

Contents Preface | VII Preface to the first edition | IX 1 1.1 1.1.1 1.2 1.2.1 1.3 1.4 1.5 1.5.1 1.6 1.6.1 1.6.2 1.7 1.7.1

The Poisson integral and Hardy spaces | 1 Preliminaries | 1 Green’s formulas | 3 The Poisson integral | 5 Borel measures and the space h1 | 6 The spaces hp and Lp (𝕋) (p > 1) | 12 The space hp (p < 1) | 14 Harmonic conjugates | 21 Privalov–Plessner’s theorem and the Hilbert operator | 22 Hardy spaces: basic properties | 26 Radial limits and mean convergence | 28 The space H1 | 31 The Riesz projection theorem | 33 Aleksandrov’s theorem: Lp (𝕋) = Hp (𝕋) + Hp (𝕋) | 37 Further notes and results | 39

2 2.1 2.1.1 2.2 2.3 2.3.1 2.4 2.4.1 2.5 2.6 2.6.1 2.6.2 2.7 2.7.1 2.7.2 2.7.3 2.8 2.9

Subharmonic functions and Hardy spaces | 45 Basic properties of subharmonic functions | 45 The maximum principle | 47 Properties of the mean values | 48 The Riesz measure | 52 Riesz’ representation formula | 53 Factorization theorems | 56 Inner–outer factorization | 57 Some sharp inequalities | 59 Hardy–Stein identities | 63 Lacunary series | 66 Application to absolutely summing operators | 67 The subordination principle | 69 Nevanlinna counting function and the subordination principle | 71 Composition with inner functions | 74 Approximation with inner functions | 78 The theorem of Burkholder, Gundy, and Silverstein | 79 Addendum: Weak Hardy spaces | 82 Further notes and results | 85

XII | Contents 3 3.1 3.2 3.3 3.3.1 3.4 3.5 3.6 3.6.1

Subharmonic behavior and Bergman-type spaces | 93 Quasinearly subharmonic functions | 94 Regularly oscillating functions | 95 Mixed-norm spaces. Definition and basic properties | 102 Harmonic conjugates and self-conjugate spaces | 112 Embedding theorems | 113 Fractional integration | 116 Reproducing kernels and atomic decomposition | 120 The Coifman–Rochberg theorem | 121 Further notes and results | 126

4 4.1 4.2 4.3 4.4 4.5 4.5.1 4.6

Mixed-norm spaces with nonstandard weights | 133 Classes of real functions | 133 Bergman spaces with rapidly decreasing weights | 136 Mixed-norm spaces with subnormal weights | 139 Lq -integrability of power series with positive coefficients | 140 Lacunary series with complex-valued coefficients | 144 Lacunary series in mixed norm spaces | 146 Remarks on weights | 147 Further notes and results | 152

5 5.1 5.2 5.2.1 5.2.2 5.3 5.3.1 5.4

Taylor coefficients and maximal functions | 157 Using Green’s formula | 157 Using interpolation of operators on Hp | 160 An embedding theorem | 163 The case of monotone coefficients | 168 Strong convergence in H1 | 171 Generalization to (C, α)-convergence | 173 A (C, α)-maximal theorem | 175 Further notes and results | 177

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.7.1 6.8 6.8.1

Besov spaces | 181 Decomposition of Besov spaces: case 1 < p < ⬦ | 181 A maximal function | 184 Decomposition of Besov spaces: case 0 < p ≤ ∞ | 187 Duality in the case 0 < p ≤ ∞ | 189 Embeddings between Hardy and Besov spaces | 195 Best approximation by polynomials | 200 “Normal” Besov spaces | 202 Isomorphisms between Besov spaces | 203 Inner functions in Besov and Hardy–Sobolev spaces | 205 Approximation of a singular inner function | 206

Contents | XIII

6.8.2 6.8.3 6.9

p

Hardy–Sobolev space HS1/p | 211 f-property and K-property | 212 Addendum: Radial limits of Hardy–Bloch functions | 213 Further notes and results | 217

7 7.1 7.2 7.2.1 7.3 7.4

The dual of H 1 , and some related spaces | 221 The norms on BMOA | 221 Garsia’s and Fefferman’s theorems | 225 Fefferman’s duality theorem | 230 Vanishing mean oscillation | 230 p BMOA and B1/p | 232

8 8.1 8.2 8.2.1 8.2.2 8.3 8.4 8.4.1

Lipschitz spaces of first order | 265 Definitions and basic properties | 265 Lipschitz spaces on the circle and disc | 271 Lipschitz spaces of analytic functions | 272 Mean Lipschitz spaces | 273 Lipschitz condition for the modulus | 275 Composition operators | 277 Mean Lipschitz condition for the class RO | 280 Further notes and results | 283

9 9.1 9.2 9.3 9.4 9.4.1 9.4.2 9.5

Lipschitz spaces of higher order | 287 Moduli of smoothness and related spaces | 287 Lipschitz spaces and spaces of harmonic functions | 290 Conjugate functions | 298 Integrated mean Lipschitz spaces | 301 Generalized Besov–Lipschitz spaces | 304 Radial Lipschitz conditions | 307 The Poisson integral and the moduli of Lipschitz functions | 312

7.4.1 7.5 7.6 7.7 7.8 7.8.1 7.8.2 7.8.3 7.9 7.10

p

Tauberian nature of B1/p | 234 Coefficients of BMOA-functions | 235 The Bloch space | 236 Mean growth of Hp -Bloch functions | 239 Composition operators on B and BMOA | 241 Composition operators | 241 Compact composition operators on BMOA | 243 Weighted Bloch spaces | 244 Proof of the bi-Bloch lemma | 251 Addendum: Carleson measures | 254 Further notes and results | 256

XIV | Contents 9.5.1 9.6

Division and multiplication by inner functions | 317 Invariant Besov spaces | 320 Further notes and results | 323

10 Littlewood–Paley theory | 329 10.1 Vector maximal theorems and Calderon’s area theorem | 329 10.2 The Littlewood–Paley g-theorem | 331 10.3 Applications of the (C, m)-maximal theorem | 335 10.4 A generalization of the g-theorem | 340 10.5 A generalized version of the Luzin area theorem | 342 10.6 Littlewood–Paley-type inequalities | 346 10.7 A proof of Calderón’s theorem and generalizations | 351 10.7.1 Hyperbolic Hardy classes | 360 10.8 Integration operators on Hardy spaces | 362 10.8.1 A generalized integration operator | 366 10.9 Addendum: Tents and Triebel–Lizorkin spaces | 370 Further notes and results | 377 11 One-to-one mappings | 385 11.1 Integral means of univalent functions | 385 11.1.1 Distortion theorems | 386 11.2 Hardy–Prawitz theorem and applications | 389 11.3 A Littlewood–Paley theorem for univalent functions | 393 11.4 Quasiconformal harmonic mappings of the disc | 396 11.4.1 Boundary behavior of QCH homeomorphisms of the disc | 396 11.5 Hp theory of QCH mappings | 404 Further notes and results | 410 12 Multipliers on H p , BMOA, and Besov spaces | 417 12.1 Multipliers on abstract spaces | 417 12.1.1 Compact multipliers | 421 12.2 Multipliers for Hardy and Bergman spaces | 422 12.2.1 Multipliers from H1 to BMOA | 425 12.3 Solid spaces | 427 12.3.1 Solid hull of Hardy spaces (0 < p < 1) | 429 12.4 Multipliers between Besov spaces | 431 12.4.1 Monotone multipliers | 434 12.5 Some applications to composition operators | 436 Further notes and results | 438 13

Decompositions of spaces with subnormal weights and applications | 443

Contents | XV

13.1 13.2 13.2.1 13.3 13.3.1 13.4 13.5

Decompositions | 443 Duality | 447 Jackson spaces | 448 Reformulation of the duality theorems | 450 Duality for Bergman spaces | 453 Multipliers on spaces with subnormal weights | 455 Harmonic Besov spaces. Case 0 < p < 1 | 458 Further notes and results | 459

A A.1 A.2 A.3 A.4 A.4.1 A.5 A.6 A.6.1

Quasi-Banach spaces | 461 Quasi-Banach spaces | 461 q-Banach envelops | 462 The closed graph theorem | 465 F -spaces | 468 The Nevanlinna class | 468 The spaces ℓp | 469 Lacunary series in quasi-Banach spaces | 470 Lp -integrability of lacunary series on (0, 1) | 471 Further notes and results | 472

B B.1 B.2 B.2.1 B.3

Bounded vector-valued analytic functions | 475 Some properties of homogeneous spaces | 475 Subharmonic behavior of ‖F (z)‖X | 482 The Banach envelope of Hp (X ), 0 < p < 1 | 485 Linear operators on Hardy and Bergman spaces | 487 Further notes and results | 493

C C.1 C.2 C.3 C.4 C.5 C.6 C.7 C.8

Lebesgue spaces: Interpolation and maximal functions | 495 The Riesz–Thorin theorem | 495 Weak Lp -spaces and Marcinkiewicz’s theorem | 497 Classical maximal functions | 501 The Rademacher functions and Khintchine’s inequality | 507 Nikishin’s theorem | 509 Nikishin–Stein’s theorem | 511 Banach’s principle and the theorem on a. e. convergence | 514 Vector-valued maximal theorem | 516 Further notes and results | 517

Bibliography | 519 Index | 545

1 The Poisson integral and Hardy spaces This chapter contains basic properties of the Poisson integral of an L1 -function and, more generally, of a complex measure on the circle 𝕋. Fatou’s theorem on radial limits, the Privalov–Plessner theorem on radial limits of the conjugate function, the Fefferman–Stein theorem on subharmonic behavior of |f |p , and the Riesz projection theorem are some of the most important results of the chapter. Also, we present the well-known connection between the harmonic Hardy space hp (1≤ p ≤ ∞) and the Lebesgue space Lp (𝕋) with a proof for p = 1. In Section 1.4, we briefly discuss hp for p < 1. In the last section, we present a quick introduction to basic properties of (analytic) Hardy spaces. Our approach differs from those in other texts [166, 198, 321, 510, 517, 629] because we first prove the Hardy–Littlewood decomposition lemma and then deduce the radial limit theorem and some other fundamental results due to F. and M. Riesz, Smirnov, Szegö, Kolmogorov, et al., without using Blaschke products. At one place, we use the Hardy–Littlewood complex maximal theorem although we consider the maximal functions in Appendix C, Section C.3. However, the reader can treat Section C.3 as part of this chapter inserted before considering Hardy spaces.

1.1 Preliminaries Some notation We denote by ℝ, ℂ, ℤ, and ℕ the real line, the complex plane, the set of all integers, and the set of nonnegative integers, respectively. By ℝ+ and ℕ+ we denote the set of positive real numbers and the set of positive integers. If dμ is a finite positive measure on a sigma-algebra of subsets of a set S, then we write − f dμ = ∫ S

1 ∫ f dμ μ(S) S

and, in particular, 2π

− f (eiθ ) dθ = ∫ 0

2π

1 ∫ f (eiθ ) dθ, 2π 0

− f dA = ∫ 𝔻

1 ∫ f dA, π 𝔻

where dA is the Lebesgue measure on ℂ, and 𝔻 = {z ∈ ℂ: |z| < 1}. Similarly, − f (ζ )|dζ | = ∫ 𝕋 https://doi.org/10.1515/9783110630855-001

1 ∫ f (ζ )|dζ |, 2π 𝕋

where 𝕋 = 𝜕𝔻.

2 | 1 The Poisson integral and Hardy spaces We denote the arc-length measure on 𝕋 by dl, and so ∫ f (ζ )|dζ | = ∫ f dl. 𝕋

𝕋

We denote the two-dimensional measure of a measurable set G ⊂ ℂ by |G|. Similarly, |S| denotes the arc-length measure of S ⊂ 𝕋. Möbius transformations of the unit disc Every biholomorphic mapping (Möbius transformation) φ from 𝔻 onto 𝔻 can be represented as φ(z) = bσa (z), where |b| = 1 and σa (z) =

a−z , 1 − az̄

|a| < 1, |z| ≤ 1.

These transformations form a group, called the Möbius group and denoted by Möb(𝔻), with respect to composition of mappings. The functions σa have important properties: – σa−1 = σa , where σa−1 denotes the inverse mapping. – –

2

1−|a| , |a| < 1, |z| ≤ 1. σa󸀠 (z) := (σa )󸀠 (z) = − (1− ̄ 2 az) We have

2 2 󵄨2 (1 − |a| )(1 − |z| ) 󵄨󵄨 󸀠 󵄨󵄨 󵄨 1 − 󵄨󵄨󵄨σa (z)󵄨󵄨󵄨 = = 󵄨󵄨σa (z)󵄨󵄨(1 − |z|2 ) 2 ̄ |1 − az|

and, more generally, 1 − σa (z)σa (w) = –

–

(1 − |a|2 )(1 − z w)̄ . ̄ − aw)̄ (1 − z a)(1

The functional dh (a, z) = |σa (z)| (a, z ∈ 𝔻) is a metric on 𝔻 and is called the pseudohyperbolic metric. It is Möbius invariant in the sense that dh (σ(w), σ(z)) = dh (w, z) for all σ ∈ Möb(𝔻) and z, w ∈ 𝔻. The measure dτ(z) = (1 − |z|2 )−2 dA(z) is Möbius invariant, which means in particular that ∫ h ∘ σa dτ = ∫ h dτ, 𝔻

𝔻

where h ≥ 0 is a measurable function on 𝔻. Exercise 1.1. Let f be an analytic self-map of 𝔻, and let a ∈ 𝔻. Then f 󸀠 (a) = 0 if and only if 󵄨󵄨 f (z) − f (a) 󵄨󵄨 󵄨󵄨 a − z 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨≤󵄨 󵄨, 󵄨󵄨 󵄨󵄨 1 − f (a)f (z) 󵄨󵄨󵄨 󵄨󵄨󵄨 1 − az̄ 󵄨󵄨󵄨

z ∈ 𝔻.

1.1 Preliminaries | 3

1.1.1 Green’s formulas There are various formulas named “Green”. The following one plays a substantial role in several subsequent results. Theorem. If F is a C 2 -function in D = {z : |z| < R}, then π

d 1 − F(reiθ ) dθ = ∫ (ΔF)(z) dA(z) ∫ dr 2πr −π

(1.1)

|z|δ

Now we apply the inequality |tP 󸀠 (r, t)| ≤ 2P(r, t), |t| ≤ π, to obtain δ 󵄨󵄨 γ(t) 󵄨󵄨 1 󵄨󵄨 󵄨 L ≤ lim sup ∫ P(r, t)󵄨󵄨󵄨 󵄨 dt. 󵄨󵄨 t 󵄨󵄨󵄨 r→1− π −δ

Finally, the desired result follows from this and from the hypothesis γ(t)/t → 0, t → 0. As a particular case of Theorem 1.7, we have the following: Corollary 1.11. Every bounded harmonic function on 𝔻 has radial limits almost everywhere. On the other hand, this particular case is sufficient to prove the qualitative part of Fatou’s theorem, i. e., to prove the existence of the limits for h1 -functions (see the proof of Corollary 1.14). The following particular case is also of fundamental importance. Corollary 1.12. A measure μ ∈ M(𝕋) is singular if and only if limr→1− 𝒫 [μ](rζ ) = 0 almost everywhere. Exercise 1.13 ([528, Proposition 2.5]). If u is the Poisson integral of a singular measure, then 󵄨p 󵄨 lim ∫󵄨󵄨󵄨u(rζ )󵄨󵄨󵄨 |dζ | = 0 r→1

𝕋

Nontangential limits There exists a result stronger than Theorem 1.7.

for 0 < p < 1.

12 | 1 The Poisson integral and Hardy spaces Theorem 1.8. If f ∈ h1 , then for almost all ζ ∈ 𝕋, there exists the limit ∢ lim f (z) := lim f (z), z→ζ

Uζ ∋z→ζ

where Uζ is a Stoltz angle, Uζ = Uζ ,ρ = the open convex envelope of {z : |z| < ρ} ∪ {ζ }, and ρ < 1 is fixed. Theorem 1.8 can be deduced from Theorem 1.7 and the following: Theorem 1.9. Let f be a bounded analytic function in 𝔻, and let ζ ∈ 𝕋. Then the existence of the radial limit of f at ζ implies the existence of the nontangential limit at ζ . Corollary 1.14. If f ∈ H(𝔻) and u = Re f ∈ h1 , then f has nontangential limits at almost every point ζ ∈ 𝕋. Proof. We may assume that u is positive. Then the function 1/(1 + f ) is analytic and bounded in 𝔻. The result follows. Proof of Theorem 1.9. Let ζ = 1 and limr→1 f (r) = 0. If f fails to have the limit 0 within Uζ ,ρ = {tw + 1 − t: |w| < ρ, 0 < t < 1}, then there exist ε > 0 and sequences tn → 0, 0 < tn < 1, and wn , |wn | < ρ < 1, such that 󵄨 󵄨󵄨 󵄨󵄨f (tn wn + 1 − tn )󵄨󵄨󵄨 > ε

for all n.

(1.16)

Consider the functions fn (w) = f (tn w + 1 − tn ), w ∈ 𝔻. The sequence fn is uniformly bounded in 𝔻, and therefore there exists a subsequence, denote it by fn , that converges to a function g ∈ H(𝔻) uniformly on |w| < ρ. This implies fn (wn ) − g(wn ) → 0. By hypothesis we have f (tn r + 1 − tn ) → 0 for 0 < r < 1, and therefore g(r) = 0 for 0 < r < 1, whence g(w) = 0 for all w ∈ 𝔻. Thus fn (wn ) → 0, which contradicts (1.16).

1.3 The spaces hp and Lp (𝕋) (p > 1) For a Borel-measurable function f : ρ𝔻 󳨃→ ℂ, we define the integral means Mp (r, f ), 0 < r < ρ, by 2π

󵄨p 󵄨 Mp (r, f ) = ‖fr ‖p = (− ∫ 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ)

1/p

,

0

where fr (eiθ ) = f (reiθ ). The harmonic Hardy space hp (0 < p ≤ ∞) is defined by hp = {f ∈ h(𝔻): ‖f ‖p = sup Mp (r, f ) < ∞}. r 1)

| 13

In particular, h∞ = h∞ (𝔻) is the subspace of L∞ (𝔻) spanned by harmonic functions. The space hA(𝔻) consists of functions continuous on 𝔻 and harmonic in 𝔻. By Parseval’s formula we have ∞

󵄨2 󵄨 M22 (r, f ) = ∑ 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 r 2|n| , n=−∞

and, consequently, 1/2

∞

󵄨2 󵄨 ‖f ‖2 = ( ∑ 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 ) . n=−∞

Radial limits Since 1/2

∞

󵄨̂ 󵄨󵄨2 ‖ϕ‖2 = ( ∑ 󵄨󵄨󵄨ϕ(n) 󵄨󵄨 ) , n=−∞

̂ where ϕ ∈ L2 , and since f ̂(n) = ϕ(n), provided that f = 𝒫 [ϕ], we see that the Poisson integral acts as an isometric isomorphism from L2 onto h2 . It is very important that this holds 1 < p ≤ ∞. The following fact is also named “Fatou’s theorem” and is stated together with Theorem 1.7 as Theorem 5.3 in [198]. Theorem 1.10. The function f belongs to hp (1 < p ≤ ∞) if and only if it is equal to the Poisson integral of some function g ∈ Lp (𝕋). If f = 𝒫 [g], g ∈ Lp (𝕋), then π

‖f ‖p = ‖g‖p

(1 ≤ p ≤ ∞),

− |f (reiθ ) − g(eiθ )|p dθ = 0 lim ∫

r→1−

(1 ≤ p < ∞),

(1.17)

−π

and lim f (reiθ ) = g(eiθ ) almost everywhere.

r→1−

The Poisson kernel shows that an h1 -function need not be equal to the Poisson integral of the boundary function. However, (1.17) implies the following: Corollary 1.15. If f ∈ hp , p > 1, then f = 𝒫 [f∗ ], where f∗ (eiθ ) = lim− f (reiθ ). r→1

The Poisson kernel also shows that boundedness of f∗ does not imply the boundedness of f . However, if p > 1, then we have the following generalization of the maximum modulus principle.

14 | 1 The Poisson integral and Hardy spaces Corollary 1.16. If a function f ∈ h(𝔻) have radial limits f∗ (eiθ ) almost everywhere, f∗ ∈ L∞ (𝕋), and f ∈ hp for some p > 1, then f ∈ h∞ and ‖f ‖∞ = ‖f∗ ‖∞ . Exercise 1.17. If f ∈ hp (p > 1) is real valued, then there are nonnegative functions fj ∈ hp such that f = f1 − f2 and ‖f ‖p = (‖f1 ‖pp + ‖f2 ‖pp )1/p . Exercise 1.18. Let z ∈ 𝔻. The norm of the linear functional z 󳨃→ f (z) on the space hp (1 ≤ p < ∞) is equal to Kp (|z|)(1 − |z|2 )−1/p , where π

1/q

󵄨 󵄨2q−2 dt) Kp (r) = ( ∫ − 󵄨󵄨󵄨1 − reit 󵄨󵄨󵄨

(1/p + 1/q = 1).

−π

Observe that K2 (r) = (1 + r 2 )1/2 . For the case of several variables, see [290], where the norm is expressed via the Euler gamma function. Exercise 1.19. The inclusion hp ⊂ h(𝔻) (1 ≤ p ≤ ∞) is compact, i. e., every closed ball of the space hp is compact in the h(𝔻)-topology. Exercise 1.20. If p ≥ 1 and f ∈ h(𝔻), then Mp (r, f ) increases in r ∈ (0, 1). If p > 1, then Mp (r, f ) is strictly increasing unless f =const.

1.4 The space hp (p < 1) We have already defined the space hp for p < 1 in the previous section. In contrast to the case p ≥ 1, the structure of hp when p < 1 is rather mysterious and complicated. For instance, it is not easy to verify that the inclusion hp ⊂ h(𝔻) is continuous, a fact which simply holds for p ≥ 1 because |f ̂(n)| ≤ ‖f ‖p . 1.4.a (Admissible spaces). For simplicity, by the term “quasinormed space” we mean a vector space endowed with a p-norm (0 < p ≤ 1); see Section A.1. A quasinormed space X ⊂ h(𝔻) is called admissible (or h-admissible) if it is complete, h(𝔻) ⊂ X,

(1.18)

the inclusion X ⊂ h(𝔻) is continuous.

(1.19)

and

If an admissible space X satisfies sup ‖fr ‖X ≤ CX ‖f ‖X ,

0 0, choose a polynomial P such that ‖f − P‖X ≤ ε. Then ‖f − fr ‖X ≤ M‖f − P‖X + M‖P − Pr ‖X + M‖Pr − fr ‖X , where M = const. Since X is R-admissible, we have ‖Pr − fr ‖X ≤ C‖P − f ‖X ≤ Cε. Since ‖P − Pr ‖X → 0 (r → 1), we see that lim sup ‖f − fr ‖X ≤ Mε + MCε. r→1−

This completes the proof of the lemma. (If X ⊂ h(𝔻), then the proof is the same.) 1.4.c (Homogeneous spaces and the Fatou property). Many classical spaces of analytic or harmonic functions satisfies the following conditions: If f ∈ X, then fζ ∈ X (|ζ | ≤ 1) and sup ‖fw ‖X = ‖f ‖X ;

(1.23)

If sup ‖fw ‖X ≤ 1, then f ∈ X and ‖f ‖X ≤ CX .

(1.24)

w∈𝔻

w∈𝔻

If an admissible space X satisfies condition (1.23), then it is called homogeneous. The space X is said to have the Fatou property, abbreviated “X is FP” or “X is an FP-space”, if it satisfies (1.24). It is clear that a homogeneous space is R-admissible. Note two facts. If f ∈ X, where X is admissible, then fw ∈ h(𝔻) ⊂ X for w ∈ 𝔻; and if X is both homogeneous and FP, then CX = 1 in (1.24). Using the preceding results, it is not difficult to verify that the space hp (p ≥ 1) is admissible and homogeneous. However, the fact that hp is admissible for p < 1 is not so clear. The standard path begins by one of nontrivial inequalities of Hardy and Littlewood (Theorem 1.15) but, by the author’s opinion, the right and maybe a unique approach should start from the Fefferman–Stein theorem on subharmonic behavior of |u|p . Here we present one of two simple proofs from [428]. The second proof will be given later on; see Theorem 3.1. Theorem 1.11 (Fefferman–Stein). If u ∈ h(𝔻) and 0 < p < 1, then 󵄨p 󵄨󵄨 p 󵄨󵄨u(0)󵄨󵄨󵄨 ≤ Cp ∫ |u| dA. 𝔻

(1.25)

1.4 The space hp (p < 1)

| 17

Of course, if p ≥ 1, then (1.25) holds with Cp = 1/π. Theorem 1.11 is a formal consequence of the following theorem. Theorem 1.12 (Hardy–Littlewood). If p > 0, u = Re f , and Im f (0) = 0, where f is analytic in 𝔻, then ∫ |f |p dA ≤ Cp ∫ |u|p dA. 𝔻

(1.26)

𝔻

Indeed, in deducing (1.25) from this theorem, we may assume that u is real valued. Since |u(0)|p = |f (0)|p and |f |p is subharmonic, which means, in particular, that 󵄨p 1 󵄨󵄨 p 󵄨󵄨f (0)󵄨󵄨󵄨 ≤ ∫ |f | dA, π

p > 0,

𝔻

we see that (1.26) implies (1.25). However, it seems more natural to prove Theorem 1.11 directly and then deduce Theorem 1.12. The latter is postponed to Chapter 3, where a more general result will be proved (Theorem 3.8). Proof of Theorem 1.11. Assuming, as we may, that u is real valued, we start from the inequality 󵄨 󵄨 󵄨 K 󵄨󵄨 󵄨󵄨∇u(a)󵄨󵄨󵄨 ≤ sup{󵄨󵄨󵄨u(z)󵄨󵄨󵄨 : |z − a| < r}, r

(1.27)

which is valid whenever the disc

Dr (a) = {z : |z − a| < r} is contained in 𝔻. (The constant K is independent of r and a. In this case, we have K = 2.) Choose a such that 2󵄨 2󵄨 󵄨p 󵄨p (1 − |a|) 󵄨󵄨󵄨u(a)󵄨󵄨󵄨 ≥ (1 − |z|) 󵄨󵄨󵄨u(z)󵄨󵄨󵄨

for all z ∈ 𝔻.

(1.28)

From Lagrange’s theorem and inequality (1.27) it follows that 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨u(a)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 + K(t/r) sup |u| Ds (a)

(s = t + r),

provided that z ∈ Dt (a) and Ds (a) ⊂ 𝔻. Now choose t and r such that s = (1 − |a|)/2. Then u(w)p ≤ 4u(a)p for w ∈ Ds (a), and therefore u(a) ≤ u(z) + Kp (t/r)u(a),

z ∈ Dt (a),

where Kp = K ⋅ 41/p . Next, we choose t and r such that Kp (t/r) = 1/2 and find that u(a)p ≤ 2p u(z)p , z ∈ Dt (a). Integrating this inequality over z ∈ Dt (a), we get t 2 u(a)p ≤ 2p ∫ up dA ≤ 2p . Dt (a)

Since t = cp (1 − |a|), we see that u(0)p ≤ (1 − |a|)2 u(a)p ≤ 2p /cp2 , which concludes the proof.

18 | 1 The Poisson integral and Hardy spaces By translations and dilations, we obtain the following: Theorem 1.13. If f is a function harmonic in a subdomain G ⊂ ℂ and Dr (a) ⊂ G, then Cp 󵄨p 󵄨󵄨 ∫ |f |p dA 󵄨󵄨f (a)󵄨󵄨󵄨 ≤ |Dr (a)|

whenever Dr (a) ⊂ G.

Dr (a)

Applying (1.25) to the disc of radius 1 − |z| centered at z ∈ 𝔻, we get −2 󵄨p 󵄨󵄨 p 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ Cp (1 − |z|) ∫ |f | dA.

(1.29)

𝔻

This was the reason for choosing a by (1.28). The following theorem shows that the inclusion hp ⊂ h(𝔻) is continuous for p < 1. Theorem 1.14 (Hardy–Littlewood). For u ∈ hp (0 < p < 1), we have: −1/p 󵄨 󵄨󵄨 ‖u‖p 󵄨󵄨u(z)󵄨󵄨󵄨 ≤ Cp (1 − |z|) 1/p−1 󵄨󵄨 ̂ 󵄨󵄨 ‖u‖p 󵄨󵄨u(n)󵄨󵄨 ≤ Cp (|n| + 1)

(z ∈ 𝔻), (n ∈ ℤ).

Proof. By (1.29) we have 󵄨󵄨 −2 iθ 󵄨p 󵄨󵄨u(re )󵄨󵄨󵄨 ≤ Cp (1 − r)

∫ 2r−1 n0 , then Mp (r, un − um ) < ε for all r ∈ (0, 1). Letting m tend to ∞, we get Mp (r, un − u) < ε for all n > n0 . Since r is independent of ε, we see that ‖un − u‖p ≤ ε, which was to be proved. The following estimates are proved in a similar way as (1.30) and (1.31). Theorem 1.15 (Hardy–Littlewood). If u = Re f , f ∈ H(𝔻), and u ∈ hp (0 < p ≤ 1), then Mp (r, f 󸀠 ) ≤ Cp (1 − r)−1 ‖u‖p , Mp (r, f ) ≤ Cp (log

1/p

2 ) 1−r

‖u‖p

(0 < r < 1).

The first inequality holds for all 0 < p ≤ ∞. 1.4.d (The space ohp ). The space ohp is the subspace of hp consisting of u ∈ h(𝔻) such that limr→1− Mp (r, u) = 0. It is an infinite-dimensional space. Indeed, according to Exercise 1.13 (or to Proposition 1.23) ohp contains Pζ for all ζ ∈ 𝕋, where Pζ (z) = P(z, ζ ). We leave to the reader to show that ohp is closed in hp [528, Proposition 2.4]. Proposition 1.23. The Poisson kernel satisfies the following conditions: Mpp (r, P) = Mqq (r, P) (q = 1 − p) and Mp (r, P) ≍ βp (r), where βp (r) = (1 − r)

for 0 < p < 1/2;

= (1 − r)(log = (1 − r)1/p−1

2

e ) 1−r

for p = 1/2;

(1.32)

for p > 1/2.

Proof. We have 󵄨p 󵄨 Ip (r) := Mpp (r, P) = ∫ − P(r, ζ )p |dζ | = ∫ − 󵄨󵄨󵄨σr󸀠 (ζ )󵄨󵄨󵄨 |dζ |, 𝕋

𝕋

where σr ∈ Möb(𝔻) is the involution σr (ζ ) = (r − ζ )/(1 − rζ ). By the substitution ζ = σr (ξ ) we get 󵄨 󵄨p 󵄨 󵄨 Ip (r) = ∫ − 󵄨󵄨󵄨σr󸀠 (σr (ξ ))󵄨󵄨󵄨 󵄨󵄨󵄨σr󸀠 (ξ )󵄨󵄨󵄨|dξ |. 𝕋

From this, using the formula σr󸀠 (σr (ξ )) = 1/σr󸀠 (ξ ), we obtain Ip = I1−p . To estimate the integral Ip (r), we use the relation P(r, θ) ≍ (1 − r)/(1 − r + |θ|)2 , |θ| < π, r > 1/2. A simple calculation gives the result.

20 | 1 The Poisson integral and Hardy spaces 1.4.e. The notation A ≍ B, where A and B are nonnegative quantities depending on some parameters, means that there are constants c and C independent of the parameters such that cB ≤ A ≤ CB; in particular, this means that A is finite if and only if so is B. We call c and C the equivalence constants. Isomorphic copy of ℓ∞ in hp (0 < p < 1) That the space hp (p < 1) is not minimal is a consequence of the following statement (minimal spaces are separable). Theorem 1.16 (Shapiro). If 0 < p < 1, then hp contains an isomorphic copy of ℓ∞ and consequently is not separable. Proof [528]. Let h denote the h(𝔻)-topology. The space ohp is closed in hp , and by (1.30) h is weaker than the hp -topology. Moreover, since the unit ball B of ohp is h-relatively compact, we have that if h coincides with the hp -topology, then B is h-closed and hence hp -compact. However, this implies that ohp is finite-dimensional, a contradiction. Thus the inclusion map (ohp , h) 󳨃→ hp is not continuous, which implies the existence of a sequence un ∈ ohp such that un → 0 in h (n → ∞) and ‖un ‖p = 1. Now we choose an increasing sequence rn ∈ (0, 1) and an increasing sequence kn ∈ ℕ+ as follows. Since un → 0 in h, we can choose k1 such that |uk1 (z)|p < ε/2 for |z| < 1/2 = r0 . Then, since Mp (r, uk1 ) → 0 as r → 1− , we can choose r1 > r0 such that Mpp (r, uk1 ) < ε/2 for r1 ≤ r < 1. Next, we choose k2 such that |uk2 (z)|p < ε/22 and r2 > r1 such that Mpp (r, uk2 ) < ε/22 . Continuing in this way, we find sequences rn ↑ 1 and vn such that Mpp (r, vn ) < ε/2n for r ∈ [0, 1) \ [rn−1 , rn ) (n ≥ 1). Let {an } ∈ ℓ∞ and consider the formal series u = ∑∞ n=1 an vn . Fix r ∈ [0, 1) and let j be the unique integer such that rj ≤ r < rj+1 . Then ∞

∑ Mpp (r, an vn ) = |aj |p Mpp (r, vj ) + ∑ |an |p Mpp (r, un )

n=1

n=j̸

p

p

≤ |aj | + sup |an | ∑ n=j̸

n=j̸

ε 2n

󵄩p 󵄩 ≤ |aj | + sup |an | ε ≤ (1 + ε)󵄩󵄩󵄩{an }󵄩󵄩󵄩∞ . p

p

n=j̸

From this and from (1.30) it follows that the partial sums of u form an h-Cauchy sequence, and therefore the series u converges uniformly on compacts to a harmonic function; denote it by u. Incidentally, the last inequality shows that ‖u‖pp ≤ (1 + ε)‖{an }‖p∞ . In the other direction, let ε < 1 and λ > 0 be such that ε < λp < 1. Choose an index j such that |aj | > λ‖{an }‖∞ . It is also clear that there exists r ∈ [rj−1 , rj ) such that

1.5 Harmonic conjugates | 21

Mp (r, vj ) = 1. Then ‖u‖pp ≥ Mpp (r, u) ≥ |aj |p Mpp (r, vj ) − ∑ |an |p Mpp (r, vn ) n=j̸

p

≥ |aj |

‖u‖pp

󵄩p 󵄩 󵄩p 󵄩 − ε󵄩󵄩󵄩{an }󵄩󵄩󵄩∞ ≥ (λp − ε)󵄩󵄩󵄩{an }󵄩󵄩󵄩∞ .

Letting λ tend to 1, we get the desired result. Two open problems As far as the author knows, the following two problems are still unsolved. Problem 1.1. Does there exist a function u ∈ hp (p < 1) such that 1/p−1 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨u(n)󵄨󵄨 ≥ cp (|n| + 1)

(n ∈ ℤ)?

(1.33)

Problem 1.2. Does there exist a function f ∈ H(𝔻) such that u = Re f ∈ hp and Mpp (r, f ) ≥ c log

2 1−r

(0 < r < 1)?

(1.34)

Hardy and Littlewood [236] proved that the answer is positive, provided that p = 1/N, N = 2, 3, . . . ; their example is u(z) = DN P(z) =

𝜕N P iθ (re ), 𝜕θN

where P is the Poisson kernel. It is clear that u satisfies condition (1.33) for p = 1/N. Inequality (1.34) follows from the estimate (e. g., [528]) 1 − |z|2 󵄨 󵄨󵄨 N . 󵄨󵄨D P(z)󵄨󵄨󵄨 ≤ C |1 − z|N+1 It should be noted that solving Problem 1.1 leads to solution of Problem 1.2. Namely, (1.33) implies (1.34). Indeed, if (1.33) is satisfied, then |f ̂(n)| ≥ c(n + 1)1/p−1 , so the conclusion follows from the inequality ∞

󵄨p 󵄨 Mpp (r, f ) ≥ c ∑ (n + 1)p−2 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 r np , n=0

which will be proved later (Theorem 5.1).

1.5 Harmonic conjugates To each f ∈ h(𝔻), there corresponds the harmonic conjugate f ̃ ∈ h(𝔻), ∞

f ̃(reiθ ) = −i ∑ (sign n)f ̂(n)r |n| einθ . n=−∞

22 | 1 The Poisson integral and Hardy spaces If f is real valued, then f ̃ is uniquely determined by the following conditions: f ̃ is real valued, f + if ̃ is analytic, and f ̃(0) = 0. ̃ ̃f = For an arbitrary f ∈ h(𝔻), we have f ̃ = −f + f (0), and if f is analytic, then Re Im(f − f (0)). The function conjugate to 𝒫 [ϕ], ϕ ∈ L1 (𝕋), equals π

π

−π

0

iθ ̃ ̃ θ − t)ϕ(eit ) dt = ∫ ̃ t)[ϕ(θ − t) − ϕ(θ + t)] dt, P[ϕ](re )= ∫ − P(r, − P(r,

̃ denotes the conjugate Poisson kernel, where we write ϕ(x) instead of ϕ(eix ). Here P 2z 1+z ̃ P(z) = Im = Im , 1−z 1−z that is, ̃ θ) = P(re ̃ iθ ) = P(r,

2r sin θ . 1 + r 2 − 2r cos θ

This kernel does not belong to h1 . ̃ are connected by the formula The kernels P and P ̃ θ) − P(r,

1 1 − r P(r, θ) =− . tan(θ/2) tan(θ/2) 1 + r

(1.35)

̃ θ). Note that 1/tan(θ/2) = P(1, Exercise 1.24. Let f (reiθ ) be a harmonic function in 𝔻, and let ℛf = r 𝜕f and Df = 𝜕r ̃ Then ℛf and Df are harmonic in 𝔻, and ℛu = Du.

𝜕f . 𝜕θ

Exercise 1.25. If g = u ∗ v, where u and v are harmonic in 𝔻, then g̃ = u ∗ v.̃ 1.5.1 Privalov–Plessner’s theorem and the Hilbert operator The following fundamental result is due to Privalov [491] and Plessner [481]. Theorem 1.17 (Privalov–Plessner). If g ∈ L1 (𝕋), then the following two limits exist and are equal a. e.: π

iθ ̃ lim− P[g](re ) and

lim+

r→1

ε→0

1 g(θ − t) − g(θ + t) dt. ∫ π 2 tan(t/2) ε

iθ ̃ The existence of limr→1− P[g](re ) is contained in Corollary 1.14. This theorem guarantees the existence of the improper integral

g̃ (eiθ ) =

π

1 g(θ − t) − g(θ + t) dt. ∫ π 2 tan(t/2) 0+

1.5 Harmonic conjugates | 23

There exists a function g ∈ C(𝕋) such that this integral does not converge absolutely for any θ. It is even more interesting that there exists a function g ∈ C(𝕋) such that the improper integral π

∫ 0+

g(θ + t) − g(θ) dt 2 tan(t/2)

diverges for every θ ([639, p. 133–134]). The Hilbert operator The function g̃ is said to be conjugate with g, and the operator H taking g to g̃ is called the Hilbert operator.2 This operator maps L1 into Lp for every p < 1, but not into L1 , so in the general case the Poisson integral of g̃ has no sense. However, as we will prove ̃ later (Theorem 1.39), if g̃ ∈ L1 , then 𝒫 [g̃ ] = P[g]. Proof of Theorem 1.17. It suffices to prove that iθ ̃ )− lim− (P[g](re

r→1

π

g(θ − t) − g(θ + t) 1 dt) = 0 ∫ 2π tan(t/2)

(1.36)

1−r

under the hypothesis that θ is a Lebesgue point of g (p. 502 and Theorem C.8). We write the difference under limr→1− in (1.36) as I1 (r) + I2 (r), where 1−r

1 ̃ t)[g(θ − t) − g(θ + t)] dt. I1 (r) = ∫ P(r, 2π 0

̃ t)| ≤ 2/(1 − r) for |t| ≤ 1 − r, we have Since |P(r, 1−r

1 󵄨 󵄨 󵄨 󵄨󵄨 ∫ 󵄨󵄨g(θ − t) − g(θ + t)󵄨󵄨󵄨 dt → 0 󵄨󵄨I1 (r)󵄨󵄨󵄨 ≤ π(1 − r) 󵄨

(r → 1),

0

because θ is a Lebesgue point of g. In the case of the integral π

I2 (r) =

1 1 ̃ t) − ][g(θ − t) − g(θ + t)] dt, ∫ [P(r, 2π tan(t/2) 1−r

we use formula (1.35); it follows that 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 ̃ 󵄨󵄨 󵄨󵄨P(r, t) − 󵄨 ≤ const. P(r, t) (1 − r < |t| < π). 󵄨󵄨 tan(t/2) 󵄨󵄨󵄨 2 Usually, 2 tan(t/2) is replaced by t.

24 | 1 The Poisson integral and Hardy spaces Thus π

󵄨 󵄨 󵄨 󵄨󵄨 − P(r, t)󵄨󵄨󵄨g(θ − t) − g(θ + t)󵄨󵄨󵄨 dt. 󵄨󵄨I2 (r)󵄨󵄨󵄨 ≤ C ∫ −π

Now the hypothesis that θ is a Lebesgue point implies that I2 (r) → 0 (r → 1), and this completes the proof. The Privalov–Plessner theorem can be stated in the following form. Theorem 1.18. Let g ∈ L1 (𝕋), and let Φ(θ) be the indefinite integral of the function θ 󳨃→ g(eiθ ). Then the improper integral π

1 Φ(θ + t) + Φ(θ − t) − 2Φ(θ) dt − ∫ π 4 sin2 (t/2) 0+

exists for all θ and is equal to g̃ (θ) almost everywhere. Proof. By partial integration we have π

∫ ε

Φ(θ + t) + Φ(θ − t) − 2Φ(θ) 4 sin2 (t/2)

dt =

Φ(θ + ε) + Φ(θ − ε) − 2Φ(θ) 2 tan(ε/2) π

+∫ ε

g(θ + t) − g(θ − t) dt 2 tan(t/2)

for ε > 0. Now the result follows from Theorem 1.17 and the fact that lim

ε→0

Φ(θ + ε) + Φ(θ − ε) − 2Φ(θ) =0 2 tan(ε/2)

whenever Φ󸀠 (θ) = g(θ). Remark 1.26. If {ak } is a convex sequence tending to 0, then the sum of the series 1 a0 /2 + ∑∞ n=1 an cos nθ is positive for every θ ∈ (−π, π) and belongs to L (𝕋) ([302, Theorem 4.1] or [639, Ch. V(1.5)]). In particular, the function ∞

g(eiθ ) = ∑ (log(n + 2)) cos nθ −1

n=0

−1 is in L1 , whereas the function conjugate to g is equal to ∑∞ n=1 (log(n + 2)) sin nθ and is 1 not in L ([639, Ch. V, (1.14)]).

1.5 Harmonic conjugates | 25

The Riesz–Zygmund inequality As we have seen, if g ∈ h1 , then the conjugate function g̃ need not belong to h1 . A result of Riesz [504] and Zygmund [633] (cf. [639, Ch. IV, (6.28)]) states that if g ∈ h1 , then 1

it 󵄨󵄨 󵄨󵄨 g(re ) 󵄨󵄨 󵄨 ̃ ∫ 󵄨󵄨󵄨 󵄨 dr ≤ π‖g‖1 . 󵄨󵄨 r 󵄨󵄨󵄨

(1.37)

−1

In other words, we have the following: Theorem 1.19 (Riesz–Zygmund). If g ∈ h(𝔻) and 𝜕g/𝜕θ ∈ h1 , then π 󵄨󵄨 𝜕g 󵄨 󵄨󵄨 󵄨󵄨 𝜕g 1 󵄨 󵄨 󵄨󵄨 it 󵄨󵄨󵄨 ∫ 󵄨󵄨 (re )󵄨󵄨 dr ≤ sup ∫ 󵄨󵄨󵄨 (reiθ )󵄨󵄨󵄨 dθ. 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕r 2 0 1/4 to the case p > 1/2, etc. The set of all harmonic polynomials is not dense in hp for p ≤ 1. However, we have the following: Theorem 1.23. If p ∈ ℝ+ , then P is dense in H p ; in other words, H p is minimal for p ∈ ℝ+ . Proof. It follows from (1.40) and (1.41) that ‖f − fρ ‖p → 0 as ρ → 1− . From this and from 1.4.b we obtain the result.

1.6 Hardy spaces: basic properties | 29

Some simple inequalities Inequality (1.30) can be improved in the new situation. Lemma 1.31. If f ∈ H p , 0 < p ≤ ∞, then 󵄨 󵄨󵄨 2 −1/p ‖f ‖p , 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ (1 − |z| )

(1.42)

and limr→1− M∞ (r, f )(1 − r 2 )1/p = 0. Corollary 1.32. If f ∈ H p , then 1/q−1/p

Mq (r, f ) ≤ (1 − r 2 )

(q > p).

‖f ‖p

(1.43)

Proof. This follows from (1.42) and π

󵄨 q−p 󵄨 󵄨p 󵄨q−p 󵄨 󵄨 − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ ≤ (sup󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨) Mpp (r, f ). Mqq (r, f ) = ∫ θ

−π

Taking q = 1 > p and r = 1 − (1/(n + 1)) in (1.43), we get, via the inequality M1 (r, f ) ≥ |f ̂(n)|r n , one of many results of Hardy and Littlewood [237] (compare (1.31) and Problem 1.1): Theorem 1.24 (H–L (1/p − 1)-inequality). If f ∈ H p (0 < p < 1), then 󵄨󵄨 ̂ 󵄨󵄨 1/p−1 , 󵄨󵄨f (n)󵄨󵄨 ≤ Cp ‖f ‖p (n + 1) and f ̂(n) = o(n1/p−1 ), n → ∞. p − |f | dl. Proof of Lemma 1.31. Assume that f ∈ H(𝔻). Then, for p ∈ ℝ+ , we have ‖f ‖pp = ∫ By the substitution ζ = σz (ξ ) we get

󵄨 󵄨p 󵄨 󵄨 󵄨 󵄨p − 󵄨󵄨󵄨f (σz (ξ ))󵄨󵄨󵄨 󵄨󵄨󵄨σz󸀠 (ξ )󵄨󵄨󵄨|dξ | = ∫ ‖f ‖pp = ∫ − 󵄨󵄨󵄨f (σz (ξ ))(σz )󸀠 (ξ )1/p 󵄨󵄨󵄨 |dξ |. 𝕋

𝕋 1/p

The function w 󳨃→ f (σz (w))(σz ) (w) 󸀠

is analytic, and therefore, by Lemma 1.29,

󵄨p 󵄨 󵄨p 󵄨 ‖f ‖pp ≥ 󵄨󵄨󵄨f (σz (0))(σz )󸀠 (0)1/p 󵄨󵄨󵄨 = 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ), which proves the inequality. The rest holds because H p is minimal. Exercise 1.33. For a fixed z ∈ 𝔻, equality occurs in (1.42) if and only if f (w) = c(

1/p

1 − |z|2 ) ̄ 2 (1 − zw)

,

where c is a constant. Exercise 1.34. If f ∈ H p and p ≥ 1, then Mp (r, f 󸀠 ) ≤ (1 − r 2 )−1 ‖f ‖p . For the case p < 1, see Exercise C.10. Exercise 1.35. Considering the functions (1 − z)−γ , γ > 0, we can prove that the exponent 1/p − 1 in Theorem 1.24 is optimal, i. e., it cannot be replaced by a smaller one.

30 | 1 The Poisson integral and Hardy spaces The Poisson integral of log |f∗ | If f ∈ H(𝔻), then log |f | ≤ 𝒫 [log |f∗ |] because of the subharmonicity of log |f |. This continues to be true for all f ∈ H p [501, 576], but the proof is rather subtle. Theorem 1.25 (F. Riesz–Szegö). Let f ≢ 0 belong to H p (p > 0). Then log |f∗ | ∈ L1 (𝕋), and we have π

󵄨 󵄨 󵄨 󵄨 − P(z, ζ ) log󵄨󵄨󵄨f∗ (ζ )󵄨󵄨󵄨|dζ |, log󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ≤ ∫

z ∈ 𝔻.

(1.44)

−π

Before the proof, we note an important consequence. Theorem 1.26 (Riesz–Szegö uniqueness theorem). If f ∈ H p and f∗ (eiθ ) = 0 on a set of positive measure, then f (z) = 0 for every z ∈ 𝔻. We mention a deep result of Privalov [491]. If a function f meromorphic in 𝔻 has a nontangential limit zero in a set of positive measure on 𝕋, then f (z) = 0 for all z ∈ 𝔻. For a proof, see [639, Theorem XIV(1.9)]. On the other hand, there is a function f ∈ H(𝔻) such that limr→1− f (rζ ) = 0 for all ζ ∈ 𝕋 [124, p. 12]. Proof of Theorem 1.25. Let f ∈ H p and f (0) ≠ 0. From the inequality log |f∗ | ≤ |f∗ |p /p it follows that − log |f∗ | dℓ ∫ 𝕋

exists (finite or −∞). By the subharmonicity of log |f |, 󵄨 󵄨 󵄨 󵄨 − log󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨|dζ |, log󵄨󵄨󵄨f (0)󵄨󵄨󵄨 ≤ ∫

0 < r < 1.

(1.45)

𝕋

From the inequality p 󵄨 󵄨 log󵄨󵄨󵄨f (ρζ )󵄨󵄨󵄨 ≤ (Mrad f (ζ )) /p,

󵄨 󵄨 where Mrad f (ζ ) = sup 󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 0 1, this theorem is contained in Corollary 1.16, whereas in the general case, it is a consequence of (1.44) or of the weaker inequality π

󵄨p 󵄨 󵄨󵄨 iθ 󵄨p − P(r, θ − t)󵄨󵄨󵄨f∗ (eit )󵄨󵄨󵄨 dt 󵄨󵄨f (re )󵄨󵄨󵄨 ≤ ∫

(0 ≤ r < 1)

(1.46)

−π

and (1.40). It is worth noting that (1.46) can be deduced immediately from (1.41) and the inequality |f (ρz)|p ≤ 𝒫 [|fρ |p ](z). In fact, inequality (1.44), together with Jensen’s inequality for the function x 󳨃→ ex , implies a more general fact: Theorem 1.28 (Smirnov). If f ∈ H p and f∗ ∈ Lq for some q > p, then f ∈ H q . 1.6.2 The space H 1 The results of this section, due to F. and M. Riesz, Privalov, and Smirnov, show how much H 1 differs from h1 . A function belonging to h1 need not be equal to the Poisson integral of the boundary function. However, Riesz proved the following: Theorem 1.29 (F. and M. Riesz). If f ∈ H 1 , then f∗ ∈ L1 and f = 𝒫 [f∗ ]. This is easily deduced from the relation f (rz) = 𝒫 [fr ](z) (r < 1) by means of (1.41).

32 | 1 The Poisson integral and Hardy spaces Exercise 1.36 (Cauchy integral formula). If f ∈ H 1 , then f (z) =

f (ζ ) 1 dζ ∫ ∗ 2πi ζ − z

(z ∈ 𝔻),

𝕋

and moreover f∗ (ζ ) f (n) (z) 1 dζ = ∫ n! 2πi (ζ − z)n+1

(z ∈ 𝔻, n ≥ 1).

𝕋

Now we are in position to prove the famous theorem of F. and M. Riesz. Theorem 1.30. Let μ be a complex Borel measure on 𝕋 such that ∫ ζ n dμ(ζ ) = 0 𝕋

for every n = 1, 2, . . . . Then μ is absolutely continuous. ̂ Proof. Let f = 𝒫 [μ]. Then f ∈ h1 and f ̂(k) = μ(k) for every k ∈ ℤ (Proposition 1.3). Therefore the condition of the theorem implies that f is analytic. Thus f ∈ H 1 , and so, according to Theorem 1.29, we have f = 𝒫 [f∗ ]. In view of the injectivity of the Poisson integral, it follows that dμ(eiθ ) = f∗ (eiθ ) dθ. Theorem 1.31 (F. and M. Riesz). If f ∈ H 1 and if the boundary function is almost everywhere equal to a function of bounded variation, then f has absolutely continuous extension to 𝔻.3 Proof. Let f∗ = γ a. e., γ ∈ BV[−π, π]. Then f = 𝒫 [γ] by Theorem 1.29. We have g(reiθ ) :=

𝜕f = PS[γ](reiθ ) − k ⋅ P(r, θ + π), 𝜕θ

where PS[γ] is the Poisson–Stieltjes integral of γ (see (1.12) and (1.14)). Then, by the Riesz–Herglotz theorem, g ∈ H 1 , and, by Theorem 1.29, g = 𝒫 [g∗ ], and therefore g = PS[G], where θ

G(θ) = ∫ g∗ (eit ) dt

(θ ∈ ℝ).

0

Applying (1.14) again, with the obvious change of notation, since the function G is 2π-periodic because g(0) = 0, we get π

𝜕 𝜕f = PS[G](reiθ ) = − P(r, θ − t)G(t) dt. ∫ 𝜕θ 𝜕θ −π

3 i. e., a continuous extension that is absolutely continuous on 𝕋.

1.7 The Riesz projection theorem

| 33

It follows that π

f (reiθ ) = const + ∫ − P(r, θ − t)G(t) dt = const + 𝒫 [G1 ](reiθ ), −π

where G1 (eiθ ) = G(θ). Now the desired result follows from Theorem 1.2 with ϕ = G1 . In a similar way, we prove the following [491, 492]: Theorem 1.32 (Privalov). The derivative of a function f ∈ H(𝔻) belongs to H 1 if and only if f has absolutely continuous extension to 𝔻. If f 󸀠 ∈ H 1 , then the boundary function of the function (𝜕/𝜕θ)f (reiθ ) = ireiθ f 󸀠 (reiθ ) is equal to (d/dθ)f∗ (eiθ ).

1.7 The Riesz projection theorem The operator R+ acting from h(𝔻) into h(𝔻) according to the rule ∞

n ̂ (R+ u)(z) = ∑ u(n)z n=0

is called the Riesz projection. It is a direct consequence of Parseval’s theorem that R+ acts as an orthogonal projection from h2 onto H 2 . This fact was generalized by M. Riesz [506]. Theorem 1.33 (Riesz projection theorem). If 1 < p < ∞, then R+ acts as a bounded projection from hp onto H p . Remark 1.37. Hollenbeck and Verbitsky [264] proved that the best constant in the inequality ‖R+ u‖p ≤ Cp ‖u‖p is given by Cp = 1/ sin(π/p). See Further notes 1.10 for some more results of this kind. From the projection theorem and Theorem 1.10 it follows that, for every ϕ ∈ Lp (𝕋) ̂ ̂ (1 < p < ∞), there exists a unique function ψ ∈ Lp (𝕋) such that ψ(n) = ϕ(n) for n ≥ 0 p ̂ and ψ(n) = 0 for n < 0. This enables us to treat R+ as an operator from L (𝕋) to H p (𝕋), 1 < p < ∞, where ̂ H p (𝕋) = {ϕ ∈ Lp (𝕋): ϕ(n) = 0 for n < 0} (p ≥ 1).

(1.47)

However, the Riesz projection does not map L1 into H 1 (𝕋) (see Remark 1.26); furthermore, H 1 is not complemented in L1 , i. e., there is no bounded projection from L1 to H 1 (see [520, Example 5.19], where it was shown that A(𝔻) is not complemented in C(𝕋)).4 This was first proved by Newman [401] and later generalized by Rudin [516]. 4 If every subspace of a Banach space X is complemented in X, then X is isomorphic to a Hilbert space [350].

34 | 1 The Poisson integral and Hardy spaces Since the Riesz projector is connected with conjugate function in a simple way, ̃ namely R+ u = (u(0) + u + iu)/2, the Riesz theorem can be stated as follows. Theorem 1.34 (Riesz conjugate functions theorem). If u ∈ hp , 1 < p < ∞, then ũ ∈ hp , and there exists a constant Cp such that ‖u‖̃ p ≤ Cp ‖u‖p . Remark. Pichorides [478] proved that the best possible constant in the inequality ‖u‖̃ p ≤ Cp ‖u‖p with real-valued u is tan(π/2p) if 1 < p ≤ 2, Cp = { cot(π/2p) if p ≥ 2. In view of the connection between conjugate functions and the Hilbert operator (Privalov–Plessner theorem 1.17), we have the following: Theorem 1.35. The Hilbert operator maps Lp (𝕋) to Lp (𝕋) for 1 < p < ∞. In the case p = 2, Theorem 1.34 follows from Parseval’s formula; we have ‖u‖̃ 22 = − |u(0)|2 . If a proof is known either for 1 < p < 2 or for p > 2, then the general case can be treated by duality. Here we present an elementary proof due to Stein [556], based on the Hardy–Stein identities. As a consequence of Green’s formula and the identity ‖u‖22

Δ(up ) = p(p − 1)up−2 |∇u|2

(u > 0),

we have the following: Lemma 1.38. Let u > 0 belong to hp , 1 < p < ∞. Then 1 󵄨p p(p − 1) 󵄨 󵄨2 󵄨 ‖u‖pp = 󵄨󵄨󵄨u(0)󵄨󵄨󵄨 + − u(z)p−2 󵄨󵄨󵄨∇u(z)󵄨󵄨󵄨 log dA(z). ∫ 2 |z| 𝔻

Proof of Theorem 1.34. As mentioned before, it suffices to consider the case 1 < p < 2. We may assume that u is real valued and positive (see Exercise 1.17). Let u ∈ h(𝔻), and let f ∈ H(𝔻) be such that u = Re f . The function |f |p is of class C 2 (𝔻) because f has no zeros in 𝔻, so we can apply Green’s theorem to get 2 1 󵄨p p 󵄨 󵄨2 󵄨 ‖f ‖pp = 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + − |f |p−2 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 log dA(z). ∫ 2 |z| 𝔻

From this, Lemma 1.38, and the inequality up−2 ≥ |f |p−2 it follows that 󵄨p 󵄨 ‖f ‖pp − 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 ≤

p 󵄨p 󵄨 (‖u‖pp − 󵄨󵄨󵄨u(0)󵄨󵄨󵄨 ). p−1

If u ∈ hp (u > 0) is arbitrary, then we apply this inequality to the functions uρ (z) = u(ρz) and fρ (0 < ρ < 1), which belong to h(𝔻), and then let ρ tend to 1 to complete the proof.

1.7 The Riesz projection theorem

| 35

Remark 1.39. We have proved an inequality of the form A(f ) ≤ CB(f ) under the hypothesis f ∈ H(𝔻), where C was independent of f , then applied it to fρ , where f ∈ H(𝔻), ρ < 1, and then passed to the limit as ρ → 1. This approach will be used very often and will be explained only when necessary. Remark. If f is a conformal mapping of the disc onto the domain G = {z : 0 < Re z < 1}, then Re f ∈ h∞ , but f is not in H ∞ ; therefore the theorem does not hold for p = ∞. For the case p = 1, see Remark 1.26. The projection theorem has many important applications. For example, the trigonometric system is a Schauder basis in Lp (𝕋) for 1 < p < ∞; in other words, the system of the functions r |n| einθ (−∞ < n < ∞) is a (two-sided) Schauder basis in hp .5 More precisely: ikθ ̂ , where m and Theorem 1.36. Let ϕ ∈ Lp (𝕋), 1 < p < ∞, and ϕm,n (eiθ ) = ∑nk=m ϕ(k)e n are integers, m < n. Then

‖ϕm,n ‖p ≤ Cp ‖ϕ‖p ,

‖ϕ − ϕm,n ‖p → 0

as

n → ∞, m → −∞.

(1.48) (1.49)

Proof. Let ek (eiθ ) = eikθ . Then ϕm,n = em R+ (e−m ϕ) − en+1 R+ (e−n−1 ϕ). From this and from Theorem 1.33 we obtain (1.48), and from (1.48) and the Weierstrass approximation theorem we obtain (1.49). Now we can determine the dual of H p for 1 < p < ∞. Namely, as a consequence of 󸀠 the projection theorem and the duality between Lp and Lp , p󸀠 = p/(p − 1), we have the following: Theorem 1.37. If 1 < p < ∞, then (H p )󸀠 ≃ H p under the duality pairing 󸀠

∞

̂ (f , g) 󳨃→ ∫ − f (ζ )g(ζ ̄ )|dζ | = ∑ f ̂(n)g(n), n=0

𝕋

(1.50)

the series being convergent in the ordinary sense. 1.7.a (Convention). We write X ≃ Y, where X and Y are quasi-Banach spaces, if X and Y coincide as sets and if their quasinorms are equivalent. If X = Y as sets and ‖f ‖X = ‖f ‖Y for all f , then we write X ≅ Y. Exercise 1.40 (Isomorphism Lp with H p [90]). If 1 < p < ∞, then the formula ∞

∞

2n−1 2n ̂ ̂ (Tu)(z) = ∑ u(n)z + ∑ u(−n)z n=0

p

n=1

p

defines an isomorphism of h onto H . 5 However, there are spaces, e. g., Bergman, in which this system is not a basis although the analogue of the projection theorem holds. See Further notes 6.2.

36 | 1 The Poisson integral and Hardy spaces Exercise 1.41 (Parseval’s formula). If f ∈ Lp (𝕋) and g ∈ Lp (𝕋) with 1 < p < ∞, then |n| inθ ̂ ̂ the series ∑∞ e converges uniformly in 𝔻, and the Parseval’s formula n=−∞ f (n)g(n)r holds: 󸀠

π

∞

−π

n=−∞

̂ − f (eiθ )g(e−it ) dθ = ∑ f ̂(n)g(n). ∫

Kolmogorov–Smirnov theorem The Riesz projection theorem does not hold for p = 1. Instead, the following weaker version holds [317, 541]. Theorem 1.38 (Kolmogorov–Smirnov). If f ∈ H(𝔻) and Re f ∈ h1 , then f ∈ H p for all 0 < p < 1. Proof. Assume, as in the proof of Theorem 1.34, that u > 0 and u ∈ h(𝔻) and additionally that f (0) = u(0). Then Lemma 1.38 is still valid, so we have Mpp (1, u) = u(0)p −

1 p(1 − p) − |u|p−2 |∇u|2 log dA(z) ∫ 2 |z| 𝔻

p(1 − p) 1 󵄨 󵄨2 ≤ u(0) − − |f |p−2 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 log dA, ∫ 2 |z| p

𝔻

and hence 1 p(1 − p) 󵄨 󵄨2 − |f |p−2 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 log dA ≤ u(0)p = ‖u‖p1 , ∫ 2 |z| 𝔻

that is, 2 p 1 1 󵄨p p 󵄨󵄨 󵄨 󵄨2 − |f |p−2 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 log dA ≤ u(0)p + ‖u‖p = ‖u‖p , ∫ 󵄨󵄨f (0)󵄨󵄨󵄨 + 2 |z| 1−p 1 1−p 1 𝔻

as desired. (See Remark 1.39.) The Poisson integral of the conjugate function ̃ Theorem 1.39. If g ∈ L1 and g̃ ∈ L1 , then 𝒫 [g]̃ = P[g]. Proof. Assume that g is real valued. By the Kolmogorov–Smirnov theorem the function ̃ belongs to H p for p < 1. Now Smirnov’s theorem 1.28 tells us that f ∈ H 1 , f = 𝒫 [g]+iP[g] and hence f = 𝒫 [f∗ ] by Theorem 1.29. Finally, since f∗ = g+ig,̃ by the theorems of Fatou and Privalov–Plessner we see that ̃ 𝒫 [g] + iP[g] = 𝒫 [f∗ ] = 𝒫 [g] + i𝒫 [g̃ ], and the result follows.

1.7 The Riesz projection theorem

| 37

It follows from Theorem 1.38 and the complex maximal theorem (Theorem C.11) that if f ∈ H(𝔻) and Re f ∈ h1 , then ‖Mrad f ‖p ≤ Cp ‖ Re f ‖1

for every p ∈ (0, 1).

This is improved by the following result [317]. Theorem 1.40 (Kolmogorov). If f ∈ H(𝔻) and Re f ∈ h1 , then 󵄨 C 󵄨󵄨 󵄨󵄨{ζ ∈ 𝕋: Mrad f (ζ ) > λ}󵄨󵄨󵄨 ≤ ‖ Re f ‖1 , λ

λ > 0,

where C is an absolute constant. Equivalently, the operator u 󳨃→ Mrad ũ acts as a continuous operator from h1 into the weak L1 -space L1,⋆ (𝕋). For the definition and properties of weak Lp -spaces, see Section C.2. Remark. For the best constant in the inequality ‖u‖̃ 1,⋆ ≤ C‖u‖1 , see [55] and [139]. Proof. Let {rn } be the sequence of all rational numbers in (0, 1), and let Tn u(eiθ ) = ̃ n eiθ ). By Corollary 1.14 we have Tmax u(eiθ ) = Mrad u(e ̃ iθ ) < ∞ for almost all θ. Now u(r Banach’s principle (Theorem C.19) tells us that Tmax continuously maps h1 into ℒ0 (𝕋). (Here ℒ0 (𝕋) is the space of all a. e. finite measurable functions on 𝕋.) Finally, since Tmax commutes with rotations, we can apply the Nikishin–Stein theorem C.17 to conclude the proof. Corollary 1.42. If f ∈ H 1 , then ‖sn f ‖1,⋆ ≤ C‖f ‖1 for n ≥ 1, where C is an absolute constant, and n

sn f (z) = ∑ f ̂(k)z k . k=0

(1.51)

Proof. The inequality ‖sn f ‖1,⋆ ≤ C‖f ‖1 is proved in the same way as Theorem 1.36. Since the polynomials are dense in H 1 (𝕋), this implies that ‖sn g − g‖1,⋆ → 0 as n → ∞. Exercise 1.43 (Zygmund’s theorem [632]). If g ∈ L log+ L(𝕋), then Hg ∈ L1 (𝕋). The space L log+ L(𝕋) is defined by the requirement ∫𝕋 Φ(|g|) dℓ < ∞, where Φ(t) = t log+ t, and x+ = max{0, x} for x ∈ ℝ. It coincides with the obviously defined space L log(1 + L), which is more convenient for using Green’s formula. 1.7.1 Aleksandrov’s theorem: Lp (𝕋) = H p (𝕋) + H p (𝕋) Relation (1.40) shows that Lp (𝕋) contains an isometric copy of H p (p > 0); denote this subspace by H p (𝕋). Thus H p (𝕋) = {f∗ : f ∈ H p }. If p ≥ 1, then H p (𝕋) can be described

38 | 1 The Poisson integral and Hardy spaces by (1.47). In the case p < 1, this cannot be applied, simply because the Fourier coefficients are not defined; then H p (𝕋) is equal to the Lp -closure of ̂ 𝒯+ = {ϕ ∈ 𝒯 : ϕ(−n) = 0 for n ≥ 1}, where 𝒯 is the set of all trigonometric polynomials. Let H p (𝕋) = {ϕ : ϕ ∈ H p (𝕋)}. From the projection theorem it follows that Lp (𝕋) = p H (𝕋) + H p (𝕋), 1 < p < ∞. This fact was extended by Aleksandrov [20, 21] to the case p < 1. However, in this case, the decomposition is not unique (up to an additive constant) because the intersection H p ∩ H p is equal to the linear span of the set of the functions ga (ζ ) = 1/(1 − aζ ) (a ∈ 𝕋, ζ ∈ 𝕋) (see [20, 21]). Theorem 1.41 (Aleksandrov). If f ∈ Lp (𝕋), p < 1, then there are functions f1 ∈ H p (𝕋) and f2 ∈ H p (𝕋), such that f = f1 + f2 and ‖f1 ‖p + ‖f2 ‖p ≤ Cp ‖f ‖p . Proof. Let X denote the direct sum of the spaces H p and H p . Consider the operator T : X 󳨃→ Lp , T(f1 , f2 ) = f1 + f2 . For every trigonometric polynomial f , ‖f ‖p ≤ 1, we will find (f1 , f2 ) such that f = T(f1 , f2 ), where ‖(f1 , f2 )‖ ≤ Cp (Cp depends only of p), and then the result will follow from Theorem A.2. Let f = ∑|k|≤n ak eikθ and γ(ζ ) = ζ n . Then γf and γ f ̄ belong to H p . Put φ = γ/(γ − 1). Then φ ∈ H p ∩ H p and φ + φ̄ = 1. Now let φt (ζ ) = φ(ζeit ), t ∈ ℝ, and gt = fφt , ht = f ̄φt . Then gt and ht belong to H p , and f = gt + h̄ t . Routine calculation shows that π

− ‖gt ‖pp dt = ‖φ‖pp ‖f ‖pp , ∫ −π

which means that there exists t such that ‖gt ‖p ≤ ‖φ‖p . For this value of t, we have ‖ht ‖pp ≤ ‖f ‖pp + ‖gt ‖pp ≤ 1 + ‖φ‖pp . Finally, we take f1 = gt and f2 = h̄ t , and this completes the proof. As an application of Aleksandrov’s theorem, we have the following result of Shapiro [528]. Corollary 1.44. Let hpP (0 < p < 1) denote the closure in hp of the set of all harmonic polynomials. Then hpP /ohp ≃ Lp (𝕋). Proof. If u ∈ hpP , then ‖u − ur ‖hp → 0 as r → 1− , where ur (z) = u(rz), |z| < 1/r. Hence there exists a function g ∈ Lp (𝕋) such that ur |𝕋 → g in Lp (𝕋). We define the operator S : hpP 󳨃→ Lp (𝕋) by Su = g. We have ‖Su‖Lp ≤ ‖u‖hp by Fatou’s lemma, and the kernel of S is ohp . Since obviously S(hpP ) ⊃ H p ∪ H p = Lp (𝕋), we find, by means of the open mapping theorem, that hpP /ohp is isomorphic to Lp (𝕋). Exercise. Aleksandrov’s theorem does not hold for p = 1.

Further notes and results | 39

Further notes and results A lot of information and references (529) before 1985 can be found in the survey paper of Schvedenko [536]. The proof of Theorem 1.11 was given by Fefferman and Stein [187] and two years later by Kuran [329]. Kuran’s proof is very complicated and cannot be used in the subharmonic context. Fefferman and Stein’s proof can be found in Koosis [321] and Garnett [198]. As far as the author knows, Hardy and Littlewood nowhere formulated Theorem 1.11. For slightly more general results, we refer to [428, 430]. Theorem 1.12 is formulated without proof in [236, Theorem 5]. In the same paper (Theorem 1), Theorem 1.14 was proved. Besides Theorem 1.16, Shapiro [528] proved that ohp contains an isomorphic copy of c0 , the space of sequences tending to zero. Theorem 1.22, as well as some other fundamental theorems on Hardy spaces, is due to F. Riesz [501], who introduced the term “Hardy spaces”. Theorems 1.29, 1.30, and 1.31, due to the brothers Riesz, were proved in their amazing paper [505]. “This is a paper every analyst should read!” writes Koosis [321, Ch. II, Sec. A], and the original proof of Theorem 1.30 is immediately after this sentence. 1.1 (The Cauchy transform). For a measure μ ∈ M(𝕋), we define the Cauchy transform Kμ as (Kμ)(z) = ∫ − 𝕋

dμ(ζ ) , 1 − ζ ̄z

z ∈ 𝔻.

The space 𝒦 of all Cauchy transforms becomes a Banach space when endowed with the norm ‖g‖𝒦 = inf{‖μ‖: g = Kμ}. The following fact can be easily deduced from the general theory of the Banach space duality; see, e. g., the book Cima, Matheson, and Ross [122, Theorem 4.2.2]. Theorem. The space 𝒦 is isomorphically isometric to the dual of A(𝔻) with the dual pairing [f , Kμ] = ∫ − f (ζ ̄ ) dμ(ζ ),

f ∈ A(𝔻).

𝕋

Note a subtle fact: By the Lebesgue decomposition theorem we have dμ = dμa + dμs , where dμa and dμs are the absolutely continuous and singular parts of dμ, respectively; moreover, ‖μ‖ = ‖μa ‖ + ‖μs ‖. This implies that 𝒦 = 𝒦a ⊕ 𝒦s , where 𝒦a and 𝒦s are defined via the absolutely continuous and singular parts of dμ, respectively. It is interesting that if dν is singular, then ‖Kν‖𝒦 = ‖ν‖. The space 𝒦a is separable (𝒦s is not) and is equal to 𝒦P [122, Proposition 4.1.21]. Besides, the dual of 𝒦a can be identified with H ∞ [122, Theorem 4.1.22]. The theory of the Cauchy integral is so beautiful and deep that anyone should read at least the expository paper [121] of the aforementioned authors.

40 | 1 The Poisson integral and Hardy spaces 1.2. The proof of Theorem 1.1, although very short, has a disadvantage in that it is based on very special properties of the Poisson kernel. The standard proof depends on (1.8) and the following property: limr→1 supδ 0 in 𝔻: 1 − |u(z)| 󵄨 󵄨󵄨 . 󵄨󵄨∇u(z)󵄨󵄨󵄨 ≤ 2 1 − |z|2

(1.53)

It is easy to verify that (1.52) and (1.53) are incomparable. Both (1.52) and (1.53) are deduced from the case z = 0 by an application of this case to the function u ∘ σz . Chen [118] improved the Kalaj–Vuorinen result as follows. Theorem (Chen). Under the hypotheses of the Kalaj–Vuorinen theorem, we have π 󵄨 4 cos( 2 u(z)) 󵄨󵄨 . 󵄨󵄨∇u(z)󵄨󵄨󵄨 ≤ π 1 − |z|2

(1.54)

The inequality is sharp for any fixed z ∈ 𝔻. The proof is based on the fact that the function tan( π4 f (z)), where f is the analytic “completion” of u, maps 𝔻 into 𝔻. Chen’s result can be stated in the (equivalent) form 󵄨󵄨 z − a 󵄨󵄨 π π 󵄨󵄨 󵄨 dh (tan( u(z)), tan( u(a))) ≤ dh (z, a) = 󵄨󵄨󵄨 󵄨, 󵄨󵄨 1 − az̄ 󵄨󵄨󵄨 4 4

z, a ∈ 𝔻.

(1.55)

From this inequality we can deduce various consequences. For example, if we set a = 0 and u(0) = 0, then we immediately get the inequality of Heinz [252] 󵄨 4 󵄨󵄨 󵄨󵄨u(z)󵄨󵄨󵄨 ≤ arctan |z|, π

(1.56)

which can also be proved by using the subordination principle; see Further notes 2.2. 1.7 (Conformal mappings and H 1 ). The following theorem is due to Smirnov. Theorem. Let f be a conformal mapping of 𝔻 onto a domain G whose boundary 𝜕G is a Jordan curve. Then f 󸀠 ∈ H 1 if and only if 𝜕G is rectifiable. If 𝜕G is rectifiable, then π

|𝜕G| = ∫ |f 󸀠 (eiθ )| dθ,

(1.57)

−π

where |𝜕G| is the length of 𝜕G. It follows from (1.57) and the inequality |f ̂(n)| ≤ π‖f ‖1 , n+1 n=0 ∞

∑

(1.58)

̂ due to Hardy (see Theorem 2.15), that |𝜕G| ≥ 2 ∑∞ n=0 |f (n)|. For further information concerning Hardy spaces and conformal mappings, see Zygmund [639, Ch. VII, Sec. 10].

42 | 1 The Poisson integral and Hardy spaces 1.8 (Kalton’s theorem). Let 𝒟αp , α > −1, denote the (Dirichlet) space of functions f ∈ H(𝔻) such that α 󵄨p 󵄨 (∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z))

1/p

< ∞,

𝔻 p normed in one of obvious ways. We shall prove later on that 𝒟p−1 ⊂ H p (0 < p < 2) (see Theorem 10.15). Moreover, the inclusion is proper, which is proved by using lacunary series, and therefore the following result of Kalton [296, Theorem 8.1] improves Theorem 1.41. p p Theorem (Kalton). If f ∈ Lp (𝕋), p < 1, then there are functions g ∈ 𝒟p−1 and h ∈ 𝒟p−1

such that f = g∗ + h∗ and ‖g‖𝒟p + ‖h‖𝒟p ≤ Cp ‖f ‖p . p−1

p−1

The proof is based on the integration theory for functions with values in a quasiBanach space, a complicated and extensive topic (see [296] and references therein). 1.9 (Boundary decay of harmonic functions). Aleksandrov [23] defined the space Hp (𝕋) (0 < p ≤ 1) by the requirement that u ∈ Hp (𝕋) if and only if u ∈ h(𝔻) and ‖u‖Hp := sup

0 1/3). 1.10. Let 1 < p < ∞, and let p̄ = max{p, p/(p − 1)}. Verbitsky [589] proved the following result: If f = u + iv ∈ H p and v(0) = 0, then ̄ ̄ sec(π/(2p))‖v‖ p ≤ ‖f ‖p ≤ csc(π/(2p))‖u‖ p,

(1.60)

and both estimates are sharp. This result improves the sharp inequality ‖v‖p ≤ cot(π/(2p))‖u‖p

(1.61)

found by Pichorides; see Remark following Theorem 1.34. For a harmonic function f = g + h ∈ hp , (hg)(0) = 0, Kalaj [284] defines the norm |||⋅|||p by 1/p

󵄨2 p/2 󵄨2 󵄨 󵄨 |||f |||p = (∫(󵄨󵄨󵄨g∗ (ζ )󵄨󵄨󵄨 + 󵄨󵄨󵄨h∗ (ζ )󵄨󵄨󵄨 ) |dζ |)

.

𝕋

Then he finds the best constants Ap and Bp in the inequalities |||f |||p ≤ Ap ‖f ‖p and ‖f ‖p ≤ Bp |||f |||p . His main results (Theorems 2.1 and 2.3) state that Ap =

1 , (1 − | cos πp |)1/2

Bp = √2 cos

π . 2p̄

Taking g = h, we get (1.60). There are several other sharp inequalities in that paper. Using these inequalities, Kayumov et al. [303] proved a sharp form of the Fejér– Riesz inequality for the space hp , 1 < p ≤ 2. Several variables Many results of this chapter remain true, mutatis mutandis, in the case n ≥ 3. For example, the Poisson integral of a function g ∈ L1 (𝜕𝔹n ) is defined by 𝒫 [g](x) = ∫ P(x, y)g(y) dσn (y), 𝕊n

where P(x, y) =

1 − |x|2 , |x − y|n

x ∈ 𝔹n , y ∈ 𝕊n ,

44 | 1 The Poisson integral and Hardy spaces is the Poisson kernel for the unit ball 𝔹n ⊂ ℝn . Here dσn is the normalized surface measure on 𝕊n = 𝜕𝔹n . The Poisson integral acts as an isometrical isomorphism from Lp (𝕊n ) (1 < p ≤ ∞) onto hp (𝔹n ) = {u ∈ h(𝔹n ): ‖u‖hp = sup Mp (r, u) < ∞}, 0 0 such that {z : |z − a| < R} ⊂ D and π

u(a) ≤ ∫ − u(a + ρeiθ ) dθ

for every 0 < ρ < R.

(2.2)

−π

Upper semicontinuity implies boundedness from above, which guarantees the existence of the integral in (2.2). The boundedness can be easily proved by using the fact that u is upper semicontinuous if and only if for each λ ∈ ℝ, the set {z : u(z) < λ} is open. If K ⊂ D is a compact set, then u attains its maximum on K, which along with (2.2) shows that, for every a ∈ D, there exists a sequence zn → a such that u(a) ≤ u(zn ), which implies lim supz→a u(z) = u(a). In particular, u is continuous at a if u(a) = −∞. There are discontinuous subharmonic functions, e. g., the function ∞

log |z − 2−n | 2n n=1

u(z) = ∑

is subharmonic in the entire plane and is discontinuous at zero. https://doi.org/10.1515/9783110630855-002

46 | 2 Subharmonic functions and Hardy spaces

–

–

–

–

–

We give a list of well-known properties of subharmonic functions: In the case of C 2 -functions, there is a simple criterion of subharmonicity deduced from Green’s formula: A function u ∈ C 2 (D) is subharmonic if and only if Δu ≥ 0 in D. From the formula Δ(u ∘ φ)(z) = (Δu)(φ(z))|φ󸀠 (z)|2 , where φ is an analytic function, and u is C 2 , we get: The composition u ∘ φ is subharmonic if u is subharmonic and φ is analytic. In the general case, this assertion can be reduced to the “smooth” case by approximating an arbitrary subharmonic function by smooth ones; see Theorem 2.6. An important example of a subharmonic function taking the value −∞ is the function log |z − a|. More generally, if f is analytic in D, then the function log |f | is subharmonic in D, and |f |p is subharmonic for every p > 0. The sum and the maximum of a finite sequence of subharmonic functions are subharmonic functions. The same holds for the limit of a decreasing sequence of subharmonic functions. Let ϕ be an increasing continuous convex function on an interval I ⊂ [−∞, +∞). If v is subharmonic and takes its values in I, then the function u = ϕ(v) is subharmonic. In particular, u is subharmonic in the following cases: (i) u = |h|p , where p ≥ 1, and h is harmonic; (ii) u = vp , p ≥ 1, where v is subharmonic and nonnegative.

We call a function u ≥ 0 log-subharmonic if log u is subharmonic. Lemma 2.1 (Log-subharmonicity lemma). Let F : 𝔻 × [a, b] 󳨃→ [0, ∞) ([a, b] ⊂ ℝ) be a continuous function such that F(z, t) is log-subharmonic in z ∈ 𝔻 for every t ∈ [a, b]. b Then the function u(z) = ∫a F(z, t) dt is log-subharmonic in 𝔻. Proof. The function u is continuous and therefore it suffices to prove that it is a limit of a sequence of log-subharmonic functions. We take un = ∑nj=1 un,j , where un,j (z) = F(z, a + (b − a)j/n)/n. Thus it suffices to prove that a finite sum of log-subharmonic functions is log-subharmonic or, equivalently, that if v1 , . . . , vn are subharmonic, then so is n

v = log(∑ evj ). j=1

This implication is easy to deduce from the fact that the function f (x) = log(ex1 + ⋅ ⋅ ⋅ + exn ),

x = (x1 , . . . , xn ) ∈ ℝn ,

is convex on ℝn and increasing in each variable separately (see [265, Exercise 3.2.6]). The convexity follows from the inequality f ((x + y)/2) ≤ (f (x) + f (y))/2, which, after the substitutions exj = tj , reduces to an elementary inequality. In proving that v is

2.1 Basic properties of subharmonic functions | 47

subharmonic, we can assume that vj (z) > −∞ for all z and j; otherwise, we replace vj by max{vj , −k} and let k tend to +∞. 2.1.a. A function u is called superharmonic if −u is subharmonic. Exercise 2.2. Let h ≢ 0 be a real-valued harmonic function, and let 0 < p < 1. The function |h|p is subharmonic if and only if h is constant. The function |h|p is superharmonic if and only if h has no zeros. If h > 0 is harmonic and p < 0, then hp is subharmonic. Exercise 2.3. A C 2 function u ≥ 0 is log-subharmonic if and only if up is subharmonic for every p > 0. Analogously, a real function ϕ > 0 is log-convex (i. e., log ϕ is convex) if and only if ϕp is convex for every p > 0.

2.1.1 The maximum principle The simplest variant of the maximum principle says: Theorem 2.1 (Maximum principle). A nonconstant subharmonic function cannot attain its maximum inside the domain. In particular, a nonconstant harmonic function attains neither the maximum nor the minimum inside the domain. Proof. Let u be subharmonic in a domain D, and let M denote the set of points in D where u attains its maximum. Because of the semicontinuity, M is closed. We will prove that M is open, which will imply M = D or M = 0, so the proof will be finished. Let a ∈ M. Then (2.2) implies that, for R small enough and for all ρ < R, we have u(a) = u(a + ρζ ) almost everywhere on the circle |ζ | = 1. From this and from (2.1) it follows that u(a) ≤ u(a + ρζ ) everywhere; thus, u(a) = u(a + ρζ ) everywhere, i. e., {z : |z − a| < R} ⊂ M. Corollary 2.4. If u is upper semicontinuous on D and subharmonic in D, then max{u(z) : z ∈ D} = max{u(ζ ) : ζ ∈ 𝜕D}. Corollary 2.5. Let D be a bounded domain, let u be a function subharmonic in D and upper semicontinuous on D, and let h be a real-valued function harmonic in D and continuous on D. If u ≤ h on 𝜕D, then u ≤ h in D; besides, u < h if u is not harmonic. Local integrability If u is subharmonic in D ⊂ ℂ, then u(a) ≤

1 πρ2

∫ |z−a| −∞. (Otherwise, b is in the interior of the complement of E.) The disc G0 = {|z − a| < ε/2} contains b, and u is integrable on G0 because of (2.3). Hence b ∈ E, which means that E is closed. This was to be proved.

2.2 Properties of the mean values Convexity and monotonicity 2.2.a. By definition a real function φ(r), r > 0, is convex of log r if the function x 󳨃→ φ(ex ) is convex. In other words, φ(r) is convex of log r if φ(r11−λ r2λ ) ≤ (1 − λ)φ(r1 ) + λφ(r2 ),

0 < λ < 1.

(2.4)

If φ is of class C 2 , then it is convex of log r if and only if φ󸀠󸀠 (r) + φ󸀠 (r)/r ≥ 0. Theorem 2.3. Let u be subharmonic in the disc |z| < R. Then the function π

I(r, u) := ∫ − u(reiθ ) dθ

(0 < r < R)

−π

is finite, increasing, and convex of log r. The same holds for the function I∞ (r, u) = max u(reit ). 0≤t≤2π

Proof. If u is subharmonic, then so is the function π

I(z) := ∫ − u(zeiθ ) dθ = I(|z|, u) (|z| < R). −π

The upper semicontinuity is proved by Fatou’s lemma. To prove the sub-mean-value property, we consider, as before, the sequence un = max{−n, u} and apply Fubini’s theorem; then we obtain the desired inequality by an application of the monotone convergence theorem.

2.2 Properties of the mean values |

49

For fixed 0 < r1 < r2 , define the harmonic function h(z) = a log |z| + b by h(rj ) = In (rj ), where In (r) = I(r, un ), and n ∈ ℕ+ is fixed. Since I(z) = h(z) on the boundary of the annulus r1 ≤ |z| ≤ r2 , we have I(z) ≤ h(z) for r1 < |z| < r2 . From this we infer that In (r) is convex of log r; the same holds for I(r) because In (r) → I(r) as n → ∞. The function I(r, u) increases with r since the function φ(x) = I(r, ex ) is convex and bounded from above for −∞ < x < log R, and this completes the proof in case of I(r, u). In case of I∞ the proof is similar; we define h by h(z) = I∞ (rj ) for |z| = rj . As an application of the preceding theorem, we get the following useful fact [384, 539]. Lemma 2.6. If p > 0 and f (z) = ∑nk=m ak z k , then for 0 < p ≤ ∞, we have r n Mp (1, f ) ≤ Mp (r, f ) ≤ r m Mp (1, f ),

0 < r < 1.

Proof. It is easy to show that Mp (r, f ) = Mp (1/r, g)r n , where n

g = ∑ ā k en−k . k=m

The function g is analytic, and therefore |g|p is subharmonic. Hence Mp (1/r, g) ≥ Mp (1, g) = Mp (1, f ) because 1/r > 1. On the other hand, 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩n−m 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 Mp (r, f ) = r m 󵄩󵄩󵄩 ∑ r k−m ak ek 󵄩󵄩󵄩 = r m 󵄩󵄩󵄩 ∑ am+k r k ek 󵄩󵄩󵄩 ≤ r m Mp (1, f ), 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩p 󵄩k=m 󵄩p 󵄩 k=0 which completes the proof. Lemma 2.7. If f is a polynomial of degree ≤ n, then ‖f ‖q ≤ C(n + 1)1/p−1/q ‖f ‖p ,

0 < p < q ≤ ∞,

where C depends only on p and q. Proof. This can be deduced from Lemma 2.6 and Corollary 1.32 by taking r = 1−1/(n+1). The function I(r, u) need not be increasing if u is defined and subharmonic in an annulus; a simple example is the function u(z) = − log |z|, which is subharmonic (and harmonic) in the annulus ℂ \ {0}.

50 | 2 Subharmonic functions and Hardy spaces Theorem 2.4. Let u be subharmonic in the annulus ρ < |z| < R. Then the function π

I(r, u) = ∫ − u(reiθ ) dθ

(ρ < r < R)

−π

is finite and convex of log r. The same holds for the function I∞ (r, u). The proof is similar to that of Theorem 2.2 and is omitted here. Logarithmic convexity of the integral means A positive real function φ is said to be logarithmically convex if log φ is convex. A necessary and sufficient condition for φ to be logarithmically convex in (a, b) is that, for every c ∈ ℝ, the function ecx φ(x) is convex in (a, b); see Exercise 2.10. 2.2.b. We say that φ(r) is logarithmically convex of log r if φ(r) > 0 and log φ is convex of log r, which can be expressed as φ(r11−λ r2λ ) ≤ φ(r1 )1−λ φ(r2 )λ ,

0 < λ < 1.

If φ is continuous, then the validity of this for λ = 1/2 is sufficient for φ to be log-convex of log r. Theorem 2.5. Let u be subharmonic in the annulus ρ < |z| < R, ρ ≥ 0. Then the function I(r, eu ) is logarithmically convex of log r in the interval ρ < r < R. Proof. The function v(z) = ec log |z|+u(z) is subharmonic in the annulus ρ < |z| < R for every c ∈ ℝ. Hence the function I(ex , v) = ecx I(ex , eu ) is convex for every c ∈ ℝ. The result follows. Corollary 2.8. If f is a function analytic in the annulus ρ < |z| < R, then the function Mp (r, f ) is logarithmically convex of log r in the interval ρ < r < R for every 0 < p ≤ ∞. Remark 2.9. Doubtsov [158] proved a very interesting fact. Let 0 < p ≤ ∞ and let ω(r) be a positive increasing function on (0, 1) that is logarithmically convex of log r. Then there exists a holomorphic function f on 𝔻 such that ω(r) is comparable to Mp (r, f ). Exercise 2.10. The fact that if (a) φ > 0 is logarithmically convex, then (b) ecx φ(x) is convex for all c ∈ ℝ is simple because (a) implies that log(ecx φ(x)) is convex. To prove that (b) implies (a), we start from the inequality 2stφ(

x1 + x2 ) ≤ s2 φ(x1 ) + t 2 φ(x2 ), 2

where s = ecx1 /2 φ(x1 ), t = ecx2 /2 φ(x2 ), which implies that the discriminant of a certain quadratic form is nonpositive. Also, an interesting proof can be given in the case where φ is C 2 .

2.2 Properties of the mean values | 51

Exercise 2.11. If u ≥ 0 is subharmonic in the annulus ρ < |z| < R and p > 1, then the function Mp (r, u) is convex of log r for ρ < r < R. The following statement is useful in reducing some proofs for log-subharmonic functions to the case of the moduli of analytic functions. Lemma 2.12. If u ≢ 0 is log-subharmonic in 𝔻 and upper semicontinuous in 𝔻, then there is a zero-free analytic function f ∈ H ∞ such that u ≤ |f | in 𝔻 and |f∗ | = u a. e.on 𝕋. The desired function is defined as |f | = exp(𝒫 [log u∗ ]), where u∗ = u|𝕋 . It should be noted that the function log u∗ is in L1 (𝕋) because log u∗ is bounded above and − log u(ζ /2)|dζ | > −∞. − log u∗ dl ≥ ∫ ∫ 𝕋

𝕋

Approximation by smooth functions Theorem 2.6. Let u ≢ −∞ be subharmonic in a domain D. Then there exist an increasing sequence of open sets Dn , whose union is D, and a decreasing sequence of subharmonic functions un ∈ C ∞ (Dn ) tending to u. Proof. Let ω(z) = ω0 (|z|) be a nonnegative “radial” function of class C ∞ (ℂ) with compact support in 𝔻 such that 1

∫ ω(w) dA(w) = 2π ∫ rω0 (r) dr = 1. 0

𝔻

For ε > 0 small enough consider the sets Dε = {z : dist(z, ℂ ∖ D) > ε} and the functions uε (z) = ∫ ω(w)u(z + εw) dA(w) = ∫ ε−2 ω((w − z)/ε)u(w) dA(w), ℂ

ℂ

where u ≡ 0 outside of D. Then uε is finite (because of the local integrability of u), subharmonic, and of class C ∞ in Dε . From the formula 1

π

0

−π

uε (z) = ∫ rω0 (r) dr ∫ u(z + rεeit ) dt and the inequality π

π

∫ u(z + rεe ) dt ≤ ∫ u(z + rδeit ) dt, −π

it

δ −∞. Then π

− u(reiθ ) dθ − u(0) = ∫ −π

1 r dμ(z) ∫ log 2π |z|

(2.6)

|z| r. Choose two (bounded) sequences of C 2 functions gm and hm with compact support in (−R, R) such that gm (|z|) ≤ Gρ (z) ≤ hm (|z|) and limm→∞ (hm (|z|) − gm (|z|)) = 0. Then I(r, un ) − I(ρ, un ) ≥

1 1 ∫ gm Δun dA = ∫ un Δgm dA. 2π 2π DR

DR

Here we apply the dominated convergence theorem, which is possible because |un | ≤ |u| + |u1 | and the functions u and u1 are locally integrable, to get I(r, u) − I(ρ, u) ≥

1 1 ∫ uΔgm dA = ∫ gm dμ. 2π 2π DR

DR

Now letting m tend to ∞, we obtain I(r, u) − I(ρ, u) ≥

1 ∫ Gρ (z) dμ(z). 2π DR

The reverse inequality is proved in a similar way. Finally, let ρ tend to 0 to finish the proof. If F is a C 2 -function in a neighborhood of 𝔻, then an application of the formula (1.2) to the function f ∘ σa , a ∈ 𝔻, leads to formula (1.4). The analogous formula for subharmonic functions reads − u(ζ )P(z, ζ )|dζ | − u(z) = ∫ 𝕋

1 1 dμ (w), ∫ log 2π |σw (z)| u

z ∈ 𝔻,

(2.7)

𝔻

if u(z) > −∞. This formula, which we call the Riesz representation formula, holds if u is subharmonic in a neighborhood of 𝔻 (and if u satisfies weaker conditions, which are not important in this text). Jensen’s formula. The well-known Jensen formula is a particular case of (2.6). Namely, if a function f is analytic in DR and f (0) ≠ 0, then the Riesz measure of the function

2.3 The Riesz measure |

55

log |f (z)| is equal to 2π ∑k δak , where ak are the zeros of f , and δa denotes the Dirac measure concentrated at a. This and (2.6) give Jensen’s formula π

r 󵄨 󵄨 󵄨 󵄨 − log󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ = log󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + ∑ log , ∫ |ak | |a | 0 and r = √p/q. If q/p is an integer and f ∈ H p , then Mq (r, f ) ≤ ‖f ‖p . Moreover, if u is a function log-subharmonic in 𝔻 and upper semicontinuous on 𝔻, then Mq (r, u) ≤ Mp (1, u). Proof. Let q/p = N. Assuming that p = 2 and ‖f ‖2 = 1, we have Mqq (r, f )

2π

∞

0

n=0

󵄨2N 󵄨 =∫ − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ = ∑ |cn |2 N −n ,

where cn is given by (2.12). On the other hand, I :=

∞ N − 1 󵄨󵄨 (N − 1)!n! N−2 󵄨2N . dA(z) = ∑ |cn |2 ∫󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) π (N + n − 1)! n=0 𝔻

Since N −n ≤

(N − 1)!n! , (N + n − 1)!

which can be proved by induction on n, we have Mqq (r, f ) ≤ I, so Theorem 2.14 concludes the proof. The last statement then follows from Lemma 2.12. Remark. Lemma 2.21 is true for all q > p > 0. A proof can be found in Weissler [596], who attributes it to S. Janson. The inequality is sharp. Inequalities of Fejér–Riesz and Hilbert If g is a function analytic in 𝔻, then a particular case of the Riesz–Zygmund theorem states that 1

󵄩 󵄩 󵄨 󵄨 ∫ 󵄨󵄨󵄨g 󸀠 (r)󵄨󵄨󵄨 dr ≤ π 󵄩󵄩󵄩g 󸀠 󵄩󵄩󵄩1 . −1

Replacing g 󸀠 with f and using Riesz’ factorization, we get the Fejér–Riesz inequality [188] 1

󵄨p 󵄨 ∫ 󵄨󵄨󵄨f (r)󵄨󵄨󵄨 dr ≤ π‖f ‖pp

(f ∈ H p , p > 0).

(2.13)

−1 2n In particular, if p = 2 and f (z) = ∑∞ n=0 an z (an ≥ 0), then (2.13) yields ∞ am an ≤ π ∑ |an |2 . m + n + (1/2) n=0 m,n≥0

∑

(2.14)

62 | 2 Subharmonic functions and Hardy spaces This inequality, known as Hilbert’s inequality, can be deduced immediately from the equality 1

2

π

2

∫ f (r) dr = i ∫ f (eiθ ) eiθ dθ, 0

−1

a consequence of Cauchy’s integral theorem. From (2.14) it follows that 1/2

1/2

∞ ∞ am bn ≤ π( ∑ |an |2 ) ( ∑ |bn |2 ) . ∑ m + n + (1/2) m,n≥0 n=0 n=0

(2.15)

Hardy’s inequality From Hilbert’s inequality we can obtain a slightly improved version of Hardy’s inequality (1.58). Theorem 2.15 (Hardy’s inequality). If f ∈ H 1 , then ∞

|f ̂(n)| ≤ π‖f ‖1 . n + (1/2) n=0 ∑

(2.16)

Proof. Let f ∈ H 1 , and let f = Bg be the Riesz factorization of f . Then the functions ̂ F = Bg 1/2 and G = g 1/2 belong to H 2 , and ‖f ‖1 = ‖F‖22 = ‖G‖22 . Let ak = |F(k)| and ̂ bk = |G(k)|. Then we have n ∞ am bn |f ̂(n)| 1 ≤ ∑ . ∑ ak bn−k = ∑ n + (1/2) n + (1/2) m + n + (1/2) n=0 m,n≥0 n=0 k=0 ∞

∑

Now we use (2.15) to get ∞

|f ̂(n)| ≤ π‖F‖2 ‖G‖2 = π‖f ‖1 , n + (1/2) n=0 ∑

as desired. Remark 2.22. In the case p = 2 the isoperimetric inequality (2.14) can be written as |f ̂(n)|2 ≤ ‖f ‖21 . n + 1 n=0 ∞

∑

(2.17)

It is interesting to compare this inequality with (2.16). In general, the convergence of the series ∑ |f ̂(n)|/(n + 1) with f ∈ H(𝔻) does not imply the convergence of ∑ |f ̂(n)|2 / (n + 1). However, if f ∈ H 1 , then |f ̂(n)| ≤ ‖f ‖1 , and therefore (2.16) implies a weak form of (2.17): |f ̂(n)|2 ≤ π‖f ‖21 . n + 1 n=0 ∞

∑

2.6 Hardy–Stein identities | 63

On the other hand, (2.17) implies ‖f ‖21 −|f (0)|2 ≥ (1/2)|f 󸀠 (0)|2 , which cannot be deduced from (2.16). p Theorem 2.16 (Hardy M∞ -inequality). If f ∈ H p , p ∈ ℝ+ , then 1

p (r, f ) dr ≤ π‖f ‖pp . ∫ r −1/2 M∞ 0

Proof. Let f = Bg 2/p , where B is a Blaschke product. Then 1

∫r

−1/2

p M∞ (r, f ) dr

1

≤ ∫r

0

0

−1/2

2 M∞ (r, g) dr

1

≤ ∫r 0

∞

−1/2

∞

2

󵄨 ̂ 󵄨󵄨 n ( ∑ 󵄨󵄨󵄨g(n) 󵄨󵄨r ) dr n=0

̂ ̂ |g(m)|| g(n)| ≤ π‖g‖22 = π‖f ‖pp , m + n + 1/2 m,n=0

= ∑ as desired.

2.6 Hardy–Stein identities By Theorem 2.7 the Riesz measure of a function v subharmonic in 𝔻 is uniquely determined by the requirement that ∫ φ dμv = ∫ vΔφ dA 𝔻

(2.18)

𝔻

for all φ ∈ Cc2 (𝔻). We can suppose that φ ≥ 0 because if φ is arbitrary, then we can choose φ1 ∈ Cc2 (𝔻) such that φ1 > max{0, φ} and then apply (2.18) to φ1 − φ. Theorem 2.17. (i) If v = |f |p , p > 0, and f ∈ H(𝔻), then μv is absolutely continuous, and 󵄨2 󵄨p−2 󵄨 󵄨 dμv (z) = p2 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 dA(z). (ii) If v = |u|p , p > 1, with real-valued u ∈ h(𝔻), then μv is absolutely continuous, and 󵄨2 󵄨p−2 󵄨 󵄨 dμv (z) = p(p − 1)󵄨󵄨󵄨u(z)󵄨󵄨󵄨 󵄨󵄨󵄨∇u(z)󵄨󵄨󵄨 dA(z). Proof. (i) Let f ∈ H(𝔻) and φ ∈ Cc2 (𝔻). We start from the relation ∫ |f |p Δφ dA = lim+ ∫ gε Δφ dA, 𝔻

ε→0

𝔻

(2.19)

64 | 2 Subharmonic functions and Hardy spaces where gε = (|f |2 + ε)p/2 . Since gε ∈ C 2 , we have p/2−2

∫ gε Δφ dA = ∫ φΔgε dA = p ∫ φ(|f |2 + ε) 𝔻

𝔻

󵄨 󵄨2 [p|f |2 + 2ε]󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 dA.

𝔻

Now applying Fatou’s lemma together with (2.19), we obtain 󵄨 󵄨2 ∫ |f |p Δφ dA ≥ p2 ∫ φ|f |p−2 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 dA. 𝔻

𝔻

This shows that the function φ|f | φ(|f |2 + ε)

p/2−2

p−2

󸀠 2

|f | is integrable on 𝔻. Since

p/2−1 󵄨 󸀠 󵄨2 󵄨 󵄨2 󵄨󵄨f 󵄨󵄨 ≤ 2φ|f |p−2 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨2 , [p|f |2 + 2ε]󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 ≤ 2φ(|f |2 + ε) 󵄨 󵄨 󵄨 󵄨

we can apply the dominated convergence theorem to conclude the proof in the case p < 2. If p ≥ 2, then the function |f |p is of class C 2 , so we have ∫ |f |p Δφ dA = ∫ φΔ(|f |p ) dA, 𝔻

𝔻

which immediately gives the result (take ε = 0 in gε ). The proof of statement (ii) is similar, and we omit it. Combining Theorems 2.8 and 2.17, we get the following (see also Further notes 2.8): Theorem 2.18 (Hardy–Stein identities [228, 556]). If f ∈ H(𝔻), then 2 r 󵄨p p 󵄨 󵄨2 󵄨p−2 󵄨 󵄨 Mpp (r, f ) = 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + dA(z) ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 log 2π |z|

(∗)

p2 d p 󵄨2 󵄨p−2 󵄨 󵄨 Mp (r, f ) = ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 dA(z) dr 2πr

(†)

|z| 0. Now the result follows from the inequality M1 (r, f 󸀠 ) ≥ n|f ̂(n)|r n−1 . kn Theorem 2.23 (Paley). Let the series f (z) = ∑∞ n=1 an z , where {kn } is a lacunary se2 quence, converge in 𝔻. Then f ∈ H p (p ∈ ℝ+ ) if and only if ∑∞ n=1 |an | < ∞. There exists a constant C = Cp > 0 such that

󵄩 󵄩 󵄩 󵄩 C −1 󵄩󵄩󵄩{an }󵄩󵄩󵄩2 ≤ ‖f ‖p ≤ C 󵄩󵄩󵄩{an }󵄩󵄩󵄩2 .

(2.20)

2.6 Hardy–Stein identities | 67

Proof. In the case 1 ≤ p < 2 the result is an immediate consequence of Paley’s theorem. Let 0 < p < 1. Let f be analytic in a neighborhood of the closed disc. Then by the Cauchy–Schwarz inequality we get (2−p)/2 (2−p)/2 ‖f ‖1 = ∫ − |f |p/2 |f |1−p/2 dℓ ≤ ‖f ‖p/2 ≤ ‖f ‖p/2 . p ‖f ‖2−p p ‖f ‖2 𝕋

Since ‖f ‖1 ≥ c‖f ‖2 , we see that c‖f ‖p/2 ≤ ‖f ‖p/2 p . If f is arbitrary, then we apply this 2 inequality to the functions fρ (ρ → 1), and this completes the proof in the case 0 < p < 2. Let 2 < p < ∞ and q = p/(p − 1). It follows from Paley’s theorem that the operator P defined by (Pf )(z) = ∑ f ̂(kn )z kn is bounded from H q to H 2 . The adjoint P ⋆ is formally equal to P, and since (H q )󸀠 = H p (Theorem 1.37), we have ‖Pf ‖p ≤ Cp ‖f ‖2 for f ∈ H 2 . Hence ‖f ‖p ≤ Cp ‖f ‖2 if Pf = f . Exercise 2.25. If 0 < p ≤ 2 and f (z) = ∑ an z kn , where {kn } is lacunary, then ∞ d 2 Mp (r, f ) ≤ C ∑ kn |an |2 r 2kn −1 . dr n=1

If 2 ≤ p < ∞, then the reverse holds.

2.6.2 Application to absolutely summing operators A bounded operator T : X 󳨃→ Y, where X and Y are Banach spaces, is said to be ρ-summing (ρ ≥ 1) if there is a constant C > 0 such that n

(∑ k=1

1/ρ ρ ‖Tfk ‖Y )

n

1/ρ

󵄨ρ 󵄨 ≤ C sup ( ∑ 󵄨󵄨󵄨Λ(fk )󵄨󵄨󵄨 ) Λ∈B(X 󸀠 ) k=1

󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩 = C sup 󵄩󵄩 ∑ ak fk 󵄩󵄩󵄩 󵄩󵄩 󵄩 󸀠 󵄩X 󵄩k=1 a∈B(ℓρ )󵄩

for every finite sequence {fk } ⊂ X. Here B(X 󸀠 ) and B(ℓρ ) stand for the unit balls of X 󸀠 󸀠 (dual of X) and ℓρ , respectively. The smallest C is denoted by πp (T). We refer the reader to the book of Diestel et al. [148] for a beautiful exposition of the theory of these operators. The case of absolutely summing operators (ρ = 1) is the most interesting because these operators transform unconditionally convergent series into absolutely convergent ones. It should be remarked that if dim(X) = ∞ and {λn } ∈ ℓ2 , then, according to the Dvoretzky–Rogers theorem, there is an unconditionally convergent series ∑ fn in X such that ‖fn ‖ = |λn | for all n; see [148, p. 2]. Following Pełczyński [467], here we use Paley’s theorems and the Kolmogorov– Smirnov theorem to prove a famous theorem of Grothendieck. 󸀠

68 | 2 Subharmonic functions and Hardy spaces First, we state the following: Proposition 2.26. The operator Tf = {f ̂(2j )}∞ j=0 is an absolutely summing surjection from A(𝔻) onto ℓ2 . Proof. By Paley’s theorem we have n

n

n

󵄨 󵄨 󵄨 󵄨 − 󵄨󵄨󵄨fk (ζ )󵄨󵄨󵄨|dζ | ≤ C sup ∑ 󵄨󵄨󵄨fk (ζ )󵄨󵄨󵄨 ∑ ‖Tfk ‖2 ≤ C ∑ ∫

k=1

ζ ∈𝕋 k=1

k=1 𝕋

󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 = C sup sup 󵄨󵄨󵄨 ∑ aj fk (ζ )󵄨󵄨󵄨 󵄨󵄨 󵄨 ∞ ζ ∈𝕋 a∈B(ℓ )󵄨󵄨k=1 󵄨 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 . = C sup 󵄩󵄩󵄩 ∑ aj fk (ζ )󵄩󵄩󵄩 󵄩 󵄩󵄩󵄩A(𝔻) a∈B(ℓ∞ )󵄩 󵄩k=1 Thus T is absolutely summing. To prove that T is a surjection, it suffices to prove that ‖T ⋆ g‖A(𝔻)󸀠 ≥ c‖g‖ℓ2 (see [520, Theorem 4.13]), where T ⋆ is the adjoint of T. We identify ℓ2 with the space of analytic functions of the form ∞

n

g(z) = ∑ bn z 2 . n=0

Then we define the duality pairing ⟨f , g⟩ = ∫ − f (ζ )g(ζ ̄ ) |dζ |, 𝕋

where g is a polynomial. Since ⟨Tf , g⟩ = ⟨f , g⟩, we have T ⋆ g = g, g ∈ ℓ2 . By the Kolmogorov–Smirnov theorem and Paley’s theorem we have 󵄩 ⋆ 󵄩 󵄩󵄩 ⋆ 󵄩󵄩 󵄩󵄩T g 󵄩󵄩A(𝔻)󸀠 = 󵄩󵄩󵄩T g 󵄩󵄩󵄩𝒦 = ‖g‖𝒦 ≥ c‖g‖H 1/2 ≍ ‖g‖2 . This was to be proved. Note that the use of the relation A(𝔻)󸀠 = 𝒦 can be avoided. It can be proved that A(𝔻)󸀠 ⊂ H p for p < 1 and then use Paley’s theorem as before. Now we state Grothendieck’s theorem, proved in his famous Résumé [217]. Theorem (Grothendieck). Every bounded linear operator from ℓ1 to ℓ2 is absolutely summing. Proof [467]. Let S : ℓ1 󳨃→ ℓ2 be a bounded operator, and let T : A(𝔻) 󳨃→ ℓ2 be an absolutely summing surjection. By the open mapping theorem there is a constant K > 0 such that, for each g ∈ ℓ2 , there is f ∈ A(𝔻) with ‖f ‖ ≤ K‖g‖ and Tf = g. In particular, for each j ∈ ℕ, there is fj ∈ A(𝔻) such that Tfj = Sej and ‖fj ‖ ≤ K‖S‖. Now define ̃ ̃ ̃ S̃ : ℓ1 󳨃→ A(𝔻) by S({a j }) = ∑ aj fj . Then S = T S and ‖S‖ ≤ K‖S‖. It follows that π1 (S) ≤ Kπ1 (T)‖S‖.

2.7 The subordination principle | 69

2.7 The subordination principle A function f defined on 𝔻 is said to be univalent if it is analytic and one-to-one. The leading example is the Köbe function f (z) = z/(1 − z)2 mapping 𝔻 to ℂ slit from −1/4 to −∞ along the real axis. The class of all such functions is denoted by 𝒰 . Let F ∈ 𝒰 . A function f analytic in 𝔻 is said to be subordinate to F if f (𝔻) ⊂ F(𝔻) and f (0) = F(0). In other words, f is subordinate to F if f (z) = F(ω(z)), where |ω(z)| ≤ |z|, z ∈ 𝔻, and ω is analytic. In this form the notion of subordination is defined for arbitrary functions. This notion is important because of the following theorem of Littlewood [356]. Theorem 2.24 (Subordination principle). If a function u is subordinate to a subharmonic function U, then π

π

−π

−π

− u(reiθ ) dθ ≤ ∫ − U(reiθ ) dθ ∫

(0 < r < 1).

(2.21)

In the simplest case ω(z) = ρz, 0 < ρ < 1, this theorem reduces to Theorem 2.3. Proof. 1 We can assume that U is continuous. Let h be a function harmonic in Dr = {|z| < r}, continuous on the closure, and equal to U on the boundary. Then U ≤ h on Dr , and hence u(z) = U(ω(z)) ≤ h(ω(z)) for |z| = r. It follows that π

π

−π

−π

− u(reiθ ) dθ ≤ ∫ − h(ω(reiθ )) dθ = h(ω(0)) ∫ π

= h(0) = ∫ − h(reiθ ) dθ. −π

This concludes the proof because h(reiθ ) = U(reiθ ). Example 2.27 (A proof of the Kolmogorov–Smirnov theorem). We may assume that f (0) ∈ ℝ. Assume first that Re f > 0. Then f is subordinate to the univalent function F(z) = c(1 + z)/(1 − z), c = Re f (0). Applying the subordination principle, we get the following: π π π 󵄨󵄨 1 + reiθ 󵄨󵄨p 󵄨󵄨 1 + eiθ 󵄨󵄨p 󵄨 󵄨p 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 p − 󵄨󵄨󵄨 − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ ≤ cp ∫ dθ ≤ c − ∫ ∫ 󵄨󵄨 󵄨 󵄨 dθ 󵄨󵄨 1 − reiθ 󵄨󵄨󵄨 󵄨󵄨 1 − eiθ 󵄨󵄨󵄨 −π −π −π

= cp

π/2

u(0)p 2 ∫ (cot θ)p dθ = π cos(pπ/2) 0

1 This proof was invented by F. Riesz [503].

(0 < p < 1).

70 | 2 Subharmonic functions and Hardy spaces If f is arbitrary, then we can use Theorem 1.4(ii) to reduce the proof to the preceding case. Exercise 2.28. It follows from Exercise 2.17 that if f ∈ H p and 1/f ∈ H p for some p > 0, then f is outer. By the subordination principle, f is in H p if Re f > 0 and p < 1. Since Re(1/f ) = Re f /|f |2 > 0, we see that f is outer if Re f > 0. Our next example is the case p ≤ 1 of the following theorem [8]. Theorem 2.25 (Ahern). If f ∈ H(𝔻) and 0 < |f (z)| < 1 for all z ∈ 𝔻, then for every p > 0, we have π

󵄨p 󵄨 − (1 − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨) dθ ≥ cp (1 − r)1/2 , ∫ −π

where cp is a positive constant. Ahern’s proof is based on a nontrivial analysis of singular measures. Here we use the subordination principle to prove the theorem in the case p ≤ 1 and even improve it for p ≤ 1/2. However, it seems that application of this principle is limited to the case p ≤ 1. Theorem 2.26. Let 0 < p ≤ 1. Under the hypotheses of the previous theorem, we have π

󵄨p 󵄨 − (1 − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨) dθ ≥ cp γp (r), ∫ −π

where (1 − r)1/2 , 1/2 < p ≤ 1, { { { 2 1/2 γp (r) = {(1 − r) log 1−r , p = 1/2, { { p 0 < p < 1/2. {(1 − r) , Proof. The analytic function a(z) = − log f (z) maps 𝔻 into the right half-plane. Replacing f by ζf for a suitable ζ ∈ 𝕋, we may assume that a(0) > 0. It follows that f (z) is subordinate to Aλ (z) = exp(−λ

1+z ), 1−z

where λ > 0. The function −(1 − |z|)p is subharmonic for p ≤ 1, and therefore, by the subordination principle, π

π

−π

−π

󵄨p 󵄨 󵄨p 󵄨 − (1 − 󵄨󵄨󵄨Aλ (reiθ )󵄨󵄨󵄨) dθ. − (1 − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨) dθ ≥ ∫ ∫

2.7 The subordination principle

| 71

To estimate this integral, we use the inequalities x 2x ≤ 1 − e−x ≤ , 1+x 1+x

x > 0.

It follows that π

π

−π

0 π

p

λP(r, θ) 󵄨p 󵄨 ) dθ − (1 − 󵄨󵄨󵄨Aλ (reiθ )󵄨󵄨󵄨) dθ ≍ ∫( ∫ 1 + λP(r, θ) ≍∫ 0

(1 − r)p dθ. (1 − r + θ2 )p

Introducing the change θ = t √1 − r and computing the resulting integral, we obtain the desired inequality. Corollary 2.29. If 0 < p ≤ 1 and f is an inner function with nonconstant singular factor, then 2π

󵄨p 󵄨 ∫ (1 − 󵄨󵄨󵄨I(eiθ )󵄨󵄨󵄨) dθ ≥ cγp (r). 0

Exercise 2.30. If u is a positive harmonic function, then Mp (r, u) ≥ cp βp (r), where βp has been defined in Proposition 1.23. Problem 2.4. Concerning this exercise, it seems that it is not known if the previous estimates hold for all harmonic functions. Aleksandrov [23] proved that they hold if Mp (r, u) is replaced with 1

1/p

1 𝒥p (r, u) = ( ∫ Mpp (ρ, u)ρ dρ) 1−r

,

r

where u is arbitrary. From this we can conclude that if Mp (r, u) = o(βp (r)) (r ↑ 1), then u ≡ 0.

2.7.1 Nevanlinna counting function and the subordination principle To present the idea of Littlewood’s proof of the subordination principle, suppose that F is a subharmonic function of class C 2 in a neighborhood of 𝔻. Let φ be an analytic self-mapping of 𝔻 such that φ(0) = 0. By Green’s formula we have 1 󵄨2 󵄨 dA(z). − F(φ(ζ ))|dζ | − F(0) = 2 ∫ − (ΔF)(φ(z))󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨 log ∫ |z|

𝕋

𝔻

(2.22)

72 | 2 Subharmonic functions and Hardy spaces If φ is univalent, then we can write the right-hand side as I :=

2 1 dA(w), ∫ ΔF(w) log π |ψ(w)| φ(𝔻)

where ψ is the inverse of φ. Since |ψ(w)| ≥ |w|, we get I ≤ 2∫ − ΔF(w) log 𝔻

1 dA(w) = ∫ − F(ζ )|dζ | − F(0), |w| 𝕋

whence − F(φ(ζ ))|dζ | ≤ ∫ − F(ζ )|dζ |. ∫ 𝕋

𝕋

In the general case, after deleting the zeroes of φ󸀠 , we can divide the resulting domain into a disjoint countable union of semiclosed polar rectangles Dj such that φ is oneto-one on Dj . Then we have 1 1 󵄨2 󵄨 dA(z) = ∑ ∫ ΔF(w) log dA(w), ∫(ΔF)(φ(z))󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨 log |z| |ψ (w)| j j φj (𝔻)

𝔻

where ψj is the inverse of the restriction of φj to Dj . Denoting by χj the characteristic function of φ(Dj ), we have 1 1 󵄨2 󵄨 dA(z) = ∫ ΔF(w) ∑ χj (w) log dA(w). ∫(ΔF)(φ(z))󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨 log |z| |ψ (w)|

𝔻

𝔻

j

j

A careful examination shows that the inner sum is equal to the Nevanlinna counting function Nφ (w) = ∑ log(1/|zj |), j

w ∈ ℂ \ {0},

where zj ∈ 𝔻 are the zeroes of the equation φ(z) = w, repeated according their multiplicities (see Shapiro [529]). Thus we see that an improvement of Schwarz’s lemma is needed, namely Nφ (w) ≤ log(1/|w|). Before stating this improvement, due to Littlewood [356], we mention the partial counting functions. For a function φ ∈ H(𝔻), we define the partial counting functions by n(r,w)

Nφ (r, w) = ∑ log(r/|zj |), j=1

w ∈ ℂ \ {φ(0)},

where zj are as before but |zj | < r, and n(r, w) is the number of such zj . The function Nφ (r, w) increases with r, and it follows from the monotone convergence theorem that it tends to Nφ (1, w) = Nφ (w).

2.7 The subordination principle

| 73

Lemma 2.31 (Littlewood). Let φ : 𝔻 󳨃→ 𝔻 be an analytic function. Then for every w ∈ 𝔻 \ {φ(0)}, we have 󵄨󵄨 1 − wφ(0) 󵄨󵄨 1 󵄨 󵄨󵄨 Nφ (w) ≤ log󵄨󵄨󵄨 . 󵄨 = log 󵄨󵄨 φ(0) − w 󵄨󵄨󵄨 |σφ(0) (w)| In particular, if φ(0) = 0, then Nφ (w) ≤ log

1 . |w|

(2.23)

Proof. We will consider the case where φ(0) = 0, leaving the general case to the reader. Write Jensen’s formula (2.8) as π

󵄨 󵄨 󵄨 󵄨 Nf (r, 0) = ∫ − log󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ − log󵄨󵄨󵄨f (0)󵄨󵄨󵄨. −π

Then take f (z) = φw (z) = σw (φ(z)) with w ≠ 0 to obtain π

󵄨 󵄨 Nφ (r, w) = ∫ − log󵄨󵄨󵄨φw (reiθ )󵄨󵄨󵄨 dθ − log |w| ≤ − log |w|. −π

Letting r tend to 1 , we get (2.23). −

Inequality (2.23) improves Schwarz’s lemma, which can be seen by writing it in the form |w| ≤ ∏{|z|: φ(z) = w}, z

whilst Schwarz’s lemma asserts that if φ(z) = w, then |w| = |φ(z)| ≤ |z|. The function Nφ has an interesting property. The partial counting function can be written as ̄ ||dζ | − log |1 − wrζ Nφ (r, w) = ∫ − log |rζ − w||dζ | − ∫ 𝕋

𝕋

󵄨 󵄨 󵄨 󵄨󵄨󵄨. ̄ − log󵄨󵄨󵄨w − φ(0)󵄨󵄨󵄨 + log󵄨󵄨󵄨1 − wφ(0) 󵄨 This function is not subharmonic in 𝔻 because Nφ (r, φ(0)) = +∞. On the other hand, the first integral is subharmonic in 𝔻, the second integral is equal to zero, and w 󳨃→ ̄ log |1 − wφ(0)| is harmonic in 𝔻. It follows that N(r, w) is subharmonic in 𝔻 \ {φ(0)} for fixed r ∈ (0, 1). Consequently, we have Nφ (r, a) ≤

1 ∫ Nφ (r, w) dA(w), |D| D

where D ⊂ 𝔻 is a disc centered at a. Letting r tend to 1− , we get a particular case of [529, 4.6].

74 | 2 Subharmonic functions and Hardy spaces Lemma 2.32. For any disc D ⊂ 𝔻 \ {φ(0)} centered at a, we have Nφ (a) ≤

1 ∫ Nφ (w) dA(w). |D| D

This does not mean that Nφ is subharmonic. It is lower semicontinuous as an increasing limit of continuous functions. For further information, see [529] and references therein. Finally, we give a change-of-variable formula, which follows from the preceding consideration. Lemma 2.33. If g is a nonnegative function in 𝔻, then 1 󵄨2 󵄨 dA(w) = ∫ g(w)Nφ (w) dA(w). ∫ g(φ(w))󵄨󵄨󵄨φ󸀠 (w)󵄨󵄨󵄨 log |w|

𝔻

(2.24)

𝔻

2.7.2 Composition with inner functions In this section, we consider nonconstant inner functions. If ω is such a function, then we put ω∗ (ζ ) = ∢ lim ω(z) z→ζ

for those ζ ∈ 𝕋 for which this limit exists and belongs to 𝕋; then we extend ω∗ to a function from 𝕋 to 𝕋 in an arbitrary way. Our main purpose is to prove the validity of the relations f ∈ Hp ⇔ f ∘ ω ∈ Hp and ‖f ∘ ω‖p = ‖f ‖p

if ω(0) = 0,

due to Stephenson [558] (see Theorems 2.29 and 2.30). These relations as well as all other statements in this section become obvious when specialized to the case ω(z) = z n , n ≥ 1. In general, the composition of two Lebesgue-measurable functions need not be measurable. Proposition 2.34. Let ω be an inner function such that |ω(0)| < 1. (i) If E ⊂ 𝕋 is of measure zero, then so is ω−1 ∗ (E). (ii) If g : 𝕋 󳨃→ ℂ is Lebesgue measurable, then so is g ∘ ω∗ . (iii) If ϕ and g are Lebesgue-measurable functions on 𝕋 such that ϕ = g a. e., then ϕ ∘ ω∗ = g ∘ ω∗ a. e.

2.7 The subordination principle

| 75

(iv) If ϕn are Lebesgue-measurable functions on 𝕋 such that limn ϕn = f a. e., then limn ϕn ∘ ω∗ = f ∘ ω∗ a. e. Proof. (i) If ω(0) = a, then ω = σa ∘ (σa ∘ ω) and σ ∘ ω(0) = 0. Therefore we can assume that ω(0) = 0. We have to prove that the set F = {eiθ ∈ 𝕋: ω∗ (eiθ ) ∈ E} is of measure zero. To prove this, let ε > 0, let Eε = ⋃n In , where In ⊂ 𝕋 are closed arcs such that E ⊂ Eε , ∑n |In | < ε, and let Fε = {eiθ ∈ 𝕋: ω∗ (eiθ ) ∈ Eε }. We will prove that 2π

2π

iθ

∫ Kn (ω∗ (e )) dθ = ∫ Kn (eiθ ) dθ = |In |, 0

(2.25)

0

where Kn is the characteristic function of In . This implies that 2π

|Fε | ≤ ∑ ∫ Kn (ω∗ (eiθ )) dθ < ε, n

0

and this implies that F is of measure zero, because F ⊂ Fε for all ε > 0. To prove (2.25), when n is fixed, we choose a sequence ϕj ∈ C(𝕋) such that ϕj (eit ) tends to Kn (eit ) and |ϕj (eit )| ≤ 2 for every t. Then ϕj (ω∗ (eit )) → Kn (ω∗ (eit )) as j → ∞, so we can apply the dominated convergence theorem to reduce the proof to the formula 2π

iθ

2π

∫ ϕ(ω∗ (e )) dθ = ∫ ϕ(eiθ ) dθ, 0

ϕ ∈ C(𝕋),

0

recalling that ω(0) = 0. Finally, this is reduced to the case where ϕ is a trigonometric polynomial. The details are left to the reader. (ii) Let G be an Borel subset of ℂ. We have −1 (g ∘ ω∗ )−1 (G) = ω−1 ∗ (g (G)).

The set g −1 (G) is Lebesgue measurable, and therefore there exist a Borel set F and a set −1 −1 −1 E of measure zero such that g −1 (G) = F ∪ E. Now ω−1 ∗ (g (G)) = ω∗ (F) ∪ ω∗ (E). The set −1 −1 ω∗ (F) is Lebesgue measurable, and ω∗ (E) is of measure zero, and thus (g ∘ ω∗ )−1 (G) is Lebesgue measurable, which leads to the desired conclusion. (iii) Let ϕ(ζ ) = g(ζ ) for ζ ∈ 𝕋 \ E, where |E| = 0. Then ϕ(ω∗ (ζ )) = g(ω∗ (ζ )), −1 provided that ω∗ (ζ ) ∈ 𝕋 \ E, i. e., ζ ∈ ω−1 ∗ (𝕋) \ ω∗ (E), which proves the result because −1 −1 |ω∗ (E)| = 0 and |ω∗ (𝕋)| = |𝕋|. (iv) Let E be a set of measure zero such that limn ϕn (ζ ) = f (ζ ) for ζ ∈ 𝕋 \ E. We define the functions ψn by ψn = ϕn on 𝕋 \ E and ψn = f on E. By (iii), ψn ∘ ω∗ = ϕn ∘ ω∗ , and consequently ϕn ∘ ω∗ → f ∘ ω∗ on ω−1 ∗ (𝕋 \ E). This completes the proof.

76 | 2 Subharmonic functions and Hardy spaces Theorem 2.27 ([519, 558]). If ϕ ∈ L1 (𝕋) and ω is an inner function with ω(0) = 0, then ϕ ∘ ω∗ ∈ L1 (𝕋), and 2π

iθ

2π

∫ ϕ(ω∗ (e )) dθ = ∫ ϕ(eiθ ) dθ. 0

0

Proof. To reduce the proof to the case ϕ ∈ C(𝕋), we can suppose that ϕ is a positive real function. The sequence min{ϕ(ζ ), n} increases to ϕ(ζ ) everywhere, so the proof reduces to the case where ϕ is bounded. If ϕ is bounded, then we choose a bounded sequence ϕn ∈ C(𝕋) such that ϕn → ϕ a. e.; by Proposition 2.34 we have ϕn ∘ ω∗ → ϕ ∘ ω∗ a. e. The result follows. Theorem 2.28. If ϕ ∈ L1 (𝕋) and ω is an inner function such that |ω(0)| < 1, then 𝒫 [ϕ ∘ ω∗ ] = 𝒫 [ϕ] ∘ ω. Proof. It suffices to consider the case where ϕ ∈ C(𝕋). Then the functions 𝒫 [ϕ ∘ ω∗ ] and 𝒫 [ϕ] ∘ ω are harmonic and bounded, so it suffices to prove that their boundary functions coincide almost everywhere. Since 𝒫 [ϕ] is continuous on the closed disc, we have limr→1 𝒫 [ϕ](ω(reiθ )) = ϕ(ω∗ (eiθ )) a. e. On the other hand, limr→1 𝒫 [ϕ ∘ ω](reiθ ) = (ϕ ∘ ω∗ )(eiθ ) a. e., and this completes the proof. Corollary 2.35. If ω and I are inner functions, then so is the composition I ∘ ω, and (I ∘ ω)∗ = I∗ ∘ ω∗ a. e. If, in addition, I is singular, then so is I ∘ ω. Proof. This follows from the relations 𝒫 [I∗ ∘ ω∗ ] = 𝒫 [I∗ ] ∘ ω = I ∘ ω and 𝒫 [(I ∘ ω)∗ ] = I ∘ ω. Stephenson’s theorems Combining the previous results, we easily prove the following: Theorem 2.29 (Stephenson [558]). If f ∈ H p , p > 0, and ω is inner with ω(0) = 0, then f ∘ ω ∈ H p , (f ∘ ω)∗ = f∗ ∘ ω∗ , and ‖f ∘ ω‖p = ‖f ‖p . If f = IF is the inner–outer factorization of f , then f ∘ ω = (I ∘ ω)(F ∘ ω) is the inner–outer factorization of f ∘ ω. We conclude this section by proving the implication f ∘ ω ∈ H p ⇒ f ∈ H p . Theorem 2.30 (Stephenson [558]). If f ∈ H(𝔻) and ω is an inner function, then f ∘ ω ∈ H p implies f ∈ H p . Proof. Assume that ω(0) = 0. Let f ∘ ω ∈ H p , p > 0,

u = |f |p ,

v = |f ∘ ω|p ,

and h = 𝒫 [v∗ ].

We know that v ≤ h; see (1.46). Let r be fixed. Then u ≤ M on r𝔻 for some constant M < ∞. Let 󵄨 󵄨 Ωr = ω−1 (r𝔻) = {z : 󵄨󵄨󵄨ω(z)󵄨󵄨󵄨 < r}

2.7 The subordination principle

| 77

and 󵄨 󵄨 Gρ = Ωr ∩ ρ𝔻 = {z ∈ 𝔻: max{󵄨󵄨󵄨ω(z)󵄨󵄨󵄨/r, |z|/ρ} < 1}. For 0 < ρ < 1, let 󵄨 󵄨 Eρ = {ζ ∈ 𝕋: 󵄨󵄨󵄨ω(ρζ )󵄨󵄨󵄨 < r}. Since 1 − |ω(ρζ )| → 0 a. e. as ρ → 1, we see, by using Egorov’s theorem (“a. e. convergence implies convergence in measure”), that limρ→1 |Eρ | = 0. Hence we can choose ρ < 1 so that the following is true: (A) There exists a function φ that is bounded on ρ𝔻 and harmonic ρ𝔻, whose values are M on ρEρ and 0 on the rest of ρ𝕋, and such that φ(0) < 1. Since u is subharmonic and continuous in 𝔻, there exists a function u1 ∈ C(r𝔻) harmonic in r𝔻 such that u ≤ u1 ≤ M on r𝔻 and u1 (z) = u(z) for |z| = r. By the mean value property π

π

󵄨p 󵄨 u1 (0) = ∫ − u(re ) dθ = ∫ − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ. −π

iθ

(2.26)

−π

The function gρ (z) := max{|ω(z)|/r, |z|/ρ} is continuous, which implies that 󵄨 󵄨 𝜕Gρ ⊂ {z: max{󵄨󵄨󵄨ω(z)󵄨󵄨󵄨/r, |z|/ρ} = 1}. Consider the function U(z) := u1 (ω(z)) − h(z) on the closure of Gρ . Let z ∈ 𝜕Gρ . There are two cases: (1) |ω(z)|/r ≥ |z|/ρ, which implies |ω(z)| = r and |z| ≤ ρ; (2) |ω(z)|/r < |z|/ρ, which implies |z| = ρ and |ω(z)| < r. In case (1), we have u1 (ω(z)) = u(ω(z)) ≤ h(z) (by the definition of u1 ), and hence U(z) = u1 (ω(z)) − h(z) ≤ 0. In case (2), we have z ∈ ρEρ , which implies u1 (ω(z)) − h(z) ≤ u1 (ω(z)) ≤ M because ω(z) ∈ r𝔻. This shows that U − φ ≤ 0 on 𝜕Gρ , where φ is the function from (A). The set Gρ is compact, and the function U − φ is upper semicontinuous on it, so we can apply the maximum modulus principle (Corollary 2.5) to conclude that U(0) − φ(0) = u1 (0) − h(0) − φ(0) ≤ 0, which implies, via (2.26), π

󵄨p 󵄨 − 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨 dθ ≤ h(0) + 1. ∫ −π

Since h(0) + 1 is a constant independent of r, we see that f ∈ H p .

78 | 2 Subharmonic functions and Hardy spaces It remains to prove that U − φ is upper semicontinuous, which reduces to proving that φ is lower semicontinuous. This follows from two facts the proofs of which are left to the reader: (i) the characteristic function of an open set (in our situation, this is ρEρ ) is lower semicontinuous, and (ii) the Poisson integral of a lower semicontinuous function on 𝕋 is lower semicontinuous on 𝔻. For (i), see statement (a) after Definition 2.8 in [517]; for (ii), see Further notes 1.2.

2.7.3 Approximation with inner functions Let ω be an inner function with ω(0) = 0. If f ∈ H 2 , then, by Rogosinski’s theorem (see Further notes 2.5) and Stephenson’s theorem, we have ∞

∞

k=n

k=n

󵄨 ̂ 󵄨󵄨2 󵄨2 󵄨 ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 ≥ ∑ 󵄨󵄨󵄨F(k) 󵄨󵄨 ,

where f = F ∘ ω. This fact can be expressed in terms of best approximation. Let, as before, Pn , n ≥ 0, denote the set of all analytic polynomials of degree at most n. For a function f ∈ H p , let En (f )p = En (f )H p = infg∈Pn ‖f − g‖p . Then the last inequality can be stated as En (f ∘ ω)2 ≥ En (f )2 , f ∈ H 2 . It is interesting that this extends to the case 1 ≤ p ≤ ∞. Theorem 2.31 (Approximation with inner functions [387]). Let 1 ≤ p ≤ ∞, f ∈ H p , and let ω be an inner function with ω(0) = 0. Then En (f ∘ ω)p ≥ En (f )p . Proof. Let π

(f , h) = ∫ − f (eiθ )h(eiθ ) dθ. −π

We identify H p with H p (𝕋). From the general theory of best approximation in Banach spaces (Duren [167, Ch. VII]) we know that 󸀠 󵄨 󵄨 En (f )p = sup{󵄨󵄨󵄨(f , h)󵄨󵄨󵄨: h ∈ Lp (𝕋), (Pn , h) = 0, ‖h‖p󸀠 ≤ 1},

where (Pn , h) = 0 means that (g, h) = 0 for every g ∈ Pn . Now we apply this formula to f ∘ ω and use the following facts: (i) If h ∈ Lq (𝕋), then h ∘ ω ∈ Lq (𝕋) and ‖h ∘ ω‖q = 1, and (ii) if (Pn , h) = 0, then (Pn , h ∘ ω) = 0. Fact (i) follows from Theorem 2.27, whereas the proof of (ii) is straightforward: it suffices to observe that (h, Pn ) = 0 if and only if ̂ = 0 for 0 ≤ j ≤ n. We get h(j) 󵄨 󵄨 En (f ∘ ω)p ≥ sup{󵄨󵄨󵄨(f ∘ ω, h ∘ ω)󵄨󵄨󵄨: h ∈ Lq (𝕋), (Pn , h) = 0, ‖h‖q ≤ 1}. This concludes the proof because |(f ∘ ω, h ∘ ω)| = |(f , h)| by Theorem 2.27.

2.8 The theorem of Burkholder, Gundy, and Silverstein

| 79

Applying the same method to the pair H p , H p (1 < p < ∞), we prove the following [387, Proposition 1.1]: 󸀠

Lemma 2.36. Let 1 < p < ∞. If f ∈ H p and I is an inner function, then En (fI)p ≥ cp En (f )p ,

n ∈ ℕ.

Proof. We have 󸀠 󵄨 󵄨 En (fI)p ≥ cp sup{󵄨󵄨󵄨(fI, h)󵄨󵄨󵄨: h ∈ H p (𝕋), (Pn , h) = 0, ‖h‖p󸀠 ≤ 1}.

Since (Pn , h) = 0 and ‖h‖p󸀠 ≤ 1 imply (Pn , hI) = 0 and ‖hI‖p󸀠 = ‖h‖p ≤ 1, we have 󸀠 󵄨 󵄨 En (fI)p ≥ cp sup{󵄨󵄨󵄨(fI, hI)󵄨󵄨󵄨: h ∈ H p , (Pn , h) = 0, ‖h‖p󸀠 ≤ 1} = cp En (f ).

The last equality holds because (fI, hI) = (f , h). Problem 2.5. If we change the notation and denote by En (f )p the best Lp approximation of f ∈ Lp (𝕋) by trigonometric polynomials of degree ≤ n, then Theorem 2.31 remains valid. However, the above method heavily depends on the Hahn–Banach theorem and cannot be applied to the case p < 1. It would be interesting to study this case.

2.8 The theorem of Burkholder, Gundy, and Silverstein The following theorem, due to Burkholder, Gundy and Silverstein [106] enables us to treat H p as a space of harmonic functions and can be used to extend H p -theory to several real variables (cf. [187]). Theorem 2.32. Let 0 < p < ∞. A function f = u + iv ∈ H(𝔻) belongs to H p if and only if the function Mrad u belongs to Lp (𝕋); we have the inequality 󵄨p 󵄨 (1/Cp )‖f ‖H p ≤ ‖Mrad u‖Lp + 󵄨󵄨󵄨v(0)󵄨󵄨󵄨 ≤ Cp ‖f ‖H p .

(2.27)

The right-hand side inequality is a consequence of the complex maximal theorem. In the case p > 1 the left-hand side inequality is a consequence of the radial maximal theorem and Riesz’s inequality (1/Cp )‖f ‖H p ≤ ‖u‖Lp + |v(0)|p . On the other hand, if 1 < p < 2, then Riesz’s inequality follows from (2.27). Proof. We proceed in a similar way as Koosis [321] in the case of a half-plane (cf.Garnett [198, Theorem 3.1]). In fact, our proof is a modification of the proof given in Garnett’s book on pp. 114–115 and is a result of the author’s insufficient understanding of his proof.

80 | 2 Subharmonic functions and Hardy spaces By the preceding remarks it suffices to prove the inequality 2π

2π

0

0

p 󵄨p 󵄨 ∫ 󵄨󵄨󵄨v(eiθ )󵄨󵄨󵄨 dθ ≤ Cp ∫ (u+ (θ)) dθ,

0 < p < 2,

where u+ (θ) = Mrad u(eiθ ), supposing that f is a polynomial and f (0) = 0. By Theorem C.13 we can replace u+ by u∗ defined as u∗ (θ) = supζ ∈U |u(ζeiθ )|, where U = convex hull of {1} ∪ {ζ : |ζ | ≤ 1/√2}. So it suffices to prove 2π

2π

p 󵄨p 󵄨 ∫ 󵄨󵄨󵄨v(eiθ )󵄨󵄨󵄨 dθ ≤ Cp ∫ (u∗ (θ)) dθ,

0 < p < 2,

(2.28)

0

0

where u + iv is a polynomial. Let Vλ = {θ ∈ [0, 2π]: u∗ (θ) > λ}, Eλ = {θ ∈ [0, 2π]: u∗ (θ) ≤ λ}, and m(λ) = |Vλ |. Assume that we have proved ∫ v2 dθ ≤ ∫ u2 dθ + 2λ2 m(λ), Eλ

λ > 0.

(2.29)

Eλ

Multiplying this by qλ−q−1 , q = 2 − p > 0, then integrating from λ = 0 to ∞, and finally using Fubini’s theorem, we get 2π

2

∗ −q

∫ v (u )

2π

2

∗ −q

dθ ≤ ∫ u (u )

0

0

2π

2q 2−q dθ + ∫ (u∗ ) dθ. 2−q 0

Hence 2π

p−2

∫ v2 (u∗ )

2π

p

dθ ≤ Cp ∫ (u∗ ) dθ,

0

p < 2,

(2.30)

0 2π

where Cp = 1 + 2(2 − p)/p. To obtain (2.28), assume that ∫0 (u∗ )p dθ = 1. Then by Jensen’s inequality we have 2π

p

2/p

{ ∫ |v|p (u∗ ) (u∗ ) dθ} −p

0

2π

2π

0

p−2

p

(u∗ ) dθ

0

= ∫ |v|2 (u∗ ) and now (2.28) follows from (2.30).

−p 2/p

≤ ∫ {|v|p (u∗ ) }

dθ,

2.8 The theorem of Burkholder, Gundy, and Silverstein

| 81

Proof of (2.29). Let Hλ = ⋃{eiθ U: θ ∈ Eλ } and Γλ = (𝜕Hλ ) ∖ Eλ . Let L denote the arc-length of the arc {ζ ∈ 𝕋: Re ζ > 1/√2}. Assume first that every subarc of Vλ has length ≤ L.

(∗)

Then Γλ contains no point of the circle S = {z : |z| = 1/√2}. By Cauchy’s integral formula we have 1 f (z)2 dz = f (0)2 = 0, ∫ 2πi z 𝜕Hλ

whence 2

2

− ∫ f (eiθ ) dθ = ∫ f (reiθ ) (dθ + idr/r). Eλ

Γλ

Taking the real parts, we get ∫(v2 − u2 ) dθ = ∫(u2 − v2 ) dθ − ∫ 2uv dr/r. Γλ

Γλ

Eλ

Since |dr/rdθ| = 1 on Γλ and 2|uv| ≤ u2 + v2 , we conclude that ∫ v2 dθ ≤ ∫ u2 dθ + ∫ 2u2 dθ. Γλ

Eλ

Eλ

On the other hand, from the definitions of u and Eλ it follows that |u| ≤ λ along Γλ . Hence ∗

∫ u2 dθ ≤ ∫ λ2 dθ = ∫ λ2 dθ = λ2 m(λ). Γλ

Γλ

Vλ

This completes the proof of the theorem in case (∗). If some subarc J of Vλ has the length > L, then we can extract a few points from J and so divide it to small subarcs and then apply case (∗). Remark 2.37. Theorem 2.27 states that if u is a real-valued harmonic function, then ‖u‖̃ hp ≤ C‖u‖H+p , where the space H+p is defined by the requirement ‖u‖H+p = ‖u‖p+ = ‖Mrad u‖Lp (𝕋) < ∞.

(2.31)

In this form the theorem holds for complex-valued functions. Further, we have ‖R+ u‖p ≤ C‖u‖p+ ,

0 < p < ∞,

where R+ u is the Riesz projection of u. If f = h + g,̄ where h and g are analytic and g(0) = 0, then an equivalent norm on H+p can be given by 1/2 󵄩 1/2 󵄩 sup Mp (r, (|h|2 + |g|2 ) ) = 󵄩󵄩󵄩(|h∗ |2 + |g∗ |2 ) 󵄩󵄩󵄩Lp (𝕋) .

0 0. The space Lp,⋆ consisting of all g for 0 is defined to be the subspace of L which 1/p

lim λ(μ(g, λ))

λ→∞

= 0.

(2.33)

The set 𝒯 of all trigonometric polynomials is dense in Lp,⋆ 0 . Indeed, 𝒯 is dense in C(𝕋), ∞ p,⋆ the set C(𝕋) is dense in L (𝕋) in the topology of L , and the set L∞ (𝕋) is dense in Lp,⋆ 0 . (This explanation is due to Aleksandrov [24].) Let p > 1. We identify Lp,⋆ with the space of harmonic functions u ∈ hp0 , 1 < p0 < p, such that 1/p

sup λ(μ(u∗ , λ)) λ>0

< ∞.

p0 Also, we treat Lp,⋆ 0 as the space of functions u ∈ h such that (2.33) is satisfied with g = u∗ . It is clear from our consideration that

(Lp,⋆ )P = Lp,⋆ 0 .

Further notes and results | 85

Since Lp,⋆ is R-admissible, by Theorem 2.33, we can apply Lemma 1.21 to conclude that u ∈ Lp,⋆ 0 if and only if lim ‖u∗ − ur ‖X = 0,

r→1−

(2.34)

where X = Lp,⋆ . This proves the case p > 1 of the following theorem. p/2 . Theorem 2.35. If p > 0, then (𝕏p1 )P = Lp,⋆ 0 ∩H

Proof. Consider, for example, the case p = 1. The space 𝕏11 is R-admissible, so we can apply Lemma 1.21. To this end, we first note that if f ∈ 𝕏11 and B is the corresponding Blaschke product, then f = (B−1)f /2B+(B+1)f /2B, and therefore f can be decomposed as f = g 2 + h2 , where g, h ∈ 𝕏21 . Thus in proving the inclusion 1/2 L1,⋆ ⊂ (𝕏11 )P , 0 ∩H

(2.35)

we can assume that f = g 2 for g ∈ 𝕏21 . Then, with X = 𝕏11 , Y = 𝕏21 , and δ > 0, we have ‖f∗ − fr ‖X = sup λμ(g∗2 − gr2 , λ) λ>0

= sup λμ((g∗ − gr )(g∗ + gr ), λ) λ

≤ sup λμ((g∗ − gr ), √λ/δ) + sup λμ((g∗ + gr ), √λδ) λ>0

= δ‖g∗ − ≤ δ‖g∗ −

gr ‖2Y gr ‖2Y

+ (1/δ)‖g∗ +

+ (C/δ)‖g‖2Y .

λ>0 gr ‖2Y

Since ‖g∗ − gr ‖Y → 0 as r → 1, we get lim sup ‖f∗ − fr ‖X ≤ (C/δ)‖g‖2Y . r→1

Letting δ tend to ∞, we obtain limr→1 ‖f∗ − fr ‖X = 0. This proves the inclusion (2.35). The (easier) proof of the reverse inclusion is left to the reader. p/2 Exercise 2.43. A function f ∈ H(𝔻) belongs to Lp,⋆ if and only if 0 ∩H

lim λμ(M∗ f , λ)1/p = 0.

λ→∞

Further notes and results In the case p = ∞, Corollary 2.8 is known as Hadamard’s three-circle theorem. The case p < ∞ was studied by Hardy [228] in the first paper from “the theory of Hardy spaces”. Carleman [111] proved Theorem 2.14 in the case N = 2 under the additional hypothesis that f is in H(𝔻) and has a finite number of zeros, because at that time, Blaschke’s

86 | 2 Subharmonic functions and Hardy spaces products had not been invented yet. Observe, however, that they can be avoided at least in proving inequality (2.11). The author does not know who was the first to prove (2.11) (N = 2) for an arbitrary f ∈ H p . Theorem 2.14 for N > 2 was proved by Burbea [105]. Subsection 2.7.1 is strongly influenced by the famous Shapiro’s paper [529]. 2.1. The paper [61] by Bayart et al.is full of interesting results. One of them reads as follows. Theorem. Let α0 =

1 + √17 = 1.280776 . . . 4

Let α ≥ α0 and 0 < p < ∞. For every f ∈ Apα (𝔻), ‖f ‖Ap(α+1)/α (𝔻) ≤ ‖f ‖Apα (𝔻) . α+1

Moreover, if α > α0 , we have equality if and only if there exist constants C ∈ ℂ and ξ ∈ 𝔻 such that f (w) =

C . (1 − ξ ̄ w)2α/p

In that paper the (Bergman) space Apα (𝔻) is defined by ‖f ‖Apα (𝔻) = (

α − 1 󵄨󵄨 α−2 󵄨p ∫󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z)) π

1/p

< ∞.

𝔻

Observe that this theorem does not cover the case α → 1+ , in which it reduces to the classical isoperimetric inequality. 2.2 (Harmonic Schwarz lemma. II). If u ∈ h(𝔻) is real valued, |u| ≤ 1, and u(0) = 0, then u is subordinate to the function U(z) =

2 1+z arg . π 1−z

Hence |u(z)| ≤ π4 arctan |z| (Heinz [252]), which also holds for complex-valued functions and moreover for a harmonic function with values in a (real) Banach space X. Namely, if ‖u(z)‖ ≤ 1 for z ∈ 𝔻 and u(0) = 0, then |Φ(u(z))| ≤ 1 for all Φ ∈ B(X 󸀠 ), where B(X 󸀠 ) is the unit ball of the dual space X 󸀠 . Since Φ ∘ u is harmonic real valued and Φ(u(0)) = 0, we have |Φ(u(z))| ≤ (4/π) arctan |z|, which gives ‖u(z)‖ ≤ (4/π) arctan |z|. From this we can deduce that (A) if ‖u‖p ≤ 1 with p ≥ 1 and u(0) = 0, then Mp (r, u) ≤ (4/π) arctan r. The analogous statement for analytic functions reads:

(B) If f ∈ H p , p > 0, and f (0) = 0, then Mp (r, f ) ≤ r‖f ‖p .

Further notes and results | 87

2.3 (Hilbert matrix operator). The Hilbert matrix operator of a function f ∈ H(𝔻) is defined by 1

ℋf (z) = ∫ 0

∞ ∞ f (r) f ̂(j) dr = ∑ z n ∑ , 1 − rz j+n+1 n=0 j=0

where the integral (or the series) is somehow defined. It is easy to see that the adjoint 󸀠 of the restriction R : H p 󳨃→ Lp (0, 1), 1 ≤ p < ∞, is equal to R⋆ : Lp (0, 1) 󳨃→ (H p )󸀠 , where 1

R⋆ g(z) = ∫ 0

g(r) dr. 1 − rz

From this and from (2.13) we find that R maps Lq (0, 1) into H q for 1 < q < ∞ and that ℋ maps H q into H q ; the latter is a result from [146]. Dostanić et al. [157] proved that ⋆

‖ℋf ‖q ≤ Cq ‖f ‖q ,

1 < q < ∞,

(†)

with the best constant Cq = π/sin(π/q). In proving the inequality, they used the Hollenbeck–Verbitsky result (see Remark 1.37). Here we note that the latter and the 󸀠 Fejér–Riesz inequality first give ‖R⋆ g‖q ≤ π 1/q / sin(π/q)‖g‖Lq (0,1) , and then (†) follows from the Fejér–Riesz inequality. However, the proof that the constant Cq is optimal is complicated. The operator ℋ does not map H 1 into H 1 . Cima [120] proved that it maps H 1 into the space of Cauchy transforms; see Further notes 1.1. Moreover, ℋ is injective and maps H 1 to the space of Cauchy transforms of measures on the circle that have no point masses. In [339], it was proved that ℋ maps the space of H 1 -functions such that π

π 󵄨 󵄨 dθ < ∞ ∫ 󵄨󵄨󵄨f (eiθ )󵄨󵄨󵄨 log |θ|

−π

into H 1 . 2.4 (Hankel matrix operator). In [207] the reader will find a lot of information and valuable results concerning the Hankel matrix operator. It is defined by ∞

∞

ℋμ f (z) = ∑ ( ∑ μn+k f ̂(k))z

n

n=0 k=0

whenever the right-hand side makes sense and defines an analytic function in 𝔻. In [207] the numbers μj are defined by μj = ∫[0,1) t j dμ(t), where μ is a positive finite Borel measure on [0, 1). If μ is the Lebesgue measure, then obviously ℋμ coincides with the Hilbert matrix operator. There is a monograph (almost 800 pp.) of Peller [468] devoted to Hankel operators and their applications.

88 | 2 Subharmonic functions and Hardy spaces 2.5 (Rogosinski’s theorem [167, Sec. 6.2]). Let f (z) = F(ω(z)), where F is analytic in k ̂ 𝔻, and ω is maps 𝔻 into 𝔻, with ω(0) = 0. Let Fn (z) = ∑nk=0 F(k)z and fn (z) = n k n+1 ∑k=0 f ̂(k)z . Then Fn (ω(z)) = fn (z) + 𝒪(z ). Therefore by the subordination principle and Parseval’s formula we have: If f is subordinate to F ∈ H(𝔻), then n

n

k=0

k=0

󵄨2 󵄨 󵄨 ̂ 󵄨󵄨2 2k ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 r 2k ≤ ∑ 󵄨󵄨󵄨F(k) 󵄨󵄨 r ,

0 < r < 1, n ≥ 0.

2.6 ([167]). If U = |f |p , p ∈ ℝ+ , then strict equality holds for 0 < r < 1 in (2.21) unless f is constant or ω(z) = αz, |α| = 1. 2.7. Let R > 0 and ρ > 0. Let U be a nonnegative function subharmonic in a neighborhood of DR = {z : |z| ≤ R}, and let u = U ∘ ω, where ω : Dρ 󳨃→ DR is a function analytic in a neighborhood of Dρ . Then 2π

2π

R + |ω(0)| − U(Reiθ ) dθ. − u(ρe ) dθ ≤ ∫ ∫ R − |ω(0)| iθ

0

0

2.8 (The Hardy–Stein identity. II). It is possible to prove the Hardy–Stein identity (Theorem 2.19) without appealing to the existence of the Riesz measure. If p ≥ 2, then the function |f |p is of class C 2 , so we can apply Green’s formula to |f |p . In the case p < 2, we apply this formula to the functions gε = (|f |2 + ε)p/2 , ε > 0; using the formula Δ(uα ) = α(α − 1)uα−2 |∇u|2 + αuα−1 Δu, where u > 0 is of class C 2 (𝔻), we get Δgε =

p p p/2−2 󵄨 󵄨󵄨∇(|f |2 )󵄨󵄨󵄨2 + p (|f |2 + ε)p/2−1 Δ(|f |2 ) ( − 1)(|f |2 + ε) 󵄨 󵄨 2 2 2 p/2−2

= p(p − 2)(|f |2 + ε) p/2−2

= p(|f |2 + ε)

p/2−1 󵄨 󸀠 󵄨2 󵄨 󵄨2 󵄨󵄨f 󵄨󵄨 |f |2 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 + 2p(|f |2 + ε) 󵄨 󵄨

󵄨 󵄨2 [(p − 2)|f |2 + 2|f |2 + 2ε]󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 ,

whence p/2−2

Δgε = p(|f |2 + ε)

󵄨 󵄨2 [p|f |2 + 2ε]󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 .

Now proceed in a similar way as in proving Theorem 2.17 to finish the proof. 2.9. Hardy and Littlewood proved the inequality ‖f ‖2p

1

󵄨2 󵄨 ≤ C 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C ∫ Mp2 (r, f 󸀠 )(1 − r) dr, 0

0 0. This serves as a motivation for introducing a class of functions with subharmonic behavior.

3.2 Regularly oscillating functions | 95

3.1.a (Quasinearly subharmonic functions). Let K be a positive constant, and let QNSK (Ω) denote the class of nonnegative locally bounded Borel-measurable functions u on a domain Ω ⊂ ℂ such that u(z) ≤

K ∫ u dA whenever Dr (z) ⊂ Ω. |Dr (z)|

(3.3)

Dr (z)

If u ≥ 0 is subharmonic, then (3.3) holds with K = 1. The functions of the class QNS(Ω) := ⋃K>0 QNSK (Ω) are called quasinearly subharmonic functions. Note that (3.3) implies sup u ≤ K1 ∫ u dA,

Dε (0)

Dδ (0)

where 0 < ε < δ < 1, and K1 depends only on K, ε, and δ. The class QNSK is invariant under translations and dilations, so the proof of Theorem 3.1 gives the following result. Theorem 3.2. Let p ∈ ℝ+ . If u ∈ QNS, then up ∈ QNS. More precisely, if u ∈ QNSK , then up ∈ QNSK1 , where K1 depends only on K and p.

3.2 Regularly oscillating functions In this section we present some results closely related to Theorem 1.11. Unless otherwise stated, we consider complex-valued functions defined on a proper subdomain Ω of ℂ. 3.2.a. We denote by HCK1 (Ω) the class of all locally Lipschitz functions f satisfying 󵄨 󵄨󵄨 −1 󵄨󵄨∇f (z)󵄨󵄨󵄨 ≤ Kr sup |f |, Dr (z)

Dr (z) ⊂ Ω,

(3.4)

where K ≥ 0 is a constant independent of Dr (z) ⊂ Ω, and let HC 1 (Ω) = ⋃ HCK1 (Ω). K≥0

It should be noted that a locally Lipschitz function is differentiable almost everywhere, and therefore ∇f is defined a. e. We define |∇f | everywhere by |f (z) − f (a)| 󵄨 󵄨󵄨 . 󵄨󵄨∇f (a)󵄨󵄨󵄨 = lim sup |z − a| z→a The function |∇f | is Borel measurable; see Further notes 3.9.

(3.5)

96 | 3 Subharmonic behavior and Bergman-type spaces The proof of Theorem 1.11 (p. 17) can be easily modified to obtain the following: Theorem 3.3. If f ∈ HCK1 (Ω), then f ∈ QNSC (Ω), where C depends only on K. Note that (3.4) is implied by 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨∇f (z)󵄨󵄨󵄨 ≤ K 󵄨󵄨󵄨f (z)󵄨󵄨󵄨/δΩ (z),

δΩ (z) = dist(z, 𝜕Ω),

(3.6)

which is a restriction on the growth of f and is therefore stronger than (3.4). For example, the function f (x + iy) = ex is in HC 1 (Ω), where Ω is the right half-plane, but f does not satisfy (3.6). It is a simple but important fact that condition (3.6) is satisfied if f is a positive function harmonic in Ω. This is a consequence of the following fact. Lemma 3.1. If u is a positive harmonic function in 𝔻, then 2u(z) 󵄨 󵄨󵄨 . 󵄨󵄨∇u(z)󵄨󵄨󵄨 ≤ 1 − |z|2 As a consequence of Theorem 3.3, we get the following: Corollary 3.2. Let p ∈ ℝ+ . A function f locally Lipschitz on Ω belongs to HCK1 (Ω) if and only if there is a constant C depending only on K and p such that 󵄨p 󵄨󵄨 −2−p ∫ |f |p dA, 󵄨󵄨∇f (z)󵄨󵄨󵄨 ≤ Cr

Dr (z) ⊂ Ω.

Dr (z)

3.2.b (The classes OCK1 (Ω) and RO). The class OCK1 (Ω) is the subclass of HC 1 (Ω) consisting of those f for which 󵄨 󵄨󵄨 −1 󵄨󵄨∇f (z)󵄨󵄨󵄨 ≤ Kr Of (z, r),

Dr (z) ⊂ Ω,

where Of (z, r) is the oscillation of f on Dr (z), 󵄨 󵄨 Of (z, r) = sup{󵄨󵄨󵄨f (w) − f (z)󵄨󵄨󵄨 : w ∈ Dr (z)}. 1 It is easy to see that f ∈ OCK1 implies f − c ∈ HC2K for every constant c and that if 1 1 f − c ∈ HCK for every c, then f ∈ OCK . We put RO = ⋃K≥0 OCK1 (Ω). The functions from RO are called regularly oscillating functions.

Theorem 3.4. If f ∈ OCK1 (Ω), then both |f | and |∇f | belong to QNSC (Ω), where C depends only on K. Let Op f (z, r) = {

1 󵄨p 󵄨 ∫ 󵄨󵄨󵄨f (w) − f (z)󵄨󵄨󵄨 dA(w)} |Dr (z)| Dr (z)

the Lp -oscillation of f over Dr (z).

1/p

,

3.2 Regularly oscillating functions | 97

Corollary 3.3. Let p ∈ ℝ+ . If a function f belongs to OCK1 (Ω), then 󵄨 󵄨󵄨 −1 󵄨󵄨∇f (z)󵄨󵄨󵄨 ≤ Cr Op f (z, r),

Dr (z) ⊂ Ω,

(+)

where C depends only on K and p. Conversely, if (+) holds, then f belongs to OCK1 (Ω). This is deduced from Corollary by 3.2 considering the functions f − const. Proof of Theorem 3.4. The inclusion OCK1 (Ω) ⊂ QNSC (Ω) follows from Theorem 3.3 and 1 the inclusion OCK1 ⊂ HC2K . It remains to prove that f ∈ OC 1 (Ω) implies |∇f | ∈ QNS(Ω). 1 Let f ∈ OCK (Ω). By Theorem 3.2 it suffices to prove that, for some q, the function |∇f |q belongs to QNS(Ω). This can be reduced to proving that 󵄨q 󵄨󵄨 q 󵄨󵄨∇f (0)󵄨󵄨󵄨 ≤ C ∫ |∇f | dA. 𝔻

By Corollary 3.3 we have 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨∇f (0)󵄨󵄨󵄨 ≤ K ∫󵄨󵄨󵄨f (z) − f (0)󵄨󵄨󵄨 dA(z). 𝔻

On the other hand, for a fixed z ∈ 𝔻, the function ϕ(r) = f (rz) is absolutely continuous, and hence 1

1

0

0

󵄨 󵄨󵄨 󵄨 󵄨 󸀠 󵄨󵄨f (z) − f (0)󵄨󵄨󵄨 ≤ ∫ ϕ (r) dr ≤ |z| ∫󵄨󵄨󵄨∇f (rz)󵄨󵄨󵄨 dr, whence 1

󵄨 󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨f (z) − f (0)󵄨󵄨󵄨 dA(z) ≤ ∫ dr ∫󵄨󵄨󵄨∇f (rz)󵄨󵄨󵄨|z| dA(z). 0

𝔻

𝔻

Hence, by the change z = w/r and Fubini’s theorem, 1

󵄨 󵄨 󵄨󵄨 󵄨 󵄨 −1 󵄨 −3 󵄨󵄨∇f (0)󵄨󵄨󵄨 ≤ K ∫󵄨󵄨󵄨∇f (w)󵄨󵄨󵄨 dA(w) ∫ r |w| dr ≤ K ∫󵄨󵄨󵄨∇f (w)󵄨󵄨󵄨|w| dA(w). 𝔻

|w|

𝔻

Now the required inequality is proved by Hölder’s inequality with the indices q = 3 and q󸀠 = 3/2 using the fact that the function w 󳨃→ |w|−1 belongs to the space L3/2 (𝔻, dA). Exercise 3.4 ([456]). If f ∈ C 1 (Ω) satisfies the condition 󵄨󵄨 ̄ 󵄨 −1 󵄨󵄨𝜕f (z)󵄨󵄨󵄨 ≤ Kr sup |f | whenever Dr (z) ⊂ Ω, Dr (z)

98 | 3 Subharmonic behavior and Bergman-type spaces where K is a constant, then |f | is QNS. This can be proved by using the formula 2π

f (0) = ∫ − f (eiθ ) dθ − 0

1 𝜕f̄ (z) dA(z). ∫ π z 𝔻

3.2.c (Nearly convex functions). If a function f : Ω 󳨃→ ℝ is convex, then f (z + h) − f (z) f (z + rh/|h|) − f (z) ≤ , |h| r

0 < |h| < r,

and f (z) − f (z + h) ≤ f (z − h) − f (z). This implies somewhat more than that f is RO, namely 󵄨 K 󵄨󵄨 sup (f (w) − f (z)) 󵄨󵄨∇f (z)󵄨󵄨󵄨 ≤ r w∈Dr (z)

(3.7)

(where K = 1). A function f : Ω 󳨃→ ℝ satisfying this inequality for some K > 0 is called nearly convex. The class of all such f is denoted by NCK . We use the notation NC = ⋃K>0 NCK . Clearly, every nearly convex function is regularly oscillating. Lemma 3.5. If a function u: Ω 󳨃→ ℝ is harmonic, then |u| ∈ NC12 (Ω). Indeed, using Theorem 1.5 (|∇u(0)| ≤ 2|u(0)|), we prove that 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ 2(1 − 󵄨󵄨󵄨u(0)󵄨󵄨󵄨) if |u| ≤ 1. When applied to the function w 󳨃→ u(z + rw)/M, where M = supw∈Dr , Dr (z) ⊂ Ω, this inequality becomes 󵄨 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 |∇|u󵄨󵄨󵄨(z)󵄨󵄨󵄨 = 󵄨󵄨󵄨∇u(z)󵄨󵄨󵄨 ≤ sup (󵄨󵄨u(w)󵄨󵄨󵄨 − 󵄨󵄨󵄨u(z)󵄨󵄨󵄨). r w∈Dr (z) 󵄨 Then it is easy to check that |u|s is nearly convex for s ≥ 1. Since |∇|u|s | = s|u|s−1 |∇u|, from Theorem 3.4 we see that |u|s−1 |∇u| ∈ QNS, which can be expressed as 󵄨q 󵄨p 󵄨 󵄨󵄨 p q 󵄨󵄨u(0)󵄨󵄨󵄨 󵄨󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ Cp,q ∫ |u| |∇u| dA (p, q ≥ 0). 𝔻

From this we obtain a generalization of the Fefferman–Stein theorem. Theorem 3.5. If u : Ω 󳨃→ ℝ is a harmonic function, then |u|p |∇u|q (p, q ≥ 0) is QNS. As a further example, we have that |f | ∈ NC12 (Ω) if f is analytic in Ω. Lemma 3.6 (Schwarz modulus lemma). Let, as before, Dε (z) = {w : |w − z| < ε}, 0 < ε ≤ 1 − |z|, and f ∈ H ∞ . Then 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨 2 sup (󵄨󵄨f (w)󵄨󵄨󵄨 − 󵄨󵄨󵄨f (z)󵄨󵄨󵄨) (z ∈ 𝔻). 󵄨󵄨f (z)󵄨󵄨 ≤ ε w∈Dε (z) 󵄨

3.2 Regularly oscillating functions | 99

Proof. Let Mz = sup{|f (w)|: w ∈ Dε (z)}. If z = 0 and M0 = 1, then Schwarz’s lemma gives 󵄨 󵄨 󵄨2 󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨f (0)󵄨󵄨 ≤ 1 − 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 ≤ 2(1 − 󵄨󵄨󵄨f (0)󵄨󵄨󵄨), which is the required inequality in a particular case. In the general case, we apply this case to the function F(ζ ) = f (z + εζ )/Mz , ζ ∈ 𝔻. The aforementioned fact that |u|s (u harmonic, s ≥ 1) is nearly convex is a consequence of Lemma 3.5 and the following statement. Proposition 3.7. Let u be a nonnegative nearly convex function on Ω, and let ϕ be a positive C 1 -function on (0, ∞) such that ϕ(t)/t is increasing and ϕ(t)/t β is decreasing in t for some β > 1. Then the function v(x) = ϕ(u(x)) is nearly convex. Proof. The condition on ϕ means that 1≤

tϕ󸀠 (t) ≤ β, ϕ(t)

t > 0,

which we also denote by ϕ ∈ Δ[1, β]; see Section 4.1.c. As always, it suffices to prove that 󵄨 󵄨󵄨 󵄨󵄨∇v(0)󵄨󵄨󵄨 ≤ C sup(v(z) − v(0)) z∈𝔻

under the hypothesis 𝔻 ⊂ Ω. Assuming that u(0) ≠ 0, we have ϕ(u(0)) 󵄨 󵄨 󵄨 󵄨󵄨 󸀠 (u(z) − u(0)) 󵄨󵄨∇v(0)󵄨󵄨󵄨 = ϕ (u(0))󵄨󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ Cβ u(0) = Cβ(

ϕ(u(0)) u(z) − v(0)) u(0)

for some z ∈ 𝔻. Since u(z) ≥ u(0) and ϕ(t)/t increases with t, we get ϕ(u(0)) ϕ(u(z)) ≤ , u(0) u(z) which, together with the preceding inequality, gives the desired result in the case u(0) ≠ 0. Let u(0) = 0. If supz∈𝔻 (u(z) − u(0)) = 0, then |∇u(0)| = 0, so there is nothing to prove. Otherwise, choose z ∈ 𝔻 such that u(z) > 0. Then

ϕ(t) 󵄨 󵄨 󵄨 󵄨󵄨 󸀠 󸀠 󵄨󵄨∇v(0)󵄨󵄨󵄨 = ϕ (0)󵄨󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ Cϕ (0)u(z) = Cu(z) lim+ t t→0 ϕ(u(z)) ≤ Cu(z) = Cϕ(u(z)), u(z) which completes the proof.

100 | 3 Subharmonic behavior and Bergman-type spaces Example 3.8. Let ϕ(t) = t α logγ (1 + t). Then tϕ󸀠 (t) γt =α+ . ϕ(t) (1 + t) log(1 + t) If γ ≥ 0, then this implies α≤

tϕ󸀠 (t) ≤ α + γ, ϕ(t)

and hence ϕ ∈ Δ[α, α + γ]. Thus ϕ ∈ Δ[1, α + γ] if α ≥ 1 and γ ≥ 0. By Proposition 3.7 and Lemma 3.5 the function v(z) = |u(z)|α logγ (1 + |u(z)|) with α ≥ 1 and γ ≥ 0 is nearly convex if u is real valued and harmonic. Exercise 3.9. There are functions f satisfying (3.7) with f (z) − f (w) instead of f (w) − f (z). It is reasonable to call them nearly concave. As a consequence of Lemma 3.6 and elementary properties of concave functions, if φ : 𝔻 󳨃→ 𝔻 is analytic and ω is an increasing concave function on [0, 1], then ω(1 − |φ|) is nearly concave and therefore regularly oscillating. Moduli of vector-valued functions The facts that the moduli of analytic functions are nearly convex can be extended to functions with values in a Banach space. Proposition 3.10. If f : Ω 󳨃→ X is an analytic function with values in a complex Banach space X, then the function u(z) = ‖f (z)‖X is nearly convex. Proof. Let f be analytic in 𝔻 ⊂ Ω, supz∈𝔻 ‖f (z)‖X ≤ 1. Let Λ be a linear functional on X with ‖Λ‖ ≤ 1. Then the function ϕ(z) = Λ(f (z)) is complex valued and analytic. Since |ϕ(z)| ≤ 1, the following variant of the Schwarz lemma holds: 󵄨󵄨 ϕ(z) − ϕ(a) 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ≤ 󵄨σ (z)󵄨󵄨. 󵄨󵄨 󵄨󵄨 1 − ϕ(a)ϕ(z) 󵄨󵄨󵄨 󵄨 a 󵄨 Assuming that ‖f (z)‖ > ‖f (a)‖, we get 󵄨 |Λ(f (a))| + |σa (z)| 󵄨󵄨 . 󵄨󵄨Λ(f (z))󵄨󵄨󵄨 ≤ 1 + |Λ(f (a))||σa (z)| Hence, taking the supremum over the unit ball of X 󸀠 and using that the function t 󳨃→ c+t , t > 0, c > 0, is increasing, we get 1+ct 󵄩 ‖f (a)‖ + |σa (z)| 󵄩󵄩 . 󵄩󵄩f (z)󵄩󵄩󵄩 ≤ 1 + ‖f (a)‖|σa (z)| This implies 󵄨󵄨 1 − az̄ 󵄨󵄨󵄨󵄨 ‖f (z)‖ − ‖f (a)‖ 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨 ≤ 1. 󵄨󵄨 a − z 󵄨󵄨󵄨󵄨 1 − ‖f (a)‖‖f (z)‖ 󵄨󵄨

101

3.2 Regularly oscillating functions |

Letting z tend to a, we obtain 2 󵄨 1 − ‖f (a)‖ 󵄨󵄨 . 󵄨󵄨∇‖f ‖(z)󵄨󵄨󵄨 ≤ 2 1 − |a|

(3.8)

Now, we proceed as in the case of Lemma 3.6 to finish the proof. Exercise 3.11. Let u be a real-valued function harmonic in 𝔻 such that |u| < 1. Then by Corollary 1.7 we have 1 − |u(0)| 1 + |z| ≤ 1 − |u(z)| 1 − |z|

and

1 − |u(z)| 1 + |z| ≤ . 1 − |u(0)| 1 − |z|

From this we can deduce that the same inequalities hold for harmonic functions with values in a Banach space X. Hence, if u is X-valued and ‖u(z)‖ ≤ 1 in 𝔻, then 󵄩󵄨 2(1 − ‖u(z)‖) 󵄨󵄨 󵄩󵄩 . 󵄨󵄨∇󵄩󵄩u(z)󵄩󵄩󵄩󵄨󵄨󵄨 ≤ 1 − |z|2 Thus the modulus of a vector-valued harmonic function is nearly convex. As a particular case, we again have that the modulus of a vector-valued analytic function is nearly convex. Polyharmonic functions A C ∞ -function f is polyharmonic of order k, where k is a positive integer, if Δk f ≡ 0. Here Δk denotes the kth power of the Laplacian. For the theory of polyharmonic functions, we refer to Aronszajn, Creese, and Lipkin [43]. Theorem 3.6. If f is a complex-valued function polyharmonic in Ω, then |f |, |∇f |, and the modulus of a partial derivative of any order of f are QNS. For a complex-valued function f = u + iv, we define |∇f | by (3.5) or alternatively by |∇f | = √|∇u|2 + |∇v|2 . Proof. A polyharmonic function f of order k on 𝔻 can be represented in the form k−1

f (z) = ∑ fm (z)|z|2m , m=0

where fm are harmonic functions (Almansi’s representation theorem [43]). This implies that 2π

k−1

0

m=0

− f (reiθ ) dθ = ∑ fm (0)r 2k , ∫

102 | 3 Subharmonic behavior and Bergman-type spaces and hence, in view of the inequality 1 󵄨 k−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |a0 | ≤ Ck ∫󵄨󵄨󵄨 ∑ am r 2m 󵄨󵄨󵄨r dr, 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨m=0

{am }k−1 m=0 ⊂ ℂ,

we have |f (0)| ≤ Ck ∫𝔻 |f | dA, which, via translations and dilations, implies that |f | ∈ QNS(Ω). The rest follows from this and from the fact that the first-order partial derivatives are polyharmonic.

3.3 Mixed-norm spaces. Definition and basic properties 3.3.a (The class Lq−1 ). For a measurable function F on (0, 1) and 0 < q ≤ ∞, we write F ∈ Lq−1 = Lq−1 (0, 1) if 1

‖F‖−1,q

󵄨q r dr 󵄨 ) := (∫󵄨󵄨󵄨F(r)󵄨󵄨󵄨 1 − r2

1/q

< ∞.

(3.9)

0

In the case q = ∞, this means that F ∈ L∞ (0, 1), and if q = ⬦, then we require that F ∈ L∞ (0, 1) and limr→1− |F(r)| = 0. The last condition is not preserved by passing to a function that is equal F a. e., but this is irrelevant for us because we are not interested in linear topological properties of Lq−1 . p,q p,q 3.3.b (The spaces hp,q α and Hα ). Let Lα (0 < p, q ≤ ∞, α ∈ ℝ) denote the space of all Borel-measurable functions u on 𝔻 such that the function α

F(r) = (1 − r 2 ) Mp (r, u) belongs to Lq−1 (0, 1).

(3.10)

The quasinorm of f ∈ Lp,q α is by definition equal to the quasinorm of F in X. It is easy to check that X is s-normed, where s = min{p, q, 1}. In particular, we have ‖f ‖Lp,∞ = ess sup(1 − r 2 )Mp (r, f ). α 0 0. p,⬦ p,∞ Proof. To prove that (hp,∞ and that there is a α )P = hα , assume first that f ∈ hα p,∞ polynomial P such that ‖f − P‖X < ε, where X = hα and ε > 0. This implies that

(1 − r)α Mp (r, f ) ≤ Cp (1 − r)α Mp (r, f − P) + Cp (1 − r)α Mp (r, P) ≤ Cp ε + Cp (1 − r)α ‖P‖∞ .

Hence lim supr→1− (1 − r)α Mp (r, f ) ≤ Cp ε. In the other direction, assuming that f ∈ hp,⬦ α and ε > 0, we prove that ‖f −fρ ‖X → 0 as ρ → 1; see Lemma 1.21. Choose 0 < δ < 1 such that (1 − r)α Mp (r, f ) < ε for δ < r < 1. Then ‖f − fρ ‖X ≤ sup (1 − r)α Mp (r, f − fρ ) + Cp sup (1 − r)α Mp (r, f ) 0 0, α > 0.

The analogous inclusion holds for hp,q α , p ≥ 1, because in this case, Mp (r, u), u ∈ h(𝔻), increases with r. We will prove that this inclusion remains true in the case of hp,q α for all p, q, q1 , α. Theorem 3.11 (Increasing inclusion theorem). If q < q1 ≤ ∞ and α ∈ ℝ, then hp,q α ⊂ p,q p,⬦ hα 1 . In particular, hp,q ⊂ h for q < ⬦. If α > 0, then the inclusions are proper. α α p,q1

Proof. To prove that the inclusion hp,q α ⊂ hα the functions

are proper for q < q1 and α > 0, consider

f (z) = (1 − z)−β−1 (log

4 ) 1−z

−γ

(see [355, p. 93]). Let q < q1 , and let β = α+1/p−1 and γ = 1/q. An elementary but tedious p,q computation shows that f ∈ Hα 1 \ Hαp,q . We omit the details because the usage of lacunary series gives the desired result immediately; this will be discussed later on. This rest of the theorem is a consequence of the following proposition. Proposition 3.26. Let 0 < q < q1 ≤ ∞ and α ∈ ℝ. If an upper semicontinuous function p,q1 p,q p,q u belongs to QNSK ∩ Lp,q α , then u ∈ Lα , and we have ‖u‖Lα 1 ≤ C‖u‖Lα , where C is independent of u. Proof. Let u ∈ QNSK and rn = 1 − 2−n . The desired result follows from the equivalence ‖u‖

Lp,q α

∞

≍ (∑ 2 n=0

−nαq

Mpq (rn , u× ))

1/q

=: Q,

(3.24)

114 | 3 Subharmonic behavior and Bergman-type spaces where Q for q = ⬦, ∞ is interpreted in a similar way as in the case of Lp,q α . Let p < ⬦ and q < ⬦. Then ‖u‖qLp,q α

∞

n(1−qα)

rn+1

∫ Mpq (r, u)r dr =: S.

≍ ∑2 n=0

rn

Since obviously Mp (r, u) ≤ Mp (rn , u× )(rn+1 − rn ), we have S ≤ CQ. To prove the reverse inequality, we start from the inequality u× (rn ζ )p ≤

C ∫ u(w)p dA(w), |Bn | Bn

where Bn = {w: |w − rn ζ | < rn+2 − rn = (3/4)2−n }. Since |w − ζ | ≤ |w − rn ζ | + |rn ζ − ζ | ≤ (7/4)2−n and |w| ≤ rn+2 for w ∈ Bn , we have u× (rn ζ )p ≤

C ∫ u(w)p dA(w) |Bn | Bn

≤ C2n ∫ u(w)p Bn

≤ C2n

1 − |w|2 dA(w) |w − ζ |2 u(w)p

∫ rn−2 ≤|w|≤rn+2

1 − |w|2 dA(w), |w − ζ |2

n ≥ 2.

Integrating this inequality over ζ ∈ 𝕋, we get Mpp (rn , u× ) ≤ C2n

u(w)p dA(w).

∫ rn−2 ≤|w|≤rn+2

Hence, by Jensen’s and Minkowski’s inequalities (see Remark 3.19) Mpq (rn , u× ) ≤ C2n

∫

Mpq (|w|, u) dA(w),

rn−2 ≤|w|≤rn+2

as desired. Here we can take p = ∞. Now the inequality Q ≤ CS for q < ⬦ is proved easily. In the case q = ⬦, we have to prove the equivalence Mp (r, u) = o(1 − r)−α ⇔ Mp (rn , u× ) = o(2nα )

(r → 1− , n → ∞),

or, equivalently, sup Mp (r, u) = o(2nα ) ⇔ Mp (rn , u× ) = o(2nα ).

rn ≤r≤rn+1

3.4 Embedding theorems | 115

Since suprn ≤r≤rn+1 Mp (r, u) ≤ Mp (rn , u× ), we see that “⇐” holds. The converse holds because of the inequality n

rn+2

Mp (rn , u ) ≤ C2 ∫ Mp (r, u)r dr. ×

rn−2

The case q = ∞ is treated in the same way. The proof is now complete. Theorem 3.12 (Mixed embedding theorem). Let 0 < p < s ≤ ∞, 0 < q ≤ ∞, and α ∈ ℝ. s,q Then hp,q α ⊂ hβ , where β = α + 1/p − 1/s. If α > 0, then the inclusion is strict. Corollary 3.27. Let p < 1. If either α ≤ 1 − 1/p and q < ∞ or α < 1 − 1/p and q = ∞, then hαp,q = {0}. = {0} if and the fact that h1,q Proof. This follows from the inclusion hp,q ⊂ h1,q α α+1/p−1 β either q < ∞ and β ≤ 0 or q = ∞ and β < 0. Proof of Theorem 3.12. Let en (z) = z n , n ≥ 0. The strictness of the inclusion can be obtained from the relations ‖en ‖Hαp,q ≍ n−α (α > 0) and ‖en ‖H s,q ≍ n−β by using the β

s,q closed graph theorem. The embedding hp,q α ⊂ hβ is a consequence of the following proposition.

Proposition 3.28. Let 0 < p < s ≤ ∞, 0 < q ≤ ∞, and α ∈ ℝ. If an upper semicontinus,q ous function u belongs to QNSK ∩ Lp,q α , then it belongs to Lβ , β = α + 1/p − 1/s, and we have ‖u‖Ls,q ≤ C‖u‖Lp,q , where C is independent of u. α β

Proof. Let u ∈ QNS. It is easy to prove that M∞ (r, u) ≤ C(1 − r)−1/p Bp (r, u),

r > 1/2,

where Bp (r, u) = sup2r−1≤ρ≤ 1+r Mp (ρ, u). Using this and the inequality 2

󵄨p 󵄨s−p 󵄨 󵄨 Mss (r, u) = ∫ − 󵄨󵄨󵄨u(rζ )󵄨󵄨󵄨 󵄨󵄨󵄨u(rζ )󵄨󵄨󵄨 |dζ | ≤ M∞ (r, u)s−p Mpp (r, u), 𝕋

we obtain 1

Ms (r, u) ≤ C(1 − r) s

− p1

Bp (r, u),

and hence 1

Ms (r, u) ≤ C(1 − r) s

− p1

(Mp (2r − 1, u× ) + Mp (r, u× )),

that is, 1

Ms (r, u)(1 − r) p

− s1

≤ CMp (2r − 1, u× ) + CMp (r, u× ).

Now the inclusion is obtained by Proposition 3.22.

(3.25)

116 | 3 Subharmonic behavior and Bergman-type spaces Most properties of QNS-functions are independent of the hypothesis that the function is upper semicontinuous. For instance, inequality (3.25) holds for any QNSfunction. Using the “increasing” property of the integral means of subharmonic functions, from (3.25) we obtain the following lemma. Lemma 3.29. If u is a nonnegative subharmonic function on 𝔻 and ∞ ≥ s > p ≥ 1, then 1

Ms (r, u) ≤ C(1 − r) s

− p1

Mp (

1+r , u). 2

If u is log-subharmonic, then this inequality holds for all p > 0.

3.5 Fractional integration For a positive real number s, define the operator of fractional integration of order s by 1

(𝒥s u)(z) =

1 s−1 ∫(log(1/t)) u(tz) dt, Γ(s)

z ∈ 𝔻,

0

whenever the integral is somehow defined. A simple calculation shows that if u is a harmonic function, then iθ

∞

|n| inθ

̂ 𝒥s u(re ) = ∑ (|n| + 1) u(n)r e −s

n=−∞

.

The formula 𝒥s 𝒥η f = 𝒥s+η f holds for nonnegative Borel functions f . The proof can be reduced to radial functions; then it suffices to verify the formula for f (r) = r k , k ≥ 0. Proposition 3.30 (Fractional integration proposition). Let u be an upper semicontinup,q ous function of class QNSK , and let α > 0, s > 0. Then u ∈ Lp,q α+s implies 𝒥s u ∈ Lα . , where C is independent of u. ≤ C‖u‖Lp,q Moreover, we have ‖𝒥s u‖Lp,q α α+s In proving this, we use the maximal function 󵄨 󵄨 u+ (rζ ) = Mrad u(rζ ) = sup 󵄨󵄨󵄨u(tζ )󵄨󵄨󵄨, 0≤t≤r

ζ ∈ 𝕋.

(3.26)

Proposition 3.31. Let α > 0. If an upper semicontinuous function u belongs to QNSK ∩ + p,q + p,q p,q Lp,q α , then u ∈ Lα and ‖u ‖Lα ≤ C‖u‖Lα , where C is independent of u. Proof. Let q < ⬦. We have, by Lemma 3.20, 1

∞

∫ Mpq (r, u+ )(1 − r)qα−1 dr ≤ ∑ 2−nqα Mpq (rn+1 , u+ ) 0

n=0

3.5 Fractional integration

| 117

∞

≤ CMpq (r1 , u+ ) + C ∑ 2−nqα Mpq (rn , u× ) n=0 ∞

󵄨q

󵄨 ≤ C sup 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 + C ∑ 2−nqα Mpq (rn , u× ). n=0

|z| 0. Remark 3.33. The requirement of the upper semicontinuity of u is needed only to guarantee that u× and u+ are measurable. On the other hand, the hypothesis that u is Borel measurable implies that the functions u(×) and u(⋎) , defined by replacing “sup” with “ess sup”, are measurable, so using these functions instead of u× and u+ proves the validity of Propositions 3.26, 3.28, and 3.30 for all u ∈ QNS. Then, applying this generalized Proposition 3.30 and imitating the proof of Proposition 3.16, we obtain the following:

3.5 Fractional integration

| 119

Proposition 3.34. Let u ∈ OCK1 , p > 0, q > 0, and α > 0. Then 󵄩 󵄨 󵄩 󵄨 ‖u‖Lp,q ≍ 󵄨󵄨󵄨u(0)󵄨󵄨󵄨 + 󵄩󵄩󵄩|∇u|󵄩󵄩󵄩Lp,q , α α+1 where the equivalence constants depend only on p, q, α, and K. Corollary 3.35. Let u and v be regularly oscillating functions, and let α > 0. If |∇u| ≤ |∇v| p,q and v ∈ Lp,q α , then u ∈ Lα . This corollary can be viewed as a generalization of Theorem 3.9 because if f = u+iv is analytic, then |∇u| = |∇v| = |f 󸀠 |. On the other hand, Proposition 3.34 can be applied to the case where u is a polyharmonic function. Example 3.36. Let f (z) = |u(z)| log (1 + |u(z)|), where u is real valued and harmonic in 𝔻. Then f is RO by Example 3.8. Since |∇f | = |∇u| log(1 + |u|) + |∇u|

|u| ≍ |∇u| log(1 + |u|), 1 + |u|

by Proposition 3.34 we have that if u(0) = 0, then p p 󵄨 p 󵄨 󵄨p 󵄨 ∫ |u|p [log(1 + |u|)] dA ≍ ∫󵄨󵄨󵄨∇u(z)󵄨󵄨󵄨 [log(1 + 󵄨󵄨󵄨u(z)󵄨󵄨󵄨)] (1 − |z|) dA(z). 𝔻

𝔻

Tangential derivatives For a function u ∈ h(𝔻), let Du denote its tangential derivative, Du(reiθ ) = D1 u(reiθ ) =

∞ 𝜕u iθ |n| inθ ̂ (re ) = ∑ in u(n)r e . 𝜕θ n=−∞

Hence, if N is a positive integer, then ∞

|n| inθ ̂ DN u(reiθ ) = ∑ (in)N u(n)r e . n=−∞

(3.27)

Using the preceding results, we can prove an analogue of Theorem 3.13. Theorem 3.14 (Tangential differentiation theorem). p,q N (i) If α ∈ ℝ and u ∈ hp,q α , then D u ∈ hα+N . p,q p,q (ii) If α > 0, then u ∈ hα if and only if DN u ∈ hp,q α+N . Moreover, we have ‖u − u(0)‖Lα ≍ ‖DN u‖Lp,q . α+N

In fact, using the H–L projection theorem, we can reduce the proof to the case of by D1 f ; applying this new equivalence N times, we obtain the desired result. The theorem can be proved again by using the decomposition method; see Corollary 6.8. Hαp,q . Then it is easy to see that equivalence (3.20) remains true if we replace f 󸀠

120 | 3 Subharmonic behavior and Bergman-type spaces Another important theorem was originated by Hardy and Littlewood. Here ΔNt = (Δt ) (t ∈ ℝ) denotes the symmetric difference operator of order N, and Δt g(ζ ) = g(ζeit ) − g(ζ ) for g ∈ Lp (𝕋), ζ ∈ 𝕋. N

Theorem 3.15. Let 0 < α < N, where N is a positive integer, and let X = Lp,q N−α . If f ∈ H(𝔻) N p and D f ∈ X, then f ∈ H , and 1/q

1

‖ΔNt f ‖p q dt 󵄩󵄩 N 󵄩󵄩 ) ) 󵄩󵄩D f 󵄩󵄩X ≍ (∫( tα t

=: K(f ).

0

Conversely, if f ∈ H p and K(f ) < ∞, then DN f ∈ X. If p ≥ 1, then the analogous statements for f ∈ h(𝔻) hold. See Hardy–Littlewood [231, Theorem 3], [232, Theorem 23], and [237, Theorem 48] for the case N = 1. The case p < 1 for analytic functions was treated by Gwilliam [224]. The case where N = 2 and p = q = ∞ is of special interest and was treated by Zygmund [636]. A proof in the general case can be found in Oswald [412]. We postpone the discussion on this theorem to Chapters 8 and 9, where more general results will be proved.

3.6 Reproducing kernels and atomic decomposition The formula f (z) =

s + 1 f (w)(1 − |w|2 )s dA(w), ∫ ̄ s+2 π (1 − wz)

s > −1,

(3.28)

𝔻

which holds whenever f is analytic and the obvious integrability condition is satisfied, plays an important role in various questions concerning Bergman spaces and especially in proving a decomposition theorem due to Coifman and Rochberg, probably the most important result in theory of Bergman spaces. The function K(z, w) = (s + 1)

(1 − |w|2 )s ̄ s+2 (1 − wz)

is called a reproducing kernel for clear reasons. Formula (3.28) can be proved by using Taylor series, but there exists a more interesting way. Let Ts f (z) =

(1 − |w|2 )s s+1 dA(w), ∫ f (w) ̄ s+2 π (1 − wz) 𝔻

(3.29)

3.6 Reproducing kernels and atomic decomposition

| 121

where f is any analytic function belonging to s 󵄨 󵄨 D(Ts ) = {f ∈ H(𝔻): ∫󵄨󵄨󵄨f (w)󵄨󵄨󵄨(1 − |w|2 ) dA(w) < ∞}. 𝔻

First, it is not hard to verify that Ts (f ) = f (0) if f ∈ D(Ts ). Then write Ts as Ts f (z) = ∫ f (w)Qs (z, w) dτ(w), 𝔻

where Qs (z, w) =

s+2

s + 1 1 − |w|2 ( ) ̄ π 1 − wz

.

Hence, by the invariance of dτ, Ts f (z) = ∫ f (σz (w))Qs (z, σz (w)) dτ(w). 𝔻

Now using the identity Qs (z, σz (w)) = Qs (z, w), we obtain Ts f (z) = ∫ f (σz (w))Qs (z, w) dτ(w) = f (σz (0)) = f (z), 𝔻

which implies (3.28). The operator Ts is defined on measurable functions in an obvious way. Forelli and Rudin [196] proved that Ts maps (projects) Lp (on)to Ap if and only if s > 1/p−1 (reduced to one-variable case), which was extended (with s > 2/p−2) to a class of Lp -type spaces for p < 1 [386], where the given proof was presented; see Further notes 3.13.

3.6.1 The Coifman–Rochberg theorem Theorem 3.16 (on atomic decomposition). Let 0 < p ≤ 1, β > −1, and γ > 0. (a) There exist a sequence {wn } in 𝔻 and a constant C such that every f ∈ Apβ can be represented as ∞

f (z) = ∑ an n=1

(1 − |wn |2 )γ (1 − w̄ n z)γ+(β+2)/p

(3.30)

with ‖{an }‖ℓp ≤ C‖f ‖p,β . (b) Every function f of the form (3.30) with {an } ∈ ℓp belongs to Apβ , and ‖f ‖p,β ≤ C‖{an }‖ℓp .

122 | 3 Subharmonic behavior and Bergman-type spaces A particular case, Theorem 3.17, will be proved after the following digression: Corollary 3.37. The q-Banach envelope of H p , where p < q ≤ 1 is equal to Aqβ with β = q/p − 2. Proof. The space X = H p is embedded into the q-Banach space Y = Aqβ ; see Corollary 5.2. On the other hand, every f ∈ Y can be represented as f = ∑∞ n=1 fn , where fn (z) = an

(1 − |wn |2 )γ (1 − w̄ n z)γ+1/p

and (∑ |an |q )1/q ≤ C‖f ‖Y . Since π

∫ −π

C 1 dθ ≤ , γp iθ γp+1 (1 − |w ̄ |1 − wn e | n |)

we have ∞

∞

n=1

n=1

∑ ‖fn ‖qX ≤ C ∑ |an |q ≤ C‖f ‖qY ,

which concludes the proof. (See Proposition A.6). Theorem 3.17. Let 0 < p ≤ 1 and β > −1. Then there exist a sequence {wn } in 𝔻 and a constant C such that, for every f ∈ Ap , there exists a sequence {an } ⊂ ℓp such that ∞

f (z) = ∑ an n=1

1 − |wn |2 (1 − w̄ n z)2/p+1

and ∞

β β 󵄨p 󵄨 ∑ |an |p (1 − |wn |) ≤ C ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|) dA(z).

n=1

𝔻

If {an } ∈ ℓp , then f ∈ Ap . Proof of Theorem 3.17 We leave an easy proof of the last implication to the reader. Let Kp (w, z) =

1 ̄ 2/p+1 (1 − wz)

(z, w ∈ 𝔻).

Define the measure dμp on 𝔻 by dμp (w) = cp (1 − |w|2 )2/p−1 dA(w), where p 1 2/p−1 = ∫(1 − |w|2 ) dA(w) = . cp 2 𝔻

(3.31)

3.6 Reproducing kernels and atomic decomposition

| 123

As a particular case of (3.28), we have f (z) = ∫ Kp (w, z)f (w) dμp (w). 𝔻

In addition, it is not difficult to verify that, for every z ∈ 𝔻, the function w 󳨃→ f (w)Kp (w, z) belongs to L1 (𝔻, dμp ). Lemma 3.38. Let f ∈ Ap , 0 < p ≤ 1, and let 󵄨 󵄨󵄨 󵄨 g(z) = ∫󵄨󵄨󵄨Kp (w, z)󵄨󵄨󵄨󵄨󵄨󵄨f (w)󵄨󵄨󵄨 dμp (w) (z ∈ 𝔻). 𝔻

Then g ∈ Lp (𝔻, dA), and ‖g‖Lp ≤ Cp ‖f ‖Ap . Proof. Let ℰ be a partition of the unit disc into disjoint sets E with the following properties: 1 diam(E) ≤ ≤ C (w ∈ E), C 1 − |w| |E| 1 ≤ C (w ∈ E), ≤ C (1 − |w|)2 1 − |ζ | 1 ≤ ≤ C (ζ , w ∈ E), C 1 − |w| where C is an absolute constant. Such a family consists of the sets Ej,k = {z:

1 2πj 2π(j + 1) 1 < 1 − |z| ≤ k , k+2 ≤ arg z < }, 2k+1 2 2 2k+2

where k = 0, 1, 2, . . . and 0 ≤ j < 2k+2 . Let ℰ = {En : n ≥ 1}. Then ∞

g(z) ≤ C ∑ (1 − |wn |)

2/p−1

n=1

󵄨 󵄨 ∫ 󵄨󵄨󵄨f (w)Kp (w, z)󵄨󵄨󵄨 dA(w)

En

∞

≤ C ∑ (1 − |wn |)

2/p+1

n=1

󵄨 󵄨 sup 󵄨󵄨󵄨f (w)Kp (w, z)󵄨󵄨󵄨,

w∈En

where {wn } is an arbitrary sequence such that wn ∈ En , and C is a constant depending only on p. It follows that ∞

g(z)p ≤ C ∑ (1 − |wn |) n=1

2+p

󵄨p 󵄨 sup 󵄨󵄨󵄨f (w)Kp (w, z)󵄨󵄨󵄨 .

w∈En

For a fixed z, the function 󵄨󵄨p 󵄨 f (w) 󵄨󵄨 󵄨p 󵄨󵄨 󵄨 F(w) = 󵄨󵄨󵄨f (w)Kp (w, z)󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨 󵄨󵄨 (1 − wz)̄ 2/p+1 󵄨󵄨󵄨

124 | 3 Subharmonic behavior and Bergman-type spaces is subharmonic in 𝔻 because the function f (w)/(1 − wz)̄ 2/p+1 is analytic with respect to w. Therefore F(w) ≤

1 ∫ F dA, |Dw | Dw

where Dw ⊂ 𝔻 is an arbitrary disc centered at w. From this and from the properties of the family ℰ we get sup F ≤ C En

|f (w)|p 1 dA(w), ∫ ̄ 2+p (1 − |wn |)2 |1 − wz| Bn

where Bn is the union of those E ∈ ℰ for which the set E ∩ E n is nonempty. Combining these with the inequality ∫ 𝔻

1 C , dA(z) ≤ 2+p ̄ (1 − |w|)p |1 − wz|

we get ∞

󵄨p 󵄨 ∫ g(z)p dA(z) ≤ C ∑ ∫ 󵄨󵄨󵄨f (w)󵄨󵄨󵄨 dA(w). n=1

𝔻

Bn

Now our result follows from the easily checked fact that each Bn contains at most N members of the family ℰ with N being independent of n. Lemma 3.39. There exists a constant C such that |w − ζ | 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨Kp (w, z) − Kp (ζ , z)󵄨󵄨󵄨 ≤ C 󵄨K (w, z)󵄨󵄨󵄨 diam E 󵄨 p for all E ∈ ℰ , w, ζ ∈ E, and z ∈ 𝔻. Proof. By the mean value theorem we have 󵄨 󵄨󵄨 ̄ −2/p−2 󵄨󵄨Kp (w, z) − Kp (ζ , z)󵄨󵄨󵄨 ≤ (2/p + 1)|w − ζ | sup |1 − az| a∈E

≤ (2/p + 1) ≤ Cp

|w − ζ | 󵄨 󵄨 sup󵄨󵄨K (a, z)󵄨󵄨󵄨 1 − |a| a∈E 󵄨 p

|w − ζ | 󵄨 󵄨 sup󵄨󵄨K (a, z)󵄨󵄨󵄨. diamE a∈E 󵄨 p

On the other hand, 󵄨󵄨 ̄ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 (w − a)z 󵄨󵄨󵄨󵄨 |w − a| 1 − wz 󵄨󵄨 ≤ C. 󵄨=󵄨 󵄨≤ 󵄨󵄨1 − 󵄨󵄨 1 − az̄ 󵄨󵄨󵄨 󵄨󵄨󵄨 1 − az̄ 󵄨󵄨󵄨 1 − |a| ̄ and thus |Kp (a, z)| ≤ C|Kp (w, z)| for a, w ∈ E. The result ̄ ≤ C|1 − az|, Hence |1 − wz| follows.

3.6 Reproducing kernels and atomic decomposition

| 125

Proof of Theorem 3.17. Let ε > 0. Dividing each E ∈ ℰ into N subsets, where N is a sufficiently large integer independent of ε, we can represent 𝔻 as a disjoint union D1 ∪ D2 ∪ . . . , where D1 , D2 , . . . are subsets of 𝔻 with the properties diam(Dn ) ε ≤ ≤ C1 ε C1 1 − |w|

|Dn | ε2 ≤ C1 ε 2 ≤ C1 (1 − |w|)2

and

(w ∈ Dn ).

(3.32)

Let {wn } be a sequence such that wn ∈ Dn . Define the operator T by ∞

Tf (z) = ∑ an (1 − |wn |2 )Kp (wn , z) n=1

(z ∈ 𝔻),

where an =

1 ∫ f (w) dμp (w). 1 − |wn |2 Dn

Proceeding as in the proof of Lemma 3.38, we can prove that T maps Ap into Ap . To conclude the proof, it suffices to prove that T is an isomorphism for ε small enough. The proof of the latter is easy and independent of (3.32). To prove the rest, we start from the relation ∞

f (z) − Tf (z) = ∑ ∫(Kp (w, z) − Kp (wn , z))f (w) dμp (w). n=1 𝔻

From this by Lemma 3.39 we get ∞

󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨TF(z) − f (z)󵄨󵄨󵄨 ≤ Cε ∑ ∫ 󵄨󵄨󵄨f (w)󵄨󵄨󵄨󵄨󵄨󵄨Kp (z, w)󵄨󵄨󵄨 dμp (w) n=1

Dn

󵄨 󵄨󵄨 󵄨 = Cε ∫󵄨󵄨󵄨f (w)󵄨󵄨󵄨󵄨󵄨󵄨Kp (z, w)󵄨󵄨󵄨 dμp (w). 𝔻

Now Lemma 3.38 shows that ‖Tf − f ‖ ≤ Cp ε‖f ‖. Finally, we take ε = 1/2Cp and apply Proposition A.9. Exercise 3.40. The partition ℰ can be used to reduce the proof of the following result of Duren [164] (see [166, Theorem 9.4]) to a theorem of Hardy and Littlewood (Corollary 5.2): Let dμ be a finite positive measure on 𝔻, and let 0 < p < q < ⬦. In order that there is a constant C such that 󵄨p 󵄨 ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 dμ(z) ≤ C‖f ‖qp , 𝔻

it is necessary and sufficient that μ(W(I)) ≤ C1 |I|q/p for every arc I ⊂ 𝕋.

(3.33)

126 | 3 Subharmonic behavior and Bergman-type spaces Here W(I) denotes the Carleson window over I, z ∈ I}. |z|

W(I) = {z ∈ 𝔻 : 1 − |I| < |z| < 1,

In fact, it is easy to see that the last condition is sufficient for the validity of the inequality q/p−2 󵄨q 󵄨 󵄨p 󵄨 dA(z). ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 dμ(z) ≤ C ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|) 𝔻

𝔻

Even a weaker condition is sufficient for the validity of (3.33): μ(Ej,k ) ≤ C2−kq/p , where C is independent of j, k. Duren’s theorem can be reduced to the Hardy–Littlewood theorem, as was first observed by Blasco [84]. Exercise 3.41 ([51, 244]). Let 0 < ε < 1, p > 0, and α > −1. For a positive finite measure μ on 𝔻, the following conditions are equivalent. – There is a constant C such that ‖f ‖Lp (μ) ≤ C‖f ‖p,α for all f ∈ Apα . – There is a constant C1 such that μ(Hε (a)) ≤ C1 (1 − |a|)α+2 for all a ∈ 𝔻, where Hε (a) is the hyperbolic disc centered at a. – There is a constant C2 such that μ(W(I)) ≤ C2 |I|α+2 for all arcs I ⊂ 𝕋. – There is a constant C3 such that ̄ −2α−4 dμ(z) ≤ C3 (1 − |a|) ∫ |1 − az|

−α−2

,

a ∈ 𝔻.

𝔻

Further notes and results The “classical” Bergman spaces Apα first appeared in Hardy and Littlewood’s works. In the 1933 paper [68], Bergman began the study of the “Bergman metric”, and in the book [69], he considered the “Bergman kernel” and the Hilbert space A2 (of one and two variables) with various applications. Perhaps, the term “Bergman metric” was first used by Hahn [226]. The study of the structure of Apα was initiated by Djrbashian [151, 152]. Zakharyuta and Yudovich [624] proved the boundedness of the Bergman 󸀠 projection on Ap (1 < p < ⬦) and used this to prove that (Ap )󸀠 = Ap under the pairing ∫𝔻 f (z)g(z)̄ dA. The first book containing a systematic study of the spaces Apα is by Djrbashian and (A. E.) Shamoyan [150]. The reader interested in invariant divisors, Berezin transforms, Ap -inner functions, etc. should read the very interesting and inspiring book of Hedenmalm, Korenblum, and Zhu [251]. Further references are Duren and Schuster [163], Zhu [629], and, for several variables, Zhao and Zhu [625]. Theorem 3.8 (q = ∞) was proved by Hardy and Littlewood, and the proof was “surprisingly difficult” [236]. For the other values of q, the proof was given by Flett [194]. The unified approach presented here is maybe the simplest one. The study of

Further notes and results | 127

fractional integration and differentiation was begun by Hardy and Littlewood [237] and continued by Flett [194, 195], where the operator 𝒥β was introduced. A description of QNS in terms of the quasihyperbolic metric is in [161]. A generalization of Theorem 3.1 to “locally uniformly homogeneous” spaces, which cover all concrete cases, was given in [457]. Hervé [255] introduced the notion of a nearly subharmonic function by the requirement that u is measurable and u(a) ≤

1 ∫ u dA whenever Dε (a) ⊂ 𝔻. πε2 Dε (a)

The Nevanlinna counting function is nearly subharmonic but not subharmonic; see page 73. Motivated by Hervé, Riihentaus [508] (see also [456]) introduced the term quasinearly subharmonic. The notions of regularly oscillating and nearly convex functions were introduced in [428] and [456], respectively. The recent paper of Dovgoshey and Riihentaus [162] contains a lot of information, interesting results, and references on a generalized variant of QNS. It is the idea of Coifman and Rochberg [132] to represent a member of Apβ as a sum of “atoms” by replacing the integral in (3.28) with a Riemannian sum over a sufficiently fine partition of the disc. They proved atomic decomposition theorems for every p > 0 and for a class of domains in ℂn , in particular, for balls. A proof can be found in, e. g., [629, Theorem 4.4.9] (p = 1). For Corollary 3.37, see [21, 132, 599] and [169, 170] (q = 1). 3.1. If 1 < p < ⬦, then the formulation of the Coifman–Rochberg theorem is similar to that of Theorem 3.16; we only have to replace (3.30) with ∞

f (z) = ∑ an n=1

(1 − |wn |2 )2/p , (1 − w̄ n z)2 󸀠

see [629].

3.2 (The class OC 2 (Ω)). The class OCK2 (Ω) consists of f ∈ C 2 (Ω) such that 󵄨 󵄨󵄨 −1 󵄨󵄨Δf (z)󵄨󵄨󵄨 ≤ Kr sup |∇f |. Dr (z)

Obviously, this condition is implied by 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨Δf (z)󵄨󵄨󵄨 ≤ K 󵄨󵄨󵄨∇f (z)󵄨󵄨󵄨/δΩ (z),

z ∈ Ω.

It was proved in [430] that OCK2 ⊂ OCK1 1 , where K1 depends only on K. It is easy to see that if Ω is bounded and Δf = λf for some constant λ, then f ∈ OC 2 (Ω), and therefore f is regularly oscillating. p,q 3.3 (The space hp,q α , p < 1). It follows from Corollaries 3.17 and 3.27 that hα = {0} in the following cases: 1. q ≤ ⬦, α ≤ 1 − 1/p, 1/2 ≤ p < 1;

128 | 3 Subharmonic behavior and Bergman-type spaces 2. 3. 4. 5.

q ≤ ⬦, α ≤ −1, 0 < p < 1/2; q = ∞, α < 1 − 1/p, 1/2 < p < 1; q = ∞, α < −1, 0 < p < 1/2; q = ∞, α < −1, p = 1/2.

The estimates of the integral means of the Poisson kernel (Proposition 1.23) show that in cases 1–4 the bounds of α are optimal; this means that in the following cases the space is nontrivial and in fact infinite-dimensional: (i) q ≤ ⬦, α > 1 − 1/p, 1/2 ≤ p < 1; (ii) q ≤ ⬦, α > −1, 0 < p < 1/2; (iii) q = ∞, α ≥ 1 − 1/p, 1/2 < p < 1; (iv) q = ∞, α ≥ −1, 0 < p < 1/2. = {0}. MoreUsing Aleksandrov’s results (see Problem 2.4), we can prove that h1/2,∞ −1 over, if a function u ∈ h(𝔻) satisfies M1/2 (r, u) = o(1 − r)(log

2

2 ), 1−r

r → 1− ,

then u ≡ 0. 3.4 (Polyharmonic functions). The following statement was proved in [430]. Proposition. A C 4 -function f : G → ℝ belongs to OC 2 (G) if so does Δf . Consequently, a C ∞ -function f belongs to OC 2 (G) if so does Δk f for some integer k. In particular, every polyharmonic function belongs to OC 2 and, consequently, is regularly oscillating. Let hk (𝔻) denote the class of all polyharmonic functions of order k defined on 𝔻. If u ∈ hk (𝔻), then by Almansi’s theorem k−1

j

u(z) = ∑ (1 − |z|2 ) uj (z), j=0

(3.34)

where uj are harmonic functions uniquely determined by u. Conversely, if u is given in this way, then u ∈ hk (𝔻). The following decomposition was proved in [431]. Theorem. If u is a complex-valued polyharmonic function of order k, then p,q u ∈ Lp,q α (α > −1) if and only if uj ∈ Lα+j for all j = 0, . . . , k − 1.

(3.35)

For further results in this direction, see [73], [153], and [96]. p 3.5 (Critical integrability curves for polyharmonic functions). Let PHN,α denote the space of all polyharmonic functions f of order N such that α 󵄨p 󵄨 ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z) < ∞, 𝔻

Further notes and results | 129

where α ∈ ℝ. In the aforementioned paper [96], Borichev and Hedenmalm proved the following: Theorem (BH). Let p ∈ ℝ+ . For N = 1, 2, 3, . . . and real α, we have that PH pN,α (𝔻) = {0}

⇐⇒

α ≤ min bj,N (p) =: β(N, p), j:0≤j≤N

where bj,N (p) := max{−1 − (j + N − 1)p, −2 + (j − N + 1)p},

j = 1, . . . , N,

whilst b0,N (p) := −1 − (N − 1)p. It is interesting that β(N, p) = min{b0,N (p), b1,N (p)} for p ≥ 1/3, which is expected in view of Almansi decomposition theorem and the decomposition (3.35), whereas for p < 1/3, we have β(N, p) < min{b0,N (p), b1,N (p)}, which is unexpected. “The Almansi expansion can be expressed in the form (3.34), where all the functions uj are harmonic in 𝔻. In view of (3.34), we might be inclined to believe that the function (...) u = (1 − |z|2 )

N−1

uN−1 (z),

(3.36)

should be the smallest near the boundary. To our surprise, we find that this is not true (...) with 0 < p < 31 . Somehow the functions u0 , . . . , vN−1 can cooperate to produce nontrivial functions which decay faster than functions of the type (3.36)” – cited from [96]. The curve p 󳨃→ β(N, p) is called the critical integrability type curve for obvious reasons. It follows from Further notes 3.3 that β(1, p) = min{−1, max{p − 2, −1 − p}}. It should be noted that decomposition (3.35) is not suitable for finding β(N, p) because of the restriction α > −1. In [96] the following was proved. Theorem (The cellular decomposition theorem). Let α be real, and let p be positive. Then, for N = 1, 2, 3, . . ., every u ∈ PH pN,α (𝔻) has a unique decomposition u = w0 + M[w1 ] + ⋅ ⋅ ⋅ + MN−1 [wN−1 ], where each term Mj [wj ] is in PH pN,α (𝔻), whereas the functions wj are (N − j)-harmonic and solve the partial differential equation LN−j−1 [wj ] = 0 on 𝔻 for j = 0, . . . , N − 1. Here M[v](z) = (1 − |z|2 )v(z) and Lθ [u] = (1 − |z|2 )

2θ+1

∇ ⋅ {(1 − |z|2 )

−2θ

∇u} − 4θ2 u.

130 | 3 Subharmonic behavior and Bergman-type spaces 3.6 (Polyanalytic functions). A C ∞ -function f on 𝔻 is said to be polyanalytic of order k if 𝜕̄ k f ≡ 0 in 𝔻. Denote the set of such functions by Hk (𝔻). It is clear that Hk (𝔻) ⊂ hk (𝔻). If f ∈ Hk (𝔻), then there are unique functions fk ∈ H(𝔻) such that k−1

f (z) = ∑ z̄j fk (z), j=0

z ∈ 𝔻.

2 This can be written as f (z) = Q(z) + ∑k−1 j=0 |z| gj (z), where Q ∈ Hk (𝔻) is a polynomial,

g0 = f0 , and gj (z)/z j ∈ H(𝔻), and further

k−1

j

f (z) = Q(z) + ∑ (1 − |z|2 ) vj (z), j=0

s where vj = (−1)j ∑k−1 s=j ( j )gs . Then (3.35) becomes p,q f ∈ Lp,q α (α > −1) if and only if vj ∈ Lα+j for all j.

(3.37)

Since the integral means of vj have the “increasing” property, we see that Hk (𝔻)∩Lp,q α = {0}, q ≤ ⬦, if and only if α ≤ k − 1. This holds for all p > 0. An excellent exposition on the importance of polyanalytic functions is given by Rozenblum and Vasilevski [512]. The reader will find there many relevant results and references. 3.7 (Korenblum’s maximum principle for Bergman spaces). It was conjectured by Korenblum that, for every p ∈ ℝ+ , there is a constant 0 < c < 1 such that if f , g ∈ H(𝔻) and |f (z)| ≤ |g(z)| for c < |z| < 1, then ‖f ‖Ap ≤ ‖g‖Ap . Korenblum proved this for p = 2 and c < 21 e−2 under the additional hypothesis that g/f is analytic in 𝔻. Subsequently, it was proved that Korenblum’s conjecture holds for p ≥ 1. A long list of the authors who considered this theme can be found in Božin and Karapetrović [100], who recently disproved the conjecture for p < 1: Theorem (BK). Let 0 < p < 1. If 0 < c < 1, then there are functions f , g ∈ Ap such that |f (z)| ≤ |g(z)| for c < |z| < 1 but ‖f ‖Ap > ‖g‖Ap . 3.8 (Maximal functions). The proofs of Propositions 3.22 and 3.31 can be modified to show that they remain valid if we replace u× and u+ by the functions 󵄨 󵄨 u# (z) = sup 󵄨󵄨󵄨u(w)󵄨󵄨󵄨, w∈Dε (z)

where ε = δ(1 − |z|)

(0 < δ < 1),

and 󵄨 󵄨 u& (z) = sup 󵄨󵄨󵄨u(w)󵄨󵄨󵄨, w∈Bz

Brζ = {w : |w| < (1 + r)/2} ∩ Uζ ,c ,

Further notes and results | 131

where, as before, Uζ ,c is the open convex hull of the set c𝔻∪{ζ }, ζ ∈ 𝕋. These functions are measurable for any u (measurable or not); moreover, they are lower semicontinuous, that is, the sets {z: u# (z) > λ} and {z: u& (z) > λ} (λ ∈ ℝ) are open, and therefore, in this situation, the additional hypothesis on the semicontinuity of u is superfluous. 3.9. The function |∇f | is Borel measurable. To prove this, let {Bk } be a sequence of open balls such that G = ∪Bk , and let λ ≥ 0. Then there is a decreasing sequence {δj } tending to 0 such that, for each k, the set {z ∈ Bk : |∇f (z)| < λ} is the union of the closed sets 󵄨 󵄨 {x ∈ Bk : 󵄨󵄨󵄨f (z + h) − f (z)󵄨󵄨󵄨 ≤ λ(1 − 1/j)|h| for all h s. t. |h| < δj , h + z ∈ Bk }, where j ≥ 2. Several variables The definitions of quasinearly subharmonic and regularly oscillating functions in a domain Ω ⊂ ℝn are similar to those in the case n = 2: the disc Dr (a) should be replaced by the ball Br (a) = {x ∈ ℝn : |x−a| < r}, the unit disc by the unit ball 𝔹n , and the measure dA by the Lebesgue measure dVn . The other classes are defined in the same way, and the most results, in particular, Theorems 1.11 and 3.4 and Corollary 3.3, remain to be true in this general situation. Concerning the “invariant potential” theory in 𝔹n ⊂ ℝn , we refer to Stoll’s book [568]. 3.10. In a recent preprint [359], Liu et al. consider the higher-dimensional analogue of Borichev and Hedenmalm [96] (see Further notes 3.5) and find that the situation is p quite “normal” for p ≥ n−2 . They prove that if n ≥ 3, then PHN,α (BN ) = {0} if and only n−1 if α ≤ β(N, p), where ≤ p < n−1 , −1 − Np if n−2 { n−1 n { { n−1 β(N, p) = {−n − (N − n)p if n ≤ p < 1, { { {−1 − (N − 1)p if p ≥ 1.

(3.38)

However, nothing can be said in the case p < (n − 2)/(n − 1), in which |∇u| (u harmonic) need not be subharmonic. The reader is referred to Aleksandrov’s paper [23] for a deep analysis of harmonic functions of several variables. 3.11 (A Hardy–Stein-type characterization of Bergman spaces). The following statement was proved in [334] under the hypothesis f (0) = 0 and in [458] for arbitrary f . Theorem (KPZ). Let p ∈ ℝ+ , 0 < q < p + 2, β > −1, and f ∈ H(𝔻). Then β+q β 󵄨p 󵄨 󵄨q 󵄨 ∫ |f |p (1 − |z|2 ) dA ≍ 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + ∫ |f |p−q 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 (1 − |z|2 ) dA, 𝔻

𝔻

where the equivalence constants are independent of f .

132 | 3 Subharmonic behavior and Bergman-type spaces In fact, in [458] an analogue for several variables was proved, which included the ordinary and invariant gradients. The paper [445] contains an analogous result for (complex-valued) harmonic functions on the real ball. 3.12. Quasinearly subharmonic functions found an unexpected application in partial differential equations [19]. The authors gave an interesting characterization of QNS. Let η ≥ 0 be a radial C ∞ -function on ℝn , of integral one, such that η(x) = 1 for |x| < 1/3 and η(x) = 1 for |x| ≥ 1. For a function u, the convolution uε is defined by uε (x) =

1 x−y ) dy, ∫ η( εn ε X

where X is the domain of u. Lemma. Let u : X → [0, +∞) be a Borel-measurable function. Then u is quasinearly subharmonic if and only if for every O ⋐ X, there exist M and ε0 such that, for any 0 < ε < ε0 , u(x) ≤ Muε (x) for x ∈ O. 3.13. If p < 1, then there is no bounded projection from Lp = Lp (𝔻) onto Ap because the dual of Lp is trivial. In [386] the following substitute for Lp was defined. Let ε = 1/2 and m(f , z) = ∫H (z) |f |p dτ, where, as before, Hε (z) denotes the pseudohyperbolic ball ε

of radius ε. The space L p consists of Borel-measurable functions f on 𝔻 for which m(f , ⋅) ∈ Lp . The dual of L p separates points, and Ap = H(𝔻) ∩ L p for p ∈ ℝ+ . The following statement holds [386, Theorem 3.1]: Let 0 < p < 1. The operator Ts maps L p onto Ap if and only if s > 2/p − 2.

This can be used to prove Theorem 1.12 (p < 1). The spaces L p (p > 0) were earlier implicitly defined in Luecking’s paper [361] and later were used by Li and Luecking [347].

4 Mixed-norm spaces with nonstandard weights In this chapter, we consider spaces with “nonstandard” weights, e. g., ϕ(t) = t a (log(2/t)) ,

0 < t < 1,

−γ

(1)

and β

ϕ(t) = t b e−c/t ,

(2)

p,q defined by replacing (1 − r)α with ϕ(1 − r) in the definition of hp,q α and Hα . In the case where ϕ is normal, e. g., a > 0 in (1), we show how to modify the proofs from the is self-conjugate in preceding chapter to obtain analogous results. For instance, hp,q ϕ this case. We also consider another type of notation, which is more convenient from p,q the technical point of view: The space H[ϕ] consists of those f for which the function

ϕ(1 − r)Mp (r, f ), 0 < r < 1, belongs to Lq (dmϕ ), where dmϕ (r) = ϕ󸀠 (1 − r)dr/ϕ(1 − r). If p,q ϕ is normal, then Hϕp,q = H[ϕ] , and the converse is true. 2,q The case p = 2 is specific because f ∈ H[ϕ] if and only if 1

∞

0

n=0

q/2

∫( ∑ an r 2n )

ϕ(1 − r)q dmϕ (1 − r) < ∞,

where an = |f ̂(n)|2 . This leads us to considering some Lq -integrability lemmas for power series with positive coefficients. Since lacunary series arise naturally in this situation, we consider them as well. In the case of functions with rapidly decreasing weights, such as (2) (with c > 0 and b > 0), we state several results without proofs.

4.1 Classes of real functions 4.1.b (Almost monotone functions). According to Bernstein [72], a real function ψ is said to be almost increasing if x < y implies ψ(x) ≤ Cψ(y), where C is a constant independent of x and y. An almost decreasing function is defined similarly. We will use positive measurable real functions defined on J = (0, 1], [1, ∞), or (0, ∞). If ψ is such a function, then in most cases, we assume that there are real numbers α and β (0 ≤ α ≤ β) such that (†) ψ(x)/xα is almost increasing and (‡) ψ(x)/xβ is almost decreasing for x ∈ J. We leave to the reader to verify that if (†) holds, then the condition supx∈J ψ(2x)/ψ(x) < ∞ is equivalent to (‡); this “Δ2 -condition” occurs naturally in the theory of Orlicz spaces [326]. The following statement enables us to replace ψ with a function that has better properties. https://doi.org/10.1515/9783110630855-004

134 | 4 Mixed-norm spaces with nonstandard weights Proposition 4.1. If ψ possesses properties (†) and (‡), then there exists a function ψ2 ≍ ψ on J such that ψ2 (x)/xα is increasing and ψ2 (x)/xβ is decreasing on J.

(4.1)

Proof. Consider the case J = [1, ∞). We define ψ1 (x) = xβ sup t≥x

ψ(t) ψ(tx) = inf β , β t≥1 t t

ψ0 (x) = xα inf t≥x

ψ (xt) ψ1 (t) = inf 1 α . α t≥1 t t

It is easily checked that the desired function is ψ2 = ψ0 . If J = (0, 1], then we consider the function ψ(1/x), x ≥ 1. 4.1.c (The classes Δ[α, β]). We write “ψ ∈ Δ[α, β] on J” (α ≤ β) if ψ satisfies (4.1); here we can allow α < 0. We write ψ ∈ Δ[α, β) (resp., ψ ∈ Δ(α, β]) if ψ ∈ Δ[α, β1 ] for some β1 < β (resp., ψ ∈ Δ[α1 , β]) for some α1 > α. We also write Δ(α, β) = Δ[α, β) ∩ Δ(α, β]. If ψ ∈ Δ[α, β], then ψ is absolutely continuous, and we have α≤

xψ󸀠 (x) ≤β ψ(x)

for a. e. x ∈ J.

(4.2)

If an absolutely continuous function satisfies (4.2), then it belongs to Δ[α, β]. 4.1.d (Normal and subnormal functions). The function ψ is said to be normal on J if ψ ∈ Δ[α, β] for some α > 0 and β ≥ α; this notion was introduced by Shields and Williams [532]. We call ψ subnormal on [0, 1] if ψ ∈ Δ[0, β] for some β > 0 and ψ(0+) = 0. If ψ is defined on [1, ∞), then it is called subnormal if ψ ∈ Δ[0, β] on [1, ∞) and limt→∞ ψ(t) = ∞. The notions of almost normal and almost subnormal functions are defined in an obvious way. A normal function ϕ can be further replaced by an equivalent function with additional regularity properties. We begin with a lemma. Lemma 4.2. If ϕ ∈ Δ[α, β] on J, 0 < α ≤ β < ∞, then there is a convex function ϕ1 such that ϕ(x) ≍ ϕ1 (xα ) and the function ϕ1 (xα/β ) is concave. Proof. Let J = [0, 1] and define ϕ1 by x

ϕ1 (x) = ∫ 0

ϕ(t 1/α ) dt, t

0 < x ≤ 1.

The function ϕ1 is convex because ϕ(t 1/α )/t is increasing, and it is easy to see that ϕ(x) ≍ ϕ1 (xα ). Let γ = α/β. Then, by the substitution t = sγ , γ

x

ϕ1 (x ) = γ ∫ 0

ϕ(t 1/β ) dt. t

Since ϕ(t 1/β )/t is decreasing, we have that ϕ1 (xγ ) is concave. This proves the lemma.

4.1 Classes of real functions | 135

As a consequence, we have the following: Lemma 4.3. Let ψ(x) = ϕ1 (xα ), where the function ϕ1 has the properties described in the preceding lemma. Then ψ󸀠 ∈ Δ[α − 1, β − 1]. Proof. We have ψ󸀠 (x)/xα−1 = αϕ󸀠1 (xα ). This implies that ψ󸀠 (x)/xα−1 is increasing in x because ϕ1 is convex. On the other hand, since ψ1 (x) := ϕ1 (xα/β ) is concave, we see that ψ󸀠1 (x) =

α 󸀠 α/β α/β−1 ϕ (x )x β 1

is decreasing. Rewriting this as ψ󸀠1 (t β ) =

1 α ϕ1 (t α )t α−1+1−β = ψ󸀠 (t)t 1−β , β β

we find that ψ󸀠 (t)/t β−1 is decreasing, which completes the proof. 4.1.e (The Hardy field). Let (ℋ) be the class of expressions composed from the set {xn , log x, ex , constants} (n an integer) by successive applications of the arithmetic operations and substitutions. The class of functions defined by such expressions is denoted by (ℋ) and is called the Hardy field.1 A remarkable result of Hardy (see [97, Ch. V, Appendix]) states that if a function ψ ∈ (ℋ) is defined near ∞, then it is of constant sign near ∞. Since ψ󸀠 ∈ (ℋ), we see that ψ is monotone, and therefore limx→∞ ψ(x) exists (finite or infinite). This implies that Lψ := limx→∞ xψ󸀠 (x)/ψ(x) exists. If Lψ > 0 is finite, then ψ is normal near infinity and can be changed on some [0, t0 ] to be normal on [0, ∞). 4.1.f. For a positive continuous function ϕ on (0, 1], define the space Lp,q by replacϕ

ing (1 − r 2 )α in (3.10) with ϕ(1 − r). The corresponding spaces of analytic or harmonic functions are defined by Hϕp,q = Lp,q ∩ H(𝔻) and hp,q = Lp,q ∩ h(𝔻). ϕ ϕ ϕ

(4.3)

In this context the function ϕ is called a weight. If ϕ(x) = xα for some α > 0, then ϕ is called a standard weight, and the corresponding spaces are called standard mixednorm spaces, so the Hardy space H p = H0p,∞ is not a standard mixed-norm space. If ϕ is normal (see 4.1.c), then most of the results of the preceding chapter remain valid. In particular, we have the following: Theorem 4.1. If f ∈ H(𝔻) and ϕ is normal, then the following conditions are equivalent: (a) f ∈ Hϕp,q ; (b) f 󸀠 ∈ Hψp,q , where ψ(x) = xϕ(x); and (c) Re f ∈ hp,q . ϕ 1 The “Hardy field” as defined in [97] is much larger.

136 | 4 Mixed-norm spaces with nonstandard weights Corollary 4.4. If ϕ is normal, then hp,q is self-conjugate. ϕ Corollary 4.5. If ϕ is normal, then the Riesz projection is bounded on hp,q . ϕ The implication (c) ⇒ (b) is independent of the hypothesis that ϕ is normal. Namely, we can assume that ϕ ∈ Δ[α, η] (see 4.1.c). Then from (3.14) we get q

Mpq (|a|, h)(1 − |a|2 ) ≤ C ∫ Mpq (|z|, g) dτ(z), Hε (a)

where h(z) = |f 󸀠 (z)|ϕ(1−|z|) and g(z) = u(z)ϕ(1−|z|). Now the desired result is obtained by application of the “local-to-global” estimates; see Remark 3.18. To prove that (b) implies (a) with the hypothesis that ϕ is normal, we need to make somewhat more changes of the proof of Theorem 3.8. In fact, we need to show that Lemma 3.20 remains true if 2−nδ is replaced with ϕ(2−n ), and then to replace 2−nα with ϕ(2−n ) in the chain of inequalities on page 109. That is all. We can prove that Propositions 3.22 and 3.31 hold for Lp,q by application of these ϕ γ statements to the QNS-function u(z)ϕ(1 − |z|)/(1 − |z|) , where 0 < γ < α. A generalization of Proposition 3.30 reads as follows. with ψ(t) = Proposition 4.6. If an upper semicontinuous QNS-function u belongs to Lp,q ψ ϕ(t)t s , where ϕ is normal and s > 0, then 𝒥s u belongs to Lp,q . ϕ

Proof. Let ω(t) = ϕ(t)/t γ , 0 < γ < α, where ϕ(t)/t α increases with t. Then ω(t) is increasing, and so 1

1 s−1 ω(1 − r)𝒥s u(rζ ) ≤ ∫(log(1/t)) ω(1 − rt)u(rtζ ) dt. Γ(s) 0

Now the result follows from Proposition 3.30 (applied to Lp,q γ ) and the fact that the function u(z)ω(1 − |z|) is QNS. From this we can deduce a generalized version of Theorem 3.13. Theorem 4.2. Let ϕ be a normal function, s > 0, and ψ(t) = t s ϕ(t). Then a function . if and only if 𝒥s u ∈ hp,q . Equivalently, u ∈ hp,q if and only if 𝒥 s u ∈ hp,q u ∈ h(𝔻) is in hp,q ψ ψ ϕ ϕ Exercise. If u ∈ QNS(𝔻) and ϕ is subnormal on [0, 1], then u(z)ϕ(1 − |z|) ∈ QNS(𝔻).

4.2 Bergman spaces with rapidly decreasing weights A function φ ∈ L1 (0, 1) such that φ > 0 a. e. is called a Bergman weight. For 0 < q < ⬦, we define the (generalized) Bergman space Ap,q φ as the class of those f ∈ H(𝔻) for

4.2 Bergman spaces with rapidly decreasing weights | 137

which 1

∫ Mpq (r, f )φ(r)r dr < ∞. 0

If p = q, this is equivalent to 󵄨p 󵄨 ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 φ(|z|) dA(z) < ∞, 𝔻 p,q and then we write Apφ = Ap,p φ . The harmonic Bergman space Aφ is defined in the same

way. The “standard” Bergman space Apα (α > −1) is defined by choosing φ(r) = (1 − r 2 )α . The function 1

Dφ (r) =

1 ∫ φ(t) dt φ(r)

(4.4)

r

is called the distortion function of φ. Assume that 1

φ󸀠 (r) φ󸀠 (r) sup Dφ (r) = sup ∫ φ(x) dx ≤ L, 2 0 0 are constants. By l’Hôpital’s rule we obtain Dφ (r) ≍ (1 − r)α+1 . Thus, for example, 1

∫Mpq (r, f ) exp(− 0

κ ) dr (1 − r)α

1

κ q 󵄨p 󵄨 ) dr. ≍ 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + ∫(Mp (r, f 󸀠 )(1 + r)α+1 ) exp(− (1 − r)α 0

This is unusual because in “usual” cases, Mp (r, f ) behaves like Mp (r, f 󸀠 )(1−r), whereas here Mp (r, f ) is transformed into Mp (r, f 󸀠 )(1 − r)α+1 . It is not easy to guess what is the analogue to Theorem 4.3 when q = ∞. An earlier result of the author helps to reformulate Theorem 4.3 in, maybe, clearer way. Theorem 4.4 (Pavlović [433]). Let 0 < p ≤ ∞, and let ω > 0 be a C 2 -function on [0, 1), strictly increasing near 1, such that lim sup r→1−

ω󸀠󸀠 (r)ω(r) < ∞. ω󸀠 (r)2

(4.8)

For f ∈ H(𝔻), the following conditions are equivalent: (a) Mp (r, f ) = 𝒪(ω(r)) (r → 1 ); (b) Mp (r, f 󸀠 ) = 𝒪(ω󸀠 (r)) (r → 1− ). −

It is a reasonable to call ω a majorant. Given a Bergman weight φ ∈ C 1 (0, 1), let 1

1 = q ∫ φ(x) dx, ω(r)q r

0 < q < ⬦.

4.3 Mixed-norm spaces with subnormal weights | 139

Then condition (4.6) is equivalent to (4.8). Define the measure dνω on (0, 1) by dνω (r) =

ω󸀠 (r) dr. ω(r)

Now Theorems 4.3 and 4.4 can be stated in a unique way. Theorem 4.5 (with the hypotheses of Theorem 4.4). For a function f ∈ H(𝔻), let F1 (r) = Mp (r, f )/ω(r) and

F2 (r) = Mp (r, f 󸀠 )/ω󸀠 (r).

Then 󵄨 󵄨 ‖F1 ‖Lq (dνω ) ≍ 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + ‖F2 ‖Lq (dνω ) . The proof of this fact is rather long and is based on similar ideas as the proof of Theorem 3.8, but a more complicated technique is involved, and for this reason, we omit it. It was proved in [433] and [455] that under condition (4.8) the space Aφp,q is selfconjugate for q ≥ 1. If (4.8) is strengthened to 󵄨󵄨 ω󸀠󸀠 (r)ω(r) 󵄨󵄨 󵄨󵄨 󵄨 lim sup󵄨󵄨󵄨 󵄨 < ∞, 󸀠 2 󵄨󵄨 󵄨 r→1− 󵄨󵄨 ω (r) then the space is self-conjugate for all q > 0. In particular, Aφp,q with φ given by (4.7) (κ > 0) is self-conjugate.

4.3 Mixed-norm spaces with subnormal weights Let ϕ be a subnormal weight on (0, 1). We consider two types of spaces. 4.3.g. The spaces hp,q = h(𝔻) ∩ Lp,q and Hϕp,q ∩ Lp,q have been defined in Section 4.1.f; ϕ ϕ ϕ see (4.3). These spaces occur in the study of Lipschitz spaces in a natural way (see Chapter 9). If ϕ is normal, then they are nontrivial and moreover infinite-dimensional because 1

(∫ ϕ(t)q 0

1/q

dt ) t

−1, then the Bergman space Apα (X) consists of analytic functions f : 𝔻 󳨃→ X such that α 󵄩p 󵄩 ∫󵄩󵄩󵄩f (z)󵄩󵄩󵄩 (1 − |z|) dA(z) < ∞. 𝔻 − ‖f (rζ )‖ |dζ |. From the inequalities Let Mp (r, f ) = ∫ 𝕋 ∞

󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩f ̂(n)󵄩󵄩󵄩r n ≤ Mp (r, f ) ≤ ∑ 󵄩󵄩󵄩f ̂(n)󵄩󵄩󵄩r n n≥0

n=0

and Lemma 4.10 we see that a complex sequence λn such that |λn | ≤ C/(n + 1) is a (small) multiplier of Apα (X) in the sense that if f ∈ Apα (X), then the function ∑ λn f ̂(n)z n belongs to Apα (X). Buckley et al. [102] and Rydhe [523] call this fact the small multiplier property of Bergman spaces. Lemma 4.9 is a particular case of the following lemma because ϕ ∈ Δ[α, β], where 0 < α < β, if and only if α ≤ tϕ󸀠 (t)/ϕ(t) ≤ β, and hence dmϕ (r) ≍

dr 1−r

(if ϕ is normal).

Lemma 4.13. The statements of Lemmas 4.8 and 4.9 remain valid if we assume that ϕ is subnormal and replace dr/(1 − r) with dmϕ (r). Proof. For the proof, we need the relation 1

∫ Ψ 󸀠 (1 − r)r x dr ≍ Ψ (1/x),

x ≥ 1,

(4.12)

0

which holds for any subnormal function Ψ ; we will prove it latter on. 1 The proof of the inequality ∫0 . . . ≥ c ∑∞ n=0 . . . is simple, and we omit it. Let q ≤ 1. Assuming that {cn } is an arbitrary sequence of complex numbers, we have 1󵄨 ∞ 󵄨󵄨

q

󵄨󵄨 󵄨󵄨 󵄨 ∫󵄨󵄨󵄨 ∑ cn r λn 󵄨󵄨󵄨 ϕ(1 − r)q−1 ϕ󸀠 (1 − r) dr 󵄨󵄨 󵄨󵄨n=0 󵄨 󵄨 0

4.4 Lq -integrability of power series with positive coefficients | 143

∞

q

1

≤ ∑ |cn | ∫ ϕ(1 − r)q−1 ϕ󸀠 (1 − r)r λn q dr n=0

0

∞

≤ C ∑ |cn |q ϕ(1/λn )q , n=0

where we have used (4.12) with Ψ = ϕq . In order to prove the lemma in the case q = ∞, we have to prove that ∞

∑ Bn r λn ≤

n=0

C , ϕ(1 − r)

0 < r < 1.

(4.13) 1

Then using the Riesz–Thorin theorem, we conclude the proof of the inequality ∫0 . . . ≤ C ∑ . . . So it remains to prove (4.12) and (4.13). By partial integration, 1

1

1

0

0

0

∫ ϕ󸀠 (1 − r)r x dr = x ∫ ϕ(1 − r)r x−1 dr = x ∫ ϕ(t)(1 − t)x−1 dt. We split the last integral at t = 1/x and use the properties of ϕ to get (4.12). To prove (4.13), let 1 − 1/λj ≤ r ≤ 1 − 1/λj+1 . Then we split the sum at n = j to get j

∞

∞

∑ Bn r λn ≤ ∑ Bn + ∑ Bn (1 −

n=0

n=0

n=j+1 ∞

≤ CBj + C ∑ Bn e−K

1

λj+1

n−j−1

n=j+1

λn

) n−j−1 −K n−j−1

≤ CBj + CBj ∑ K1

e

n=j+1

,

where K1 > 1 and K > 1 are constants. This implies that the last quantity is ≤ CBj ≤ C/ϕ(1 − r). We are done. Remark. There is a purely elementary and not too long proof of Lemma 4.13; see [421, Part I], Theorem 3.1 and the proof of Lemma 3.1. Example 4.14. Let ϕ(t) = (log(2/t))−γ for 0 < t < 1, where γ > 0. Then dmϕ (r) = γ(log

dr 2 ) . 1−r 1−r −1

n

We can take λn = 22 in Lemma 4.13 to get 1

∞

n

q

∫( ∑ an r ) (log n=0

0

n−1

where In,λ = {k: 22

n

≤ k < 22

−1

γq−1

2 ) 1−r

q

∞ dr ≍ ∑ ( ∑ ak ) 2−qγn , 1 − r n=0 k∈I n,λ

} for n ≥ 1 and I0,λ = {0, 1}.

p,q Exercise 4.15. Using (4.12), prove that if ϕ is a subnormal function, then Hϕp,q = H[ϕ] if and only if ϕ is normal.

144 | 4 Mixed-norm spaces with nonstandard weights

4.5 Lacunary series with complex-valued coefficients Lemma 4.13 is a generalization of Lemma 4.9. There is, however, a substantial improvement: Lemma 4.11 remains true if we assume that {cn } is a sequence of complex numbers (cn should be replaced with |cn |); this is essentially due to Gurariy and Matsaev [222], although their theorem reads somewhat different and is stated for p ≥ 1. Here we state a generalized version of the Gurariy–Matsaev theorem. λn Theorem 4.6. Let ϕ be normal function on [0, 1], and let ℒ(r) = ∑∞ n=0 cn r (cn ∈ ℂ) be a series converging for r ∈ (0, 1), where {λn } (λ0 ≥ 1) is a lacunary sequence of real numbers. Then

󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩ℒ(r)ϕ(1 − r)󵄩󵄩󵄩Lp ≍ 󵄩󵄩󵄩{ϕ(1/λn )cn }󵄩󵄩󵄩ℓp , −1 where the equivalence constants are independent of {cn }. An even more general fact holds; see Theorem A.8. As a particular case of the latter we have the following: Theorem 4.7. Let G ∈ Δ[γ, c] on [0, ∞), 0 < γ < c, and let ϕ ∈ Δ[α, a] and ψ ∈ Δ[β, b] on [0, 1]. If α + βγ > 0 (and hence a + bc > 0), then 1

∫ G(ψ(1 − r)ℒ(r)) 0

∞ ϕ(1 − r) dr ≍ ∑ G(ψ(1/λn )|cn |)ϕ(1/λn ). 1−r n=0

Lacunary series in C[0, 1] As before, we denote by {cn } a sequence of complex numbers and consider the series ∞

λ

ℒ(r) = ∑ cn r n , n=0

where λn is a lacunary sequence, i. e., a sequence satisfying inf

n≥0

λn+1 = ρ > 1. λn

(4.14)

The following theorem was established by Hardy and Littlewood [229]. Theorem 4.8. If there exists the finite limit S := limr→1− ℒ(r), then the series ∑∞ n=0 cn converges, and its sum is equal to S. Proof. First, we prove that the hypotheses imply sup |cn | ≤ CM, n≥0

(4.15)

4.5 Lacunary series with complex-valued coefficients | 145

where M = sup0 1 such that, for any integer λ > λ0 , relation (4.18) with kn = λn holds. The proofs in [38] are formidably complicated.

4.6 Remarks on weights (1 ≤ Lusky [366, 367] studied a seemingly more general class of Bergman spaces Ap,q dμ p ≤ ∞, 1 ≤ q < ⬦) defined by 1

∫ Mpq (r, f ) dμ(r) < ∞ (f ∈ H(𝔻)), 0

where dμ is a finite measure on [0, 1] such that μ([r, 1]) → 0 as r → 1− . The space Ap,∞ dμ is defined by the requirement sup0 0 such that ψ(t)/t β is almost decreasing. Hence by Proposition 4.1 there exists a function ψ1 such that ψ1 (t) is increasing and ψ1 (t)/t β is decreasing. Thus Ap,q = Ap,q , where ν(r) = ψ1 (1 − r). dμ dν

p,q = H[ϕ] , where ϕ(t) = (ν(1 − r))1/q . We also easily see that Now we easily check that Ap,q dμ ϕ is normal if and only if μ satisfies both (⋆) and (⋆⋆). The case q = ∞ is of course simpler. In [81] and [87] an increasing function ϕ(t), 0 ≤ t ≤ 1, is said to be a Dini weight3 if x

∫ 0

ϕ(t) dt ≤ Cϕ(x), t

0 < x ≤ 1,

(4.19)

where K is independent of x. According to Janson [271], a weight ϕ(t) is said to be a bm -weight (m > 0) if 1

∫ x

ϕ(t) ϕ(t) dt ≤ C m , x t m+1

0 < x ≤ 1.

(4.20)

It turns out that condition (4.19) forces ϕ to have a better “increasing” property, whereas the “bm -condition” forces ϕ(t) to grow strictly slower than t m . 3 We use the term “Dini” in a different sense; see p. 266.

4.6 Remarks on weights | 149

Proposition 4.16. (i) The function ϕ satisfies (4.19) if and only if there is a real constant α > 0 such that ϕ(t)/t α is almost increasing in t. (ii) The function ϕ is a bm -weight if and only if there is a constant β < m such that ϕ(t)/t β is almost decreasing. Consequently, if ϕ satisfies (4.19) and (4.20), then there is ϕ1 ∈ Δ(0, m) such that ϕ1 ≍ ϕ (see Section 4.1.c). Such assertions are often encountered in the theory of regularly varying functions (cf. [35] and [527]). Proof of (i). “If” part is simple. In the other direction, (4.19) implies ϕ(λx) log(1/λ) ≤ Cϕ(x),

0 < λ < 1, 0 < x ≤ 1.

Choose λ such that C/ log(1/λ) = e−1 . Then ϕ(λn x) ≤ e−n ϕ(x) for all n ∈ ℕ. If η ∈ (0, 1) is arbitrary, then we choose n such that λn+1 ≤ η ≤ λn . It follows that n+1 ≥ (log η)/(log λ), and hence 1/ log λ

ϕ(ηx) ≤ ϕ(λn x) ≤ ee−(n+1) ϕ(x) ≤ e(e− log η )

ϕ(x) = eηα ϕ(x),

where α = −1/ log λ > 0. This concludes the proof of (i). Proof of (ii). Again, “if” part is simple. In the other direction, we have 1

1

x

x

ϕ(t) 1 F(x) := ∫ m+1 dt ≥ ϕ(x) ∫ t −m−1 dt ≥ ϕ(x)x−m , m t

0 < x ≤ 1.

This shows that F(x) ≍ ϕ(x)x−m , so it suffices to find b > 0 such that F(x)xb decreases in x. A simple calculation of (xb F(x))󸀠 shows that we can take b = 1/C, where C satisfies (4.20). Properties of Bergman weights Let φ ∈ L1 (0, 1) be a Bergman weight, i. e., φ ≥ 0 a. e. We take (ϕq )󸀠 (1 − r) = φ(r) to get 1

̂ := ∫ φ(t) dt ϕ(1 − r)q = φ(r) r p,q and Ap,q φ = H[ϕ] .

The class 𝒟̂ consists by definition of those φ for which the doubling condition ̂ ̂ + r)/2) is satisfied. This is equivalent to the Δ2 -condition ϕ(2t) ≤ Cϕ(t), φ(r) ≤ C φ((1

150 | 4 Mixed-norm spaces with nonstandard weights which means that ϕ is almost subnormal. The class 𝒟̌ consists of those φ for which there exist constants K > 1 and B > 1 such that ̂ ≥ Bφ(1 ̂ − φ(r)

1−r ), K

0 < r < 1.

(4.21)

In our language, this reads ϕ(t) ≥ Bϕ(t/K), 0 < t < 1, which means that ϕ(t)/t α is almost increasing for some α > 0. The latter follows from Proposition 4.16(i) and the inequality x/K n−1

x

∞ ϕ(t) ϕ(t) dt = ∑ ∫ dt ∫ t t n=1 0

x/K n

∞

≤ ∑ ϕ(x/K n−1 ) log K n=1 ∞

ϕ(x) log K n−1 n=1 B

≤∑

≤ Cϕ(x),

0 < x < 1.

Thus if φ ∈ 𝒟 := 𝒟̂ ∩ 𝒟̌ , then by Proposition 4.1 the function ϕ is equivalent to a normal one. This implies that the function φ̂ is almost normal near 1, which means that there ̂ ̂ are constants a > 0 and b > 0 such that φ(r)/(1 − r)a and φ(r)/(1 − r)b are almost increasing and decreasing in r ∈ (0, 1), respectively. The class ℛ (of “regular” weights) consists of those φ for which φ(r) ≍

̂ φ(r) , 1−r

0 < r < 1.

This means that φ̂ is normal near 1, which implies that φ(r)/(1−r)a is almost decreasing and φ(r)/(1 − r)b is almost increasing in r for some constants a > −1 and b > −1. p,q p,q , 4.6.a (Improving properties of 𝒟-weights). By Proposition 8.4 we have H[ϕ] = H[ψ] ψ ≍ ϕ, where ψ is a subnormal function for which there exist a concave function ψ0 and a real number m ≥ 1 such that ψ = ψm 0 . Then a simple analysis shows that

ψ󸀠 (Kx) ≤ K m−1 ψ󸀠 (x) for all K > 1 and 0 < x < 1/K, which means that ψ󸀠 (x)/xm−1 is decreasing in x ∈ (0, 1), and, as a consequence, we p,q have that if φ ∈ 𝒟̂ , then there is φ1 ∈ 𝒟̂ such that φ̂ ≍ φ̂ 1 , Ap,q φ = Aφ1 , and φ1 (r)/(1 − r)β

is increasing in r ∈ (0, 1) for some constant β > −1.

(4.22)

It is the question whether for φ ∈ 𝒟, there is a function φ1 with the above properties such that φ1 (r)/(1 − r)α

is decreasing in r ∈ (0, 1) for some constant α > −1.

This can be resolved by translating Proposition 4.7. We obtain the following:

(4.23)

4.6 Remarks on weights | 151

Proposition 4.17. If φ ∈ 𝒟, then there exists a weight φ1 satisfying (4.22) and (4.23) and p,q such that φ̂ ≍ φ̂ 1 and Ap,q φ = Aφ . 1

QK -spaces Finally, we mention a rather complicated condition used in the study of the so-called QK -spaces, where K is an positive increasing function on (0, ∞) such that K(x) = K(1) for x ≥ 1. The space QK consists of those f ∈ H(𝔻) for which 󵄨2 󵄨 󵄨2 󵄨 sup ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 K(1 − 󵄨󵄨󵄨σa (z)󵄨󵄨󵄨 ) dA(z) < ∞. a∈𝔻

𝔻

For information, results, and references, we refer the reader to [604] or the monograph by Wulan and Zhu [605]. We note that, under the assumed conditions on K, the space QK contains all polynomials; see at least the first page of [604]. Given such K, we define the auxiliary function φK by φK (s) = sup

0 0. Although these conditions look nice, their verification can cause technical difficulties. However, it is not hard to prove that (4.24)(1) is equivalent to the existence of ε > 0 such that K(x)/xγ−1+ε is almost increasing in x ∈ (0, 1], whereas (4.24)(2) is equivalent to the existence of β < γ such that K(x)/xβ is almost decreasing in x. In both cases, “if” part is easy. In proving “only if” part in the case of, e. g., (1), we start from the inequality 1

∫ 0

K(st) ds ≤ CK(t), sγ

0 < t ≤ 1,

which is implied by (1), then make the substitution s = x/t and proceed similarly as in the proof of Proposition 4.16(i). Observe that if γ < 1, then (1) is satisfied for any K because φK ≤ 1, so we can take ε = 1 − γ. As a consequence, we have: If K satisfies (1), with γ ≥ 1 and if φK (2) < ∞, then there is a normal function K1 (t), 0 < t ≤ 1, such that K1 ≍ K; if γ < 1, then we can only guarantee that K1 is subnormal. We refer the reader to Blasco’s paper [85] for various types of weight functions and a list of papers where they were introduced.

152 | 4 Mixed-norm spaces with nonstandard weights

Further notes and results As far as the author knows, the first results concerning the spaces Hϕp,q and hp,q ϕ were obtained by Shields and Williams [532–534] in some particular cases. The study of Hϕp,q in the general case, with subnormal ϕ, was begun in [384]. On the other hand, it seems that the spaces with exponential weights occur for the first time in [33] and [433]. Lemma 4.13 was proved by the author in 1986 [421, Theorem 3.2] and also in the first edition of this book. It was recently rediscovered in [464, Proposition 9] although [421] is on the list of references in [464]. Their proof shows that in the analysis of spaces with radial weights, the Besov-type notation is more suitable than the Bergman-type notation. Although there are Bergman weights ω with irregular behavior, everything 1 ̂ depends only on ω(r) = ∫r ω(t) dt; an example from [464]: 󵄨󵄨 1 󵄨󵄨󵄨󵄨 󵄨 ω(r) = 󵄨󵄨󵄨sin(log )󵄨v(r) + 1, 󵄨󵄨 1 − r 󵄨󵄨󵄨

1 < α < ∞,

(4.25)

where v(r) = ((1 − r)(log

α −1

e ) ) , 1−r

1 < α < ∞.

The function ω is “rapidly increasing” in the sense that lim r→1

̂ ω(r) = ∞. (1 − r)ω(r)

On the other hand, the weights like (4.25) are rather artificial, and it would be interesting to find a situation where they appear in a natural way. 4.1 (Spaces with rapidly decreasing weights). One of the first results on spaces with rapidly decreasing weights is Theorem 4.4 in [433], where it was also proved that the space hp,∞ [ω] := {u ∈ h(𝔻): Mp (r, u) = 𝒪(ω(r))} is self-conjugate if (4.8) holds. The notion of the distortion function was introduced by Siskakis [537], who proved Theorem 4.3 for 1 ≤ p = q < ⬦ but under three hypotheses, the most complicated of which is that, for all sufficiently small δ > 0, lim sup r→1−

φ(r) < ∞, φ(r + δDφ (r))

where Dφ(r) is defined by (4.4). The other two read 1

A B φ(r), ∫ φ(x) dx ≤ φ(r) and φ󸀠 (r) ≤ 1−r 1−r r

(4.26)

Further notes and results | 153

where A and B are positive constants independent of r ∈ (0, 1). It is easy to show that (4.26) implies (4.5) with L = B/A. a 4.2 (Duality and projections in spaces with exponential weights). Let ω(r) = exp(− 1−r ) 2 2 with a > 0. Let Pω be the orthogonal projection from Lω := L (𝔻, ω(r)dr) onto A2ω . Dostanić [155] proved that Pω is bounded from Lpω to Apω only for p = 2, and he posed the related problem of identifying the duals of Apω for p ≥ 1, p ≠ 2. In [137], this problem was solved by Constanin and Peláez, who proved the following:

Theorem (CP). The operator Pω is bounded from Lpωp/2 to Apωp/2 whenever 1 < p < ⬦, and the dual of Apωp/2 can be identified with Ap p󸀠 /2 . 󸀠

ω

There are more general results in [137]. 4.3 (Projections in Bergman spaces with normal weights). Let Pω be as in Further notes 4.2. Then the reproducing kernel Bω,z (w) is uniquely defined by the relation f (z) = ∫ f (w)Bω,z (w)ω(w) dA(w). 𝔻

Peláez and Rättyä [463] considered various properties of Bω and Pω under the hypotĥ − t), esis that ω and φ are regular radial weights. In our language, this means that ω(1 0 < t < 1, is normal in (0, 1) or, equivalently, that ω(1 − t) belongs to Δ[α, β] for some β > α > −1. One of the most interesting results reads as follows. Theorem (PR). Let 1 < p < ⬦, and let ω and φ be regular weights. Then the following conditions are equivalent: (a) Pω+ : Lpφ → Lpφ is bounded; (b) Pω : Lpφ → Lpφ is bounded;

(c) (ω/φ)p φ is a regular weight. 󸀠

Here 󵄨 󵄨 Pω+ f (z) = ∫󵄨󵄨󵄨f (w)Bω,z (w)󵄨󵄨󵄨ω(w) dA(w). 𝔻

In [463] the reader will find subtle estimates for the reproducing kernels. In the case of a regular weight ω, it is shown that the dual of Apω (1 < p < ⬦) is

isomorphic to Apω under the pairing 󸀠

∫ f (z)g(z)ω(|z|) dA(z). 𝔻

The dual of A1ω is equal to the Bloch space under the same pairing.

154 | 4 Mixed-norm spaces with nonstandard weights Compare with our Theorems 13.4 and 13.6, where the dual of hp,q is given in the [ϕ]

case of subnormal (in our terminology) weights. Since (in [463]) Apω = hp,p , where [ϕ] ̂ − x)p and ϕ is normal (see Section 4.3.h), it turns out that our results are ϕ(x) = ω(1 more general. 4.4 (Bergman spaces with nonradial weights). The author does not know what is the stage of knowledge in the area of Bergman spaces with nonradial weights. The analytic Bergman spaces Apμ are defined by 󵄨p 󵄨 ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 dμ(z) < ∞,

p ∈ ℝ+ ,

𝔻

where μ is a positive measure on 𝔻. We can take, in particular, dμ(z) = ω(z) dA(z), where ω is not radial. The same holds for the weighed space Hω∞ defined by |f (z)| ≤ Cω(z). Concerning Bergman spaces, the interested reader should read the aforementioned thesis [45]. Bonet and Vukotić [95] studied the completeness of Hω∞ defined over an arbitrary subdomain of ℂ. There are useful references in that paper; some of them concern the completeness of Apμ . According to Peláez and Rättyä [461], there are positive weights ω such that the polynomials are not dense in Apω . This book contains a few results in this direction (e. g., Lemmas 9 and 10) and numerous references concerning “nonradial” spaces. Bao et al. [58] consider the Bergman space Apω where ω is a positive nonradial weight that has the Harnack property, which means that there are positive constants C and ε < 1 such that ω(z1 )/ω(z2 ) ≤ C whenever dh (z1 , z2 ) < ε. They proved that then p 󵄨p 󵄨p 󵄨 󵄨 ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|) ω(z) dA(z) ≤ C ∫󵄨󵄨󵄨f (z) − f (0)󵄨󵄨󵄨 ω(z) dA(z). 𝔻

(∗)

𝔻

The reverse inequality holds under some additional, rather complicated, conditions. We note that, for any fixed p, there are radial weights ω that have the Harnack property but for which the reverse of (∗) does not hold. For example, if we take ϕ(t) = p,p coincides with Apω , where 1/ log(2/t), then the space H[ϕ] ω(z) = (log

2 ) 1 − |z|

−p−1

1 1 − |z|

has the Harnack property. On the other hand, the condition p 󵄨p 󵄨 ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|) ω(z) dA(z) < ∞ 𝔻

means that f 󸀠 belongs to the space Hψp,p , where 2 ψ(t) = t(log ) t

−(p+1)/p

.

(∗∗)

Further notes and results | 155

p,p The weight ψ is normal, and we can use Lemmas 4.13 and 4.8 to show that H[ϕ] ≠ Hψp,p for p = 2. These spaces are different for all p, which can be seen by using Theorems 13.1 and 6.15.

4.5 (Spaces with nonradial weights. II). Arroussi and Pau [46] consider the space Apω , where ω = e−2φ , φ ∈ C 2 , Δφ > 0, (Δφ(z))−1/2 ≍ τ(z), the function τ is Lipschitz in 𝔻, and 0 < τ(z) ≤ C(1 − |z|) for z ∈ 𝔻. If in addition for each m ≥ 1, there are constants bm > 0 and 0 < tm < 1/m such that τ(z) ≤ τ(ξ ) + tm |z − ξ |,

for |z − ξ | > bm τ(ξ ),

then ω is said to be of class ℰ . An example of a nonradial weight of class ℰ is given by ωp,f (z) = |f (z)|p ω(z), where p > 0, ω is a radial weight of class ℰ , and f is a nonvanishing analytic function in Apω . The class ℰ also contains exponential weights such as ωα (z) = exp(

−c ), (1 − |z|2 )α

α > 0, c > 0.

Theorem (AP1). Let Kz be the reproducing kernel of A2ω , where ω is a weight in the class ℰ . For each M ≥ 1, there exists a constant C > 0 (depending on M) such that, for all z, ξ ∈ 𝔻, we have M

min(τ(z), τ(ξ )) 1 1 󵄨 󵄨󵄨 ω(z)−1/2 ω(ξ )−1/2 ( ) . 󵄨󵄨Kz (ξ )󵄨󵄨󵄨 ≤ C τ(z) τ(ξ ) |z − ξ | Using this and some other facts, the following is proved. Theorem (AP2). Let 1 ≤ p < ∞ and ω ∈ ℰ . The Bergman projection Pω : Lpωp/2 󳨀→ Apωp/2 is bounded. There are theorems on complex interpolation and duality. For example, the dual 󸀠 is isomorphic to Ap p󸀠 /2 , 1 < p < ⬦. This generalizes the result of [137] mentioned ω in Further notes 4.2. of Apωp/2

4.6. Zygmund’s theorem, mentioned after formulation of Theorem 4.11, was generalized by Gnuschke and Pommerenke in the following way [209]. ∞ kn Theorem (GP). If f (z) = ∑∞ n=1 cn z is a lacunary series, then ∑n=1 |cn | < ∞ if and only if 󸀠 there is a curve γ with γ ∩ 𝕋 ≠ 0 such that ∫γ |f (z)| |dz| < ∞.

5 Taylor coefficients and maximal functions In this chapter, we present two proofs of the Hardy–Littlewood “∑ np−2 ” inequality on the Taylor coefficients of an H p -function. One proof is elementary and is based on Green’s formula, that is, on the Hardy–Stein identities. The second proof is based on a Marcinkiewicz-type interpolation theorem for Hardy spaces. We give applications of this and other inequalities to the partial sums and the Cesàro means of the Taylor series. In particular, we use the inequality |f ̂(n)|p ≤ c(n + 1)1−p ‖f ‖pp (0 < p < 1) and the Hardy–Littlewood complex maximal theorem to prove the Hardy–Littlewood– Sunouchi theorem. We also present some improvements of the coefficient theorems. The “∑ np−2 ” inequality is used to prove an inequality that leads us to the conjecture that the Fourier series of an Lp function, 1 < p ≤ 2, is (C, α)-summable for α > 1/p − 1; since α < 0, this “improves” the Carleson–Hunt convergence theorem.

5.1 Using Green’s formula Besides the Housdorff–Young theorem and the inequality 󵄨󵄨 ̂ 󵄨󵄨 1/p−1 ‖f ‖p , 󵄨󵄨f (n)󵄨󵄨 ≤ Cp (n + 1)

f ∈ H p, 0 < p ≤ 1

(5.1)

(see Corollary 1.24), there is another very important result on the coefficients of an H p -function due to Hardy and Littlewood [230]. Theorem 5.1 (Hardy–Littlewood Σ-inequality). If f ∈ H p , 0 < p ≤ 2, then ∞

󵄨p 󵄨 Kp (f ) := ∑ (n + 1)p−2 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 ≤ Cp ‖f ‖pp . n=0

(5.2)

If Kp (f ) < ∞ for some p ≥ 2, then f ∈ H p and ‖f ‖pp ≤ Cp Kp (f ). The inequality ‖f ‖pp ≤ Cp Kp (f ) is, of course, obtained from (5.2) by a duality argument. We have already proved (5.2) for p = 1 with Cp = π, and because obviously the inequality holds for p = 2 with Cp = 1, it is reasonable to expect that the case 1 < p < 2 can be resolved by means of interpolation. In turns out that this is possible because a Riesz–Thorin-type theorem (due to Zygmund) for H p -spaces holds. We postpone this approach to the next section. In this section, we present a proof based on Green’s formula. This proof was given 1983 in the author’s PhD thesis [420] and in the recent paper [452]. Proof. Let 0 < p < 2, Φ(t) = t p , f (0) = 0. Denote Q(t) = Φ(√t), https://doi.org/10.1515/9783110630855-005

∞

n

2

ω(r) = ( ∑ |an |r ) , n=1

158 | 5 Taylor coefficients and maximal functions ∞

σ(r) = ∑ n|an |2 r 2n−1 = n=1

1 󵄨󵄨 󸀠 󵄨󵄨2 ∫ 󵄨f 󵄨 dA, πr 󵄨 󵄨 r𝔻

where an = f ̂(n). Let rEΦ (r, f ) = ∫ r𝔻

Φ(|f (z)|) 󵄨󵄨 󸀠 󵄨󵄨2 󵄨f (z)󵄨󵄨 dA(z). |f (z)|2 󵄨

(5.3)

Since |f (z)|2 ≤ ω(r) (|z| < r) and the function Q(t)/t is decreasing, we have rEΦ (r, f ) ≥

Q(ω(r)) Q(ω(r)) 󵄨󵄨 󸀠 󵄨󵄨2 σ(r). ∫ 󵄨󵄨f 󵄨󵄨 dA = πr ω(r) ω(r) r𝔻

It follows that KIΦ (f ) =

K‖f ‖pp

1

≥∫ 0

Q(ω(r)) σ(r) dr. ω(r)

On the other hand, Q(x) y ≥ Q(y) − Q(x), x

x > 0, y > 0,

because the function Q(t)/t is decreasing. The last two inequalities along with the Hardy–Stein identity imply 1

1 KIΦ (f ) ≥ ∫[Q(Cσ(r)) − Q(ω(r))] dr, C

(5.4)

0

where C > 1 is a constant which will be chosen later on. Applying Lemma 4.8, we get 1

∞

0

n=0

∫ Q(σ(r)) dr ≥ K1 ∑ 2−n Q(2n ∑ |ak |2 ), 1

k∈In

2

∞

∫ Q(ω(r)) dr ≤ K2 ∑ 2−n Q[( ∑ |ak |) ], n=0

0

k∈In

where In = {k: 2n−1 ≤ k < 2n }, I0 = {0}, and K1 and K2 are positive constants depending only on p. Using this and the inequality 2

( ∑ |ak |) ≤ 2n ∑ |ak |2 , k∈In

k∈In

5.1 Using Green’s formula

| 159

we get 1

1

1

∫ Q(ω(r)) dr ≤ K3 ∫ Q(σ(r)) dr ≤ 0

0

1 ∫ Q(Cσ(r)) dr, 2 0

where C is chosen so that C p/2 ≥ 2K3 . From this and from (5.4) we infer 1

∞

0

n=0

KIΦ (r) ≥ ∫ Q(σ(r)) dr ≍ ∑ 2−n Q(2−n ∑ k 2 |ak |2 ). k∈In

(5.5)

Since the function Q is concave, we have Q(2−n ∑ k 2 |ak |2 ) ≥ 2−n ∑ Q(k 2 |ak |2 ) = 2−n ∑ Φ(k|ak |), k∈In

k∈In

k∈In

which along with (5.5) concludes the proof in the case p < 2. If p > 2, then we use the inequality Q(x) y ≤ Q(y) + Q(x) x and proceed as before. This proof can be easily extended to Hardy–Orlicz spaces, which can be defined in various ways. For example, if Φ is an arbitrary increasing continuous function from [0, ∞) to [0, ∞) with Φ(0) = 0, we set 󵄨 󵄨 IΦ (f ) = sup ∫ − Φ(󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨)|dζ |. 0 α > 0, then we define the function t

s

N(t) = ∫(∫ 0

0

1 1

ds Φ(x) Φ(sx) dx) = ∫∫ ds dx. x s sx 0 0

This function is of class C 2 on (0, ∞) and “lies” between t α and t β , and therefore we have the Hardy–Stein identity 2πr

d 󵄨 󵄨2 I (r, f ) = ∫ (N 󸀠󸀠 (|f |) + N 󸀠 (|f |)/|f |)󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 dA dr N r𝔻

(5.6)

160 | 5 Taylor coefficients and maximal functions (see [484, Lemma 5.1]). Simplifying the expression under the integral, we get 2π

d I (r, f ) = EΦ (r, f ), dr N

(5.7)

where EΦ (r, f ) is defined by (5.3). On the other hand, using the inequalities (sx)β Φ(t) ≤ Φ(sxt) ≤ (sx)α Φ(t), we get Φ(t)/(β2 ) ≤ N(t) ≤ Φ(t)/(α2 ), and hence 1

K −1 IΦ (f ) ≤ ∫ EΦ (r, f ) dr ≤ KIΦ (f ),

(5.8)

0

where K is a constant depending only on α and β.

5.2 Using interpolation of operators on H p In this section, we use a Marcinkiewicz-type theorem to deduce (5.2) from (5.1) for all p < 2. Theorem 5.2 (Kislyakov–Xu [313]). Let μ be a sigma-finite measure over a set Ω, let 0 < p < q < ⬦, and let T be a quasilinear operator from H p to the set of all nonnegative μ-measurable functions. Assume that there exist constants C1 and C2 independent of f such that ‖Tf ‖p,⋆ ≤ C1 ‖f ‖p ,

‖Tf ‖q,⋆ ≤ C2 ‖f ‖q ,

f ∈ H p,

(5.9)

f ∈H .

(5.10)

q

Then for every s ∈ (p, q), there exists a constant C independent of f such that ‖Tf ‖s ≤ C‖f ‖s ,

f ∈ H s.

As before, we use ‖ ⋅ ‖p,⋆ to denote the quasinorm in Lp,⋆ . “Quasilinear” means T(f + g) ≤ K(Tf + Tg); see p. 498. Observe that the case q = ∞ is now excluded. In that case the things are deeper, as we can see in [280] (cf. [65, Ch. 5]). Proof. The idea is the same as in the case of the classical Marcinkiewicz theorem (see Theorem C.2 and its proof). The obstacle is in that we cannot use the “old” decomposition of f because gλ need not be analytic. Fortunately, we have the decomposition f = gλ + hλ , where gλ and hλ are analytic, and ‖gλ ‖pp ≤ A ∫ |f |p dl |f |>λ

5.2 Using interpolation of operators on Hp

|

161

and ‖hλ ‖qq ≤ A ∫ |f |q dl + Aλ2q ∫ |f |−q dl |f |>λ

|f |≤λ

where A = const (Lemma 5.1). Assuming that T(f + g) ≤ Tf + Tg and C1 = C2 = 1, we have μ(Tf , λ) ≤ μ(Tgλ , λ/2) + μ(Thλ , λ/2) ≤ A(2/λ)p ∫ |f |p dl + A(2/λ)q ∫ |f |q dl |f |>λ

|f |≤λ

+ A(2λ)q ∫ |f |−q dl = I1 (λ) + I2 (λ) + I3 (λ). |f |>λ

Now we multiply this by sλs−1 and integrate these three summands from λ = 0 to ∞. For instance, we have |f |

∞

s ∫ I3 (λ)λs−1 dλ = A2q ∫ |f |−q dl ∫ λq λs−1 dλ = 0

0

𝕋

A2q ∫ |f |−q |f |q+s dl. q+s 𝕋

In the case of I1 and I2 , we proceed similarly. Lemma 5.1 (Bourgain). If f ∈ H p (p ∈ ℝ+ ) and λ > 0, then there are functions h ∈ H ∞ and g ∈ H p such that f = g + h, |f∗ | λ , ), and λ |f∗ | 󵄨p 󵄨󵄨 ∫ 󵄨󵄨f∗ (ζ )󵄨󵄨󵄨 |dζ |,

|h∗ | ≤ Cλ min( ‖g‖pp ≤ C

ζ ∈𝕋, |f∗ (ζ )|>λ

where C depends only on p. Proof. Let λ > 0 and define the functions A on 𝕋 and ϕ on 𝔻 by A = max(1, (

p/2

|f∗ | ) λ

)

and ϕ =

1

̃ 𝒫 [A] + iP[A]

.

Since 𝒫 [A] ≥ 1 in 𝔻, we have 0 < |ϕ| ≤ 1 in 𝔻. Therefore the function ψ = 1 − (1 − ϕ4/p )

2/p

is well-defined, analytic, and bounded in 𝔻. We claim that the functions h = ψf and g = (1 − ψ)f satisfy the desired conditions.

162 | 5 Taylor coefficients and maximal functions Since |ϕ∗ | ≤ 1/A and |ψ| ≤ C|ϕ|4/p (by Schwarz’s lemma), we have |h∗ | ≤ C|f∗ | min(1, (

−2

|f∗ | ) ), λ

and this gives the desired estimate for h. On the other hand, 󵄨󵄨 1 − A + iH(A) 󵄨󵄨2/p 󵄨󵄨 󵄨 |g∗ | ≤ C|1 − ϕ∗ |2/p |f∗ | = C 󵄨󵄨󵄨 󵄨 |f∗ | 󵄨󵄨 A + iH(A) 󵄨󵄨󵄨 2/p

A − 1 |H(A − 1)| + ) |f∗ | A A 󵄨2/p 󵄨 ≤ C(1 − 1/A)2/p |f∗ | + Cλ󵄨󵄨󵄨H(A − 1)󵄨󵄨󵄨 ,

≤ C(

where H is the Hilbert operator; we have used that H(1) = 0 and A−2/p ≤ λ/|f∗ |. Since A = 1 on the set {|f∗ | ≤ λ} and H is bounded on L2 , we see that 󵄨2 󵄨 ‖g‖pp ≤ C ∫(1 − 1/A)2 |f∗ |p dl + Cλp ∫󵄨󵄨󵄨H(A − 1)󵄨󵄨󵄨 dl 𝕋

𝕋

≤ C ∫ |f∗ |p dl + Cλp ∫(A − 1)2 dl |f∗ |>λ

𝕋

≤ C ∫ |f∗ |p dl. |f∗ |>λ

This completes the proof. Proof of Theorem 5.1.1 Define the measure μ on ℕ by μ({n}) = (n + 1)−2 and the operator (Tf )(n) = (n + 1)|f ̂(n)|. Let 0 < p < s < 2. Since ‖Tf ‖2,⋆ ≤ ‖Tf ‖2 = ‖f ‖2 , we see that condition (5.10) is satisfied (q = 2). It remains to prove that 󵄨 󵄨 aλ := μ({n: (n + 1)󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 > λ}) ≤ (C2 /λ)p ‖f ‖pp ,

λ > 0.

We may assume that λ > 1 because μ is finite. Let ‖f ‖p = 1. We have, by (5.1), aλ ≤ μ({n: K(n + 1)1/p > λ}) = μ({n: (n + 1) > (λ/K)p }) ≤

∑ (n + 1)−2 ≤ C3 (K/λ)p .

n≥(λ/K)p

Hence (5.9) holds with C2 = C3 K p . Now Theorem 5.2 gives us that ∞

󵄨s 󵄨 ∑ ((n + 1)󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨) (n + 1)−2 ≤ C‖f ‖ss ,

n=0

as desired. 1 See the proof of Theorem C.6.

5.2 Using interpolation of operators on Hp

|

163

If 0 < p < 1, then both inequalities (5.1) and (5.2) are contained in the following [381]: Theorem 5.3. If f ∈ H p , 0 < p < 1, then ∞

󵄨p 󵄨 ∑ (n + 1)p−2 sup 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 ≤ Cp ‖f ‖pp . 0≤k≤n

n=0

Proof. Define the measure μ as before and the operator T as 󵄨 󵄨 (Tf )(n) = (n + 1) sup 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 0≤k≤n

and then use (5.1) twice to show (5.9) and (5.10) for 0 < p < q < 1. The result follows.

5.2.1 An embedding theorem We have proved that if f ∈ H p , p < ∞, then Mq (r, f ) = 𝒪((1 − r)1/q−1/p ) (r → 1)

for q > p;

(5.11)

see (1.43). Then, since the polynomials are dense in H p , we can prove that (5.11) remains valid if we replace “𝒪” by “o”. This is further improved by the following: Theorem 5.4 (Hardy–Littlewood Mqp -theorem). If f ∈ H p , p ∈ ℝ+ , and ∞ ≥ q > p, then 1

∫ Mqp (r, f )(1 − r)−p/q dr ≤ Cp,q ‖f ‖pp .

(5.12)

0

q,p . From this and from Theorem 3.11 we obtain This can be written as H p ⊂ H1/p−1/q the following:

Theorem 5.5. If f ∈ H p , p ∈ ℝ+ , ∞ ≥ q > p, and s ≥ p, then 1

∫ Mqs (r, f )(1 − r)sα−1 dr ≤ C‖f ‖sp , 0

where α = 1/p − 1/q (> 0), and C is independent of f . In the case s = q, this can be written in the following form. Corollary 5.2. If f ∈ H p , p ∈ ℝ+ , and ⬦ > q > p, then q/p−2 󵄨q 󵄨 dA(z) ≤ Cp,q ‖f ‖qp . ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) 𝔻

164 | 5 Taylor coefficients and maximal functions Proof of Theorem 5.4. We can suppose that q is finite because M∞ (r, f ) ≤ (1 − r)−1/q Mq (√r, f ). Define the (quasilinear) operator T: H ρ 󳨃→ C(0, 1) by (Tf )(r) = (1 − r)−1/q Mq (r, f ),

0 < r < 1,

where ρ is chosen so that 0 < ρ < p. We will prove that T maps H s into Ls,⋆ (0, 1) for s = ρ and s = q. By Theorem 5.2 this will imply that T maps H p into Lp (0, 1) for ρ < p < q, which will conclude the proof. In the case s = ρ, using inequality (1.43), we get Tf (r) ≤ A(1 − r)−1/ρ ‖f ‖ρ ,

(5.13)

where A is a positive constant. If ‖f ‖ρ = 1, then it follows from (5.13) that 󵄨 󵄨 󵄨 󵄨󵄨 −1/ρ > λ}󵄨󵄨󵄨 󵄨󵄨{r ∈ (0, 1): Tf (r) > λ}󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨{r ∈ (0, 1): A(1 − r) = min{1, A/λρ }, which proves that T is of weak type (ρ, ρ). In the case s = q the desired conclusion follows from the inequality Mq (r, f ) ≤ ‖f ‖q . This completes the proof. Remark. If f is harmonic in 𝔻, then (5.12) holds for q > p > 1. This was proved by Flett [193] using Marcinkiewicz’s theorem. Remark 5.3. Using the H–L decomposition lemma, we can reduce the proof of Theorem 5.4 to the case q = 2. In this case the theorem was proved in the preceding section; see (5.5). On the other hand, the proof can be reduced to the case p = 2; see Further notes 5.1. Some improvements of Theorem 5.1 To express some inequalities in the form of embedding theorems, we use a class of mixed-norm sequence spaces. The Kellogg spaces. The space ℓαp,q (0 < p, q ≤ ∞), α ∈ ℝ, introduced by Kellogg [304], consists of complex sequences {ak }∞ 0 such that 1/p ∞

{2kα (∑ |aj |p ) j∈Ik

The quasinorm in X =

ℓαp,q

}

k=0

∈ ℓq .

is given by

1/p ∞ 󵄩 󵄩󵄩 󵄩󵄩 󵄩 kα 󵄩󵄩 󵄩󵄩 p 󵄩󵄩{aj }󵄩󵄩X = 󵄩󵄩󵄩{2 (∑ |aj | ) } 󵄩󵄩󵄩 . 󵄩󵄩 󵄩 ℓq k=0 󵄩 j∈Ik

It follows that ℓ0p,p is identical with ℓp .

5.2 Using interpolation of operators on Hp

165

|

These spaces often appear in the theory of coefficient multipliers, which will be considered later on. As a particular case of Kellogg’s theorem, we have the following: Lemma 5.4. Let 1/s󸀠 + 1/s = 1 for 1 ≤ s ≤ ∞ and s󸀠 = ∞ for 0 < s < 1. If 0 < q ≤ ⬦, then p󸀠 ,q󸀠 the dual of ℓαp,q is isometrically isomorphic to ℓ−α under the pairing ∞

⟨{an }, {bn }⟩ = ∑ an bn , n=0

the series being absolutely convergent. Recall that {an } ∈ ℓαp,⬦ if (∑k∈In |ak |p )1/p = o(2−nα ). In the case 1 < p < 2, there is an improvement of Theorem 5.1; see Theorem C.5. The following theorem contains, however, an improvement of (5.2) in other direction. Theorem 5.6. Let 1 < p < 2 and 0 < q < p󸀠 . If f ∈ H p , then p/q

∞

󵄨q 󵄨 ∑ 2n(p−1) (2−n ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 )

n=0

k∈In

≤ Cp,q ‖f ‖pp ,

(5.14)

or, equivalently, q,p H p ⊂ ℓ1/p 󸀠 −1/q .

Before proving this theorem, note that if q = p, then (5.14) reduces to (5.2). On the other hand, combining the Hausdorff–Young inequality and Theorem 2.21, we get ‖f ‖2p

1

∞

0

n=0

2/p󸀠

󵄨p󸀠 󸀠 ≥ cp ∫(1 − r)( ∑ n 󵄨󵄨f ̂(n)󵄨󵄨󵄨 r np ) p󸀠 󵄨󵄨

dr

and hence, by Lemma 4.10, H p ⊂ ℓ0p ,2 󸀠

(1 ≤ p < 2), 2

which improves, up to a multiplicative constant, the classical Hausdorff–Young inp󸀠 ,2 q,p are incomparable. The space Y is not equality. The spaces X = ℓ1/p 󸀠 −1/q and Y = ℓ0 n

a subset of X, which can be deduced from the fact that the function f (z) = ∑ cn z 2 belongs to H p if and only if f ∈ Y. On the other hand, it is easy to see that if f (z) = ∑ bn z n , bn ↓ 0, then the condition {bn } ∈ X is equivalent with ∑ 2n(p−1) bp2n < ∞, which is equivalent with f ∈ H p by Theorem 5.9 below. 2 This inclusion was proved by Kellogg [304].

166 | 5 Taylor coefficients and maximal functions Proof of Theorem 5.6. Since (2−n ∑In |f ̂(k)|q )1/q increases with q, we may assume that 2 < q < p󸀠 . Then we have p < q󸀠 < 2, so we can apply Theorem 5.4 and the Hausdorff– Young inequality to obtain 1

‖f ‖pp ≥ c ∫(1 − r)−p/q Mqp󸀠 (r, f ) dr 󸀠

0

1

−p/q󸀠

≥ c ∫(1 − r)

󵄨q 󵄨 ( ∑ 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 r nq )

p/q

n=0

0 ∞

∞

p/q

∞

󵄨q 󵄨 ≥ c ∑ 2n(p/q −1) ∑ ( ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 ) 󸀠

n=0

n=0 k∈In

.

The result follows. In the same way, we prove the following: Theorem 5.7 (Mateljević–Pavlović [381]). If f ∈ H 1 , then 1/q

∞

󵄨q 󵄨 ∑ (2−n ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 )

n=0

k∈In

≤ C‖f ‖1 ,

0 < q < ⬦,

(5.15)

where In = {k : 2n ≤ k < 2n+1 } for n ≥ 1, and I0 = {0, 1}. Relationship with the Hardy–Littlewood–Sobolev theorem Let α > 0. Recall that, for a function f ∈ C(𝔻), we defined 𝒥α f by 1

α−1

1 1 𝒥α f (z) = ∫(log ) Γ(α) ρ

f (ρz) dρ,

z ∈ 𝔻.

0

If f is analytic in 𝔻, then ∞

n

𝒥α f (z) = ∑ (n + 1) f ̂(n)z . −α

n=0

The following theorem was proved by Hardy and Littlewood [237] and Sobolev [545]. Theorem 5.8. Let 0 < p < q < ⬦ and α =

1 p

− q1 . Then 𝒥α maps H p into H q .

It may be interesting that the operator 𝒥α is not compact; see Theorem 12.19. Proof [21]. Assuming, as we may, that f is a polynomial and that ‖f ‖p = 1, we have |f (ρeiθ )| ≤ (1 − ρ)−1/p and |f (ρeiθ )| ≤ Mrad f (eiθ ), that is, 󵄨󵄨 −1/p iθ 󵄨 , Mrad f (eiθ )}. 󵄨󵄨f (ρe )󵄨󵄨󵄨 ≤ min{(1 − ρ)

5.2 Using interpolation of operators on Hp

|

167

1/2

Since also log(1/ρ) ≍ 1 − ρ as ρ → 1− and ∫0 (log 1/ρ)α−1 dρ < ∞, we have 1

󵄨󵄨 α−1 −1/p iθ 󵄨 , M} dρ, 󵄨󵄨𝒥α f (e )󵄨󵄨󵄨 ≤ CM∞ (1/2, f ) + C ∫(1 − ρ) min{(1 − ρ) 0

where M = Mrad f (eiθ ) + 1. Hence 1

1−M −p

󵄨󵄨 α−1−1/p iθ 󵄨 dρ + C ∫ (1 − ρ)α−1 M dρ 󵄨󵄨𝒥α f (e )󵄨󵄨󵄨 ≤ C1 + C ∫ (1 − ρ) 0

1−M −p

C C ≤ C1 + M p/q + M −pα M q α = C1 + C2 (Mrad f (eiθ ) + 1)

p/q

.

It follows that p 󵄨󵄨 iθ iθ 󵄨q 󵄨󵄨𝒥α f (e )󵄨󵄨󵄨 ≤ C3 + C4 (Mrad f (e ) + 1) .

Now the desired result is obtained by integration an using the complex maximal theorem. Taking q = 2, we get the following: Corollary 5.5. If 0 < p < 2 and f ∈ H p , then ∞

󵄨2 󵄨 Kp󸀠 (f ) := ∑ (n + 1)1−2/p 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 ≤ Cp ‖f ‖2p .

(5.16)

n=0

Consequently, by duality, ∞

󵄨2 󵄨 ‖f ‖2q ≤ Cq ∑ (n + 1)1−2/q 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 , n=0

2 ≤ q < ⬦.

(5.17)

It is interesting to compare (5.16) with (5.2). Let Xp (resp., Yp ) be the space of the sequences {an } such that Kp󸀠 ({an }) < ∞ (resp., Kp ({an }) < ∞; see (5.2)) normed in the obvious way. First, we show that they are incomparable (treated independently of H p -spaces). Assuming that Yp ⊂ Xp , we have, by the closed graph theorem, ‖en ‖Xp =

(n + 1)1/2−1/p ≤ Cp ‖en ‖Yp = Cp (n + 1)1−2/p , that is, (n + 1)1/p−1/2 ≤ Cp , which is not true.

On the other hand, the sequence an = n1/p−1 /(log n)1/p , n ≥ 2, belongs to Xp \ Yp . However, if p ≤ 1, then (5.16) is a consequence of (5.1) and (5.2). Indeed, if ‖f ‖p = 1, then ∞

∞

n=0

n=0

2−p

∑ (n + 1)1−2/p |an |2 ≤ C ∑ (n + 1)1−2/p |an |p ((n + 1)1/p−1 )

168 | 5 Taylor coefficients and maximal functions ∞

= C ∑ (n + 1)p−2 |an |p . n=0

It is not clear whether such a simple trick is possible when 1 < p < 2. In this case, we can deduce (5.16) from (5.14). To see how it goes, we take q = 2 and rewrite (5.14) as p/2

∞

󵄨2 󵄨 ∑ (2n(1−2/p) ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 )

n=0

k∈In

≤ Cp ‖f ‖pp .

Now we use the inequality ∑n xnp/2 ≥ (∑n xn )p/2 (xn ≥ 0) to obtain (5.16). In a similar way, starting from (5.14) and adding the Hausdorff–Young inequality, we obtain ∞

󵄨q 󵄨 ∑ (n + 1)q−q/p−1 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 ≤ Cp,q ‖f ‖qp ,

n=0

1 < p < 2, p ≤ q ≤ p󸀠 ,

(5.18)

which unifies (5.2) and (5.16); see [230, p. 161]. Remark 5.6. If 0 < p ≤ 1, then Theorem 5.8 holds for q = ∞. To see this, we write 𝒥1/p f = 𝒥1 𝒥1/p−1 f . It follows from Theorem 5.8 that 𝒥1/p−1 f ∈ H 1 if f ∈ H p . Then, by Hardy’s inequality (Theorem 2.15) we have ∞

󵄨 󵄨 ∑ (n + 1)−1/p 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 ≤ C‖f ‖p ,

n=0

which implies 𝒥1/p f ∈ H ∞ . On the other hand, if 1 < p < ∞, then the theorem does not hold for q = ∞. This can be deduced from the fact that if f ̂(n) decreases to zero, then f ∈ H p if and only if ∑(n+1)p−2 |f ̂(n)|p < ∞ (see Theorem 5.9). In particular, the function n 1/p−1 n z / log n belongs to H p , whereas 𝒥1/p f (z) = ∑∞ f (z) = ∑∞ n=2 z /(n log n) is not in n=2 n ∞ H . It should be noted that in the case p = 1 and q = ∞, Privalov’s theorem 1.32 can be restated to get a result stronger than that 𝒥1 maps H 1 into H ∞ : actually, 𝒥1 maps H 1 into the subclass of A(𝔻) consisting of absolutely continuous functions on 𝕋. Exercise 5.7 ([276, Theorem 12.1.2]). Using Aleksandrov’s proof of the Hardy–Littlewood–Sobolev theorem, show that: If f ∈ H p and f (z) = 𝒪((1 − |z|)−γ ), where 0 < γ ≤ 1/p, then 𝒥β f ∈ H q , where q = γp/(γ − β) and 0 < β < γ.

5.2.2 The case of monotone coefficients If the sequence {f ̂(n)} is decreasing, then both implications in Theorem 5.1 become equivalences.

5.2 Using interpolation of operators on Hp

|

169

Theorem 5.9 (H–L monotone-coefficients theorem [234]). Let 1 < p < ⬦, f ∈ H(𝔻), and let the sequence {f ̂(n)} (→ 0) be real and decreasing. Then f ∈ H p if and only if ∞

Kp (f ) := ∑ (n + 1)p−2 f ̂(n)p < ∞, n=0

and we have ‖f ‖p ≍ Kp (f ). In view of Theorem 5.1, we have to prove two inequalities: ‖f ‖p ≤ CKp (f ),

(5.19)

1 < p < 2,

Kp (f ) ≤ C‖f ‖p ,

2 < p < ∞.

Let In = {k : 2n ≤ k < 2n+1 } for n ≥ 1 and I0 = {0, 1}, and let Δn f (z) = ∑ f ̂(k)z k . k∈In

To prove (5.19), we use the Riesz projection theorem, Lemmas 2.6 and 4.9, and an inequality of Littlewood and Paley (which will be proved in Chapter 6, Theorem 6.13). We get ‖f ‖pp

1

󵄨p 󵄨 ≤ C 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C ∫(1 − r)p−1 Mpp (r, f 󸀠 ) dr 0 ∞

󵄩p 󵄩 󵄨p 󵄨 ≤ C 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C ∑ 2−n 󵄩󵄩󵄩Δn (f 󸀠 )󵄩󵄩󵄩p . n=1

Hence, by Exercise A.14, ∞

‖f ‖pp ≤ C ∑ ‖Δn f ‖pp , n=0

1 < p < 2.

(5.20)

p p Hence, by duality, ∑∞ n=0 ‖Δn f ‖p ≤ C‖f ‖p , 2 < p < ∞. To finish the proof, we need another lemma. k Lemma 5.8. If Pn (z) = ∑2n−1 k=n z , then, for p > 1, 1− p1

‖Pn ‖p ≍ n

,

n ≥ 1.

Proof. By Corollary 1.32 we have, for r = 1 − 1/n, 1

1

‖Pn ‖p ≥ M∞ (r, Pn )(1 − r 2 ) p ≥ r 2n−1 ‖Pn ‖∞ (1 − r 2 ) p 1− p1

≥ (1 − 1/n)2n−1 n(1/n)1/p ≥ cn

.

170 | 5 Taylor coefficients and maximal functions On the other hand, let fr (z) = (1 − rz)−1 . Then, by Exercise A.14, Lemma 2.6, and the projection theorem, 󵄩 󵄩 1 󵄩󵄩󵄩󵄩2n−1 k 󵄩󵄩󵄩󵄩 k r e ∑ 󵄩 k󵄩 󵄩󵄩 (ek (z) = z ) r 2n−1 󵄩󵄩󵄩󵄩 k=n 󵄩󵄩 1 C C −1 1− 1 ≤ 2n−1 ‖fr ‖p ≤ 2n−1 (1 − r) p ≤ Cn p , r r

‖Pn ‖p ≤

and the claim is proved. We continue the proof of the theorem. Let 1 < p < 2, and let the sequence {f ̂(n)} be decreasing. Then, by (5.20), Lemma 5.8, and Exercise A.14, ∞

p 󵄨p 󵄨 ‖f ‖pp ≤ C 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C ∑ f ̂(2n ) ‖P2n ‖pp n=0 ∞

p p−1 󵄨p 󵄨 ≤ C 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C ∑ f ̂(2n ) (2n ) n=0

∞

≤ C ∑ (n + 1)p−2 f ̂(n)p . n=0

This completes the proof for 1 < p < 2. The case p > 2 is discussed similarly. In view of the fact that the sequence en is not a Schauder basis in H 1 , it is perhaps surprising that Theorem 5.9 continues to be true for p = 1. Theorem 5.10. If f ̂(n) ∈ ℝ and f ̂(n) ↓ 0, then f ̂(n) , n+1 n=0 ∞

‖f ‖1 ≤ C ∑ where C is an absolute constant. Proof. We write f (z) as ∞

n

n=0

k=0

f (z) = ∑ (f ̂(n) − f ̂(n + 1)) ∑ z k , which holds because f ̂(n) → 0. It follows that 󵄨󵄨 󵄨󵄨 n ∞ 󵄨󵄨 󵄨󵄨 M1 (r, f ) ≤ ∑ (f ̂(n) − f ̂(n + 1)) ∫ − 󵄨󵄨󵄨 ∑ r k ζ k 󵄨󵄨󵄨|dζ | 󵄨󵄨 󵄨 󵄨 n=0 󵄨 𝕋 󵄨k=0 ∞

n

∞ 1 f ̂(k) =C∑ , k+1 k+1 k=0 k=0

≤ C ∑ (f ̂(n) − f ̂(n + 1)) ∑ n=0

where C is an absolute constant.

5.3 Strong convergence in H1

| 171

5.3 Strong convergence in H 1 For a function f analytic in 𝔻, let Pn f =

1 n 1 s f, ∑ Ln j=0 j + 1 j

n

1 j+1 j=0

where Ln = ∑

(n ∈ ℕ),

where sj f are the partial sums of the Taylor series of f . It is well known that ‖sn f ‖ ≤ CLn ‖f ‖ and that Ln is “best possible”. A direct consequence is that 1 n 1 ‖s f ‖ ≤ C‖f ‖ log n ∑ log n j=0 j + 1 j

(n ≥ 2),

(5.21)

where C is an absolute constant. It turns out, however, that the stronger inequality holds, namely 1 n 1 ‖s f ‖ ≤ C‖f ‖ (f ∈ H 1 , n ≥ 2). ∑ log n j=0 j + 1 j

(5.22)

Moreover, we have the following characterization of the space H 1 . Theorem 5.11 ([429, 542]). A function f ∈ H(𝔻) belongs to H 1 if and only if one of the following two conditions is satisfied: sup n

1 n 1 ‖s f ‖ < ∞; ∑ Ln j=0 j + 1 j

sup ‖Pn f ‖ < ∞. n

(5.23) (5.24)

Remark 5.9. It follows from the proof that the quantities occurring in (5.23) and (5.24) are equivalent to the original norm in H 1 ; in particular, (5.22) holds. Since the polynomials are dense in H 1 , we have the following consequence. Theorem 5.12. If f ∈ H 1 , then lim n

1 n 1 ‖f − sj f ‖ = 0, ∑ Ln j=0 j + 1

and, consequently, lim n

1 n 1 ‖s f ‖ = ‖f ‖. ∑ Ln j=0 j + 1 j

Corollary 5.10 ([429]). If f ∈ H 1 , then lim infn→∞ ‖f − sn f ‖ = 0.

172 | 5 Taylor coefficients and maximal functions There are functions ϕ ∈ L1 such that limn ‖ϕ − sn ϕ‖ = ∞; such an example is given −1/2 cos jθ. Since the sequence (log j)−1/2 is convex, the function by ϕ(eiθ ) = ∑∞ j=2 (log j)

belongs to L1 (see Remark 1.26). Furthermore, we can show that ‖f − sn f ‖ ≥ c(log n)1/2 , c = const > 0. We omit the details. By means of Fatou’s lemma from Corollary 5.10 we obtain the following: Corollary 5.11. If ϕ ∈ H 1 (𝕋), then lim infn→∞ |ϕ(eiθ ) − sn ϕ(eiθ )| = 0 a. e.

On the other hand, there exists a function ϕ ∈ H 1 (𝕋) whose Fourier series diverges almost everywhere (see [639, Ch. VIII (3.6)]). This result is due to Hardy and Rogosinski [241]; see Theorem 5.16. Proof of Theorem 5.11. It is obvious that (5.23) implies (5.24). To prove that f ∈ H 1 implies (5.23), let f ∈ H 1 and for fixed n ≥ 2 and w ∈ 𝔻, define the function g ∈ H 1 by g(z) = (1 − rz)−1 f (rwz) (|z| ≤ 1), j j where r = 1 − 1/n. We have g(z) = ∑∞ j=0 sj f (w)r z . Applying Hardy’s inequality (Theorem 2.15), we get ∞ 1 󵄨󵄨 1 󵄨󵄨 󵄨j ̂ 󵄨󵄨󵄨󵄨 ≤ π‖g‖. 󵄨󵄨sj f (w)󵄨󵄨󵄨r = ∑ 󵄨󵄨g(j) j + 1 j + 1 j=0 j=0 ∞

∑

Since r j = (1 − 1/n)j ≥ c for 0 ≤ j ≤ n, where c > 0 is an absolute constant, we have 2π

n

1 󵄨󵄨 󵄨 󵄨 it 󵄨 it 󵄨−1 󵄨 󵄨󵄨sj f (w)󵄨󵄨󵄨 ≤ (π/c)‖g‖ = (1/2c) ∫ 󵄨󵄨󵄨1 − re 󵄨󵄨󵄨 󵄨󵄨󵄨f (rwe )󵄨󵄨󵄨dt. j + 1 j=0 ∑

0

Integrating this inequality over the circle |w| = 1, we find n

2π

1 󵄨 󵄨−1 ‖sj f ‖ ≤ (1/2c)‖f ‖ ∫ 󵄨󵄨󵄨1 − reit 󵄨󵄨󵄨 dt, ∑ j+1 j=0 0

where we have used Fubini’s theorem. Finally, using the estimate 2π

󵄨 󵄨−1 ∫ 󵄨󵄨󵄨1 − reit 󵄨󵄨󵄨 dt ≤ C log 0

1 = C log n, 1−r

we see that (5.22) holds, and therefore (5.23) is implied by f ∈ H 1 . Let f be analytic in 𝔻. From the uniform convergence of sn f on compact sets it follows that Pn f 󴁂󴀱 f . Assuming that ‖Pn f ‖ ≤ 1 for each n, we have M1 (r, Pn f ) ≤ 1 for all n and r < 1. This implies, via the uniform convergence of Pn f on the circles |z| = r, that M1 (r, f ) ≤ 1 for every r < 1, which means that ‖f ‖ ≤ 1. Thus we have proved that (5.24) implies f ∈ H 1 , and this completes the proof.

5.3 Strong convergence in H1

| 173

5.3.1 Generalization to (C, α)-convergence The Cesàro means of order α > −1 of an analytic function f are defined by σnα f (z) =

n Γ(n + 1) Γ(α + n + 1 − k) ̂ f (k)z k , ∑ Γ(α + n + 1) k=0 Γ(n + 1 − k)

where Γ is the Euler gamma function. This can be written as 1 n α ̂ ∑ A f (k)z k , Aαn k=0 n−k

σnα f (z) = where Aαn = (

n+α ) ≍ (n + 1)α , n

n ≥ 0.

(5.25)

In particular, n

σn1 f (z) = ∑ (1 − k=0

k )f ̂(k)z k . n+1

The function f and the sequence σnα f are connected by the formula ∞

(1 − z)−α−1 f (ζz) = ∑ Aαn σnα f (ζ )z n , n=0

z ∈ 𝔻, ζ ∈ 𝕋.

(5.26)

This means that the sequence Aαn σnα f (ζ ) (n ≥ 0) coincides with the sequence of the Taylor coefficients of the function z 󳨃→ (1 − z)−α−1 f (ζz). From this and from (5.2) and (5.25) we obtain ∞

󵄨p 󵄨 ∑ (n + 1)p−2 (n + 1)αp 󵄨󵄨󵄨σnα f (ζ )󵄨󵄨󵄨 r np

n=0

2π

󵄨 󵄨−(α+1)p 󵄨󵄨 iθ 󵄨p ≤ Cp,α ∫ 󵄨󵄨󵄨1 − reiθ 󵄨󵄨󵄨 󵄨󵄨f (ζre )󵄨󵄨󵄨 dθ,

0 < r < 1.

0

After integration over ζ ∈ 𝕋, we obtain ∞

󵄩 󵄩p ∑ (n + 1)p−2+αp 󵄩󵄩󵄩σnα f 󵄩󵄩󵄩p r np ≤ Cp,α Mpp (r, f )γ(r),

n=0

where −(α+1)p+1 , α > 1/p − 1, {(1 − r) { { 2 γ(r) = {log 1−r , α = 1/p − 1, { { α < 1/p − 1. {1,

The most interesting case is α = 1/p − 1. Then we have the following:

174 | 5 Taylor coefficients and maximal functions Theorem 5.13. Let f ∈ H p , 0 < p ≤ 2, and α = 1/p − 1. Then 1 n 1 󵄩󵄩 α 󵄩󵄩p p ∑ 󵄩σ f 󵄩 ≤ Cp ‖f ‖p , Ln k=0 k + 1 󵄩 k 󵄩p

(5.27)

lim

(5.28)

1 n 1 󵄩󵄩 α 󵄩p ∑ 󵄩󵄩σk f − f 󵄩󵄩󵄩p = 0, n→∞ L k + 1 n k=0 󵄩 󵄩 lim inf󵄩󵄩󵄩σnα f − f 󵄩󵄩󵄩p = 0, n→∞

󵄨 󵄨 lim inf󵄨󵄨󵄨σnα f (ζ ) − f (ζ )󵄨󵄨󵄨 = 0 n→∞

for a. e. ζ ∈ 𝕋.

(5.29)

Remark 5.12. In the case 2 ≤ p ≤ ∞ the above proof shows only that (5.29) remains true with α = −1/2. Relation (5.29) deserves to be commented because of the following result. Theorem 5.14 (Zygmund [634]). If 0 < p < 1, then (5.29) remains true if we replace n ̂ lim inf with lim sup, that is, the series ∑∞ n=0 f (n)ζ is (C, α)-summable for a. e. ζ ∈ 𝕋. If p = 1, then by the Hardy–Rogosinski theorem (Theorem 5.16 below) there exists f ∈ H p such that sn f (ζ ) = σn0 f (ζ ) diverges a. e. on 𝕋. On the other hand, by the Carleson–Hunt theorem, if f ∈ H p and p > 1, then sn f (ζ ) → f (ζ ) a. e. on 𝕋. Since (C, α)-summability implies (C, β)-summability for α < β (see [639, Ch. III (1.1)]) and 1/p − 1 < 0 for 1 < p ≤ 2, the following question naturally arises. Problem 5.6. Let 1 < p ≤ 2 and α = 1/p − 1. Are the following assertions true: β (i) If β > α, then σn f (ζ ) → f (ζ ) for a. e. ζ ∈ 𝕋. (ii) There is a function f ∈ H p (𝕋) such that σnα f (ζ ) diverges a. e. (iii) There exists a function f ∈ C(𝕋) such that σn−1/2 f (ζ ) diverges a. e. Of course, (iii) imply (ii) for p = 1/2. Proof of (5.28). Let f ∈ H p , ε > 0, and choose a polynomial P of degree s such that ‖f − P‖ < ε. Then we have 𝒮n f :=

=

1 n 1 󵄩󵄩 α 󵄩p ∑ 󵄩σ f − f 󵄩󵄩󵄩 Ln k=0 k + 1 󵄩 k

1 m 1 n ∑ ... + ∑ . . . =: Qn + Rn , Ln k=0 Ln k=m+1

n > m > s,

where m is a fixed integer, which will be chosen later on. We easily check that limn→∞ Qn = 0. To estimate Rn we start from the inequality Rn ≤ 2p−1

1 󵄩󵄩 α 1 n 󵄩p 󵄩p 󵄩 (󵄩󵄩σk (f − P)󵄩󵄩󵄩 + 󵄩󵄩󵄩σkα P − P 󵄩󵄩󵄩 + ‖f − P‖p ) ∑ Ln k=m+1 k + 1

󸀠󸀠󸀠 = R󸀠n + R󸀠󸀠 n + Rn .

5.4 A (C, α)-maximal theorem | 175

By (5.27) we have R󸀠n ≤ C2p−1 ‖f − P‖p < C2p−1 εp and, obviously, Sn󸀠󸀠󸀠 ≤ 2p−1 εp , so it remains to deal with Sn󸀠󸀠 . Since P is a polynomial, we have 󵄩 󵄩 lim 󵄩󵄩󵄩σkα P − P 󵄩󵄩󵄩 = 0. k→∞ p−1 p Now choose m such that ‖σkα P−P‖ < ε for k > m, which implies R󸀠󸀠 ε , concluding n ≤2 the proof.

5.4 A (C, α)-maximal theorem In contrast to the case 1 < p < ⬦, the sequence {en } is not a Schauder basis in H p for p ∈ (0, 1]. Hardy and Littlewood [238] proved that this sequence is a (C, α) basis in H p for α > 1/p − 1 (p ≤ 1). Define the maximal operator σ∗α by 󵄨 󵄨 (σ∗α f )(ζ ) = sup󵄨󵄨󵄨σnα f (ζ )󵄨󵄨󵄨 (ζ ∈ 𝕋). n The nontangential maximal function M∗ f is dominated by a constant multiple of σ∗α f ; in the case α = 1, this follows from the inequality |1 − z|2 1 󵄨 󵄨󵄨 (σ f )(ζ ), 󵄨󵄨f (ζz)󵄨󵄨󵄨 ≤ (1 − |z|)2 ∗ whereas the latter follows from ∞

f (ζz) = (1 − z)2 ∑ (n + 1)(σn1 f )(ζ )z n . n=0

Theorem 5.15 (Hardy–Littlewood–Sunouchi). If f ∈ H p , p ∈ ℝ+ , and α > max{0, 1/p − 1}, then 󵄩󵄩󵄩σ α f 󵄩󵄩󵄩 ≤ Cp ‖f ‖p , (5.30) 󵄩 ∗ 󵄩p 󵄩 󵄩 lim 󵄩󵄩f − σnα f 󵄩󵄩󵄩p = 0,

(5.31)

lim σ α f (eiθ ) n→∞ n

(5.32)

n→∞󵄩

iθ

= f (e ) a. e.

Proof. The beginning of our proof differs from that of Oswald [412, pp. 410–411]. We start from identity (5.26). By the inequality 󵄨󵄨 ̂ 󵄨󵄨s 1−s s s 󵄨󵄨g(n)󵄨󵄨 ≤ C(n + 1) ‖g‖s , g ∈ H , s ≤ 1, we have 󵄨s 󵄨 r ns 󵄨󵄨󵄨Aαn σnα f (ζ )󵄨󵄨󵄨

1−s

≤ C(n + 1)

π

󵄨 󵄨−(α+1)s 󵄨󵄨 it 󵄨s ∫ 󵄨󵄨󵄨1 − reit 󵄨󵄨󵄨 󵄨󵄨f (re ζ )󵄨󵄨󵄨 dt −π π

≤ C(n + 1)1−s ∫ (|1 − r + |t|) −π

󵄨󵄨f (reit ζ )󵄨󵄨󵄨s dt, 󵄨 󵄨

−(α+1)s 󵄨

1/2 ≤ r < 1, s < 1.

176 | 5 Taylor coefficients and maximal functions From now on our proof is almost identical to Oswald’s proof. We take r = 1−1/(n+2) and use (5.25) to obtain 󵄨s 󵄨󵄨 α 󵄨󵄨σn f (ζ )󵄨󵄨󵄨 ≤ C(n + 1)

󵄨󵄨 it 󵄨s 󵄨󵄨f (rζe )󵄨󵄨󵄨 dt

∫ |t| 1) and then the complex maximal theorem to obtain (5.30). Relation (5.31) is a consequence of (5.30) and the density of polynomials in H p . Finally, we obtain (5.32) by using the theorem on a. e. convergence (Theorem C.22).

Further notes and results | 177

A proof of the Hardy–Rogosinski theorem Theorem 5.16. The set of functions f ∈ H 1 (𝕋) such that the Fourier series of f diverges almost everywhere is of the second category in H 1 (𝕋). ikθ ̂ Proof [197]. Let sn h(eiθ ) = ∑nk=−n h(k)e . By the Sawier–Stein theorem (Theorem C.21) iθ we have that either (a) smax f (e ) < ∞ a. e. for every f ∈ H 1 (𝕋) or (b) smax f (eiθ ) = ∞ on a subset of the second category in H 1 (𝕋). Assume that (a) holds. Then, by Banach’s principle (Theorem C.19) smax acts as a continuous operator from H 1 (𝕋) into ℒ0 (𝕋). Given a trigonometric polynomial N

g(eiθ ) = ∑ ck eikθ , k=−N

we consider the polynomial f ∈ H 1 (𝕋), f (eiθ ) = eiNθ g(eiθ ). We have sn g(eiθ ) = e−iNθ (sN+n f (eiθ ) − sN−n f (eiθ )),

0 ≤ n ≤ N.

This implies |smax g| ≤ 2|smax f |, so we have 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨{ζ ∈ 𝕋 : smax g(ζ ) > λ‖g‖1 }󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨{ζ ∈ 𝕋 : smax f (ζ ) > λ‖f ‖1 /2}󵄨󵄨󵄨, where we have used the relation ‖g‖1 = ‖f ‖1 . Since, by our hypothesis smax continuously maps H 1 (𝕋) into ℒ0 (𝕋), we have, by Lemma C.12, 󵄨 󵄨󵄨 󵄨󵄨{ζ ∈ 𝕋 : smax g(ζ ) > λ‖g‖1 }󵄨󵄨󵄨 ≤ c(λ),

where c(λ) → 0 as λ → ∞,

for every trigonometric polynomial g. Since these polynomials are dense in L1 (𝕋), we see that the above relation holds for all g ∈ L1 (𝕋), which implies that smax continuously maps L1 (𝕋) to ℒ0 . However, this is impossible because there exists a function h ∈ L1 (𝕋) such that smax h(eiθ ) = ∞ for a. e. θ.

Further notes and results For further and deeper results and information on interpolation of operators on Hardy spaces, we refer the reader to [312, 311]. Lemma 5.1 is essentially due Bourgain [99, Lemma 1.5, Prop. 1.6], whilst its presentation is taken from Kislyakov–Xu [313, Lemma 5]. The reader should read the Hardy–Littlewood’s memoir [230], where further interesting and deep results can be found. On page 162, they wrote: “Our proof of Theorem 5 (= Theorem 5.1, 1 < p < 2) is of the same character as Hausdorff’s original proof of his theorem, but is decidedly more difficult, and it seems to us unlikely that there is any really easy proof.” For a long time, it is known that there is an “easy” proof based on the classical Marcinkiewicz theorem (see [639, Ch. XII (3.19)]), but Hardy and

178 | 5 Taylor coefficients and maximal functions Littlewood wrote their paper thirteen years before the appearance of Marcinkiewicz’s theorem. However, as we have seen, it is possible to prove Theorem 5.1 without using interpolation theorems. See also Further notes 5.1 and 5.2. Hardy and Littlewood [238] proved (5.31) for all α > 1/p−1 (0 < p < 1) but (5.32) only for α > [ p1 ]. As remarked after Theorem 5.13, Zygmund [634]3 proved a much stronger result, the validity of (5.32) for α = 1/p − 1 (0 < p < 1). The maximal inequality (5.30) was established by Sunouchi [574]. (See also Flett [191, 192].) Of course, in the case p > 1, Theorem 5.15 states that the Fourier series of f (eiθ ) is (C, ε)-summable for all ε > 0. It is interesting that Zygmund in [634] at the beginning of p. 327 wrote: “The problem whether in this result we may replace summability (C, ε) by ordinary convergence remains open, but the answer is probably negative.” As we already mentioned, Carleson proved that the answer is positive. Concerning other results on the Cesàro means, we refer to Stein’s paper [549]. 5.1 (Elementary proof of Theorem 5.4). Theorem 5.4, from which we very simply deduced Theorem 5.1(a), can be proved in an elementary way. Namely, applying the Hardy–Littlewood decomposition lemma, we reduce the proof to the case p = 2. So we have to prove that 1

∫ Mq2 (r, f )(1 − r)−2/q dr ≤ C‖f ‖22 ,

f ∈ H 2 , q > 2.

0

Write f as f (z) = ∑∞ k=0 Pk (z), where 2k −1

Pk (z) = ∑ f ̂(j)z j j=2k−1

(k ≥ 1),

P0 (z) = f (0).

Then, by the L2 -integrability lemma, Lemma 2.6, and Lemma 2.7, 1

∞

0

k=1

󵄨2 󵄨 ∫ Mq2 (r, f )(1 − r)−2/q dr ≤ C1 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C2 ∑ 2k(2/q−1) ‖Pk ‖2q ∞

2 󵄨2 󵄨 ≤ C1 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C2 ∑ 2−k(1−2/q) (2k(1/2−1/q) ‖Pk ‖2 ) k=0 ∞

󵄨 󵄨2 = C1 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + C2 ∑ ‖Pk ‖22 ≤ C‖f ‖22 . k=0

5.2. Theorem 5.3 is easily deduced from Theorem 5.4. Take q = 1 in (5.12), then use the inequality M1 (r, f ) ≥ c supn |f ̂(n)|r n and the Lp -integrability lemma 4.8. 3 This paper should be read by everyone who wants to see how subtle and deep the Fourier analysis can be.

Further notes and results | 179

5.3. Kolmogorov’s theorem is usually stated in the following way (see Further notes 1.3): (a) there exists h ∈ L1 (𝕋) such that the Fourier series of h diverges a. e., although he proved that (b) there is h such that smax h(eiθ ) = ∞ a. e. (see [639, pp. 305–306]). However, Theorem C.23 shows that (a) implies (b). 5.4 (Konyagin’s theorem). Konyagin [320] proved the following improvement of Kolmogorov’s theorem. Theorem (Ko). If {ψ(m)} is a sequence of positive numbers such that ψ(m) = o(√ln m/√ln ln m) as m → ∞, then there exists a function ϕ ∈ L1 (𝕋) such that lim sup sm ϕ(eiθ )/ψ(m) = ∞ m→∞

for all θ ∈ 𝕋.

5.5. Inequality (5.21) is optimal in L1 in the sense that log n cannot be replaced by any ψ(n) (independent of f ) such that ψ(n) = o(log n). To see this, take f to be the Poisson kernel, then let r tend to 1 and use the norm estimate for the Dirichlet kernel. 5.6. Using Fejér’s theorem, we can show, by summation by parts, that if f ∈ h1 , then supn ‖Pn f ‖ < ∞, where Pn is extended to harmonic functions in the obvious way. Conversely, if f is harmonic in 𝔻 and supn ‖Pn f ‖ < ∞, then f ∈ h1 . 5.7 (Littlewood’s conjecture). In connection with the so-called Littlewood’s conjecture, 2π󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨 ∑ eiλk θ 󵄨󵄨󵄨 dθ ≥ c log n, 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨k=1

(†)

the following extension of Theorem 5.1 (p = 1) was proved by McGehee, Pigno and Smith [389]. Theorem (MPS). If {λn }∞ 1 is a strictly increasing sequence of positive integers, and f ∈ H 1 is such that supp f ̂ ⊂ {λn : n ≥ 1}, then ∞

󵄨 󵄨 ∑ n−1 󵄨󵄨󵄨f ̂(λn )󵄨󵄨󵄨 ≤ C‖f ‖1 ,

(5.34)

n=1

where C is independent of {λn } and f . Taking f (z) = ∑nk=1 z λk , we get (†). Wittman [598] improved (5.34) by proving that ∑nj=0 |f ̂(j)|2 󵄨 󵄨󵄨f ̂(n)󵄨󵄨󵄨 ≤ 66‖f ‖1 , 󵄨 n 2󵄨 ̂ n=0 (∑j=0 |f (j)|) ∞

∑

f ∈ H 1.

6 Besov spaces In this chapter, we define the analytic Besov space Bp,q α (α ∈ ℝ) as a subset of H(𝔻) by the requirement that, for some (equivalently, for all) s > α, the function (1 − r)s−α Mp (r, 𝒥 s f ), 0 < r < 1, belongs to Lq−1 , where 𝒥 s is the radial derivative of order s. p,q (α < 0), Hardy–Bloch The scale of Besov spaces contains the mixed-norm spaces H−α spaces (α = 0), and Lipschitz spaces of functions on the unit circle (α > 0). We prove q that Bp,q α can be decomposed into an ℓ -sum of finite-dimensional Hardy spaces if q 1 < p < ∞ and into “quasi”-ℓ -sum in the other cases, and we use these decompositions to describe the dual of Bp,q α . In Section 6.5, we consider 3 × 2 important embedding theorems between Hardy and Besov spaces due to Hardy–Littlewood, Littlewood–Paley, and Flett. Section 6.6 is devoted to characterizations of Bp,q α (α > 0) p via the best polynomial approximation in the H -norm. In Section 6.8, we give optimal estimates for best approximations of a singular inner function and use them to discuss the membership of an inner function in Besov spaces. In the last section, we consider the existence of the radial limits of Hardy–Bloch functions.

6.1 Decomposition of Besov spaces: case 1 < p < ⬦ 1

Various results of Hardy, Littlewood, and Paley show that quantities of the form ∫0 (1 −

r)β Mpq (r, f (ν) ) dr occur in a natural way. This serves as a motivation for introducing the mixed-norm spaces hp,q α in Section 3.3. An application of the maximum modulus prinp,q ciple shows that the original norm in X = hp,q α (inherited from Lα ) is equivalent to ‖f ‖󸀠X

1

:=

(∫ Mpq (r, f )(1

qα−1

− r)

1/q

dr)

.

0

Although the original quasinorm is more natural, the quasinorm ‖ ⋅ ‖󸀠 is sometimes more convenient in calculations. p,q The subspace of hp,q α consisting of analytic functions is denoted by Hα . For 0 < p, q ≤ ∞ and α ∈ ℝ, we define the harmonic Besov space Bαp,q to be the subclass of h(𝔻) that consists of those f for which 𝒥 s f ∈ hp,q s−α , where s is any real number greater than α, and ∞

σ σ iθ |n| ikθ 𝒥 f (re ) = ∑ (|k| + 1) f ̂(k)r e , k=−∞

σ ∈ ℂ.

p,q In particular, we have hp,q = B−β (β > 0). This definition is independent of the choice β of s > α, as can be deduced from Theorem 3.13. Since the space Bαp,q is “self-conjugate” (Theorem 3.8), we can reduce the proof of various facts concerning Bαp,q to the case of analytic Besov spaces. https://doi.org/10.1515/9783110630855-006

182 | 6 Besov spaces p,q The analytic Besov space Bp,q α is the subspace of Bα spanned by analytic functions. It is possible to give two alternative definitions of Bp,q α . One of them uses the ordinary derivative f (s) (s is a nonnegative integer), and the second uses the operator ℛs defined for all s ∈ ℝ by ∞

s n s ℛ f (z) = ∑ n f ̂(n)z .

(6.1)

n=1

Observe that if N is an integer, then DN f = iN ℛN f ; see (3.27). The equivalence of these definitions will be clear from a decomposition theorem; see Corollary 6.8. Let In = {k : 2n ≤ k < 2n+1 } for n ≥ 1 and I0 = {0, 1}, and let Δn f (z) = ∑ f ̂(k)z k . k∈In

We define Δp,q α to be the space of functions f ∈ H(𝔻) for which ∞

q

nα

( ∑ (2 ‖Δn f ‖p ) ) n=0

1/q

(6.2)

0, b − a ≥ 1, and Np > 1. Then, under the hypotheses of Lemma 6.3, there is a constant C = Cp,N such that 2π

󵄩 󵄩p 󵄨p 󵄨 ∫ 󵄨󵄨󵄨WΨ (eiθ )󵄨󵄨󵄨 dθ ≤ C 󵄩󵄩󵄩Ψ(N) 󵄩󵄩󵄩∞ (b − a)Np+p−1 . 0

Another useful lemma is a consequence of Lemma 6.2. Lemma 6.5. Let σ = s + it be a complex number, and let p > 0. Then 󵄩󵄩 󵄩󵄩 4n 󵄩󵄩 󵄩󵄩 4n 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ∑ k σ ak ek 󵄩󵄩󵄩 ≍ ns 󵄩󵄩󵄩 ∑ ak ek 󵄩󵄩󵄩 , 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩p 󵄩󵄩k=n 󵄩󵄩p 󵄩󵄩k=n

n ≥ 1.

Proof. This is proved by taking ψ(x) = xσ φ(x), where φ is a C ∞ -function such that supp(φ) ⊂ (0, ∞) and φ(x) = 1 for x ∈ [1, 4].

186 | 6 Besov spaces The operator Wmax For a function f ∈ H(𝔻), we define the maximal function Wmax f by 󵄨 󵄨 ψ (Wmax f )(ζ ) = (Wmax f )(ζ ) = sup󵄨󵄨󵄨Wnψ ∗ f (ζ )󵄨󵄨󵄨 n

(ζ ∈ 𝕋).

Lemma 6.6. If 0 < q ≤ 1, then (Wmax f )q ≤ Cq ℳ(|f |q ),

f ∈ Hq.

Proof. Let supp(ψ) ⊂ [−2, 2]. Then 󵄨󵄨 π 󵄨󵄨 2n 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 − f (rζeit ) ∑ r −k ψ(k/n)e−ikt dt 󵄨󵄨󵄨 󵄨󵄨Wn ∗ f (ζ )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 ∫ 󵄨󵄨 󵄨󵄨󵄨−π k=−2n 󵄨 π 󵄨󵄨 󵄨 󵄨 2n 󵄨󵄨 r −2n 󵄨󵄨 󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨f (rζeit )󵄨󵄨󵄨󵄨󵄨󵄨 ∑ r 2n−k ψ(k/n)e(2n−k)it 󵄨󵄨󵄨 dt. 󵄨󵄨 󵄨󵄨−2n 2π 󵄨 󵄨 −π For fixed ζ and n, put 2n

g(z) = f (zζ ) ∑ ψ(k/n)z 2n−k k=−2n

and rewrite the preceding inequality as |Wn ∗ f (ζ )| ≤ r −2n M1 (r, g). The function g is analytic, and hence M1 (r, g) ≤ (1 − r 2 )1−1/q Mq (1, g) (see Corollary 1.32). Putting r = 1 − 1/(n + 1), we get π

󵄨 iθ 󵄨q 󵄨q 󵄨󵄨 1−q 󵄨󵄨Wn ∗ f (ζ )󵄨󵄨󵄨 ≤ Cn ∫ 󵄨󵄨󵄨g(e )󵄨󵄨󵄨 dθ −π π

󵄨q 󵄨q 󵄨 󵄨 = Cn1−q ∫ 󵄨󵄨󵄨f (ζeiθ )󵄨󵄨󵄨 󵄨󵄨󵄨Wn (eiθ )󵄨󵄨󵄨 dθ. −π

Hence, by Lemma 6.2, 󵄨󵄨 1−q it 󵄨q 󵄨󵄨Wn ∗ f (e )󵄨󵄨󵄨 ≤ Cn

󵄨 󵄨q ∫ 󵄨󵄨󵄨f (ei(θ+t) )󵄨󵄨󵄨 nq dθ

|θ| 0, then each f ∈ Bp,q α has radial limits almost everywhere. Moreover, it is known that then f is in Bp,q α if and only if the function t 󳨃→ ‖Δnt f ‖p t −α belongs to Lq ((0, 1), dt/t), which follows from Theorem 3.15 and the H–L conjugate-function theorem. From Lemma 6.5 and Theorem 6.4 we obtain the following: Corollary 6.8. Let σ = s + iη ∈ ℂ and s > α. A function f ∈ H(𝔻) belongs to Bp,q α if and p,q σ only if ℛ f ∈ Hs−α . If N is a nonnegative integer, then the equivalence remains true if we replace ℛN with f (N) . ϱ,q

Exercise 6.9. If λ ∈ ℝ+ and q ∈ {⬦, ∞}, then B ∩ Bλ/ϱ ⊂ Bp,q for 0 < ϱ < p < ⬦. λ/p The following statement is an immediate consequence of Theorem 6.4 although can be easily deduced from the definition of Bp,q α . p,⬦ Corollary 6.10. If q < ⬦ and α ∈ ℝ, then Bp,q α ⊂ Bα . s,q Theorem 6.5 (Mixed embedding theorem). If s > p, then Bp,q α ⫋ Bβ , where β = α + 1/s − 1/p.

This is a reformulation of Theorem 3.12 but can also be deduced from Theorem 6.4 and Lemma 2.7. As an application of Theorem 6.4 and Lemma 6.5 we have a fact, which is again a reformulation of a result obtained by means of quasinearly subharmonic functions (Theorem 3.13): Theorem 6.6. Let γ ∈ ℝ. Then for all α ∈ ℝ, we have that 𝒥 γ f ∈ Bp,q α if and only if γ f ∈ Bp,q , and the corresponding norms are equivalent. Moreover, 𝒥 acts as an isomorα+γ p,q phism from Bp,q onto B . α+γ α

6.4 Duality in the case 0 < p ≤ ∞

| 189

For a substantial generalization of this theorem in the case γ > 0, see Further notes 6.1. The most simple case occurs if α < 0 and γ = 1. Then we again get (3.20). The case α > 0 of Theorem 3.11 is easily deduced from Theorem 6.4 (and the “selfconjugacy” of hp,q α ). In the terms of Besov spaces, it reads as follows. p,q1

Theorem 6.7 (Increasing inclusion theorem). We have Bp,q α ⫋ Bα α ∈ ℝ.

for q1 > q and all

The inclusion is proper as can be verified by using lacunary series: kn Theorem 6.8. Let f (z) = ∑∞ n=0 cn z be a function from H(𝔻) such that {kn } is lacunary. α p,q Then f ∈ Bα if and only if {kn cn } ∈ ℓq .

Proof. This follows from Theorem 4.10.

6.4 Duality in the case 0 < p ≤ ∞ p,q 󸀠 If p ∈ ̸ (1, ⬦), then the duality between Bp,q α and (Bα ) cannot be expressed in the form (6.3) with the series converging in the ordinary sense since otherwise {en } is a Schauder basis in the space; for the case p = 1, see Further notes 6.2. However, it is possible to use the Abel summability (for q ≤ ⬦). Before stating the duality theorem for p ∈ ̸ (1, ⬦), we discuss in some details the notion of the Abel dual of an admissible space.

The Abel dual The Abel dual X A of an admissible space X is, by definition, the set of all g ∈ h(𝔻) such that the finite limit ∞

n ̂ ⟨f , g⟩ := lim− ∑ f ̂(n)g(n)r r→1 n=−∞

(6.11)

exists for all f ∈ X. Theorem 6.9. Let X be a minimal space. Then: (i) If g ∈ X A , then the functional f 󳨃→ ⟨f , g⟩, f ∈ X, belongs to X 󸀠 . (ii) Conversely, if Φ ∈ X 󸀠 , then there is a unique function g ∈ X A such that Φ(f ) = ⟨f , g⟩. We have ‖Φ‖ = sup{|⟨f , g⟩| : f ∈ X, ‖f ‖X ≤ 1}. Proof. Statement (i) is proved by using the Banach–Steinhaus principle. To prove (ii), ̂ define g by g(n) = Φ(en ). Since limr→1− ‖f − fr ‖ = 0, we have ∞

Φ(f ) = lim− Φ(fr ) = lim− Φ( ∑ f ̂(n)en r n ) r→1

r→1

∞

n=−∞

= lim− ∑ f ̂(n)Φ(en )r n = ⟨f , g⟩. r→1 n=−∞

190 | 6 Besov spaces n ̂ The series ∑∞ n=−∞ f (n)en r converges in X for 0 < r < 1 because of (1.21). The last statement is obvious.

There are several ways to express ⟨f , g⟩. For example, by Parseval’s formula we have ∞

2n ̂ ⟨f , g⟩ = lim− ∑ f ̂(n)g(n)r − f (rζ )g(zζ̄ )|dζ |. = lim− ∫ r→1 n=−∞

r→1

(6.12)

𝕋

To represent ⟨f , g⟩ in another way, we use Green’s formula (1.1). If we take F = uv, where u and v are real-valued functions harmonic in 𝔻, then Δ(uv) = 2∇u∇v, so that − u(ρζ )v(ρζ ̄ )|dζ | = u(0)v(0) + ∫

ρ 1 dA(z). ∫ ∇u(z)∇v(z)̄ log π |z| |z| 2 and λk = k 1/p−1 (log k)−1/2 . Then by Lemma 6.20 the function f (z) = k p,2 ∑∞ n=2 λk z is not in B . On the other hand, we have ∞

∞

n=2

n=2

∑ np−2 λnp = ∑ n−1 (log n)−p/2 < ∞

because p/2 > 1. Thus f ∈ H p by the Hardy–Littlewood inequality. This proves that inclusion (1) is proper. The proof that inclusion (2) is proper is similar, and we omit the details. There is an interesting fact concerning Bp,p . Although Bp,2 increases as p decreases, the “function” Bp,p is not monotone: Proposition 6.22. If p < q, then Bp,p ⊄ Bq,q and Bq,q ⊄ Bp,p . n

−1/p 2 z belongs to Bq,q but not to Bp,p . Proof. By Theorem 6.8 the function f (z) = ∑∞ n=1 n ∞ By Lemma 6.20 the function g(z) = ∑n=1 n1/q−1 z n belongs to Bp,p but not to Bq,q .

. ⊄ Bp,2 and Bp,2 ⊄ Bq,p Proposition 6.23. If 2 < p < ⬦ and q < p, then Bq,p 1/q−1/p 1/q−1/p The same holds if p < 2 and q > p. n

2 Proof. Let 2 < p < ⬦ and q < p. Let f (z) = ∑∞ n=1 (1/n)z . By Theorem 6.8, f beq,p p,2 longs to B but not to f ∈ B1/q−1/p because the inequality 1/q − 1/p > 0 implies np(1/q−1/p) −p n = ∞. By Lemma 6.20 the function ∑∞ n=1 2 ∞

∑ n1/p−1 (log n)−1/2 z n

n=2

but not to Bp,2 . In the case p < 2, we proceed in a similar way, but belongs to Bq,p 1/q−1/p slightly different functions are to be used.

6.6 Best approximation by polynomials p As noted in remark to Corollary 6.7, the space Bp,q α , α > 0, consists of H -functions satisfying a Lipschitz condition. Here we prove a characterization via the best approximation by polynomials.

Lemma 6.24. Let 0 < p ≤ ∞. Let Rn f = ∑∞ k=n+1 Vk ∗ f . Then cp ‖Rn f ‖p ≤ E2n (f )p ≤ ‖Rn−1 f ‖p ,

n ≥ 1.

Proof. We have f −Rn f = ∑nk=0 Vk ∗f = P+ W2n ∗f ; see (6.9). From the definition of W2n it follows that it is a polynomial of degree 2n+1 , which implies the right-hand inequality.

6.6 Best approximation by polynomials | 201

̂2n (k) = 1 for 0 ≤ k ≤ 2n , we see that if P is a polynomial of degree ≤ 2n , then Since W W2n ∗ P = P, and so 󵄩 󵄩 ‖Rn ∗ f ‖p = 󵄩󵄩󵄩Rn ∗ (f − P)󵄩󵄩󵄩p ≤ Cp ‖f − P‖p . This completes the proof.

Theorem 6.14. Let 0 < p ≤ ∞, 0 ≤ q ≤ ∞, and α > 0. Then f ∈ Bp,q α if and only if one of the following conditions are satisfied: ∞

p

1/q

q −1

α

f ∈ H and ( ∑ [n En (f )p ] n ) n=1 ∞

p

f ∈ H and ( ∑ 2

nqα

E2n (f )qp )

nqα

‖Rn f ‖qp )

n=0 ∞

p

f ∈ H and ( ∑ 2 n=0

< ∞,

(6.33)

1/q

< ∞,

(6.34)

< ∞.

(6.35)

1/q

Remark 6.25. If q = ∞, then condition (6.33) is interpreted as En (f )p = 𝒪(n−α ); if q = ⬦, then “𝒪” should be replaced by “o”. Proof. Let 0 < q < ∞. The sequence En (f )p (n ≥ 1) is decreasing, which shows that (6.33) is equivalent to (6.34). By Lemma 6.24, to prove that the condition f ∈ Bp,q α is equivalent to (6.34), it suffices to prove that f ∈ Bp,q if and only if (6.35) holds. α Since ∞

Vn ∗ f = Vn ∗ ∑ Vj ∗ f , j=n−1

n ≥ 0,

we have 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 ‖Vn ∗ f ‖p ≤ C 󵄩󵄩󵄩 ∑ Vj ∗ f 󵄩󵄩󵄩 , 󵄩󵄩 󵄩󵄩 󵄩p 󵄩j=n−1

from which, by Theorem 6.4, the inequality ‖f ‖qBp,q ≤ C ∑ 2nqα ‖Rn f ‖qp follows. α In other direction, we start from the inequality ∞

‖Rn f ‖sp ≤ ∑ ‖Vk ∗ f ‖sp , k=n+1

where s = min{p, 1}.

Now the following lemma shows that f ∈ Bp,q α . Lemma 6.26. Let {cn }∞ 0 be a scalar sequence such that cn = 0 for n large enough, and let β, γ be positive real numbers. Then ∞

∑2

n=0

󵄨󵄨 ∞ 󵄨󵄨γ ∞ 󵄨󵄨 nβγ γ 󵄨 󵄨󵄨󵄨 ∑ ck 󵄨󵄨󵄨 ≤ C ∑ 2 |cn | , 󵄨󵄨k=n 󵄨󵄨 n=0

nβγ 󵄨󵄨󵄨

where C is a constant independent of {cn }.

202 | 6 Besov spaces Proof. First, we rewrite the inequality in the equivalent form ∞

∑2

n=0

󵄨󵄨γ 󵄨󵄨 ∞ ∞ γ −kβ 󵄨󵄨󵄨 󵄨󵄨 ∑ 2 ck 󵄨󵄨 ≤ C ∑ |cn | , 󵄨󵄨 󵄨󵄨 n=0 󵄨 󵄨k=n

nβγ 󵄨󵄨󵄨

(†)

Let γ ≤ 1. Then ∞

∑2

n=0

󵄨󵄨γ ∞ 󵄨󵄨 ∞ ∞ −kβγ nβγ −kβ 󵄨󵄨󵄨 |ck |γ 󵄨󵄨 ∑ 2 ck 󵄨󵄨 ≤ ∑ 2 ∑ 2 󵄨󵄨 󵄨󵄨 n=0 k=n 󵄨 󵄨k=n

nβγ 󵄨󵄨󵄨

∞

k

∞

k=0

n=0

k=0

= ∑ 2−kβγ |ck |γ ∑ 2nβγ ≤ C ∑ |ck |γ , which proves (†) in this case. Let 1 ≤ γ ≤ ∞. Define the operator T by βn T({cn }∞ 0 ) = {2 an }0 , ∞

∞

where an = ∑ 2−βk ck , k=n

and consider the action of T on the spaces Lγ (μ, ℕ), where μ({n}) = 2nβ . By the preceding case, T acts from the subset of Lγ (ℕ, μ), consisting of “finite” sequences, to ℓγ for γ = 1. It is easily verified that the same holds for γ = ∞. Therefore, by the Riesz–Thorin theorem, T maps ℓγ into ℓγ for 1 < γ < ∞. Since the norms are independent of {cn }, we get (†), which completes the proof. Exercise. The lemma can be proved in an elementary way, that is, without appealing to the Riesz–Thorin theorem. Remark 6.27. The proof of Theorem 6.14 and the Riesz projection theorem show that p if 1 < p < ∞, then f ∈ Bp,q α if and only if f ∈ H and ∞

qα−1

( ∑ (n + 1) n=0

‖f −

1/p q sn f ‖p )

< ∞.

Remark 6.28. The analogue of Theorem 6.14 for harmonic Besov spaces is valid if we assume that p ≥ 1 and interpret En (g)p , g ∈ Lp (𝕋), as the best approximation by trigonometric polynomials of order n. In the case p < 1, this is not clear, and we will return to this question in Section 13.5.

6.7 “Normal” Besov spaces Let ϕ ∈ Δ[α, β](J), J = [0, 1], where α, β ∈ ℝ (α ≤ β). Recall that this means that ϕ(t)/t α (resp., ϕ(t)/t β ) is increasing (resp., decreasing) in t ∈ [0, 1]. We define the space

6.7 “Normal” Besov spaces | 203

Bp,q = {f ∈ H(𝔻) : . . .} by the requirement that for some s > β, ϕ s

p,q

𝒥 f ∈ Hϕ , s

where ϕs (t) =

ts ; ϕ(t)

see Chapter 4. (The harmonic Besov space Bϕp,q is defined in the same way.1 ) More precisely, f ∈ Bp,q if ϕ 1

q

∫[ϕs (1 − r)Mp (r, 𝒥 s f )] 0

dr β.

Here it is important that the function ϕs is normal and, more precisely, belongs to the class Δ[s − β, s − α]. Working exactly as in the case where ϕ(t) = t α , we arrive at the following generalization of Theorem 6.4. if and only if Theorem 6.15. A function f ∈ H(𝔻) belongs to Bp,q ϕ {[1/ϕ(2−n )]‖Vn ∗ f ‖p }n∈ℕ ∈ ℓq . In the proof, Lemma 4.10 is to be used. As a corollary, we have that the definitions of these spaces are independent of s > β. By using Theorem 6.15 we can generalize some of the preceding assertions. We note two theorems. Theorem 6.16. If q ≤ ⬦ and ϕ is normal, then (Bp,q )A = Bp1/ϕ,q , where ϕp = ϕ for p ≥ 1 ϕ 󸀠

and ϕp (t) = ϕ(t)t 1−1/p for p < 1.

󸀠

p

Theorem 6.17. Let ϕ be normal. Then Theorem 6.14 remains true if we replace nqα−1 and 2nqα with ϕ(1/n)−q /n and (ϕ(2−n ))−q , respectively. In particular, f ∈ Bp,∞ if and only if ϕ En (f ) = 𝒪(ϕ(1/n)), n → ∞. 6.7.1 Isomorphisms between Besov spaces The following theorem can be viewed as an extension of Theorem 6.6. 1 An alternative way is using the tangential derivative Ds .

204 | 6 Besov spaces Theorem 6.18. There exists a function λ ∈ H(𝔻), independent of p and q, such that (i) ̂ 1/λ(n) ≍ ϕ(1/(n + 1)) and (ii) f ∈ Bp,q if and only if λ ∗ f ∈ Bp,q . Moreover, the operator ϕ

f 󳨃→ λ ∗ f acts as an isomorphism from Bp,q onto Bp,q . ϕ

Proof. Since ‖𝒥 γ Vn ∗ f ‖p ≍ 2nγ ‖Vn ∗ f ‖p , we can assume that ϕ is normal. We define ̂ λ(n) = 1/Ψ(n), where 1

Ψ(t) = ∫ r t 0

ϕ(1 − r) dr, 1−r

t ≥ 0.

It is easy to verify that Ψ(t) ≍ ϕ(1/(t + 1)); see the proof of Lemma 4.13. Since 1

1

0

0

󵄨󵄨 (m) 󵄨󵄨 t m ϕ(1 − r) dr ≍ ∫ r t (1 − r)m−1 ϕ(1 − r) dr 󵄨󵄨Ψ (t)󵄨󵄨 = ∫ r | log r| 1−r for positive integers m, we have Ψ(t) 󵄨󵄨 (m) 󵄨󵄨 , 󵄨󵄨Ψ (t)󵄨󵄨 ≤ C (t + 1)m

t ≥ 0.

Using this and the identity (m)

Ψ(1/Ψ)

m−1 m = − ∑ ( )Ψ(m−j) (1/Ψ)(j) , j j=0

we obtain, by induction, 1/Ψ(t) 󵄨 󵄨󵄨 (m) , 󵄨󵄨(1/Ψ) (t)󵄨󵄨󵄨 ≤ C (t + 1)m

t ≥ 0.

Now we can copy the proof of Lemma 6.19 to obtain the desired conclusion. Remark 6.29. Taking ϕ(t) = st s , s > 0, we can obtain the original version of the Hardy–Littlewood fractional integral theorem [237]. They defined the operators D[s] and D[−s] = D[s] as D[s] f (z) =

∞ 1 Γ(n + s + 1) ̂ f (n)z n , ∑ Γ(s + 1) n=1 Γ(n + 1) ∞

Γ(n + 1) ̂ f (n)z n . Γ(n + s + 1) n=0

D[s] f (z) = Γ(s + 1) ∑ Since 1

s ∫ r n (1 − r)s−1 dr = sB(n + 1, s) = 0

Γ(n + 1)Γ(s + 1) , Γ(n + 1 + s)

(6.36) (6.37)

6.8 Inner functions in Besov and Hardy–Sobolev spaces | 205

we have, as a consequence of Theorem 6.18 and its proof, another result of Hardy and Littlewood: p,q D[s] f ∈ Bp,q α if and only if f ∈ Bα+s

(s ∈ ℝ).

The most important property of D[s] is D[s] f (z) = (1 − az)−s−1 ,

where f (z) = (1 − az)−1 .

Note also that D[s] D[s] f = D[s] D[s] f = f . Using the formula (cf. [402, Appendix A, (22)]) m cj Γ(n + s + 1) = (n + 1)s (c0 + ∑ + 𝒪((n + 1)−m−1 )) j Γ(n + 1) (n + 1) j=1

(6.38)

(where c0 ≠ 0) for every integer m > 0 and then the decomposition, we can easily prove that 𝒥 s can be replaced with D[s] in the definition of Besov spaces. Moreover, every multiplier transform of the form ∞

Tf (z) = ∑ λn f ̂(n)z n n=0

such that λn is equal to the right side of (6.38) for s ∈ ℝ and sufficiently large m can be used instead of 𝒥 s (almost) everywhere. See [275, Theorem 2.6] for a more general class of operators or [276, Lemma 7.4.2].

6.8 Inner functions in Besov and Hardy–Sobolev spaces For 0 < p ≤ ∞, we define the Hardy–Sobolev space HSpα (α > 0) by HSpα = {f ∈ H(𝔻): 𝒥 α f ∈ H p }

(6.39)

with the norm ‖f ‖HSpα = ‖𝒥 α f ‖p . The decomposition theorem 6.4, together with the density of P in HSpα (p ≤ ⬦), can be used to show that HSpα ⊂ Bp,⬦ α (p ≤ ⬦). In this section, we detect the spaces X ∈ {HSpα , Bp,q } for which the following holds: If an inner α function I belongs to X, then I is a Blaschke product. We present two methods. The first one is based on the best approximation of singular inner functions, and the second, due to Ahern and Jevtić [11], is based on a simple but powerful inequality combined with the complex maximal theorem; see Lemma 6.32.

206 | 6 Besov spaces 6.8.1 Approximation of a singular inner function Recall that Aλ (z) = exp(−λ

1+z ), 1−z

where λ > 0.

Here we use Theorem 2.31 and Lemma 2.36 to prove the following: Theorem 6.19. If I is an inner function with a singular inner factor then, for p ∈ ℝ+ , En (I)p ≥ cp,I n−1/2p ,

n ≥ 1.

The exponent −1/2p is the best possible. Lemma 6.30. If p > 1, then En (Aλ )p ≍ n−1/2p . Proof. First, we write Aλ as Aλ (z) = e−λ exp(

−2λz ). 1−z

From the theory of orthogonal polynomials (Szegö [577]) we know that ∞

n Aλ (z) = e−λ ∑ L(−1) n (2λ)z , n=0

where Ln(−1) are the Laguerre polynomials (a particular case). By Fejér’s theorem [577, Theorem 8.22.1] we have, for a fixed x ∈ ℝ+ , −3/4 cos(2√nx + π/4) + 𝒪(n−1 ), L(−1) n (x) = Cx n

n ≥ 1.

It follows that ∞

2 󵄨 󵄨2 (En (Aλ )2 ) = ∑ 󵄨󵄨󵄨L(−1) (2λ)󵄨󵄨󵄨 k k=n+1 ∞

󵄨2 󵄨 ≥ cλ ∑ k −3/2 󵄨󵄨󵄨cos(2√2λk + π/4)󵄨󵄨󵄨 − kn−1 . k=n+1

Solving the inequality | cos(2√2λk + π/4)| ≥ 1/√2, we have k ∈ Am := [K(m − 1/2)2 , Km2 ],

where K = π 2 /8λ, and m is an integer.

A segment [a, b] ⊂ ℝ contains at least b−a integers. Hence Am contains at least K(2m− 1/2)/2 integers, so we can choose m0 such that Am ≠ 0 for m ≥ m0 . For a sufficiently large n, choose m such that n + 1 ∈ Am . Since Am are mutually disjoint, we have 2

(En (Aλ )) ≥ cλ ∑ ∑ k −3/2 (1/2) − cn−1 j=m+1 k∈Aj

6.8 Inner functions in Besov and Hardy–Sobolev spaces | 207 ∞

≥ cλ ∑ (Kj2 )

−3/2

j=m+1

(1/2)K(2j − 1/2) − cn−1

∞

≥ cλ󸀠 ∑ j−2 − cn−1 ≥ cλ󸀠󸀠 (m + 1)−1 − cn−1 ≥ c/√n. j=m+1

Since the reverse inequality obviously holds, we see that En (Aλ )2 ≍ n−1/4 ,

(6.40)

where the equivalence constants depend only on λ. To treat the case p ≠ 2, we need the following estimates for the νth derivative of Aλ : (1 − r)1/2p−ν , ν > 1/2p, { { { 2 2ν Mp (r, A(ν) ) ≍ {(log 1−r ) , ν = 1/2p, λ { { ν < 1/2p. {1,

(6.41)

From this we infer that Aλ ∈ Bp,∞ for p > 0, and hence, by Theorem 6.14, 1/2p ‖Rn Aλ ‖p ≤ C2−n/2p

if p > 0.

Let 1 < p < ⬦. Then, by (6.41) and (6.40), c2−n/2 ≤ ‖Rn Aλ ‖22 ≤ ‖Rn Aλ ‖p ‖Rn Aλ ‖p󸀠 ≤ C‖Rn Aλ ‖p 2−n/2p , 󸀠

and hence ‖Rn Aλ ‖p ≥ c2−n/2p . This proves the lemma. (See Lemma 6.24.) Remark. Estimates (6.41) can be obtained in a similar way as those used in the proof of Theorem 2.26, so we omit the details; see [382] or treat this as an exercise. Proof of Theorem 6.19. Let 1 < p < ⬦, and let S be the singular factor of I. Then by Lemma 2.36 we have En (I)p ≥ cEn (S)p . The function S is subordinate to Aλ , where, as we may assume, λ > 0. Hence, by Theorem 2.31 and Lemma 6.30, En (I)p ≥ c1 En (S)p ≥ c2 n−1/2p for p > 1. Let p ≤ 1. Then ‖Rn I‖pp = ∫ − |Rn I|−1 |Rn I|p+1 dl ≥ c ∫ − |Rn I|p+1 dl ≥ c2−n/2 . 𝕋

𝕋

The exponent −1/2p is optimal, which follows from Theorem 6.14 and the fact that, . This concludes the proof. according to (6.41), Aλ ∈ Bp,∞ 1/2p As an immediate consequence of Theorems 6.19 (see Remark following it) and 6.14, we have the following: Corollary 6.31. Let I be an inner function with a singular factor, and let p ∈ ℝ+ . If either 0 < q ≤ ⬦ and α ≥ 1/2p, or q = ∞ and α > 1/2p, then I does not belong to Bαp,q . The result is the best possible in the sense that the atomic (singular) function Aλ belongs to p,∞ Bp,q for α ≤ 1/2p. α for q ≤ ⬦ and α < 1/2p and to Bα

208 | 6 Besov spaces The last statement follows from relations (6.41). Thus we have the following theorem. Theorem 6.20. Let p ∈ ℝ+ . If an inner function I belongs to Bp,q α , where either q ≤ ⬦ and α ≥ 1/2p, or q = ∞ and α > 1/2p, then I is a Blaschke product. The following fact is in fact an improvement of the preceding theorem. Theorem 6.21 (Ahern–Jevtić [11]). If I is an inner function such that Mpp (r, 𝒥 1/2p I) = o(log

1 ) 1−r

for some p ∈ ℝ+ ,

then I is a Blaschke product. Lemma 6.32 ([11]). If 0 < α < β < ⬦ and f ∈ H ∞ with ‖f ‖∞ ≤ 1, then |𝒥 α f (z)| ≤ Cα,β M α/β (z, 𝒥 β f ), where M(z, f ) = sup0 1. We need another fact. Lemma 6.33. If 1 < p < ⬦ and I is an inner function with nontrivial singular factor, then ‖I − Ir ‖p ≥ c(1 − r)1/2p ,

0 < r < 1.

Proof. For r ∈ (0, 1), choose a positive integer n such that 1−1/n ≤ r ≤ 1−1/(n+1). Then, ̂ by the Riesz projection theorem, ‖I −Ir ‖p ≥ cp ‖ℛn I − ℛn Ir ‖p , where ℛn f = ∑∞ k=n+1 f (k)ek . Hence ‖I − Ir ‖p ≥ cp (‖ℛn I‖p − ‖ℛn Ir ‖p ) ≥ cp (‖ℛn ‖p − r n+1 ‖ℛn I‖p ),

6.8 Inner functions in Besov and Hardy–Sobolev spaces | 209

where we have used the inequality ‖ℛn Ir ‖p ≤ r n+1 ‖ℛn I‖p , which is proved in the same way as Lemma 2.6. Since r n+1 ≤ (1 − 1/(n + 1))n+1 < e−1 , we arrive at the inequality ‖I − Ir ‖p ≥ cp (1 − 1/e)‖ℛn I‖p . Finally, since ‖ℛn I‖p ≍ En (I)p , we see from Theorem 6.19 that ‖I − Ir ‖p ≥ cn−1/2p , from which the result follows. Proof of Theorem 6.21. In view of (6.42), we may assume that p > 1. Let q > 1. Then, by the previous lemma, 1/2

c(1 − r)

≤ ‖I −

Ir ‖qq

2π

1

0

r

q

󵄨 󵄨 ≤∫ − (∫󵄨󵄨󵄨I 󸀠 (ρeiθ )󵄨󵄨󵄨 dρ) dθ.

Let 0 < α < 1. Then, by Hölder’s inequality, 1

q

q

1

󵄨 󵄨 󵄨 󵄨 (∫󵄨󵄨󵄨I 󸀠 (ρeiθ )󵄨󵄨󵄨 dρ) = (∫󵄨󵄨󵄨I 󸀠 (ρeiθ )󵄨󵄨󵄨(1 − ρ)α (1 − ρ)−α dρ) r

r

1

q−1 󵄨q 󵄨 ≤ Cq ∫󵄨󵄨󵄨I 󸀠 (ρeiθ )󵄨󵄨󵄨 (1 − ρ)αq dρ((1 − r)1−αq/(q−1) ) r

1

󵄨q 󵄨 ∫󵄨󵄨󵄨I 󸀠 (ρeiθ )󵄨󵄨󵄨 (1 − ρ)αq dρ.

q−1−αq

= Cq (1 − r)

r

From this and from (6.43) we obtain 1

c(1 − r)−1 ≤ (1 − r)q−5/2−αq ∫ Mqq (ρ, I 󸀠 )(1 − ρ)αq dρ. r

Since also Mq (ρ, I 󸀠 ) ≤ C(1 − ρ)1/2p−1 Mq ((1 + ρ)/2, 𝒥 1/2p I), we see that 1

c(1 − r)−1 ≤ (1 − r)q−5/2−αq ∫ Mqq (ρ, 𝒥 1/2p I)(1 − ρ)q/2p−q+αq dρ. r

Now we take q = 2p and α = 1 − 1/2p, and this inequality becomes −1

c(1 − r)

−3/2

≤ (1 − r)

1

2p (ρ, 𝒥 1/2p I) dρ. ∫ M2p r

Integrating this inequality from 0 to t and using Fubini’s theorem, we obtain t

ρ

0

0

1 2p ≤ ∫ M2p c log (ρ, 𝒥 1/2p I) dρ ∫(1 − r)−3/2 dr 1−t

(6.43)

210 | 6 Besov spaces t 2p 1/2p I) dρ ∫(1 ∫ M2p (ρ, 𝒥 t 0 1

+

− r)−3/2 dr

t

2p = C ∫ M2p (ρ, 𝒥 1/2p I)(1 − ρ)−1/2 dρ 0

−1/2

+ C(1 − t)

1

2p (ρ, 𝒥 1/2p I) dρ ∫ M2p t

= CI1 (t) + C(1 − t)−1/2 I2 (t). We have 1

2π

󵄨2p 󵄨 I2 (t) = ∫ dt ∫ − 󵄨󵄨󵄨𝒥 1/2p I(ρeiθ )󵄨󵄨󵄨 dθ t

1

0

2π

󵄨p 󵄨 ≤ C ∫ dt ∫ − 󵄨󵄨󵄨𝒥 1/2p I(ρeiθ )󵄨󵄨󵄨 (1 − ρ)−1/2 dρ t

1

0

= C ∫ Mpp (ρ, 𝒥 1/2p I)(1 − ρ)−1/2 dt. t

Now we estimate I1 (t) by using Lemma 6.32. For a fixed θ, we have t

λ

t

󵄨2p 󵄨 I1 (t, θ) := ∫󵄨󵄨󵄨𝒥 1/2p I(ρeiθ )󵄨󵄨󵄨 (1 − ρ)−1/2 dρ = ∫ . . . dρ + ∫ . . . dρ 0

0

λ

λ

2p

1

≤ C ∫(1 − ρ)−3/2 dρ + [M(teiθ , 𝒥 1/2p I)] ∫(1 − ρ)−1/2 dρ 0 −1/2

≤ C(1 − λ)

iθ

+ 2[M(te , 𝒥

1/2p

2p

λ

I)] (1 − λ)1/2 .

If M := M(teiθ , 𝒥 1/2p I) ≤ 1, then we take λ = 0 to obtain I1 (t) ≤ C. If M > 1, then we choose λ = 1 − M −2p and obtain I1 (t, θ) ≤ CM p , and hence, by integration and the complex maximal theorem, I1 (t) ≤ C + CMpp (t, 𝒥 1/2p I). Altogether, log

1 ≤ C + CMpp (t, 𝒥 1/2p I) 1−t −1/2

+ C(1 − t) The result follows.

1

∫ Mpp (ρ, 𝒥 1/2p I)(1 − ρ)−1/2 dρ. t

6.8 Inner functions in Besov and Hardy–Sobolev spaces | 211

Theorem 6.21 is sharp in the sense that the atomic function satisfies Mpp (r, 𝒥 1/2p Aλ ) = 𝒪(log

1 ). 1−r

By means of (6.42) this is reduced to the case where 1/2p is an integer; then (6.41) is used. As a consequence, we have the following: Corollary 6.34. If an inner function I belongs to HSpα , where p ∈ ℝ+ and α ≥ 1/2p, then I is a Blaschke product. This result is also sharp because Aλ ∈ HSpα for α < 1/2p, which can be proved by using (6.41) and (6.42). See Further notes 6.4. In the case p = 1/2, we can take α = 1 to get the following result of Ahern and Clark [13]. Corollary 6.35. If an inner function I is such that I 󸀠 ∈ H p , p ≥ 1/2, then I is a Blaschke product. It should be noted that if I 󸀠 ∈ H 1 , then I is continuous up to the boundary (Privalov’s theorem 1.32), and hence it is a finite Blaschke product. See Further notes 6.5 for the case of the weak H 1 space. p

6.8.2 Hardy–Sobolev space HS 1/p Problem 6.7. The Hardy–Littlewood–Sobolev theorem can be expressed via Hardy– Sobolev spaces in a symmetric form: HSp1/p ⊂ HSq1/q , 0 < p < q < ⬦. If p = 1 or 2, then HSp1/p is Möbius invariant in the sense that ‖f ∘σ −f (σ(0))‖X ≤ C‖f ‖X for all σ ∈ Möb(𝔻), where C is independent of f and σ. Therefore it is natural to ask: what about the other values of p? Some interpolation theorem for the pair (HS11 , HS21/2 ) would solve this problem for 1 < p < 2, but the author does not know any such theorem. The spaces HSp1/p have some nice properties, which suggest that they might be invariant. To show some of them, we use the (Möbius) duality pairing (f , g)m =

1 ∫ f 󸀠 (z)g 󸀠 (z) dA(z), π

(6.44)

𝔻

where the integral is somehow defined. It is easy to check that (f ∘ σ, g ∘ σ)m = (f , g)m

for all σ ∈ Möb(𝔻).

Therefore, if X is invariant and Y is isomorphic to X 󸀠 with respect to the Möbius pairing, then Y is also invariant. Denote Y by X m and call it the Möbius dual of X. Then the three

212 | 6 Besov spaces duality theorems for H p can be reformulated as m (HSp1/p )

B for p < 1, { { { = {BMOA for p = 1, { { p󸀠 {HS1/p󸀠 for 1 < p < ⬦.

(6.45)

Thus, if HSp1/p is invariant for 1 < p < 2, then it is invariant for 2 < p < ⬦. Is this a coincidence? 6.8.3 f-property and K-property In view of Lemma 2.36, we can ask whether En (fI)𝒦 ≥ En (f )𝒦 , where 𝒦 = A(𝔻)A is the space of Cauchy transforms (see Further notes 1.1), but the author could not prove this; Vinogradov [590] proved that 𝒦 has the f-property (Havin [247]). An admissible space X has this property if the following holds: If f ∈ H s for some s > 0 and fI ∈ X, then f ∈ X. It is easier to prove that the space 𝒦a has the f-property because, as mentioned before, (𝒦a )A ≅ H ∞ ; can we use this to prove that En (fI)𝒦a ≥ En (f )𝒦a ? The following result is a consequence of Theorem 6.14 and Lemma 2.36. Theorem 6.22. Let α > 0, 1 < p < ⬦, and 0 < q ≤ ∞. Then X = Bp,q α has the f-property. The first result of this kind (excluding Hardy spaces) was proved by Carleson [113] (Dirichlet spaces). Gurariy [223] proved that ℓ1 does not have the f-property, and this was the first example of such a space; now there many such examples, e. g., H p ∩ B [205]. Kahane [282] was the first who used the best approximation in proving that some spaces have the f-property. In [247] Havin considered the case q = p = ∞ of Theorem 6.22 and some other spaces and proved that all they have the so-called K-property, which means that the Töplitz operators Tφ (f )(z) = ∫ − 𝕋

φ(ζ )f (ζ ) |dζ | 1 − ζ ̄z

map the space under consideration into itself for all φ ∈ H ∞ . Havin noted that Korenblum [322] was the first who observed that the K-property implies the f-property. The spaces Bp,q α (p ≥ 1, q ≥ 1, α > 0) have the K-property, which can also be deduced from the three facts: (a) Tφ is the adjoint to the operator of pointwise multi-

plication by φ; (b) Bpα ,q (q ≠ ⬦) is equal to the dual of the Bergman space Hαp,q (q ≤ ⬦); (c) Tφ maps P into P . It follows that Theorem 6.22 holds for p ∈ {1, ∞} as well. One of powerful methods developed by Dyn’kin [180, 181] in detecting the K-property and in various other questions is that of pseudoanalytic continuation. However, there are many important classes, for example, H 1 , HS11 , A(𝔻), that have f-property without having the K-property. In such cases the proofs can be much more 󸀠

󸀠

6.9 Addendum: Radial limits of Hardy–Bloch functions | 213

delicate. We refer the reader to the Shirokov’s book [535] for further results on various spaces of this kind and for references before 1988. Among the papers after 1988, we mention [591], where Vinogradov considered the problem of division and multiplication by inner functions in the space Bp,p , 0 < p < 2. He proved that in this space, we “can multiply” but “cannot divide” by the atomic inner function and also that there exists a Blaschke product in H p \ Bp,p . In the case p = 1 the latter was proved by Rudin [514] 40 years earlier. As far the author knows, the most recent expository paper concerning the f- and K-properties was written by Girela, González, and Peláez [206].

6.9 Addendum: Radial limits of Hardy–Bloch functions By the Littlewood–Paley theorem (see, e. g., Theorem 6.13) we have Bp,p ⊂ H p for p ≤ 2 and H p ⊂ Bp,p for p > 2, and, by Theorem 2.21, we have H p ⊂ Bp,2 for p < 2 and Bp,2 ⊂ H p for p ≥ 2. It follows that if f ∈ Bp,p , p ≤ 2, or f ∈ Bp,2 , p ≥ 2, then f has radial limits a. e. Therefore it is natural to ask for which p, q we have the implication f ∈ Bp,q ⇒ f has radial limits a. e.

(6.46)

First, note that Bp,q ⊂ Bp,2 for q ≤ 2, and hence if f ∈ Bp,q for q ≤ 2 ≤ p, then this implication holds. Also, since Bp,q ⊂ Bp,p for q ≤ p ≤ 2, we see that the implication holds in this case as well. Theorem 6.23. Implication (6.46) does not hold if and only if one of the following two conditions is satisfied: 1° 2 < q ≤ ∞, 2° p < q ≤ 2. Moreover, in these cases, there is a subset X of the second category in Bp,q such that lim supr→1− |f (reiθ )| = ∞ a. e. for all f ∈ X. In particular, implication (6.46) holds if and only if either 0 < q ≤ p ≤ 2 or q ≤ 2 < p. Remark. In [451] the condition q ≤ 2 < p was omitted. For the case p = q > 2, see [309], where the existence of a function f for which radial limits exist almost nowhere was given based on a different idea. Proof. Consider case 1°. Let (6.46) hold. Define the operators Tn : Bp,q 󳨃→ ℒ0 (𝕋) by Tn f (eiθ ) = f (rn eiθ ), where rn ↑ 1. Because of (6.46), we have 󵄨 󵄨 Tmax f = sup󵄨󵄨󵄨Tn f (eiθ )󵄨󵄨󵄨 < ∞ n

a. e.

for every f ∈ Bp,q . Hence by Banach’s principle (Theorem C.19) the sublinear operator Tmax continuously maps Bp,q into ℒ0 (𝕋). The hypotheses of the Nikishin–Stein theorem (C.17) are satisfied with X = Bp,q and T = Tmax , because every s-Banach space

214 | 6 Besov spaces (0 < s ≤ 1) is of type s. It follows from this and from Corollary C.13 that Tmax maps Bp,q into Lϱ for some ϱ > 0, which means that Bp,q ⊂ H ϱ . Let q > 2. Then the preceding inclusion does not hold, which can be seen by conn sidering lacunary series; for instance, the function f (z) = ∑(n + 1)−1/2 z 2 belongs to Bp,q (see Theorem 6.8), whereas f ∈ ̸ H ϱ because of Paley’s theorem. Thus there is at least one function f ∈ Bp,q such that Tmax f = ∞. Now the existence of a set X satisfying the desired property follows from Theorem C.21. Case 2° is not so easy. We first observe that Bp,q is of type p, i. e., that 1

󵄩󵄩p 󵄩󵄩 󵄩 󵄩 ∫󵄩󵄩󵄩 ∑ rj (t)fj 󵄩󵄩󵄩 dt ≤ K ∑ ‖fj ‖p 󵄩󵄩 󵄩󵄩 j j≥0

(6.47)

0

for every finite sequence {fj } ⊂ Bp,q , where rj are the Rademacher functions. This is well known (see any book on the geometry of Banach spaces) and is a consequence of the same fact for mixed Lp,q -spaces.2 (In the case p ≤ 1, this is a consequence of the fact that Bp,q is a p-Banach space.) Then we apply the Nikishin–Stein theorem to conclude that Bp,q ⊂ Lp,⋆ , where Lp,⋆ (⊃ Lp ) is the weak Lp space. However, the author encountered the problem of finding a counterexample. The functions fξ (z) = (1−ξz)−1/p (0 < ξ < 1) constitute a typical bounded family in Lp,⋆ . Then the author decided to experiment with interpolation spaces. The idea is simple and well known. Lemma 6.36. If the identity operator maps Bp,q into Lp,⋆ , where p < q ≤ 2, then it maps Bϱ,q into H ϱ , where 1 1 1 = + . ϱ 2p 2q Before proving this lemma, we continue the proof of the theorem. As noted before, the identity operator Id maps Bp,q into Lp,⋆ . Hence, by Lemma 6.36, Id maps Bϱ,q into H ϱ (continuously). To see that this is impossible, consider the functions fξ (z) = (1 − ξz)−1/ϱ ,

0 < ξ < 1.

It is easy to check that 1/ϱ

4 ) 1−ξ

‖fξ ‖ϱ ≍ (log

.

On the other hand, since Mϱ (r, fξ󸀠 ) ≤ C(1 − rξ )−1 , we have 1

q−1

‖fξ ‖Bϱ,q ≤ C(∫(1 − r) 0

2 See Remark 6.37.

1/q −q

(1 − rξ )

dr)

≤ C(log

1/q

4 ) 1−ξ

.

6.9 Addendum: Radial limits of Hardy–Bloch functions | 215

From this and from the inclusion Bϱ,q ⊂ H ϱ we find that 1/ϱ

(log

4 ) 1−ξ

≤ C(log

1/q

4 ) 1−ξ

,

which does not hold because 1/ϱ > 1/q. Proof of Lemma 6.36. To prove this “non-lemma”, we need some facts from the theory of interpolation spaces. Given two quasi-Banach spaces X0 and Y0 that are continuously embedded into a linear topological space, we use the K-method (of the real interpolation) to construct the spaces [X0 , Y0 ]ϱ,η (0 < ϱ < ∞, 0 < η < 1). Let X1 , Y1 be another couple of such spaces. Then the following statement holds: If T is a linear operator that continuously maps X0 into Y0 and X1 into Y1 , then T maps [X0 , Y0 ]ϱ,η into [X1 , Y1 ]ϱ,η . It is known that if 1 1−η η = + , ϱ p q then [H p , H q ]ϱ,η = H ϱ

and [Lp,⋆ , Lq,⋆ ]ϱ,η = Lϱ .

(6.48)

For the first relation in a more general situation, see Kislyakov and Xu [312]. The second relation is well known and can be found in many books, for example, [67]. Another fact is needed (see [67, Theorem 5.6.2]): [ℓq (X), ℓq (Y)]ϱ,η = ℓq ([X, Y]ϱ,η ). From (6.48) and (6.49) it follows that [ℓq (H p ), ℓq (H q )]ϱ,η = ℓq (H ϱ ). ∞ q p Now let F = {fj }∞ 0 ∈ ℓ (H ). Consider the operator TF = ∑j=0 Vj ∗ fj . Since

Vn ∗ Vj = 0,

|j − n| ≥ 2

(V−1 := 0),

we have Vn ∗ TF = Vn ∗ (fn−1 ∗ Vn−1 + fn ∗ Vn + fn+1 ∗ Vn+1 ), which implies ‖Vn ∗ TF‖p ≤ C(‖fn−1 ‖p + ‖fn ‖p + ‖fn+1 ‖p ).

(6.49)

216 | 6 Besov spaces Hence TF ∈ Bp,q . For the same reason, T maps ℓq (H q ) into Bq,q . Now the hypothesis of the lemma implies that TF ∈ Lp,⋆ and ‖TF‖p,⋆ ≤ C‖F‖ℓq (H p ) . On the other hand, from the Littlewood–Paley inequality it follows that ‖TF‖q,⋆ ≤ C‖TF‖q ≤ C‖TF‖Bq,q ≤ C‖F‖ℓq (H q ) . Now we apply interpolation to conclude that T maps the space [ℓq (H p ), ℓq (H q )]ϱ,1/2 = ℓq ([H p , H q ]ρ,1/2 ) = ℓq (H ϱ ) into [Lp,⋆ , Lq,⋆ ]ϱ,1/2 = Lϱ . This means in particular that ∞

‖TF‖ϱ ≤ C( ∑

n=0

1/q q ‖fn ‖ϱ ) .

Take Qn = Vn−1 + Vn + Vn+1

(V−1 = 0) and fn = Qn ∗ f ,

where f ∈ H ϱ .

Since Qn ∗ Vn = Vn , we have TF = f , and therefore ∞

‖f ‖ϱ ≤ C( ∑ ‖Qn ∗ f ‖qϱ ) n=0

1/q

≤ C‖f ‖Bϱ,q .

This completes the proof. Remark. The reader will found results, explanations, and references concerning interpolation for Hardy and other spaces in Kislyakov’s paper [311]. Remark 6.37. The fact that Bp,q is of type p when 1 < p < q ≤ 2 can be proved by using Kahane’s inequality 1/q 1/p 1󵄩 N 1󵄩 N 󵄩󵄩q 󵄩󵄩p 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 (∫󵄩󵄩󵄩 ∑ rk (t)fk 󵄩󵄩󵄩 dt) ≤ C(∫󵄩󵄩󵄩 ∑ rk (t)fk 󵄩󵄩󵄩 dt) ; 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩X 󵄩X 0 󵄩k=0 0 󵄩k=0

see [283] and [353, p. 74]. Here X is a Banach space, and q > p > 1.

Further notes and results | 217

Problem 6.8. It may be of some importance to characterize Bp,q via the boundary functions (when they exist). In the case p = q = 1, this was done by de Souza and Sampson [144]. They connected B1,1 with the space B defined as ∞

B = {f : 𝕋 󳨃→ ℝ | f (eiθ ) = ∑ cn bn (t)}. n=1

Each bn is a special atom, that is, a real-valued function b defined on 𝕋 such that either 1R 1L b(t) ≡ 1/2π or b = − |I| + |I| , where I is an interval on 𝕋, and where R is the right half of I, and L is the left half. It was shown in [144] that B can be identified with the boundary values of B1,1 in the sense that if f ∈ B1,1 , then limr→1− Re f (reiθ ) = g(eiθ ) belongs to B, and that if g ∈ B, then f (z) = ∫ − 𝕋

ζ +z g(ζ )|dζ | ζ −z

1,1

belongs to B .

Further notes and results Theorem 6.11, sometimes in a different form, was proved and reproved by several authors: Zakharyta and Yudovich [624] (1 < p = q < ∞), Taibleson [578] (1 < p, q < ∞) (nonperiodic spaces), Flett [195] (0 < p ≤ ∞, 0 ≤ q ≤ 1), Anderson, Clunie and Pommerenke [37] (the case of the little Bloch space b := B∞,⬦ ), and Ahern and Jevtić [10] (p = 1 and 1 < q < ∞). Our approach is borrowed from [421]. We will again use this approach in Chapter 13. Theorem 6.14 is well known at least for p ≥ 1, but it is not easy to locate the paper where it was first formulated; at least for p ≥ 1, it can be deduced from Theorem 3.15 and some results of Soviet mathematicians on relationships between moduli of smoothness and best approximation by polynomials (see, e. g., [59], [583]). The case 0 < p < 1 (and p ≥ 1) is treated in [412]. For information and references in the nonperiodic case, we refer to Nikol’skiĭ’s monograph [405]. Corollary 6.31 is a result of Ahern [8], who proved it by using his Theorem 2.25; the proof in the text is based on the ideas from [387]. 6.1. Theorem 6.6 has a striking generalization obtained by using a molecular decomposition of Bp,q α ; see Cleanthous et al. [126, Theorem 4.2]. Theorem (CGN). Let p > 0, q > 0, and let k > min{1, 1/p} be an integer. Let ψ ∈ C k (ℝ+ ) be such that 󵄨󵄨 (j) 󵄨󵄨 γ−j 󵄨󵄨ψ (t)󵄨󵄨 ≤ Cj (1 + t) ,

t > 0,

p,q k p,q ̂ for 1 ≤ j ≤ k. Then the operator Tf (z) = ∑∞ n=0 ψ(k)f (k)z maps Bα+γ into Bα . (The same p holds for the Triebel–Lizorkin spaces Fq,α for q ∈ ℝ+ ; see page 372 for the definition.)

218 | 6 Besov spaces 6.2 (Non-Schauder bases in Hβ1,q ). An interesting difference between Hardy and Ber-

gman spaces should be noted. The spaces hp (1 < p < ⬦) and h1,q are self-conjugate. β In the case of hp , this was used to show that the sequence {en } is a Schauder basis in H p . In contrast, this sequence is not a Schauder basis in Hβp,q , p ≤ 1. To check this, let

X = Hβp,q , and assume that, for instance, q < ⬦. Define two spaces: 1

Xβq = {f ∈ H(𝔻) : ∫ ‖fρ ‖qX (1 − ρ)qβ−1 dρ < ∞} and 0

∞

ΔXβq = {f ∈ H(𝔻) : ∑ 2−nβq ‖Δn f ‖qX < ∞}. n=0

p,q A relatively simple calculation shows that Xβq = H2β and ΔXβq = Δp,q ; see (6.2). As2β suming that {en } is a Schauder basis in X, we can use Lemmas 2.6 and 4.11 to show that p,q . ⊂ Δp,q H2β 2β

(†)

We will prove that this is impossible and moreover that the following holds. p,q Proposition ([384]). If p ≤ 1, then H2β ⊈ Δ1,∞ γ , where γ = 2β + 1/p − 1.

This improves the negation of (†), which follows from the inclusions Δp,q ⊂ Δp,∞ ⊂ Δ1,∞ γ 2β 2β

(p ≤ 1).

Proof. Assuming that p,q Y := H2β ⊂ Δ1,∞ =: Z, γ

(‡)

we want to obtain a contradiction by use of an example due to F. Riesz [60, p. 599]: n

n

2/p

fn (z) = z 2 (1 − z)−2/p (1 − z 2 )

n

n

2/p

= z 2 (1 − z 2 )

∞

∑ ck z k ,

k=0

where ck = Γ(k + 2/p)/k!Γ(2/p) ≍ (k + 1)2/p−1 . We have Δn fn (z) = z

2n

2n −1

∑ ck z k

k=0

and hence, by Hardy’s inequality, 󵄩󵄩 󵄩󵄩2n −1 n 󵄩󵄩 󵄩󵄩 |c | ‖Δn fn ‖1 = 󵄩󵄩󵄩 ∑ cn−k ek 󵄩󵄩󵄩 ≥ ∑ n−k ≥ c(n + 1)2n(2/p−1) , 󵄩󵄩 󵄩󵄩 󵄩1 k=0 k + 1 󵄩 k=0

(+)

Further notes and results | 219

that is, ‖fn ‖Z ≥ c(n + 1)2−nγ 2n(2/p−1) = c(n + 1)2n(−2β+1/p) . On the other hand, 1 2π󵄨2n −1 󵄨󵄨2 q/p 󵄨󵄨 n 󵄨󵄨 󵄨 ‖fn ‖qY ≍ ∫(1 − r)2βq−1 r q2 ( ∫ 󵄨󵄨󵄨 ∑ r k eikθ 󵄨󵄨󵄨 ) dr dθ ≤ C2nq/p 2−2nβq , 󵄨󵄨 󵄨󵄨 󵄨 0 0 󵄨 k=0

where we have used the inequality 󵄨󵄨2 2n −1 󵄨󵄨 󵄨󵄨 k ikθ 󵄨󵄨󵄨 − 󵄨󵄨 ∑ r e 󵄨󵄨 dθ = ∑ r 2k ≤ 2n . ∫ 󵄨󵄨 󵄨󵄨 k=0 󵄨 0 󵄨 k=0 2π󵄨2n −1

From this inequality, (+), and (‡) it follows that (n + 1)2n(−2β+1/p) ≤ C2n(−2β+1/p) , a contradiction. In the case p = ∞, we can use the following fact from [66, Lemma 1.14]: There exist a constant c > 0 and a sequence of polynomials Pn of degree 2n such that ‖Pn ‖∞ = 1 and ‖sn Pn ‖∞ ≥ c log(n + 2) for all n (sn P is the nth partial sum of P). 6.3. If 1 < p < ⬦, then, due to Theorem 6.1, inequalities (6.19) and (6.20) can be expressed in the form ∞

∞

n=0

n=0

cp ∑ ‖Δn f ‖pp ≤ ‖f ‖pp ≤ Cp ( ∑ ‖Δn f ‖2p ) ∞

cp ( ∑ ‖Δn f ‖2p ) n=0

p/2

p/2

∞

≤ ‖f ‖pp ≤ Cp ∑ ‖Δn f ‖pp n=0

(2 ≤ p < ⬦), (1 < p ≤ 2).

These inequalities were deduced from the Littlewood–Paley g-theory by Sledd [538], who, however, did not recognized their connection with Theorems 2.21 and 6.13. 6.4 (Blaschke products in Hardy–Sobolev spaces). The paper [219] of Gröhn and Nicolau contains a number of deep results concerning membership of a Blaschke product in the Hardy–Sobolev space HSp1 , 1/2 < p < 1 (and a large number of references to earlier results). To state the main result of [219], let FB (ζ ) = ∑

zn ∈Uζ

1 , 1 − |zn |

ζ ∈ 𝔻,

where Uζ is the Stoltz angle with vertex ζ , and zn are the zeroes of B (Blaschke product) repeated according to their multiplicity. Then we have the following:

220 | 6 Besov spaces Theorem (GN). Let 1/2 < p < 1, and let B be a Blaschke product with zeroes {zn }. Then the following conditions are equivalent: (a) B ∈ HSp1 ; (b) There exists a constant c, 0 < c < 1, such that ∫ z∈𝔻, |B(z)| λ}󵄨󵄨󵄨 ≤ , λ

λ > 0, 0 < r < 1,

where C depends only on f ∈ H(𝔻). Since H 1,⋆ ⊂ H p for p < 1, it follows from Corollary 6.35 that if an inner function I is such that I 󸀠 ∈ H 1,⋆ , then I is a Blaschke product. On the other hand, Cima and Nicolau [123] proved a surprising result. Theorem (CN). Let B be a Blaschke product. Then B󸀠 ∈ H 1,⋆ if and only if B is an exponential Blaschke product. that

A Blaschke product B is called exponential if there exists a constant M > 0 such card{z : B(z) = 0, 2−k−1 ≤ 1 − |z| ≤ 2−k } ≤ M

for all integers k ≥ 1. Let {zn } be the zeroes of B ordered so that |zn | ≤ |zn+1 | for n ≥ 1. Then B is exponential if and only if there exist constants C > 0 and δ < 1 such that 1 − |zn | ≤ Cδn . Several variables In a recent paper [267], Ivanov and Petrushev proved that the decomposition theorem 6.4, with an appropriate modification, remains valid if we pass to the Besov (and Triebel–Lizorkin) spaces of harmonic functions on the unit ball of ℝn . As usual, analysis of harmonic function spaces of three or more variables is subtler than analysis of analytic function spaces. The paper [202] by Gergün et al. is very relevant in this context because it contains new ideas, results, and references, although the authors consider “two-parameter” spaces, that is, the spaces of the form Bαp,q , p = q (in several variables).

7 The dual of H 1 , and some related spaces We define the space BMO (of functions of bounded mean oscillation) via a Garsiatype norm, and then, following a recent paper of Knese, we prove an Lp -variant of Garsia’s theorem. The main ingredient of the proof is Uchiyama’s lemma, a relatively simple consequence of Green’s theorem. This lemma enables us to prove Fefferman’s duality theorem, (H 1 )A ≃ BMOA, without using the Carleson measures and Carleson’s theorem. In the rest of this chapter, we present, among other things, some results on the Bloch space B ⊃ BMOA and its predual B1,1 , and on the space Bp1/p = Bp,∞ ⊂ 1/p BMOA (p < ∞). In Section 7.8, we present some recent results on compact composition operators on the Bloch space and BMOA.

7.1 The norms on BMOA A function g ∈ L1 (𝕋) is called, after John and Nirenberg [278], a function of bounded mean oscillation if sup I⊂𝕋

1 ∫ |g − gI | dl = ‖g‖∗ < ∞, |I|

where gI =

I

1 ∫ g dl, |I| I

and the supremum is taken over all subarcs of 𝕋. The class BMO(𝕋) = {g: ‖g‖∗ < ∞} is normed by ‖g‖ = ‖g‖L1 + ‖g‖∗ . It is not easy to see that BMO(𝕋) coincides with the class BMO(𝕋)2 , which consists of the functions g ∈ L2 (𝕋) such that 1 󵄨 󵄨2 ‖g‖2∗∗ := sup ∫󵄨󵄨󵄨g(eit ) − gI 󵄨󵄨󵄨 dt < ∞. I I⊂𝕋 I

This is a consequence of the John–Nirenberg inequality, which says that there exist constants c, C > 0 such that, for any interval I ⊂ 𝕋, |{ζ ∈ I : |g(ζ ) − gI | > λ}| −cλ ), ≤ C exp( |I| ‖g‖∗ which in turn is implied by the strong John–Nirenberg inequality: there exists c > 0 such that ε < c/‖g‖∗ implies sup I⊂𝕋

it 1 ∫ eε|g(e )−gI | dt < ∞. |I|

I

We are mainly concerned with the intersection of BMO with H 1 (𝕋), which is denoted by BMOA. https://doi.org/10.1515/9783110630855-007

222 | 7 The dual of H1 , and some related spaces Since BMOA ⊂ H 1 (𝕋), we can treat BMOA as a subset of H(𝔻). It is known that (see [198, Ch. VI, Th. 1.2]) 󵄨 󵄨 ‖f∗ ‖∗ ≍ sup ∫ − 󵄨󵄨󵄨f∗ (ζ ) − f (z)󵄨󵄨󵄨P(z, ζ )|dζ | =: ‖f ‖∗1 . z∈𝔻

𝕋

We do not reproduce the proof of this fact here. Instead, we define BMOA by the requirement ‖f ‖∗1 < ∞ (with the norm ‖f ‖BMO1 = |f (0)| + ‖f ‖∗1 ). On the other hand, Garsia proved that the original norm in BMOA is equivalent with 󵄨 󵄨2 󵄨 󵄨 ‖f ‖BMO2 := 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + sup(− ∫󵄨󵄨󵄨f (ξ ) − f (a)󵄨󵄨󵄨 P(a, ξ )|dξ |) a∈𝔻

󵄨 󵄨 = 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + ‖f ‖∗2 ,

1/2

𝕋

(see [198, Ch. VI]). Here we start from “our” definition and prove Garsia’s theorem, and then we use it to prove Fefferman’s duality theorem. First, observe that ‖f ‖∗1 can be expressed as 󵄨 󵄨 ‖f ‖∗1 = sup ∫ − 󵄨󵄨󵄨f (σa (ζ )) − f (ζ )󵄨󵄨󵄨|dζ |. a∈𝔻

𝕋

As follows from the following lemma, the seminorm ‖f ‖∗2 can be expressed in several ways. Lemma 7.1. We have B1 (a, f ) = B2 (a, f ) = B3 (a, f ) = B4 (a, f ) = B5 (a, f ) (a ∈ 𝔻, f ∈ H 2 ), where 󵄨2 󵄨 B1 (a, f ) = ∫ − 󵄨󵄨󵄨f (ξ ) − f (a)󵄨󵄨󵄨 P(a, ξ )|dξ |, 𝕋

󵄨2 󵄨2 󵄨 󵄨 B2 (a, f ) = ∫ − (󵄨󵄨󵄨f (ξ )󵄨󵄨󵄨 − 󵄨󵄨󵄨f (a)󵄨󵄨󵄨 )P(a, ξ )|dξ |, T

󵄨2 󵄨 B3 (a, f ) = ∫ − 󵄨󵄨󵄨f (σa (ζ )) − f (a)󵄨󵄨󵄨 |dζ |, 𝕋

B4 (a, f ) =

2 󵄨󵄨 󸀠 1 󵄨2 󵄨2 󵄨 dA(z), ∫󵄨󵄨f (σa (z))󵄨󵄨󵄨 󵄨󵄨󵄨σa󸀠 (z)󵄨󵄨󵄨 log π |z| 𝔻

1 2 󵄨 󵄨2 dA(z). B5 (a, f ) = ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 log π |σa (z)| 𝔻

Recall that σa (z) =

a−z . 1 − az̄

7.1 The norms on BMOA

| 223

Proof. The identity B1 = B2 holds because 󵄨2 󵄨2 󵄨 󵄨 B1 (a, f ) = ∫ − (󵄨󵄨󵄨f (ζ )󵄨󵄨󵄨 + 󵄨󵄨󵄨f (a)󵄨󵄨󵄨 − 2 Re f (ζ )f (a))P(a, ζ )|dζ | 𝕋

and 󵄨2 󵄨 − Re(f (ζ )f (a))P(a, ζ )|dζ | = 󵄨󵄨󵄨f (a)󵄨󵄨󵄨 . ∫ 𝕋

The identity B1 = B3 is proved by the change σa (ζ ) = ξ . The equality B1 = B4 follows from the Green’s formula applied to the function f ∘ σa − f (a). Then we use the change σa (z) = w to show that B4 = B5 . It is sometimes more convenient to work with the seminorm 1/2

B6 (f ) := sup(B6 (a, f )) , a∈𝔻

where 󵄨 󵄨2 󵄨 󵄨 B6 (a, f ) = ∫󵄨󵄨󵄨f 󸀠 (σa (z))󵄨󵄨󵄨 󵄨󵄨󵄨σa󸀠 (z)󵄨󵄨󵄨(1 − |z|2 ) dA(z) 𝔻

󵄨2 󵄨 󵄨2 󵄨 = ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − 󵄨󵄨󵄨σa (z)󵄨󵄨󵄨 ) dA(z) 𝔻

2 2 󵄨2 (1 − |a| )(1 − |z| ) 󵄨 = ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 dA(z). 2 ̄ |1 − az|

(7.1)

𝔻

The relation B6 (f ) ≍ supa∈𝔻 (B4 (a, f ))1/2 = ‖f ‖∗2 is a consequence of the following lemma. Lemma 7.2. There exists a constant C independent of a and f such that B4 (a, f )/C ≤ B6 (a, f ) ≤ CB4 (a, f ). Proof. If u ≥ 0 is a subharmonic function, then using the maximum principle we show that ∫ u(z) log 𝔻

1 dA(z) ≍ ∫ u(z)(1 − |z|2 ) dA(z), |z| 𝔻

where the equivalence constants are independent of u. Taking 󵄨 󵄨2 󵄨 󵄨 u(z) = 󵄨󵄨󵄨f 󸀠 (σa (z))󵄨󵄨󵄨 󵄨󵄨󵄨σa󸀠 (z)󵄨󵄨󵄨, we get the result.

224 | 7 The dual of H1 , and some related spaces Lemma 7.3. The space BMOAp := {f ∈ H p : ‖f ‖∗p < ∞} (p = 1, 2) normed by 1/p

󵄨 󵄨p 󵄨 󵄨 ‖f ‖BMOp = 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + sup(− ∫󵄨󵄨󵄨f (ζ ) − f (a)󵄨󵄨󵄨 P(a, ζ )|dζ |) a∈𝔻

𝕋

is homogeneous and has the Fatou property.1 Proof. Consider the case p = 1. We have 󵄨 󵄨 󵄨 󵄨 − 󵄨󵄨󵄨(f (wζ ) − f (wa))P(a, ζ )󵄨󵄨󵄨|dζ | sup ‖fw ‖BMO1 = 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + sup sup ∫ a∈𝔻 |w| 101 so that ∞

∑ ϱj exp (−2−j−1/2 ϱj−1 √ϱ) < 1/100. j=1

Since 1/ϱn < 1/100, we have 1/4 − 1/ϱn − 1/(ϱ − 1) > 1/4 − 1/100 − 1/100. Combining all these estimates, we get (7.38). In the case of (7.39), we have n−1

∞

󵄨 󵄨󵄨 n μ j μ j 󵄨󵄨h2 (z)󵄨󵄨󵄨 ≥ ϱ |z| n − 1 − ∑ ϱ − ∑ ϱ |z| j ≥ ϱn (1 −

j=1 μn

1 ) μn

j=n+1

−1−

μj

∞ ϱn 1 ) − ∑ ϱj (1 − ϱ − 1 j=n+1 λn+1 ∞

≥ ϱn (1/4 − 1/ϱn − 1/(ϱ − 1)) − ∑ ϱj exp(− j=n+1

∞

λn+1

= ϱn (1/4 − 1/ϱn − 1/(ϱ − 1)) − ϱn ∑ ϱj exp(− j=1

μj

)

μj+n

λn+1

)

∞

≥ ϱn (1/4 − 1/ϱn − 1/(ϱ − 1)) − ϱn ∑ ϱj exp(−23/2−2j ϱj−1 √ϱ). j=1

Now the proof of (7.39) is completed as in the case of (7.38). Finally, we have to prove that 1 󵄨 󵄨󵄨 ), 󵄨󵄨h(z)󵄨󵄨󵄨 ≤ Cϕ( 1 − |z|

|z| < 1,

where h = h1 or h2 , but this follows from the modified Lemma 7.22. Thus the theorem is proved.

7.10 Addendum: Carleson measures Theorem 7.5 is usually proved by using the famous Carleson’s theorem on the Carleson measures. A positive measure μ on 𝔻 is called a Carleson measure if there is a constant C such that μ(W(I)) ≤ C|I|

(†)

7.10 Addendum: Carleson measures | 255

for every arc I ⊂ 𝕋, where W(I) = {z ∈ 𝔻 : 1 − |I| < |z| < 1,

z ∈ I}. |z|

The set W(I) is called a Carleson window. The smallest C satisfying (†) is called the Carleson norm of μ; denote it by ‖μ‖∗ . Theorem 7.25 (Carleson measure theorem). Let μ be a positive Borel measure on 𝔻. Then the following assertions are equivalent: (a) μ is a Carleson measure. (b) There is a constant C1 such that μ{|f | > λ} ≤ C1 |{M∗ f > λ}| for every continuous function f on 𝔻. (c) There is a constant C2 such that ∫𝔻 |f |p dμ ≤ C2 ‖f ‖pp for all (for some) p > 0 and f ∈ H p. (d) There is a constant C3 such that ∫𝔻 |σa󸀠 (z)| dμ(z) ≤ C3 for all a ∈ 𝔻. Moreover if ‖μ‖j (j = 1, 2, 3) denotes the smallest Cj ≥ 0 satisfying the corresponding inequality, then ‖μ‖∗ ≍ ‖μ‖j . See Carleson [112, 114]. A relatively easy, geometrically obvious proof is in Garnett [198]. Having proved Carleson’s theorem, we can easily prove Fefferman’s theorem; see [627, Section 5.4] for the case of several variables. Here we record the following characterization of BMOA. Theorem 7.26 (Carleson measure characterization of BMOA). A function f ∈ H(𝔻) belongs to BMOA if and only if the measure 󵄨2 󵄨 dmf (z) := 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|) dA(z),

z ∈ 𝔻,

is a Carleson measure. The statement remains true if we replace 1 − |z| with log(1/|z|) and f 󸀠 with ℛ1 f or 𝒥 1 f . The “only if” part follows from Lemma 7.6. The ‘iIf” part can be seen from the proof of Lemma 7.7, the Carleson measure theorem (item (c), p = 1), and Fefferman’s duality theorem 7.5. Carleson measures and the Nevanlinna counting function Consider a Carleson window of the following form: W(ζ , ε) = {z ∈ 𝔻: |z| ≥ 1 − h and | arg z − arg ζ | ≤ ε},

(7.40)

where 0 < ε < 1, and set W(ζ , ε) = 𝔻 for ε ≥ 1. For every analytic self-map φ of 𝔻, we define the measure λφ on 𝔻 by 󵄨 󵄨 2πλφ (E) = 󵄨󵄨󵄨{ζ ∈ 𝕋: φ∗ (ζ ) ∈ E}󵄨󵄨󵄨,

E ⊂ 𝔻,

256 | 7 The dual of H1 , and some related spaces and the maximal function of λφ for 0 < ε < 1 by ρφ (ε) = sup λφ (W(ζ , ε)). ζ ∈𝕋

The function ρφ is called the Carleson function of φ. There are two criteria for the compactness of the composition operator 𝒞φ on H 2 : (A) ρφ (ε) = o(ε), ε → 0 (MacCluer [127]); (B) Nφ (w) = o(1 − |w|), |w| → 1 (Shapiro [529]). This indicates that some connection between two seemingly different notions must exist. A few authors expressed this connection in a qualitative form, and we refer to Lefèvre et al. [341] for explanation and references. As noted in [341], “…the regularity of composition operators Cφ on H 2 (…) in terms of their “symbol” φ has been studied either from the point of view of Carleson measures or from the point of view of the Nevanlinna counting function, those two points of view being completely separated.” In that paper a quantitative relation was proved: Theorem 7.27. There exists a universal constant C > 1 such that, for every analytic selfmap φ: 𝔻 → 𝔻, we have (1/C)ρφ (ε/C) ≤ sup Nφ (w) ≤ Cρφ (Cε) |w|≥1−ε

(7.41)

for ε small enough. Moreover, for every ζ ∈ 𝔻, we have (1/C)λφ [W(ζ , ε/C)] ≤

sup

w∈W(ζ ,ε)∩𝔻

Nφ (w) ≤ C

λφ [W(ζ , Cε)].

The measure λφ can be used to describe ‖f ∘ φ‖p by means of an integral on 𝔻: 1/p

‖f ∘ φ‖H p = (∫ |f |p dλφ )

.

𝔻

This formula, called the transfer formula, plays an important role in the study of various properties of the composition operators. Lefèvre and Rodríguez-Piazza [343], considered the question when a Carleson embedding (in particular, 𝒞φ ) is ρ-summing; see Subsection 2.6.2 for the definition.

Further notes and results Our approach to BMOA in Section 7.2 (except Subsection 7.2.1) is essentially the same as that in Knese’s paper [314], although we have made some technical simplifications. Also, Knese does not treat the case p < 1 of Theorems 7.2 and 7.4. Beside Knese’s paper, the reader can read [444] for various, rather paradoxical characterizations of h1 .

Further notes and results | 257

A rather complicated proof of Theorem 7.4 (p ≥ 1) and Theorem 7.2 (p > 0) can be found in Girela’s survey paper [203, Theorems 4.1, 5.1, 5.4]. This paper and Baernstein’s paper [56] contain a lot of information and references. The first characterization of compact operators on B, the equivalence (a) ⇔ (c) of Theorem 7.18, was found by Madigan and Matheson [370]. The equivalence (a) ⇔ (d) is due to Maria Tjani [584], and (a) ⇔ (b) was proved by Wulan, Zheng, and Zhu [603]. The relations (a) ⇔ (b) ⇒ (c) of Theorem 7.19 were proved by Wulan, Zheng, and Zhu [603]. Their proof is based on an earlier result of Wulan [602], who proved that (a) is equivalent to (b) & (c); in [603], it is only observed that (b) implies (c). Theorem 7.20, from which the validity of (c) ⇒ (a) immediately follows, was proved by Laitila, Nieminen, Saksman, and Tylli [338]. For further results, references, and the history of the subject, see [603], [338], and [337]. Lemma 7.25 was proved in [336] and [1]; for a substantial improvement, see [220]. 7.1 (Campanato spaces, named after Campanato [109]). For a real number p, the Campanato space 𝒞p consists of g ∈ L1 (𝕋) for which 󵄨2 󵄨 ‖g‖2𝒞p := sup |I|−p ∫󵄨󵄨󵄨f (ζ ) − fI 󵄨󵄨󵄨 |dζ | < ∞, I⊂𝕋

(†)

I

where the supremum is taken over all arcs I ⊂ 𝕋. The analytic Campanato space 𝒞𝒜p is defined as 𝒞p ∩ H 1 . The following table was copied from [595]: Index p

Analytic Campanato space 𝒞𝒜p

p ∈ (−∞, 0] p ∈ (0, 1) p=1 p ∈ (1, 3]

Analytic Hardy space H2 Holomorphic Morrey space ℋℳ2,p BMOA Analytic Lipschitz space HΛ p−1

p ∈ (3, ∞)

Complex constant space ℂ

2

(The analytic Lipschitz space will be considered in Chapter 8). The Morrey space ℳp,q can be defined by (†), 0 < p < 1, as in Ye [618], but there are other possibilities; see, for example, Cascante et al. [116]. As noted in [618], “Morrey spaces were introduced in 1938 by Morrey [395] to show that certain systems of partial differential equations had Hölder continuous solutions.”. From [607] we know that if 0 < p < 2 and b ∈ 𝔻 is fixed, then the following conditions are equivalent: (a) f ∈ 𝒞𝒜p ; (b) |f 󸀠 (z)|2 (1 − |z|2 ) dA(z) is a p-Carleson measure; (c) |f 󸀠 (z)|2 (1 − |σb (z)|2 ) dA(z) is a p-Carleson measure; (d) supa∈𝔻 (1 − |a|2 )1−p ∫𝔻 |f 󸀠 (z)|2 (1 − |σa (z)|2 ) dA(z) < ∞.

258 | 7 The dual of H1 , and some related spaces A measure μ on 𝔻 is said to be a p-Carleson measure if sup μ(W(I))/|I|p < ∞. I⊂𝕋

In [595] a predual of 𝒞𝒜p was determined. Xiao and Yuan [609] considered a generalized version of Campanato spaces. For p ≥ 1 and η ≥ 0, they defined the space ℒp,η ⊂ Lp (𝕋) by the requirement 1/p

󵄨p 󵄨 ‖f ‖p,η := (sup |I|−η ∫󵄨󵄨󵄨f (ζ ) − fI 󵄨󵄨󵄨 |dζ |) I⊂𝕋

< ∞.

I

7.2 (Qp -spaces). For p ∈ ℝ+ , the space Qp ⊂ H(𝔻) is defined by the requirement 󵄨2 p 󵄨 󵄨2 󵄨 sup ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − 󵄨󵄨󵄨σa (z)󵄨󵄨󵄨 ) dA(z) < ∞. a∈𝔻

𝔻

The reader is referred to the Xiao’s books [606] and [607] for the theory, history, and references. The papers [28] and [29] of Aleman et al. are relevant in this context and should be read by everyone interested in the theory of spaces of analytic functions. Here we note that Qp = B for p > 1 and, obviously, Q1 = BMOA. Observe that f ∈ Qp if and only if |f 󸀠 (z)|2 (1 − |z|2 )p dA(z) is a p-Carleson measure. 7.3 (Reverse Carleson measures). It is interesting to characterize the measures μ on 𝔻 for which ‖f ‖pp ≤ C ∫ |f |p dμ,

f ∈ H(𝔻) ∩ C(𝔻),

(7.42)

𝔻

where p > 1 is fixed. For the first time, this was done in Lefèvre et al. [342] under the hypothesis that μ is already a Carleson measure: μ(W(I)) ≥ c|I| for every arc I ⊂ 𝕋.

(7.43)

Hartmann et al. [243] removed the extra hypothesis and proved that (7.42) is equivalent to (7.43) for an arbitrary measure μ. There are two additional equivalent conditions [243, Theorem 2.1]. 7.4 (Inner functions in the little Bloch space). There are inner functions in b that are not finite Blaschke products (Stephenson [559]). An infinite Blaschke product belonging to b was constructed by Bishop [76, 77]. Each of these constructions are highly nontrivial. See, however, Garnett [198, Ch. X, Further results 11], where a simpler proof is outlined. Aleksandrov, Anderson, and Nicolau [25, Theorem 3] substantially improved Stephenson’s theorem by proving that there is an inner function I ∈ b such that lim−

|z|→1

|I 󸀠 (z)|(1 − |z|2 ) = 0. 1 − |I(z)|2

Further notes and results | 259

As shown in [25], such a function cannot be extended analytically to any point of 𝕋; on the other hand, there is an inner function in b that extends analytically to a. e. point of 𝕋. kn 7.5 (Lacunary series in weighted Bloch spaces). A lacunary series f (z) = ∑∞ n=0 cn z ∞ (k0 ≥ 1) belongs to B if and only if {cn } ∈ ℓ ; this was first observed by Pommerenke [483] and then generalized by Yamashita [612]. This result was extended to some particular cases of weighted Bloch spaces by several authors (see, e. g., [325]). Finally, Yang and Xu [617] proved a result that covers all these particular cases. In the terms of weighted Bloch spaces, defined by (7.22), their theorem states that if ω(t)/t, 0 < t ≤ 1, is normal, then f ∈ ℬω if and only if {ω(1/kn )cn } ∈ ℓ∞ . However, as an application of Theorem 4.9, we obtain much more [447]: If ω(t)/t is normal, then

1 − r 󵄨󵄨 󵄨 󵄨f (rζ )󵄨󵄨󵄨 ≍ sup ω(1/kn )|cn |, ω(1 − r) 󵄨 0 p, then 1

Ip,q (f ) := ∫(Mp (r, f )/ log1/s 0

q

dr 2 ) 1 was established in [208]. The proof was, however, unnecessarily complicated. We give a simple proof, which also works in the case of harmonic functions of several variables. Assume that f (0) = 0 and consider the case where p ≥ 2 and q > 2. By Theorem 2.21 and Hölder’s inequality with indices q/s, q/(q − s), q/s (Mps (r, f ))

r

≤

C(∫ Mps (t, f 󸀠 )(r 0

s−1

− t)

q/s

dt)

260 | 7 The dual of H1 , and some related spaces r

≤ C ∫ Mpq (t, f )(1 − t)(s−1)q/s ϕ(t)q/s dt 0

r

× (∫ ϕ(t)

−q/(q−s)

dt)

(q−s)/s

,

0

where ϕ(t) = (1 − t)(q−s)/q log

2 . 1−t

After computing the inner integral, using the resulting inequality and Fubini’s theorem, we obtain the result. The same proof works if 1 ≤ p < 2. If p = s < 1, then (r − t)(p−1)q/p cannot be replaced with (r − t)(p−1)q/p . This technical problem can be avoided at the start by using the Littlewood–Paley inequality in the form 1

‖f ‖pp ≤ Cp ∫ Mpp (ρ, 𝒥 1/p f ) dρ,

0 < p < 2,

0

and, at the very end, Theorem 4.2. 7.8 (Bloch spaces of C 1 -functions). It was proved by Holland and Walsh [263] that a function f ∈ H(𝔻) belongs to B if and only if |f (w) − f (z)|√(1 − |z|2 )(1 − |w|2 ) < ∞. |w − z| w,z∈𝔻

β(f ) := sup

It is maybe surprising that the identity 󵄩 󵄩 β(F) = B(F) := sup(1 − |z|2 )󵄩󵄩󵄩F 󸀠 (z)󵄩󵄩󵄩

(7.44)

z∈𝔻

holds for any C 1 -function F with values in a Banach space [443]. The hypothesis F ∈ C 1 can be weakened, so we can use the relation |∇|f | | = |f 󸀠 | (f ∈ H(𝔻)) to show that B(f ) = β(|f |), for f ∈ B. Another example is sup (1 − r 2 )Mp (r, f 󸀠 ) = sup

‖fw − fz ‖p √(1 − |z|2 )(1 − |w|2 )

w,z∈𝔻

0 1, then such an inequality does not hold. It is not known what happens when a = 1. In an earlier paper [372], Makarov proved that this inequality holds for 0 < a < π 2 /64 = 0.1542 . . . . 7.13 (Cauchy transform of unimodular functions). The Cauchy transform of a function g ∈ L1 (𝕋) is defined by 𝒦[g] = 𝒦μ, where dμ(ζ ) = g(ζ )|dζ |. It is clear from the proof of Fefferman’s duality theorem that BMOA = {𝒦[g]: g ∈ L∞ (𝕋)}. In particular, if g is unimodular, that is, if |g(ζ )| = 1 for a. e. ζ ∈ 𝕋, then 𝒦[g] ∈ BMOA. We can prove that the cone generated by such Cauchy transforms is dense in BMOA. Moreover, we have the following: Theorem (A). The set {λ𝒦[f ̄/|f |]: f ∈ H 1 \ {0}, λ ∈ ℂ} is dense in BMOA. For the proof, we need some facts from the geometry of Banach spaces, one of which is a deep theorem due to Bishop and Phelps [78] (cf. [147, p. 3]). Theorem (BP). Let X be a complex Banach space. The subset of X 󸀠 consisting of nonzero functionals that attain their norm on the unit sphere S(X) of X is dense in the norm topology of X 󸀠 . A real Banach space X is said to be smooth if for each f on the unit sphere of X, there exists a unique Φ ∈ X 󸀠 such that ‖Φ‖ = 1 = Φ(f ). If X is complex, then X is smooth if it is smooth regarded as a real space. It is well known and it is not hard to prove that X is smooth if and only if its norm is Gateaux differentiable at every point at X \ {0}, which means the existence of the limit ρ󸀠 (f , h) := lim

ℝ∋t→0

‖f + th‖ − ‖f ‖ t

for all f ∈ X \ {0} and h ∈ X. It turns out the functional Φf (h) = ‖f ‖ρ󸀠 (f , h) is a unique bounded linear functional on X with ‖Φf ‖ = ‖f ‖; see [147, p. 20]. If X is a complex space, then the functional Φ0f (f ) = Φf (f ) + iΦf (if ) is a unique complex-linear functional that achieves its norm at f /‖f ‖, and ‖Φ0f ‖ = ‖Φf ‖. (This does not mean that f /‖f ‖

Further notes and results | 263

is a unique point in S(X) where Φ0f attains its norm.) Therefore, as a consequence of Theorem (BP), we have: If X is a smooth complex space, then the set {Φ0f : f ∈ X \ {0}} is dense in the norm topology of X 󸀠 . Unlike L1 , the space H 1 is smooth, and we have ρ󸀠 (f , h) = Re ∫ − 𝕋

f (ζ ) h(ζ )|dζ |. |f (ζ )|

Hence Theorem (A) is a consequence of Theorem (BP) and the following statement: If Λ ∈ (H 1 )󸀠 , Λ ≠ 0, and Λ attains its maximum on S(H 1 ), then there exists f ∈ S(H 1 ) such that Λ(h) = ‖Λ‖ ∫ − 𝕋

f (ζ ) h(ζ )|dζ |, |f (ζ )|

h ∈ H 1.

Exercise 7.27. The Bishop–Phelps theorem can be used to prove that the dual of Lp 󸀠 (1 < p < ⬦) is isometrically isomorphic to Lp . 7.14 (Hankel operators and vectorial BMOA). The reader interested in the notions of vectorial BMOA should read the excellent paper of Rydhe [523]. Here we present a small piece of the contents of that paper. The Hankel operator with symbol g ∈ H(𝔻) is defined on H(𝔻) by ∞

∞

̂ + n)f ̂(m))z n , Γg f (z) = ∑ ( ∑ g(m n=0 m=0

z ∈ 𝔻.

(7.46)

It is known that Γg extends to a bounded operator on H 2 if and only if g ∈ BMOA or, equivalently, if and only if g ∈ R+ L∞ (𝕋); the latter is known as Nehari’s theorem. (See the references in [523].) Let ℋ be a separable Hilbert space. Then (7.46) makes sense if g is an ℒ-valued analytic function and f ∈ 𝒪(ℋ), where ℒ is the space of all bounded operators on ℋ, and f is an ℋ-valued function analytic in a neighborhood of 𝔻 (abbreviated as f ∈ 𝒪(ℋ)). Nehari’s theorem was generalized by Page: Γg is H 2 (ℋ)-bounded if and only if g ∈ R+ L∞ (𝕋, ℒ), where L∞ (𝕋, ℒ) is the space of bounded Bochner–Lebesgue-measurable functions with values in ℒ. This leads to the notion of the Nehari–Page BMOA: It consists of ℒ-valued analytic functions for which the corresponding Hankel operator is bounded on H 2 (ℋ) and is denoted by BMOA𝒩 𝒫 (ℒ). The Carleson BMOA, denoted by BMOA𝒞 (ℒ), consists of ℒ-valued functions g analytic in 𝔻 for which 󵄩2 󵄩 sup ∫󵄩󵄩󵄩(𝒥 1 g)(z)f (z)̄ 󵄩󵄩󵄩ℋ (1 − |z|2 ) dA(z) < ∞,

f ∈𝒪1 (ℋ)

𝔻

(7.47)

264 | 7 The dual of H1 , and some related spaces where 𝒪1 (ℋ) is the set of H 2 (ℋ)-normalized functions from 𝒪(ℋ), and, as in the scalar case, 1

∞

n

̂ 𝒥 g(z) = ∑ (n + 1)g(n)z . n=0

It is known that BMOA𝒞 (ℒ) ≠ BMOA𝒩 𝒫 (ℒ); [523, Corollary 1.8]. Then the space BMOA𝒞 # (ℒ) is defined by the requirement that g # ∈ BMOA𝒞 (ℒ), where g # (z) = g(z)̄ ⋆ ; this function is obtained by taking the Hilbert space adjoint of each Taylor coefficient of g. It turns out that BMOA𝒞 (ℒ) ∩ BMOA𝒞 # (ℒ) ⊊ BMOA𝒩 𝒫 (ℒ). Finally, we note a surprising fact. The class, denote it by 𝒵 , defined by replacing f (z)̄ in (7.47) with f (z) differs from BMOA𝒞 (ℒ) [523, Corollary 1.11]. More precisely, there exists a bounded analytic function g : 𝔻 󳨃→ ℒ such that g ∈ 𝒵 \ BMOA𝒞 (ℒ). This is Rydhe’s negative answer to a question of Nazarov, Treil, Volberg, and Pisier, and a sufficient reason to read his paper.

8 Lipschitz spaces of first order This chapter contains not so deep but still important facts concerning relationships between the growth of the modulus of continuity of a function with the growth of its gradient or its tangential derivative. In Section 8.3, we present two theorems of Dyakonov (Theorems 8.8 and 8.9) on Lipschitz condition on the modulus of an analytic function with simple proofs due to the author. In Section 8.4, Lipschitz-type (derivative-free) characterizations of composition operators between Lipschitz spaces is given, which can be viewed as extensions of Dyakonov’s theorems or as a nonlinear counterpart to the Hardy–Littlewood theory of Lipschitz spaces.

8.1 Definitions and basic properties The space Λα (K). Let K be a bounded subset of ℂ. The modulus of continuity of a function g : K 󳨃→ ℂ is defined by 󵄨 󵄨 ω(g, δ; K) = sup{󵄨󵄨󵄨f (z) − f (w)󵄨󵄨󵄨 : |z − w| ≤ δ, z, w ∈ K},

δ ≥ 0.

If K is 𝔻 or 𝕋, we simplify the notation and write ω(g, δ) = ω(g, δ; 𝕋) and that

Ω(g, δ) = ω(g, δ; 𝔻).

By definition, Λα (K) (0 < α ≤ 1) is the set of complex-valued functions g on K such ω(g, δ; K) ≤ Cδα ,

δ > 0,

or, equivalently, 󵄨 󵄨󵄨 α 󵄨󵄨g(z) − g(w)󵄨󵄨󵄨 ≤ C|z − w|

(z, w ∈ K),

where C is a constant independent of z and w. The space Λω (K). More generally, let ω be a positive function on (0, t0 ], with ω(0+) = 0, where t0 is large enough. Then the space Λω (K) is defined by the requirement ω(g, δ; K) ≤ Cω(δ),

0 < δ < t0 .

(8.1)

The norm is given by Cg + ‖g‖∞ , where Cg = C (≥ 0) is the smallest constant satisfying (8.1), and ‖g‖∞ = supK |g|; with this norm, the space Λω (K) is a Banach space. If g ∈ Λω (K), then g is uniformly continuous on K and therefore has a continuous extension to K, the closure of K; moreover, we have ω(g, δ; K) = ω(g, δ; K). We will assume that ω is a majorant, that is, an increasing function on [0, t0 ] such that ω(0+ ) = ω(0) = 0 and that ω(t)/t decreases (t ∈ (0, t0 )). By Proposition 8.4 the https://doi.org/10.1515/9783110630855-008

266 | 8 Lipschitz spaces of first order hypothesis that ω(t)/t is decreasing can be replaced by an apparently weaker one: ω(t)/t is almost decreasing. The reason for which we assume that ω(t)/t is (almost) decreasing lies in the following: Proposition 8.1. The functions ω(g, δ)/δ and Ω(g, δ)/δ are almost decreasing for δ > 0 (provided they are finite). In other words, ω(g, sδ) ≤ Csω(g, δ) and

Ω(g, sδ) ≤ CsΩ(g, δ) ,

s ≥ 1, δ > 0,

where C is independent of s and δ. (Moreover, C can be chosen independently of g as well.) Proof. In the case of Ω, we use the inequality Ω(g, 2δ) ≤ 2Ω(g, δ) and the fact that Ω(δ) increases with δ. In the case of ω, we replace ω(g, δ) by ̄ δ) = sup{󵄨󵄨󵄨󵄨g(eiθ ) − g(eit )󵄨󵄨󵄨󵄨 : |t − θ| ≤ δ, t, θ ∈ ℝ}, ω(g, ̄ 2δ) ≤ 2ω(g, ̄ δ). which is equivalent to ω(g, δ), and then use the inequality ω(g, Properties of majorants Let ω be a majorant defined on the interval [0, 2]. We say that ω is a Dini majorant if 2

∫ 0

ω(x) dx < ∞. x

Following Dyakonov [177], we call a majorant ω fast if x

∫ 0

ω(t) dt ≤ Cω(x), t

0 < x < 2,

(8.2)

ω(x) ω(t) dt ≤ C , 2 x t

0 < x < 2,

(8.3)

where C is a constant. If 2

∫ x

then ω is called slow of order 1. Throughout this chapter, we abbreviate “slow of order 1” to “slow”. If ω is both slow and fast, then it is called regular. The following fact is useful in verifying whether a majorant is fast or slow. Proposition 8.2. A majorant ω is fast if and only if there exists a constant α > 0 such that the function

ω(x)/xα (0 < x < 2) is almost increasing.

(8.4)

8.1 Definitions and basic properties | 267

A majorant ω is slow if and only if there exists a constant β < 1 such that the function ω(x)/xβ (0 < x < 2) is almost decreasing.

(8.5)

Proof. This is a consequence of Proposition 4.16. The additional hypothesis that ω(t)/t is decreasing can be used to give a simpler proof of the implication (8.2) ⇒ (8.4). Remark 8.3. Note that the equivalence (8.3) ⇔ (8.5) is independent of the hypothesis that ω(t)/t is (almost) decreasing in t. Proposition 8.4. If φ is an almost increasing function on (0, t0 ), 0 < t0 ≤ ∞, such that φ(t)/t is almost decreasing and φ(0+ ) = 0, then there exists a concave function ϕ on (0, ∞) such that ϕ(t) ≍ φ(t), 0 < t < t0 . Proof. If t0 is finite, then we extend φ to the interval (0, ∞) by putting φ(t) = φ(t0 )t/t0 for t > t0 . Assuming that φ is almost increasing and φ(t)/t is almost decreasing, define the functions φ1 (x) = inft≥x φ(t) and φ2 (x) = x sup t≥x

φ1 (t) φ (tx) = sup 1 . t t t≥1

Then φ2 is increasing, φ2 (x)/x, x > 0, is decreasing, and φ2 ≍ φ. The strictly increasing function φ3 (x) = φ2 (x)+x/(x +1) has the same properties. Let Φ be the inverse function of φ3 , and let x

Φ1 (x) = ∫ 0

Φ(t) dt. t

The function Φ1 is convex (because Φ(t)/t is increasing), and we have x

Φ(x) ≥ Φ1 (x) ≥ ∫ x/e

x

1 Φ(t) dt ≥ Φ(x/e) ∫ dt = Φ(x/e). t t x/e

It follows that ϕ, the inverse of Φ1 , is concave and satisfies φ3 (x) ≤ ϕ(x) ≤ eφ3 (x). Exercise 8.5. If φ : (0, t0 ) 󳨃→ (0, ∞) is an increasing function such that φ(t)/t decreases for t ∈ (0, t0 ), then φ(x + y) ≤ φ(x) + φ(y),

x > 0, y > 0, x + y < t0 .

The following lemma generalizes the well-known fact that if |∇u| is bounded in 𝔻, then u satisfies the ordinary Lipschitz condition (ω(t) = t).

268 | 8 Lipschitz spaces of first order Lemma 8.6. Let ω be a Dini majorant, and let u: 𝔻 󳨃→ ℂ be a C 1 -function such that 󵄨󵄨 󸀠 󵄨󵄨 ω(1 − |z|) , 󵄨󵄨u (z)󵄨󵄨 ≤ 1 − |z|

z ∈ 𝔻.

(8.6)

Then u ∈ Λω[1] (𝔻), where x

ω[1] (x) = ∫ 0

ω(t) dt, t

and |u(z) − u(w)| ≤ 3ω[1] (|w − z|) (z, w ∈ 𝔻). In particular, if ω is fast and u satisfies (8.6), then u ∈ Λω (𝔻). Here |u󸀠 (z)| denotes the norm of the derivative u󸀠 (z). Proof. Let (8.6) be satisfied. Let |a| ≤ |b| ≤ 1. By Lagrange’s theorem, 󵄨 ω(1 − |c|) 󵄨󵄨 |a − b|, 󵄨󵄨u(a) − u(b)󵄨󵄨󵄨 ≤ 1 − |c| where c = (1 − λ)a + λb for some λ ∈ (0, 1). Since |c| ≤ |b| and ω(t)/t decreases, we see that ω(1 − |c|) ω(1 − |b|) ≤ , 1 − |c| 1 − |b| and hence |u(a) − u(b)| ≤ ω(|a − b|) ≤ ω[1] (|a − b|) under the condition |a − b| ≤ 1 − |b|. If 1 − |b| ≤ |a − b| ≤ 1 − |a|, then |u(a) − u(b)| ≤ |u(a) − u(b󸀠 )| + |u(b󸀠 ) − u(b)|, where 󸀠 b = (1 − δ)b/|b|, δ = |a − b|. Using Lagrange’s theorem as before, we get ω(1 − |b󸀠 |) 󵄨󵄨 ω(δ) 󵄨󵄨 󵄨󵄨 󸀠 󵄨 󸀠󵄨 󸀠󵄨 󵄨󵄨u(a) − u(b )󵄨󵄨󵄨 ≤ 󵄨a − b 󵄨󵄨󵄨 = 󵄨a − b 󵄨󵄨󵄨 ≤ ω(δ) ≤ ω[1] (δ). 1 − |b󸀠 | 󵄨 δ 󵄨 In the case of |u(b󸀠 ) − u(b)|, we have |b|

1

|b󸀠 |

1−δ

ω(1 − t) ω(1 − t) 󵄨 󵄨󵄨 󸀠 dt ≤ ∫ dt = ω[1] (δ). 󵄨󵄨u(b ) − u(b)󵄨󵄨󵄨 ≤ ∫ 1−t 1−t Finally, if δ > 1 − |a|, then we use the inequality |u(a) − u(b)| ≤ |u(a) − u(a󸀠 )| + |u(a󸀠 ) − u(b󸀠 )| + |u(b󸀠 ) − u(b)|, where a󸀠 = (1 − δ)a/|a|, and then proceed in a similar way as before. As an application of Lemma 8.6, we have the following result of Hardy and Littlewood [237].

8.1 Definitions and basic properties | 269

Theorem 8.1 (Hardy–Littlewood). Let ω be a fast majorant. A function u ∈ h(𝔻) belongs to Λω (𝔻) if and only if M∞ (r, |∇u|) ≤ C

ω(1 − r) . 1−r

Moreover, if C(u) denotes the smallest C ≥ 0 satisfying this condition, then C(u) ≍ ‖u‖Λω (𝔻) − ‖u‖L∞ (𝔻) . Proof. The “only if” part is a direct consequence of Lemma 8.6 and the hypothesis that ω is fast. The converse follows from the inequality |u(z + (1 − |z|)w) − u(z)| ω(1 − |z|) 󵄨 󵄨󵄨 ≤C , 󵄨󵄨∇u(z)󵄨󵄨󵄨 ≤ C sup 1 − |z| 1 − |z| w∈𝔻

where C is an absolute constant (C = 2 if u is real-valued). Lipschitz conditions on regularly oscillating functions For the validity of Lemma 8.6, it is not necessary to suppose that u is C 1 . It suffices to assume that it is locally Lipschitz and use, as before, the following definition of |∇u|: |u(z) − u(a)| 󵄨 󵄨󵄨 . 󵄨󵄨∇u(a)󵄨󵄨󵄨 = lim sup |z − a| z→a It turns out that some results on the Lipschitz spaces concerning harmonic or analytic functions can be extended to the class RO. For example, using the definition of RO functions and nearly convex functions (see Section 3.2.c) and Lemma 8.6, we prove the following: Lemma 8.7. Let ω be fast majorant, and let g be an RO function. Then: (i) g ∈ Λω (𝔻) if and only if 󵄨 󵄨󵄨 󵄨󵄨g(w) − g(z)󵄨󵄨󵄨 ≤ Cω(1 − |z|)

whenever |w − z| < 1 − |z|.

(8.7)

(ii) g ∈ Λω (𝔻) if and only if ω(1 − |z|) 󵄨 󵄨󵄨 , 󵄨󵄨∇g(z)󵄨󵄨󵄨 ≤ C 1 − |z|

z ∈ 𝔻.

(iii) If, in addition, g is nearly convex, then we can replace |g(w) − g(z)| with g(w) − g(z) in (i). As another property of RO, we have the following: Lemma 8.8. Let ω be fast majorant, and let g ∈ C(𝔻) be an RO function. Then the following conditions are equivalent: (a) g ∈ Λω (𝔻);

270 | 8 Lipschitz spaces of first order (b) |g(ζ ) − g(z)| ≤ Cω(|ζ − z|) for all ζ ∈ 𝕋 and z ∈ 𝔻; (c) |g(ζ ) − g(rζ )| ≤ Cω(1 − r) and |g(ζ ) − g(η)| ≤ Cω(|ζ − η|) for all 0 < r < 1 and ζ , η ∈ 𝕋. Of course, in (b) and (c), the constant C depends only on g. Proof. The implications (a) ⇒ (b) ⇒ (c) are trivial. Suppose that (c) holds, and let ζ ∈ 𝕋, z ∈ 𝔻 \ {0}, and z0 = z/|z|. Note that |z − z0 | = 1 − |z| ≤ |ζ − z|. Then 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨g(ζ ) − g(z)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨g(ζ ) − g(z0 )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(z0 ) − g(z)󵄨󵄨󵄨 ≤ Cω(|ζ − z0 |) + Cω(1 − |z|)

≤ Cω(|ζ − z|) + Cω(|z − z0 |) + Cω(1 − |z|) ≤ 3Cω(|ζ − z|).

This proves that (c) implies (b) (and the proof is independent on the hypotheses of the theorem). It remains to prove that (b) implies (a). By the previous lemma it suffices to deduce (8.7). By (b) we have 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨g(w) − g(z)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨g(w) − g(z0 )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(z0 ) − g(z)󵄨󵄨󵄨 ≤ Cω(|z0 − w|) + Cω(|z0 − z|)

≤ Cω(|w − z|) + Cω(|z − z0 |) + Cω(|z0 − z|)

≤ 3Cω(1 − |z|). This completes the proof.

Remark 8.9. We can define five seminorms on Λω (𝔻): |g(w) − g(z)| , w,z∈𝔻 ω(|w − z|) |g(w) − g(z)| , ‖g‖2 = sup w,z∈𝔻, |w−z| 0 by using the subharmonicity of the function φ(z) = ‖fw − fz ‖qp ,

p > 0, q > 0

(see Lemma 2.1 and statement (B) of Further notes 2.2). So we have ω(f , t)p ≤ Ω(f , t)p ≤ Cp ω(f , t)p ,

f ∈ H p , 0 < t < π.

(8.9)

The fact that ω(f , t)p /t 1/p , where f ∈ H p , p < 1, is almost decreasing is to be used.

8.3 Lipschitz condition for the modulus | 275

Storozhenko [572, Eq. (3.1), Theorem 6] proved the following: Theorem (Storozhenko). If f ∈ H p (p ∈ ℝ+ ), then ‖f∗ − fr ‖p ≍ ω(f∗ , 1 − r)p ,

0 < r < 1.

Although Theorem 6 of [572] is stated for p < ∞, it is clear from the author’s discussion before proving it that it holds for f ∈ A(𝔻) as well. (III) The generalized Theorem 8.6 asserts that f is in HΛpω (𝕋), where ω is fast and p > 0, if and only if Mp (r, f 󸀠 ) = 𝒪(

ω(1 − r) ). 1−r

In the case where ω is regular and p ≥ 1, this was proved by Blasco and de Souza [87, Theorem 2.1(i)] although the paper [423], published two years earlier (see Theorem 9.1), contains deeper results under the hypothesis that ω is fast only, which was subsequently generalized in [426]. In Chapter 9, we will prove a more general result (Theorem 9.9).

8.3 Lipschitz condition for the modulus A function f ∈ H(𝔻) satisfies the condition |f (z) − f (w)| ≤ |z − w| in 𝔻 if and only if |f 󸀠 | ≤ 1 in 𝔻. On the other hand, the corresponding Lipschitz condition for |f | is satisfied if and only if |∇|f | | ≤ 1. Since |∇|f | | = |f 󸀠 | (if f (z) ≠ 0), we obtain the relation f ∈ Λ1 (𝔻) ⇔ |f | ∈ Λ1 (𝔻),

f ∈ H(𝔻).

This is the simplest case of the following theorem. Theorem 8.7. Let ω be a Dini majorant. If f ∈ H(𝔻) and |f | ∈ Λω (𝔻), then f ∈ Λ(ω[1] , 𝔻). x

Recall that ω[1] (x) = ∫0 ω(t) dt. t The theorem states, in particular, that if |f | ∈ Λω (𝔻) and ω is a Dini majorant, then f ∈ A(𝔻). On the other hand, there exists a function f ∈ H(𝔻) \ A(𝔻) such that the function |f | has a continuous extension to the closed unit disc. To show this, we use the known fact that there exists a bounded analytic function u + iv such that u is continuous on 𝔻, whereas v has no continuous extension to 𝔻. Then there are a point ζ ∈ 𝕋, two sequences {zn } ⊂ 𝔻 and {wn } ⊂ 𝔻 tending to ζ , and two points a, b ∈ ℂ (a ≠ b) such that v(zn ) → a and v(wn ) → b. We can assume that eia ≠ eib since otherwise we can consider the function (u + iv)/λ for a suitable λ > 0. Then f = exp(u + iv) is the desired function. Theorem 8.7 is a direct consequence of Lemma 8.6 and the inequality 󵄨 󵄨 |∇|f |(z)| = 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 ≤ 2

sup

|w−z| 0. We can prove that the existence of such α is implied by the hypothesis that ω̄ is fast and concave. The main properties of ω̄ are given by the inequalities α

̄ ̄ ω(t) ω(t) ≤ ω̄ 󸀠 (t) ≤ , t t

0 < t < 1,

(8.11)

and ̄ ̄ ω̄ 󸀠 (x)(x − y) ≤ ω(x) − ω(y),

0 < x, y ≤ 1,

(8.12)

where ω̄ 󸀠 is, say, the left derivative of ω.̄ Theorem 8.11. For a function φ ∈ H(𝔻, 𝔻), the following conditions are equivalent: (A0) φ belongs to 𝒞 (HΛω̄ , HΛω ); (A) ω̄ 󸀠 (1 − |φ(z)|)|φ󸀠 (z)| ≤ C ω(1−|z|) ; 1−|z| ̄ − |φ|) ∈ Λω (𝔻); (B) ω(1 ̄ − |φ(z)|) − ω(1 ̄ − |φ(ζ )|)| ≤ Cω(|ζ − z|) for ζ ∈ 𝕋 and z ∈ 𝔻. (C) |ω(1 Note that this theorem can be viewed as a generalization of Theorem 8.8. For in̄ stance, if we take ω(t) = t, and f (z) = z, then the equivalence (A) ⇔ (C) says that φ ∈ HΛω (𝔻) if and only if |φ| ∈ Λω (𝔻), which coincides with the equivalence (a) ⇔ (b) of Theorem 8.8 in the case |φ| < 1. If φ ∈ A(𝔻) is arbitrary, then we apply this particular case to the function φ/M, where M > ‖φ‖∞ . Proof of Theorem 8.11. Note first that if (A0) holds, then we take f (z) = z to conclude that |φ| ∈ Λω (𝔻), which implies φ ∈ A(𝔻) by Theorem 8.7. First, we prove that (A) implies (A0). Indeed, if (A) holds and f ∈ HΛω̄ (𝔻), then, by Theorem 8.6 and (8.11), ̄ − |φ(z)|) 󵄨󵄨 󸀠 󵄨󵄨 ω(1 󵄨󵄨 󸀠 󵄨 󵄨 󵄨 󸀠 󵄨󵄨 󸀠 󵄨φ (z)󵄨󵄨 󵄨󵄨(𝒞φ f ) (z)󵄨󵄨󵄨 = 󵄨󵄨󵄨f (φ(z))󵄨󵄨󵄨󵄨󵄨󵄨φ (z)󵄨󵄨󵄨 ≤ C 1 − |φ(z)| 󵄨

8.4 Composition operators | 279

ω(1 − |z|) 󵄨 󵄨󵄨 󵄨 ≤ C ω̄ 󸀠 (1 − 󵄨󵄨󵄨φ(z)󵄨󵄨󵄨)󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨 ≤ C . 1 − |z| Applying Theorem 8.6 again, we conclude that (A0) holds. To prove that (A0) implies (A), we consider two possible alternatives: the function ̄ ̄ ≍ t, s(t) = ω(t)/t (0 < t ≤ 1) is either bounded or unbounded. If s is bounded, then ω(t) 󸀠 so HΛω̄ (𝔻) = HΛ1 (𝔻), and therefore we have to prove that (A) implies |φ (z)| ≤ Cω(1 − |z|)/(1 − |z|): it suffices to take f (z) = z. ̄ If s is not bounded, then s(t) → ∞ as t ↓ 0. Let ϕ(x) = s(1/x) = xω(1/x), x ≥ 1. This function is increasing and satisfies the condition ϕ(2x) ≤ Cϕ(x), x ≥ 1, because ω̄ is fast. As a test function, we take the function fϕ,a from Lemma 7.22, ∞

fϕ,a (z) = ∑ 2n n=0

aλn z λn +1 , λn + 1

z ∈ 𝔻, |a| = 1,

where λn is chosen as in Lemma 7.22. Since, by the same lemma, ∞ ̄ − |z|) 1 ω(1 󵄨 󵄨󵄨 󸀠 n λ )=C , 󵄨󵄨fϕ,a (z)󵄨󵄨󵄨 ≤ ∑ 2 |z| n ≤ Cϕ( 1 − |z| 1 − |z| n=0

we see that

󸀠 fϕ,a

∈ Λ(ω,̄ 𝔻). By hypothesis, 𝒞φ (fϕ,a ) belongs to Λω (𝔻), that is, 󵄨󵄨 ∞ 󵄨󵄨 ω(1 − |z|) λn 󵄨󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨 n 󸀠 , 󵄨󵄨(𝒞φ fϕ,a ) (z)󵄨󵄨 = 󵄨󵄨 ∑ 2 (aφ(z)) 󵄨󵄨󵄨󵄨󵄨󵄨φ (z)󵄨󵄨󵄨 ≤ C 󵄨󵄨n=0 󵄨󵄨 1 − |z| 󵄨 󵄨

where C is a constant independent of a and z. Taking a = φ(z)/|φ(z)|, we obtain ω(1 − |z|) 󵄨 󵄨λ 󵄨 󵄨 . ∑ 2n 󵄨󵄨󵄨φ(z)󵄨󵄨󵄨 n 󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨 ≤ C 1 − |z| n=0 ∞

It remains to apply Lemma 7.22 and (8.11) to conclude that (A) holds. ̄ − |z|). Since |∇g(z)| = ω̄ 󸀠 (1 − |φ(z)|)|φ󸀠 (z)| and g is RO, we see by Let g(z) = ω(1 means of Lemmas 8.7 and 8.8 that the equivalences (A) ⇔ (B) ⇔ (C) hold. This concludes the proof of Theorem 8.11. ̄ − |φ|) is superharmonic, and Since ω̄ is concave, we have that the function ω(1 ̄ − |φ|) − 𝒫 [ω(1 ̄ − |φ∗ |)] ≥ 0. In this case, we have the following: hence ω(1 Theorem 8.12. Let ω a regular majorant. Then condition (A0) of Theorem 8.11 is equivalent to each of the following: ̄ − |φ∗ |) ∈ Λω (𝕋), and (D) ω(1 ̄ − |φ∗ |)](z) ≤ Cω(1 − |z|); ̄ − 󵄨󵄨󵄨󵄨φ(z)󵄨󵄨󵄨󵄨) − 𝒫 [ω(1 ω(1 ̄ − |φ∗ |) ∈ Λω (𝕋), and (E) ω(1 ̄ − 󵄨󵄨󵄨󵄨φ∗ (ζ )󵄨󵄨󵄨󵄨) ≤ Cω(1 − r) ̄ − 󵄨󵄨󵄨󵄨φ(rζ )󵄨󵄨󵄨󵄨) − ω(1 ω(1

(ζ ∈ 𝕋, 0 < r < 1).

We omit the proof because it is similar to that of Theorem 8.9.

280 | 8 Lipschitz spaces of first order 8.4.1 Mean Lipschitz condition for the class RO Here we prove the following extension of Lemma 8.6 in a particular case. Proposition 8.13. Let g be a regularly oscillating function on 𝔻, p ∈ ℝ+ , and let ω be a fast majorant. Then g ∈ Λpω (𝔻) if and only if 1/p

2π

󵄨p 󵄨 ( ∫ 󵄨󵄨󵄨∇g(eiθ z)󵄨󵄨󵄨 dθ)

≤C

0

ω(1 − |z|) , 1 − |z|

z ∈ 𝔻.

(8.13)

It should be noted that if p ≥ 1, then the “if” part of this proposition holds for arbitrary locally Lipschitz function g, which follows from the fact that Lemma 8.6 holds for functions with values in a Banach space. Proposition 8.13 is a consequence of the following two lemmas (Lemmas 8.14 and 8.15). Before stating and proving them, note that the formula |∇|f || = |f 󸀠 | and the proposition lead to a generalization of Dyakonov’s theorem: If f ∈ H(𝔻) and ω is a fast majorant, then f is in Λpω (𝔻) if and only if so is |f |. This holds for real-valued harmonic functions as well. See Further notes 8.5 for the case of quasiregular harmonic functions. Lemma 8.14. Let g be a regularly oscillating function on 𝔻, p ∈ ℝ+ , and let ω be an arbitrary majorant. If g ∈ Λpω (𝔻), then (8.13) holds. Proof. Assuming that g ∈ Λpω (𝔻), we have 󵄨p 󵄨 ∫󵄨󵄨󵄨gζ (z) − gζ (w)󵄨󵄨󵄨 |dζ | ≤ Cω(δ)p 𝕋

if |w − z| < δ = 1 − |z|. Integrating this inequality over Dδ (z), we get 󵄨p 󵄨 ∫(δ−2 ∫ 󵄨󵄨󵄨g(ζz) − g(ζw)󵄨󵄨󵄨 dA(w))|dζ | Dδ (z)

𝕋

p

= ∫[Op gζ (z, δ)] |dζ | ≤ Cω(δ)p ; 𝕋

see Theorem 3.4. If g is RO, then it belongs to OCK1 for some K; see Section 3.2.b. As it is easy to see, then gζ ∈ OCK1 , and hence by Corollary 3.3 we have p 󵄨p 󵄨󵄨 −p 󵄨󵄨∇g(zζ )󵄨󵄨󵄨 ≤ Cδ [Op gζ (z, δ)] .

The result follows by integrating this inequality over ζ ∈ 𝕋. Lemma 8.15. Let g : 𝔻 󳨃→ ℝ be a regularly oscillating function, and let p ∈ ℝ+ . If g satisfies (8.13), then g ∈ Λpω (𝔻).

8.4 Composition operators | 281

Proof. Let G = |∇g|. Let Ψ (t) =

ω(t) . t

Let a, b ∈ 𝔻, |a| ≤ |b| < 1, and δ = |a−b|. As in the proof of Lemma 8.6, we consider three cases. Case 1. δ ≤ 1−|b|. Let la,b be the straight line joining a and b. By Lagrange’s theorem we have 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨g(a) − g(b)󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨∇g(z)󵄨󵄨󵄨δ ≤ sup 󵄨󵄨󵄨∇g(z)󵄨󵄨󵄨δ + sup 󵄨󵄨󵄨∇g(z)󵄨󵄨󵄨δ, z∈la,b

z∈la,c

z∈lc,b

where c = (a + b)/2. If z ∈ la,c , then |z − a| < δ/2 ≤ (1 − |a|)/2, and hence sup G(z) ≤ sup G(z) = G# (a),

z∈la,c

z∈Dε (a)

ε = (1 − |a|)/2.

This inequality remains valid when a is replaced with b. Applying these inequalities to the points aeiθ and beiθ , then integrating the resulting inequality over (0, 2π), and using the hypotheses, we get 2π

󵄨p 󵄨 ∫ 󵄨󵄨󵄨g(aeiθ ) − g(beiθ )󵄨󵄨󵄨 dθ ≤ Mpp (|a|, G# )δp + Mpp (|b|, G# )δp . 0

By Theorem 3.4 the function G is QNS, and hence the condition Mp (r, G) ≤ Ψ (1 − r) implies the same with G# instead of G (see Further notes 3.8), so we have 2π

p p 󵄨p 󵄨 ∫ 󵄨󵄨󵄨g(aeiθ ) − g(beiθ )󵄨󵄨󵄨 dθ ≤ CΨ (1 − |a|) δp + CΨ (1 − |b|) δp ≤ Cω(δ). 0

In the last step, we used the inequalities 1 − |a| ≥ 1 − |b| ≥ δ and the condition that the function Ψ is decreasing. Case 2. 1 − |b| ≤ δ ≤ 1 − |a|. Let b󸀠 = (1 − δ)b/|b| and δ󸀠 = |a − b󸀠 |. Then 󵄨 󵄨 󵄨 󵄨󵄨 󸀠 󸀠 󵄨 󵄨 󵄨󵄨g(a) − g(b)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨g(a) − g(b )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(b ) − g(b)󵄨󵄨󵄨. Since δ󸀠 ≤ 1 − |b󸀠 | ≤ 1 − |a|, we see from the previous case that 2π

p p 󵄨p 󵄨 ∫ 󵄨󵄨󵄨g(aeiθ ) − g(b󸀠 eiθ )󵄨󵄨󵄨 dθ ≤ Cω(δ󸀠 ) ≤ Cω(δ)p = Cω(|a − b|) . 0

(Here we have used the geometrically obvious inequality δ󸀠 ≤ δ.)

(8.14)

282 | 8 Lipschitz spaces of first order On the other hand, |b|

1

|b󸀠 |

1−δ

󵄨󵄨 󵄨 iθ 󸀠 iθ 󵄨 iθ iθ 󵄨 󵄨󵄨g(be ) − g(b e )󵄨󵄨󵄨 ≤ ∫ G((b/|b|)se ) ds ≤ ∫ 󵄨󵄨󵄨G((b/|b|)se )󵄨󵄨󵄨 ds.

(8.15)

Assume that b is a positive real number and 0 < p < 1, and let h(s) = G(seiθ ), where θ is fixed. Choose n ∈ ℕ so that 2−n ≤ δ < 2−n+1 , and let rn = 1 − 2−n . Then p

1

∞

∞

p

( ∫ h(s) ds) ≤ ∑ 2−kp sup h(s)p = ∑ 2−kp G× (rk eiθ ) . rk ≤s≤rk+1

k=n−1

1−δ

k=n−1

Now, arguing as in the proof of Proposition 3.26 (see (3.24)), we conclude that the last quantity 1

≤ ∫ (1 − s) 2−n+1

p−1

Mpp (s, G) ds

rn−1

≤ C ∫ t p−1 0

p

p

ω(t)p dt tp

≤ Cω(rn−1 ) ≤ Cω(δ) , where we have used the hypothesis that ω is fast. Combining this with (8.14), we get 2π

p 󵄨p 󵄨 ∫ 󵄨󵄨󵄨g(aeiθ ) − g(beiθ )󵄨󵄨󵄨 dθ ≤ Cω(|a − b|) .

(8.16)

0

If p ≥ 1, then we apply Minkowski’s inequality to (8.15) to complete the proof in Case 2. Case 3. 1 − |a| ≤ δ. In this case, we use the inequality 󵄨 󵄨 󵄨 󵄨󵄨 󸀠 󵄨 󵄨 󸀠 󸀠 󵄨 󵄨 󸀠 󵄨 󵄨 󵄨󵄨g(a) − g(b)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨g(a) − g(a )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(a ) − g(b )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(b )󵄨󵄨󵄨 − 󵄨󵄨󵄨g(b)󵄨󵄨󵄨, where a󸀠 = (1 − δ)a/|a| and b󸀠 = (1 − δ)b/|b|, and proceed as before to get (8.16). This completes the proof of the lemma. Proposition 8.13 can be used to extend Theorems 8.11 and 8.12 to the class of composition operators from HΛω̄ (𝔻) to HΛpω (𝔻). Using the bi-Bloch lemma as on page 247, we prove the following: Theorem 8.13. Let 0 < p < ⬦. Then φ is in 𝒞 (HΛω̄ , HΛpω ) if and only if there is a constant C such that 1/p

󵄨 p 󵄨 󵄨 󵄨 (∫(󵄨󵄨󵄨φ󸀠 (rζ )󵄨󵄨󵄨ω̄ 󸀠 (1 − 󵄨󵄨󵄨φ(rζ )󵄨󵄨󵄨)) |dζ |) 𝕋

≤C

ω(1 − r) , 1−r

0 < r < 1.

(8.17)

Further notes and results | 283

̄ − |φ|))| = ω̄ 󸀠 (1 − |φ|)|φ󸀠 | and the function ω(1 ̄ − |φ|) is RO, by ProposiSince |∇(ω(1 tion 8.13 we have the following: Theorem 8.14. Let 0 < p < ⬦. Then φ is in 𝒞 (HΛω̄ , HΛpω ) if and only if the function ̄ − |φ|) belongs to the Lipschitz space HΛpω (𝔻). ω(1 In this edition the proof of this statement and its equivalent formulations is omit̄ = t α and ω(t) = t β , see [442]. ted. For the case where ω(t) Exercise 8.16. Under the hypotheses of Section 8.4.a, we have HΛω̄ ⊂ HΛpω if and only ̄ ≤ Cω(t), 0 < t < 1. if ω(t)

Further notes and results Dyakonov’s proofs of Theorems 8.8 and 8.9 in [173] are complicated and are based on theorems on pseudoanalytic continuation and theorems on division by inner functions. The presented proof is from [432]; as it is seen from Lemma 3.6, the key is in connection between the modulus of the first derivative and the oscillation of the modulus of a function. 8.1. Lemma 8.6 can be stated in the following form: Let φ be an increasing concave function on (0, 2) with φ(0+ ) = 0. If |∇u(z)| ≤ φ󸀠 (1 − |z|), z ∈ 𝔻, where φ󸀠 (t) = dφ/dt, then u ∈ Λφ (𝔻); see [407]. Our proof follows the proof of Rudin [518, Lemma 6.4.8], where the case ω(t) = t α , 0 < α ≤ 1, was considered. 8.2. Theorem 8.5 is a particular case of some results from [513, 579]. Hinkkanen [257] proved the following result under the hypothesis that μ is an increasing nonnegative function on [0, ∞) such that μ(2t) ≤ 2μ(t). Theorem (H). Let G be a bounded domain in ℂ, let f be analytic in G, and let μ be as before. Fix w ∈ 𝜕G and let |f (w) − f (z)| ≤ μ(|z − w|) for all z ∈ 𝜕G. Then |f (w) − f (z)| ≤ Cω(|w − z|) for all z ∈ G, where C = 3456. If the hypothesis is made for all z, w ∈ 𝜕G, then the conclusion holds for all z, w ∈ G. If G is a disc, then the best constant is C = exp{

log 4 ∞ ∑ arctan(2−n )}; π n=0

see [256, Theorem 1].

The following theorem is proved in [201]. Theorem (GHH). If μ(t) = t α , 0 < α ≤ 1, and G is bounded (or unbounded with some additional requirements on f ), then the condition |f (z) − f (w)| ≤ μ(|w − z|) (w, z ∈ 𝜕G) implies the same condition for z, w ∈ G. The recent paper [258] of Hinkkanen contains these results and more general ones.

284 | 8 Lipschitz spaces of first order 8.3 (Lipschitz conditions for the modulus of a harmonic function). Theorems 8.8 and 8.9 do not hold for complex-valued harmonic functions. We can say in this case only that |f | ∈ Λω (𝔻) implies f ∈ Λ√ω (𝔻); see [178]. However, if u ∈ h(𝔻) is real valued, then the things become very simple [439]. For instance, for a real-valued function u ∈ hC(𝔻), the following statements hold. Theorem (A). If ω is regular and | |u(ζ )| − |u(rζ )| | ≤ ω(1 − r), ζ ∈ 𝕋, then u ∈ Λω (𝔻). Theorem (B). If ω is slow and |u∗ | ∈ Λω (𝕋), then u ∈ Λω (𝔻). Concerning complex-valued harmonic functions, the following was proved in [439, Theorem 3]. Theorem (C). If f ∈ h(𝔻), ω a fast majorant, and |f |2 ∈ Λω (𝔻) (in particular, if |f | ∈ Λω (𝔻)), then f ∈ Λ√ω (𝔻). 8.4 (Radial Lipschitz conditions for the moduli of analytic functions). It is not known whether the condition 󵄨󵄨 󵄨 󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨f (ζ )󵄨󵄨󵄨 − 󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨󵄨󵄨󵄨 ≤ ω(1 − r),

(†)

where f ∈ A(𝔻) and ω is regular, implies f ∈ Λω (𝔻). However, using Theorem (A) of Further notes 8.3 and in fact the weaker variant where |u| is replaced with u, we can prove the following: If f (z) ≠ 0 for all z ∈ 𝔻, then the implication (†) ⇒ f ∈ Λω (𝔻) holds. In fact, if (†) holds, then the function u = log |f | is harmonic, and |u(ζ ) − u(rζ )| ≤ Cω(1 − r). By Further notes 8.3 we have log |f | ∈ Λω (𝔻). Hence |f | ∈ Λω (𝔻) and thus f ∈ Λω (𝔻). 8.5 (Dyakonov theorems for quasiregular functions). A C 1 function f : G 󳨃→ ℂ, where G is a subdomain of ℂ, is said to be quasiregular if it satisfies k := ess sup z∈G

|𝜕f̄ (z)| < 1, |𝜕f (z)|

z ∈ G.

Theorem (D). Let p > 0, and let ω be a fast majorant. Then a harmonic quasiregular function f belongs to Λpω (𝔻) if and only so does |f |. Namely, by Exercise 3.11, |f | is nearly convex. On the other hand, 󵄨󵄨 𝜕 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨󵄨 |∇|f 󵄨󵄨󵄨(z)󵄨󵄨󵄨 = 2󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨󵄨󵄨󵄨, 󵄨󵄨 𝜕z 󵄨󵄨

because |f | is real valued. We have 2 whence

𝜕 󵄨󵄨 󵄨−1 𝜕(f f ̄) 󵄨󵄨 󵄨 󵄨 󵄨−1 = 󵄨󵄨f (z)󵄨󵄨󵄨 (h󸀠 (z)f (z) + g 󸀠 (z)f (z)), 󵄨󵄨f (z)󵄨󵄨󵄨 = 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 𝜕z 𝜕z

1 − k 󵄨󵄨 󸀠 󵄨󵄨 󵄨 1 − k 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨 󵄨 󵄨 (󵄨g (z)󵄨󵄨 + 󵄨󵄨h (z)󵄨󵄨) = |∇|f |(z)| ≥ 󵄨󵄨󵄨h󸀠 (z)󵄨󵄨󵄨 − 󵄨󵄨󵄨g 󸀠 (z)󵄨󵄨󵄨 ≥ 󵄨f (z)󵄨󵄨. 1+k 󵄨 1 + k󵄨

Further notes and results | 285

These facts and Proposition 8.13 show that Theorem (D) holds. For further information and references concerning the case p = ∞, see [48]. 8.6 (Carleson–Jacobs/Havin–Shamoyan theorem). Using his results on the moduli of analytic functions, Dyakonov [173] gave a new proof of the following result of Havin– Shamoyan [248] and Carleson–Jacobs (unpublished). Theorem (CJ/HS). Let ω be a majorant such that ψ(t) := ω(√t) is regular. Let φ ∈ Λω (𝕋), φ ≥ 0, and log φ ∈ L1 (𝕋). Then the outer function F(z) := exp(

ζ +z 1 log φ(ζ )|dζ |) ∫ 2π ζ − z 𝕋

belongs to Λψ (𝕋), and ‖F‖Λψ (𝕋) ≤ C‖φ‖Λω (𝕋) (1 + ∫ log 𝕋

‖φ‖∞ |dζ |). φ(ζ )

8.7. Lindström and Sanatpour [354] generalized Theorem 8.10 in various directions. Among other things, they considered the weighted composition operator g 𝒞φ , where g ∈ H(𝔻) and φ ∈ H(𝔻, 𝔻). We state one of their results (Theorem 2.6). Theorem (LS). Let 0 < α, β ≤ 1. Then g 𝒞φ maps HΛα into HΛβ if and only if g ∈ HΛβ and |g|(1 − |φ|)α ∈ Λβ (𝔻). In [354], the reader can also find a discussion on the compactness of weighted operators. It should be noted, however, that the title of this paper is not adequate because the “Zygmund spaces” coincide with the Lipschitz spaces of first order.

9 Lipschitz spaces of higher order In this chapter, we deal with “integrated” Lipschitz classes defined via the moduli of smoothness of arbitrary order. In the first three sections, we consider the spaces Λω,n (𝕋) ⊂ C(𝕋) defined by the requirement ωn (g, t) = 𝒪(ω(t)), where ω is a majorant such that ω(t)/t n is almost decreasing. We also consider the spaces h∞ (ψ)n ⊂ h(𝔻) defined by |Dn u(z)| ≤ Cψ(1/(1 − |z|)), where ψ is almost subnormal on [1, ∞), and give a necessary and sufficient condition for the validity of the relation Λω,n (𝕋) ≃ h∞ (ψ)n . Then, in Section 9.3, we use this result to generalize Privalov’s theorem by proving that if ω is fast, then Λω,n is self-conjugate if (and only if) ω(t)/t β is almost decreasing for some β < n. In Section 9.4, we extend some results of Section 9.2 to the case of p “integrated mean Lipschitz spaces” Λp,q ω,n consisting of g ∈ L (𝕋) for which the function 1/q q ωn (g, t)p /(ω(t)t ) is in L (0, 1). It turns out, in particular, that if ω(t) = t α (0 < α < p,q n), then H p ∩ Λp,q ω,n = Bα for all p, q > 0. In the penultimate section, we give new characterizations of H p ∩ Λp,q ω,n with applications to inner functions. The last section is devoted to invariant Besov spaces.

9.1 Moduli of smoothness and related spaces If h is a complex-valued function defined on ℝ, then Δnt h (n is a positive integer, t ∈ ℝ) denotes the nth symmetric difference with step t: Δ1t h(θ) = h(θ + t) − h(θ)

(θ ∈ ℝ),

and

Δnt h = Δ1t Δnt h (n ≥ 2).

In particular, Δ2t h(θ) = h(θ + 2t) − 2h(θ + t) + h(θ). In the general case, we have n n Δnt h(θ) = ∑ ( )(−1)n−k h(θ + kt). k k=0

If g is a function on the unit circle, then Δnt g is defined by Δnt g(eiθ ) = Δnt h(θ), where h(θ) = g(eiθ ). For fixed n and t, Δnt is a linear operator that preserves C(𝕋); we have ‖Δnt g‖ ≤ 2n ‖g‖, where ‖ ⋅ ‖ = ‖ ⋅ ‖∞ denotes the max-norm in C(𝕋). The modulus of smoothness of order n is defined by ωn (g, t) = sup{‖Δns g‖ : |s| < t}, t > 0, g ∈ C(𝕋). This notion was introduced by Bernstein [71] in 1912. Lemma 9.1. If g ∈ C(𝕋), then the function ωn (g, t)/t n is almost decreasing for t > 0. Proof. The lemma is easily deduced from the inequality ωn (g, 2t) ≤ 2n ωn (g, t), t > 0, whereas the latter can be proved by means of the formula n

ijt ̂ − 1) eijθ Δnt g(eiθ ) = ∑ g(j)(e |j| 0 such that 1 − 1/k ≤ r ≤ 1 − 1/(k + 1) and split the sum at n = k. Then proceed in a similar way as in the proof of Lemma 4.13; see [533]. Returning to the proof of the theorem, we consider the analytic functions ∞

Uk (z) = k −n ψ(k)z k + ∑ j−n (ψ(j) − ψ(j − 1))z j , j=k+1

z ∈ 𝔻.

(9.11)

By summation by parts we get ∞

∞

j=k+1

j=k

ψ(k)r k + ∑ (ψ(j) − ψ(j − 1))r j = (1 − r) ∑ ψ(j)r j . From this and from (9.11) we obtain ∞

∞

j=k+1

j=k

M(r, Dn Uk ) ≤ ψ(k)r k + ∑ (ψ(j) − ψ(j − 1))r j = (1 − r) ∑ ψ(j)r j , and hence, by Lemma 9.10, M(r, Dn Uk ) ≤ Cψ(1/(1 − r)). It follows that {Uk } is a norm bounded sequence in H ∞ (ψ)n . Now using the inclusion H ∞ (ψ)n ⊂ HΛω,n , we conclude that the functions Uk are continuous on the closed disc and ωn (Uk∗ , t) ≤ Cω(t),

0 < t < 1,

(9.12)

where C is independent of t, k. On the other hand, by Lemmas 9.1 and 9.2, Cωn (Uk∗ , 1/k) ≥ ωn (Uk∗ , π/k) ≥ ‖Uk∗ ‖∞ ∞

= k −n ψ(k) + ∑ j−n (ψ(j) − ψ(j − 1)). j=k+1

Now using the formula m

m

j=k+1

j=k+1

∑ aj bj = ∑ (aj − aj+1 )Bj + am+1 Bm ,

j

where Bj = ∑ bν , ν=k+1

we get m

∑ j−n (ψ(j) − ψ(j − 1))

j=k+1

m

= ∑ (j−n − (j + 1)−n )(ψ(j) − ψ(k)) + (m + 1)−n (ψ(m) − ψ(k)) j=k+1

(9.13)

9.2 Lipschitz spaces and spaces of harmonic functions | 295 m

= ∑ (j−n − (j + 1)−n )ψ(j) j=k+1

− ψ(k)((k + 1)−n − (m + 1)−n ) + (m + 1)−n (ψ(m) − ψ(k)) m

= ∑ (j−n − (j + 1)−n )ψ(j) + ψ(m)(m + 1)−n − ψ(k)(k + 1)−n j=k+1 m

≥ ∑ (j−n − (j + 1)−n )ψ(j) − ψ(k)(k + 1)−n . j=k+1

Hence m

m

k −n ψ(k) + ∑ j−n (ψ(j) − ψ(j − 1)) ≥ ∑ (j−n − (j + 1)−n )ψ(j) j=k+1

j=k+1

m

≥ n ∑ (j + 1)−n−1 ψ(j), j=k+1

and, by (9.10), (9.12), and (9.13), ∞

∫ y−n−1 ψ(y) dy ≤ Ck −n ψ(k),

k ∈ ℕ+ .

k

We easily verify that this implies (9.3). Thus ψ satisfies (9.2) for some β < n (Lemma 9.3), and this concludes the proof of the implication (b) ⇒ (c) and incidentally of Theorem 9.1. Proof of Proposition 9.6 In proving Propositions 9.6 and 9.7, we will use the inequalities M(r, Dn+1 f ) ≤ C(1 − r)−1 M((1 + r)/2, Dn u),

M(r, f (n+1) ) ≤ C(1 − r)−1 M((1 + r)/2, Dn u),

(9.14)

where u = Re f , f ∈ H(𝔻), and n ≥ 0. Equivalently: if Dn u is bounded in 𝔻, then −1 󵄩 n 󵄩 󵄨 󵄨󵄨 n+1 󵄨󵄨D f (z)󵄨󵄨󵄨 ≤ C(1 − |z|) 󵄩󵄩󵄩D u󵄩󵄩󵄩∞ ,

−1 󵄩 󵄩 󵄨󵄨 (n+1) 󵄨󵄨 (z)󵄨󵄨 ≤ C(1 − |z|) 󵄩󵄩󵄩Dn u󵄩󵄩󵄩∞ , 󵄨󵄨f

where C is independent of f and z. The proof is left to the reader as an exercise (although these inequalities in a more general form were used in the proof of Theorem 3.8; see also Exercise 3.32). In proving Proposition 9.6, we may assume that u is real valued and harmonic in a neighborhood of the closed disc. For fixed r < 1, let h(θ) = ur (θ) = u(reiθ ). Then (Δnt h)(θ) = ∫ h(n) (θ + x1 + ⋅ ⋅ ⋅ + xn ) dx1 . . . dxn , tE

(9.15)

296 | 9 Lipschitz spaces of higher order where tE is the n-dimensional cube [0, t]n . Hence h(n) (θ) = (Dn u)(reiθ )t n = (Δnt ur )(θ) − ∫(h(n) (θ + x1 + ⋅ ⋅ ⋅ + xn ) − h(n) (θ)) dx1 . . . dxn . tE

Since 󵄨󵄨 x 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 (n+1) 󵄨󵄨 (n) (n) (θ + y) dy󵄨󵄨󵄨 ≤ M(r, Dn+1 u)x, 󵄨󵄨h (θ + x) − h (θ)󵄨󵄨 = 󵄨󵄨∫ h 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 where x = x1 + ⋅ ⋅ ⋅ + xn , we get 󵄩 󵄩 M(r, Dn u)t n ≤ 󵄩󵄩󵄩Δnt ur 󵄩󵄩󵄩∞ + ∫ M(r, Dn+1 u)(x1 + ⋅ ⋅ ⋅ + xn ) dx1 . . . dxn tE

󵄩 󵄩 = 󵄩󵄩󵄩Δnt ur 󵄩󵄩󵄩∞ + (n/2)M(r, Dn+1 u)t n+1 ,

0 < r < 1, t > 0.

The function Δnt u defined by (Δnt u)(reiθ ) = (Δnt ur )(θ) is harmonic on the closed disc, and therefore ‖Δnt ur ‖∞ ≤ ‖Δnt u∗ ‖ ≤ ωn (u∗ , t), t > 0. These inequalities, together with (9.14), yield M(r, Dn u) ≤ t −n ωn (u∗ , t) + Kt(1 − r)−1 M((1 + r)/2, Dn u)

(9.16)

(t > 0, 0 < r < 1), where K depends only on n. Let A(r) = (1 − r)−n M(r, Dn u), 0 < r < 1. It follows from (9.16) that A(r) ≤ t −n (1 − r)n ω(t) + 2n Kt(1 − r)−1 A((1 + r)/2), where ω(t) = ω(u∗ , t). Choose an integer m such that 2n K ≤ (1/4)2m and take t = a(1−r), a = 2−m . Then we have A(r) ≤ a−m ω(1 − r) + (1/4)A((1 + r)/2), 0 < r < 1. Integrating this inequality from ρ (< 1) to 1 and introducing appropriate substitutions, we get 1−ρ

1

−m

∫ A(r) dr ≤ a ρ

1

∫ ω(t) dt + (1/2) ∫ A(r) dr 0

(1+ρ)/2

1−ρ

1

≤ a−m ∫ ω(t) dt + (1/2) ∫ A(r) dr. ρ

0 1

Hence, since the integral ∫ρ A(r) dr is finite, 1

1−ρ −m

(1/2) ∫ A(r) dr ≤ a ρ

∫ ω(t) dt. 0

9.2 Lipschitz spaces and spaces of harmonic functions | 297

Now (9.5) follows from the inequalities 1−ρ

∫ ω(t) dt ≤ (1 − ρ)ω(1 − ρ), 0 n+1

M(r, D

u)(1 − ρ)

n+1

1

≤ (n + 1) ∫ A(r) dr, ρ

which are valid because the functions ω and M are increasing. Thus the proof of Proposition 9.6 is finished. Proof of Proposition 9.7 Let u = Re f , where f is analytic in 𝔻. Then 1

n

1 f (k) (rz) k z (1 − r)k + ∫(1 − s)n z n+1 f (n+1) (sz) ds f (z) = ∑ k! n! k=0

(9.17)

r

(z ∈ 𝔻, 0 < r < 1). Denoting the sum by fr,n , we have 1

󵄨 1 󵄨󵄨 ∫(1 − s)n M(s, f (n+1) ) ds. 󵄨󵄨f (z) − fr,n (z)󵄨󵄨󵄨 ≤ n! r

From this and from (9.14) it follows that (9.6) implies ‖f − fr,n ‖∞ → 0 (r → 1− ). Since the functions fr,n (r < 1) are continuous on the closed disc, we see that (9.6) implies the continuity of f , and consequently of u, on the closed disc. To prove (9.7), let ur (θ) = u(reiθ ), 0 < r ≤ 1. Then (9.7) is equivalent to 1

󵄩󵄩 n 󵄩󵄩 n−1 n 󵄩󵄩Δt u1 󵄩󵄩∞ ≤ C ∫ (1 − s) M(s, D u) ds, 1−t

0 < t < 1.

(9.18)

Let r = 1 − 2t, 0 < t < 1/4. Then ‖Δnt u1 ‖ ≤ ‖Δnt (u1 − ur )‖ + ‖Δnt ur ‖. It follows from (9.15) and the “increasing” property of M(r, Dn u) that 1

󵄩󵄩 n 󵄩󵄩 n n n−1 n 󵄩󵄩Δt ur 󵄩󵄩 ≤ t M(r, D u) ≤ n ∫ (1 − s) M(s, D u) ds, 1−t

and therefore we have to prove that ‖Δnt (u1 − ur )‖ is dominated by the right-hand side of (9.18). Since ‖Δnt (u1 − ur )‖ ≤ ‖Δnt (f1 − fr )‖, it suffices to prove that 1

󵄩 󵄩󵄩 n n−1 n 󵄩󵄩Δt (f1 − fr )󵄩󵄩󵄩 ≤ C ∫ (1 − s) M(s, D u) ds. 1−t

298 | 9 Lipschitz spaces of higher order To prove this, write (9.17) in the form n

f1 (θ) − fr (θ) = H(θ) + ∑ hk (θ)(1 − r)k /k!, k=1

1

H(θ) =

1 ∫(1 − s)n ei(n+1)θ f (n+1) (seiθ ) ds, n!

where

and hk (θ) = f (k) (reiθ )eikθ .

r

We have 1

2n 󵄩󵄩 n 󵄩󵄩 n ∫(1 − s)n M(s, f (n+1) ) ds 󵄩󵄩Δt H 󵄩󵄩 ≤ 2 ‖H‖ ≤ n! 1

r

≤ C ∫(1 − s)n−1 M((1 + s)/2, Dn u) ds r

1

= 2n C ∫ (1 − s)n−1 M(s, Dn u) ds, 1−t

where we have applied (9.14). To estimate ‖Δnt hk ‖, let m = n − k + 1 (1 ≤ k ≤ n) and observe that (9.15) implies 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 k−1 m 󵄩󵄩 k−1 m 󵄩 (m) 󵄩 k−1 󵄩 m 󵄩 󵄩󵄩Δt hk 󵄩󵄩 = 󵄩󵄩Δt Δt hk 󵄩󵄩 ≤ 2 󵄩󵄩󵄩Δt hk 󵄩󵄩󵄩 ≤ 2 t 󵄩󵄩󵄩hk 󵄩󵄩󵄩. From this and from the inequality ‖h(m) ‖ ≤ C(1 − r)−1 M((1 + r)/2, Dn u) (see (9.14)) it k follows that 1

󵄩󵄩 n 󵄩󵄩 n−1 n n−k n −k 󵄩󵄩Δt hk 󵄩󵄩 ≤ Ct M(1 − t, D u) ≤ Ct ∫ (1 − s) M(s, D u) ds, 1−t

where C is independent of t. Combining all these results yields (9.7) for 0 < t < 1/4. If t > 1/4, we can apply Lemma 9.1 to reduce (9.7) to the case 0 < t < 1/4, and this completes the proof.

9.3 Conjugate functions A well-known result of Privalov [490] says that if ω is a regular majorant (of order 1), then the Hilbert operator maps Λω (𝕋) into itself (see Theorem 8.4). This is a particular case of the following result, which contains an additional information. Theorem 9.4. Let ω be a fast majorant of order n. Then the Hilbert operator maps Λω,n into Λω,n if and only if ω is a slow majorant of order n.

9.3 Conjugate functions | 299

See Further notes 9.2. Since Λω,n ≃ h∞ (ψ)n , where ψ(x) = xn ω(1/x), x ≥ 1, we can consider the equivalent question: when the space h∞ (ψ)n is self-conjugate, that is, when the operator of harmonic conjugation maps h∞ (ψ)n into h∞ (ψ)n ? This question obviously reduces to the same question for h∞ (ψ). Then Theorem 9.4 is a consequence of the following: Theorem 9.5 (Shields–Williams). Let ψ be a slow majorant on [1, ∞). Then the spaces h∞ (ψ) is self-conjugate if and only if ψ is fast. Proof. Let ψ be slow and fast. It is easy to show that u ∈ h∞ (ψ) if and only if 1 1 󵄨 󵄨󵄨 ψ( ) 󵄨󵄨∇u(z)󵄨󵄨󵄨 ≤ C 1−r 1−r

(r = |z|).

̃ Now “if” part follows from the formula |∇u| = |∇u|. In proving the “only if” part, we use the fact that, by Proposition 8.4, there is a β concave function ψ0 such that ψ0 (x)β ≍ ψ(x). Since h∞ (ψ) = h∞ (ψ0 ), we can and will assume that ψ(x) = ψ0 (x)N , where ψ0 is concave, and N is a positive integer. Let N = 1 and ∞

kψ (reiθ ) = 2 ∑ ψ(k)r k cos kθ. k=1

Assume that we have proved M1 (r, kψ ) ≤ Cψ(

1 ), 1−r

0 < r < 1,

(9.19)

and let ∞

u(reiθ ) = ∑ k −1 ψ(k)r k sin kθ. k=1

Then ∞

̃ iθ ) = − ∑ k −1 ψ(k)r k cos kθ. u(re k=1

We have ∞

M∞ (r, u)̃ ≥ ∑ k −1 ψ(k)r k

(9.20)

k=1

and, by (9.19), 2π

θ

󵄨 󵄨 󵄨󵄨 it 󵄨 it 󵄨 iθ 󵄨 󵄨󵄨u(re )󵄨󵄨󵄨 ≤ ∫󵄨󵄨󵄨Du(re )󵄨󵄨󵄨 dt ≤ ∫ 󵄨󵄨󵄨Du(re )󵄨󵄨󵄨 dt 0

=

2π

0

1 󵄨󵄨 1 󵄨 ) ∫ 󵄨k (reit )󵄨󵄨󵄨 dt ≤ Cψ( 2 󵄨 ψ 1−r 0

(θ ∈ (0, 2π)).

300 | 9 Lipschitz spaces of higher order Hence u ∈ h∞ (ψ), and therefore, by hypothesis, ũ ∈ h∞ (ψ). From this and from (9.20) we find that ∞

∑ k −1 ψ(k)r k ≤ Cψ(

k=1

1 ). 1−r

Here we take r = 1 − 1/m, where m ≥ 2 is an integer, and get m

∑ k −1 ψ(k) ≤ Cψ(m),

k=1

whence x

∫ 1

ψ(t) dt ≤ Cψ(x), t

x > 1.

From this we conclude that ψ(x)/xα is almost increasing for some α > 0 (see the proof of Proposition 8.2). Thus it remains to prove (9.19). Summation by parts gives ∞

kψ (reiθ ) = ∑ Δ2 (ψ(k)r k )(k + 1)Kk (θ), k=0

where Kk (θ) is the Fejér kernel, Kk (θ) =

1 − cos(k + 1)θ 1 k , ∑ D (θ) = k + 1 j=0 j (k + 1)(1 − cos θ) 2π

and Δ2 ak = ak − 2ak+1 + ak+2 . Since ∫0 |Kk (θ)| dθ = 2π, we have ∞

󵄨 󵄨 M1 (r, kψ ) ≤ ∑ 󵄨󵄨󵄨Δ2 (ψ(k)r k )󵄨󵄨󵄨(k + 1). k=0

Now, using the formula Δ1 (ak bk ) = (Δ1 ak )bk + ak+1 Δ1 bk , we get Δ2 (ak bk ) = (Δ2 ak )bk + (Δ2 bk )ak+2 + 2Δ1 ak+1 Δ1 bk . Hence, taking ak = r k and bk = ψ(k), we obtain Δ2 (ψ(k)r k ) = r k+2 Δ2 ψ(k) + ψ(k)(1 − r)2 r k + 2r k+1 (1 − r)Δ1 ψ(k). Since Δ2 ψ(k) ≤ 0 and ψ(k + 1) 󵄨 󵄨󵄨 1 , 󵄨󵄨Δ ψ(k)󵄨󵄨󵄨 = ψ(k + 1) − ψ(k) ≤ k+1

9.4 Integrated mean Lipschitz spaces | 301

we have ∞

M1 (r, kψ ) ≤ ∑ (−Δ2 ψ(k))(k + 1)r k+2 k=0

∞

∞

k=0

k=0

+ (1 − r)2 ∑ ψ(k)(k + 1)r k + 2(1 − r) ∑ ψ(k)r k+1 . The last two summands are less than or equal to Cψ(1/(1 − r)) by Lemma 9.10. To estimate the first summand, using suitable changes of indices, we get ∞

∞

∞

k=0

k=2

k=2

∑ Δ2 ψ(k)(k + 1)r k+2 = P(r) + (1 − r)2 ∑ kψ(k)r k − (1 − r 2 ) ∑ ψ(k)r k ,

where P is a polynomial. In view of Lemma 9.10, this completes the proof of (9.19) in the case N = 1. If N = 2, that is, if ψ = ψ20 , where ψ0 is concave, we start from the formula π

iθ

kψ (re ) = ∫ − kψ0 (√reit )kψ0 (√rei(θ−t) ) dt, −π

from which we get M1 (r, kψ ) ≤ M1 (√r, kψ0 )2 ≤ C(ψ0 (

2

1 1 )) ≤ Cψ( ). 1−r 1 − √r

The proof (for N ≥ 3) is completed by induction on N.

9.4 Integrated mean Lipschitz spaces The Lp -modulus of smoothness of order n of a function f ∈ Lp (𝕋) (p > 0) is defined by 󵄩 󵄩 ωn (f , t)p = sup󵄩󵄩󵄩Δns f 󵄩󵄩󵄩p , |s| 0.

A refinement of the proof of Proposition 9.6 leads to the following theorem; see [426]. Theorem 9.6 (Pavlović). Let f ∈ H p , 0 < p ≤ ∞, 0 < q < ∞, and let ψ ∈ L1 (0, 1) be a nonnegative function such that ψ(2x) ≤ Kψ(x) (0 < x < 1/2). Then 1

∫ Mpq (r, Dn f )ψ(1 0

1

󵄩 q 󵄩 − r) dr ≤ C ∫[t −n 󵄩󵄩󵄩Δnt f∗ 󵄩󵄩󵄩p ] ψ(t) dt,

where C depends only on p, q, n, and K.

0

302 | 9 Lipschitz spaces of higher order Concise proof. Consider the case p ≤ 1. We may assume that f ∈ H(𝔻). In the general case, we apply the result to the function fρ (z) = f (ρz), ρ < 1, let ρ tend to 1, and apply the monotone convergence theorem. We begin as in the proof of Proposition 9.6 and find that 󵄨 󵄨 󵄨 n 󵄨󵄨 n n+1 iθ 󵄨 iθ 󵄨 n sup 󵄨󵄨󵄨Dn+1 f (rei(θ+y) )󵄨󵄨󵄨, 󵄨󵄨D f (re )󵄨󵄨󵄨t ≤ 󵄨󵄨󵄨Δt fr (e )󵄨󵄨󵄨 + (n/2)t 0 0 and q > 0. Consequently, if ω is regular of order n, then HΛp,q ω,m = HΛω,n for all m ≥ n.

Proof of Theorem 9.9. Let f ∈ Bp,q ω,n , where ω is fast. Then there exists α > 0 such that p ̄ p,q ω(t) ≥ ct α , which implies f ∈ Bp,q α ⊂ H , so we have to prove that f ∈ H Λω,n . Let q < p. Then, by Theorem 9.8, 1

ωn (f , t)qp ≤ C ∫ (1 − r)nq−1 φq (r) dr, 1−t

where φ(r) = Mp (r, Dn f ).

(9.28)

Multiplying this inequality by ω(t)−q t −1 , then integrating and changing the order of integration, we obtain 1

q

1

1

0

1−r

∫(ωn (f , t)p /ω(t)) dt/t ≤ C ∫(1 − r)nq−1 φ(r)q dr ∫ ω(t)−q t −1 dt. 0

Now the result follows from the inequality 1

∫ ω(t)−q t −1 dt ≤ Cω(x)−q , x

which holds because ω is fast.

306 | 9 Lipschitz spaces of higher order Assuming that q ≥ p, we have by Jensen’s inequality 1

(αpt −αp ∫ (1 − r)(n−α)p φ(r)p (1 − r)αp−1 dr) 1−t

≤ αpt

−αp

q/p

1

∫ (1 − r)(n−α)q φ(r)q (1 − r)αp−1 dr.

1−t

From this and from (9.28) it follows that ωn (f , t)qp

1

≤ Ct ∫ (1 − r)nq−ε−1 φ(r)q dr, ε

1−t

where ε = α(q − p). Hence 1

q

1

1

0

1−r

∫(ωn (f , t)/ω(t)) dt/t ≤ ∫(1 − r)nq−ε−1 φ(r)q dr ∫ t ε−1 ω(t)−q dt. 0 α

α

Using the inequality t /ω(t) ≤ C(1−r) /ω(1−r), t ≥ 1−r, we show that the inner integral is dominated by (1 − r)ε /ω(1 − r)q . This, together with inclusions (9.27), completes the proof. p,q We can define the analogous spaces of harmonic functions, Λ̄ p,q ω,n and Λω,n . Then we have the following:

Theorem 9.11. Let p ≥ 1 and q ∈ ℝ+ . If ω is a fast majorant satisfying (9.24), then p,q p,q Λ̄ p,q ω,n ≃ Λω,n ≃ Bω,n . Consequently, if ω is a regular majorant of order n, then p,q Λp,q ω,m = Bω

for all m ≥ n.

p,∞ Remark 9.14. The spaces Xω,n are defined by the appropriate “𝒪” conditions:

Mp (r, Dn f ) = 𝒪(

ω(1 − r) ), (1 − r)n

ωn (f∗ , t) = 𝒪(ω(t)).

Condition (9.24) reads t n = 𝒪(ω(t)) and is satisfied because of the hypothesis that ω is p,∞ p,∞ p,∞ a majorant of order n. It is easy to prove that Bp,∞ ω,n = HΛω,n and Bω,n = Λω,n under the hypothesis that ω is fast. For example, if ω(t) = t(log(2/t))γ , γ > 0, and f ∈ Lp (𝕋), we have that the conditions γ

ω1 (f , t)p = 𝒪(t(log(2/t)) ), and Mp (r, D1 𝒫 [f ]) = 𝒪(log

t → 0, γ

2 ) , 1−r

r → 1,

are equivalent. The equivalence remains true if we replace “𝒪” with “o”.

9.4 Integrated mean Lipschitz spaces | 307

Remark 9.15. Let Λp∗ = Λp,∞ ω,2 , ω(t) = t; this space is called the p-Zygmund space. An old theorem of A. Timan and F. Timan [582] (p = 2) and Zygmund [637] (1 < p < ⬦) states that if g ∈ Λp∗ , then ω(g, t)p = 𝒪(t log1/2 (2/t)) for 2 ≤ p < ⬦, and ω(g, t)p = 𝒪(t log1/p (2/t)) for 1 < p ≤ 2. The latter holds for p = 1 and, if g ∈ H p (𝕋), for p < 1. (For the case of decreasing coefficients, see Aljančić [34].) This can be proved by means of (the harmonic version of) Theorem 7.16 and Theorem 9.11. More generally, p,q we can define the (p, q)-Zygmund space by Λp,q ∗ = Λ1,2 . Then we can prove that if p ≥ 1,

p,q 1/s then Λp,q ∗ ⊂ Λω,1 , where ω(t) = t(log(2/t)) , s = min{p, 2}; see Further notes 7.7. The p,q p,q p p,q inclusion HΛ∗ := H (𝕋) ∩ Λ∗ ⊂ Λω,1 holds for all p > 0. This is one of examples showing that considering the case of nonregular weights is indispensable.

9.4.2 Radial Lipschitz conditions p,q It is substantial that the definition of Bω,1 uses the tangential derivatives. However, if ω is regular, then the weight ϕ(t) = t/ω(t) is almost normal, and since p,q

1

p,q

Bω,1 = {u ∈ h(𝔻): D u ∈ hϕ }, p,q p,q we see from Corollary 4.4 that Bω,1 is self-conjugate, which implies that Bω,1 does not ∞ |k| ikθ 1 ̂ e . change when D u is replaced with ℛu, where ℛu = r𝜕u/𝜕r = ∑−∞ |k|u(k)r

Theorem 9.12. Let p ≥ 1, q > 0, u = 𝒫 [g], g ∈ Lp (𝕋), and let ω be a regular majorant. p,q Then u ∈ Bω,1 if and only if the function ‖u − ur ‖p /ω(1 − r), 0 < r < 1, belongs to Lq−1 . Recall that Lq−1 is defined by the requirement 1

󵄨q dr 󵄨 < ∞. ∫󵄨󵄨󵄨F(r)󵄨󵄨󵄨 1−r 0

In fact we have the following: Theorem 9.13. Let p ≥ 1, u = 𝒫 [g], g ∈ Lp (𝕋), and let ω a fast majorant (of order 1) such that condition (9.24) with n = 1 is satisfied. Then the function Mp (r, ℛu)(1 − r) ω(1 − r)

,

0 < r < 1,

belongs to Lq−1 if and only if the function ‖u − ur ‖p /ω(1 − r), 0 < r < 1, belongs to Lq−1 . In the simplest case p = q = ∞, this theorem says: A function g ∈ C(𝕋) belongs to Λω (𝕋) if and only if ‖u∗ − ur ‖∞ = 𝒪(ω(1 − r)). The first result of this kind is due to Hardy and Littlewood [237, Theorem 42]: g belongs to Λα (𝕋) (0 < α < 1) if and only if ‖uρ − ur ‖∞ ≤ C(ρ − r)α (0 < r < ρ < 1). Ravisankar [496] proved that “transversally Lipschitz” harmonic functions defined on a domain with C 2 -boundary are Lipschitz.

308 | 9 Lipschitz spaces of higher order 1

Theorem 9.13 is deduced from the inequality ‖u − ur ‖p ≤ 2 ∫r ‖ℛuρ ‖p dρ (r ≥ 1/2) and the following fact (see the proof of Theorem 9.9). Proposition 9.16. Let ψ ∈ L1 (0, 1) be a nonnegative function such that ψ(2x) ≤ Kψ(x), 0 < x < 1/2, where K is a constant. Let g ∈ Lp (𝕋), p ≥ 1, q > 0, and u = 𝒫 [g]. Then 1

1

∫ Mpq (r, ℛu)ψ(1 − r) dr ≤ C ∫[ 0

‖u − ur ‖p 1−r

0

q

] ψ(1 − r) dr,

(9.29)

where C depends only on p, q, and K. Proof. We can assume that u ∈ h(𝔻). The following familiar inequality will play an important role: Mp (ρ, |∇U|) ≤ Cp (s − ρ)−1 Mp (s, U),

0 < ρ < s < 1,

(9.30)

for U ∈ h(𝔻). We start from the formula ρ 󸀠

u(ρy) − u(ry) = u (ry)(ρ − r) + ∫(ρ − t)u󸀠󸀠 (ty) dt,

(9.31)

r

where, temporarily, we use the notation u󸀠 (ry) =

𝜕u u(ry), 𝜕r

u󸀠󸀠 (ry) =

𝜕2 u u(ry). 𝜕r 2

Hence ρ 󸀠

Mp (r, u )(ρ − r) ≤ ‖uρ − ur ‖p + ∫(ρ − t)Mp (t, u󸀠󸀠 ) dt.

(9.32)

r

The function r 2 u󸀠󸀠 is harmonic, and therefore r 2 Mp (r, u󸀠󸀠 ) increases with r, so we have ρ

Mp (r, u󸀠 )(ρ − r) ≤ ‖uρ − ur ‖p + 4ρ2 ∫(ρ − t)Mp (ρ, u󸀠󸀠 ) dt r

≤ ‖uρ − ur ‖p + 2(ρ − r)2 Mp (ρ, u󸀠󸀠 ),

ρ > r > 1/2.

(9.33)

On the other hand, an application of (9.30) to the harmonic function U(ry) = ru󸀠 (ry) gives Mp (ρ, u󸀠󸀠 ) ≤ K0 (s − ρ)−1 Mp (s, u󸀠 ),

s > ρ > 1/2,

where K0 is a constant. Then putting ρ = r + δ and s = ρ + ε, where δ = a(1 − r),

ε = b(1 − r),

9.4 Integrated mean Lipschitz spaces | 309

we get ‖u − ur ‖p

δ + 2K0 Mp (r + δ + ε, u󸀠 ) ε a + 2K0 Mp (r + (a + b)(1 − r), u󸀠 ). = a(1 − r) b

Mp (r, u󸀠 ) ≤

δ ‖u − ur ‖p

(9.34)

Now choose a and b so that a + b = 1/2 and get Mp (r, u󸀠 ) ≤

‖u − ur ‖p

a + 2K0 Mp ((1 + r)/2, u󸀠 ). a(1 − r) b

Hence, assuming that q ≥ 1, for example, Mpq (r, u󸀠 ) ≤ 2q−1 a−q [

‖u − ur ‖p

q

a q ] + 2q−1 (2K0 ) Mpq ((1 + r)/2, u󸀠 ). (1 − r) b

Multiplying this inequality by ψ(1 − r) and integrating, we get 1

∫ Mpq (r, u󸀠 )ψ(1

q−1 −q

− r) dr ≤ 2

a

1/2

1

∫[ 1/2

+2

q−1

‖u − ur ‖p (1 − r)

1

a q (2K0 ) ∫ Mpq ((1 + r)/2, u󸀠 )ψ(1 − r) dr b 1/2

1

= 2q−1 a−q ∫ [ 1/2

+2

q−1

‖u − ur ‖p (1 − r)

q

] ψ(1 − r) dr

1

a q (2K0 ) ∫ Mpq (s, u󸀠 )ψ(2(1 − s)) ds b 3/4

1

≤ 2q−1 a−q ∫ [ 1/2

+2

q−1

q

] ψ(1 − r) dr

‖u − ur ‖p (1 − r)

q

] ψ(1 − r) dr

1

a q (2K0 ) K ∫ Mpq (s, u󸀠 )ψ(1 − s) ds. b 3/4

Let 1

A = ∫ Mpq (s, u󸀠 )ψ(1 − s) ds. 1/2

(9.35)

310 | 9 Lipschitz spaces of higher order q

Choose a and b so that 2q−1 (2K0 ba ) K = 1/2. Then we have A≤2

q−1 −q

a

1

∫[ 1/2

‖u − ur ‖p

q

1 ] ψ(1 − r) dr + A, (1 − r) 2

whence 1

‖u − ur ‖p q 1 A ≤ 2q−1 a−q ∫ [ ] ψ(1 − r) dr 2 (1 − r) 1/2

because A < ∞. This concludes the proof of the theorem. This result can be further generalized by introducing the radial symmetric differences of order n ≥ 2, for example, n n j Δnr u(ζ ) = ∑ ( )u(ζ (1 − (1 − r))), j n j=0

ζ ∈ 𝕋, 0 < r < 1.

Remark 9.17. Theorem 9.13 and Proposition 9.16 with appropriate reformulations hold for analytic functions for every p > 0. Membership of inner functions in Lipschitz spaces In the case where ω(t) = t α , 0 < α < 1, and p > 1/2, the following theorem was proved by Ahern [8]. Theorem 9.14. If p > 0, q > 0, and ω is a fast majorant such that the function t/ω(t), 0 < t < 1, belongs to Lq ((0, 1), dt/t). Then, an inner function I belongs to HΛp,q ω,1 if and q only if the function ‖1 − |Ir |‖p /ω(1 − r), 0 < r < 1, belongs to L−1 . In particular, taking ω(t) = t and q = ∞, we see that I belongs to the Hardy–Sobolev space HS1p (p > 0) if and only if ‖1 − |Ir |‖p = 𝒪(1 − r). Proof. In one direction, we use the inequality |I 󸀠 (rζ )| ≤ (1 − |I(rζ )|2 )/(1 − r 2 ). In the other direction, we use the inequality 1 − |I(rζ )| ≤ |I(ζ ) − I(rζ )| along with Theorem 9.13 and Remark 9.17. As a consequence of this theorem and Theorems 2.25 and 2.26, we have the following: Corollary 9.18. Let ω be as in Theorem 9.14, and let p > 0 and q > 0. Let ω satisfy one of the following conditions: 1 t q/2p dt (i) ∫0 ω(t) q t = ∞, p > 1/2; 1 t q (log 2t )2q dt ω(t)q t

(ii) ∫0

= ∞, p = 1/2.

p,q If an inner function I belongs to HΛp,q ω,1 (= Bω,1 ), then I is a Blaschke product.

9.4 Integrated mean Lipschitz spaces | 311

In the case where q = ∞ (resp., q = ⬦) the condition “t/ω(t) ∈ Lq (dt/t)” is to be interpreted as t = 𝒪(ω(t)), which is always satisfied (resp., t = o(ω(t)), t → 0+ ). Conditions (i) and (ii) take the form (i2) t 1/2p ≠ 𝒪(ω(t)), p > 1/2; (ii2) t(log 2t )2 ≠ 𝒪(ω(t)), p = 1/2; replace “𝒪” with “o, t → 0” if q = ⬦. Since the function t/ω(t) is almost increasing p,⬦ in t, we have HΛp,q ω,1 ⊂ HΛω,1 for q ≤ ⬦. Example 9.19. Ahern proved that if α ≥ 1/2p, 0 < α < 1, and I belongs to HΛp,q α , then I is a Blaschke product, which of course has no sense if p ≤ 1/2. Theorem 9.18 reproduces Ahern’s theorem but provides some new examples for p = 1/2. Let γ

2 ω(t) = t(log ) , t

γ > 0.

(This majorant is not regular.) Let q < ⬦. The condition on ω in Theorem 9.14 is satisfied if and only if γ > 1/q. Condition (ii) of Theorem 9.18 is satisfied if and only if (γ−2)q ≤ 1, that is, (‡)

1/q < γ ≤ 1/q + 2.

Hence, if (‡) holds and I ∈ HΛ1/2,q ω,1 , where I is an inner function, then I is a Blaschke product. The last statement of Theorem 9.14 can be expressed in the following way. Corollary 9.20. Let p > 0, and let I be an inner function. Then 2

p 1/p

1 − |I(rζ )| 󵄩󵄩 󸀠 󵄩󵄩 ) ) 󵄩󵄩I 󵄩󵄩p ≍ ( sup ∫( 1 − r2 0 0, and let I be an inner function such that fI ∈ HΛp,q ω . p p Since HΛp,q ⊂ H , we have f ∈ L (𝕋). By Smirnov’s theorem, this and the hypothesis ∗ ω f ∈ H s imply f ∈ H p . Now the desired conclusion follows from Theorem 9.15 and the relation {|fζ Iζ | − |Ir fr |}+ = {|fζ | − |fr Ir |}+ ≥ {|fζ | − |fr |}+ . Corollary 9.24. Let ω be a regular majorant, p ≥ 1, q ≥ 1, let I be an inner function, and p,q let f ∈ HΛp,q ω . Then fI ∈ HΛω if and only if the function 󵄩 󵄩 F5 (r) := 󵄩󵄩󵄩|fr |(1 − |Ir |)󵄩󵄩󵄩p /ω(1 − r),

0 < r < 1,

belongs to Lq−1 (0, 1). Proof. Let g = fI and h = 𝒫 [|g∗ |] = 𝒫 [|f∗ |]. Then hr − |gr | = hr − |fr | + |fr |(1 − |Ir |). q In view of the hypothesis f ∈ HΛp,q ω , we have F3 ∈ L−1 by Theorem 9.17. It follows that q F5 ∈ L−1 if and only if the function (hr − |gr |)/ω(1 − r) belongs to Lq−1 . Since |g∗ | = |f∗ | ∈ Λp,q ω (𝕋), the latter means that g ∈ HΛω by Theorem 9.17.

This corollary serves only as an illustration of possible applications of Theorem 9.17; its proof is almost identical to Böe’s proof of [91, Corollary 3.2] and the proof of an earlier result of Dyakonov [173, Corollary 2]. We can prove much more. Theorem 9.18. Let p > 0, q > 0, and let I be an inner function. Let ω be a fast (not p,q q necessarily regular) majorant, and let f ∈ HΛp,q ω,1 . Then fI ∈ HΛω,1 if and only if F5 ∈ L−1 . Observe that if f = 1, then this theorem reduces to Theorem 9.14.

318 | 9 Lipschitz spaces of higher order Proof. The idea is the same as in the case of Theorem 9.14. We use the analytic variant of Theorem 9.12 (which holds for all p > 0) and Theorem 9.9. p,q p,q Let g := fI ∈ HΛp,q ω,1 = Bω,1 . Since also f ∈ Bω,1 , we see, by Theorem 9.9, that the function r 󳨃→

‖ν∗ − νr ‖p ω(1 − r)

belongs to Lq−1 ,

where ν = g or ν = f . From this and from the identity g∗ − gr = fr (I∗ − Ir ) + I∗ (f∗ − fr ) it follows that the function r 󳨃→

‖fr (I∗ − Ir )‖p ω(1 − r)

belongs to Lq−1 ,

(9.40)

and hence F5 ∈ Lq−1 because |I∗ − Ir | ≥ 1 − |Ir |. This proves the “only if” part of the theorem. q 2 2 󸀠 Assume that f ∈ Bp,q ω,1 and F5 ∈ L−1 . Since 1 − |I(rζ )| ≥ (1 − r )|I (rζ )|, for ζ ∈ 𝕋, we get that the function r 󳨃→

‖(1 − r)fr Ir󸀠 ‖p ω(1 − r)

belongs to Lq−1 .

(9.41)

On the other hand, the function ‖(1−r)fr󸀠 Ir ‖p /ω(1−r) belongs to Lq−1 because |fr󸀠 Ir | ≤ |fr󸀠 | p,q 󸀠 󸀠 󸀠 and f ∈ Bp,q ω,1 . From this and from (9.41) it follows that fI ∈ Bω,1 because (fI) = f I + fI . This was to be proved. p,q Remark 9.25. As a byproduct of the proof, we have that if f ∈ HΛp,q ω,1 , then fI ∈ HΛω,1 if and only if (9.40) holds.

Exercise 9.26. Let ω be a regular majorant, and let p, q ≥ 1. Let f = IH be an inner– p,q outer factorization of a function f ∈ H p . Then f ∈ Bp,q ω if and only if H ∈ Bω and the q function ‖|Hr |(1 − |Ir |)‖p /ω(1 − r) belongs to L−1 .

Lipschitz functions and the quotient of bounded functions ∞ Although X := HΛp,q (see Further notes 9.3), it turns out ω need not be contained in H that every function f ∈ X can be represented as the quotient of two bounded functions from X.

Theorem 9.19. If ω is regular, p ≥ 1, and f ∈ HΛp,q ω , then there are functions Φ and ϕ belonging to H ∞ ∩ HΛp,q such that f = Φ/ϕ. ω

9.5 The Poisson integral and the moduli of Lipschitz functions | 319

We will deduce this from the Theorem 9.17 by means of a lemma due to Aleman. Lemma 9.27 (Aleman [26]). Let (M, μ) be a probability space, and let Φ ∈ L1 (μ) be such that Φ > 0 μ-a. e. and log Φ ∈ L1 (μ). Let E(Φ) = ∫ Φ dμ − exp ∫ log Φ dμ. M

M

Then (i) E(min{Φ, 1}) ≤ E(Φ) and (ii) E(max{Φ, 1}) ≤ E(Φ). Proof. Let A = {x ∈ M: Φ(x) ≥ 1}. We may assume that α = μ(A) ∈ (0, 1) since otherwise the inequalities are obvious. The first inequality is equivalent to ∫(Φ − 1) dμ ≥ exp( ∫ log Φ dμ)[exp(∫ log Φ dμ) − 1] =: Q. A

A

M\A

We have 1 Q ≤ exp(∫ log Φ dμ) − 1 = exp( ∫ log(Φα ) dμ) − 1 α A

A

1 1 ≤ ∫ Φα dμ − 1 ≤ (∫(1 + α(Φ − 1)) dμ) − 1 = ∫(Φ − 1) dμ, α α A

A

A

which proves (i). The proof of (ii) is similar. Proof of Theorem 9.19. Let ψ ∈ L1 (𝕋) be such that ψ ≥ 0 and log ψ ∈ L1 (𝕋), and define the outer function 2π

Oψ (z) = exp(− ∫ 0

eiθ + z log ψ(eiθ ) dθ). eiθ − z

p If f ∈ HΛp,q ω , then f ∈ H , and therefore f = IOψ for some ψ and inner function I. By Corollary 9.23, F := Oψ ∈ HΛp,q ω . Let ψ1 = max{ψ, 1} and ψ2 = min{ψ, 1}, so that Oψ = Oψ1 Oψ2 . First, we prove that G := Oψ1 and g := Oψ2 are in HΛp,q ω . Since |G∗ | = ψ1 and | max{x, 1} − max{y, 1}| ≤ |x − y| for x, y ∈ ℝ, we have that |G∗ | is in Λp,q ω because so is |F∗ | = ψ. On the other hand, by Aleman’s lemma,

𝒫 [|G∗ |](rζ ) − 󵄨󵄨󵄨G(rζ )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝒫 [|F∗ |](rζ )󵄨󵄨󵄨 − 󵄨󵄨󵄨F(rζ )󵄨󵄨󵄨,

󵄨

󵄨

󵄨

󵄨

󵄨

󵄨

and this implies that G ∈ HΛp,q ω by Theorem 9.17. In a similar way, we prove that |g∗ | ∈ Λp,q and α 𝒫 [|g∗ |](rζ ) − 󵄨󵄨󵄨g(rζ )󵄨󵄨󵄨 ≤ 𝒫 [|F∗ |](rζ ) − 󵄨󵄨󵄨F(rζ )󵄨󵄨󵄨.

󵄨

󵄨

󵄨

󵄨

320 | 9 Lipschitz spaces of higher order The latter can be expressed as where Ψ = Ig.

󵄨 󵄨 󵄨 󵄨 0 ≤ 󵄨󵄨󵄨F(rζ )󵄨󵄨󵄨 − 󵄨󵄨󵄨g(rζ )󵄨󵄨󵄨 ≤ 𝒫 [|f∗ |](rζ ) − 𝒫 [|Ψ∗ |](rζ ),

Multiplying this by |I|, we get 𝒫 [|Ψ∗ |](rζ ) − |Ψ (rζ )| ≤ 𝒫 [|f∗ |](rζ )| − |f (rζ )|. Hence Ig ∈ p,q HΛp,q ω by Theorem 9.17. Since f = Ig/(1/G), it remains to prove that 1/G ∈ HΛω ; this 󸀠 󸀠 2 󸀠 holds because |(1/G) | = |G |/|G| ≤ |G |. The result follows.

9.6 Invariant Besov spaces The membership of a function f ∈ H(𝔻) in Bp,p ω,1 (p ∈ ℝ+ ), where ω is a fast majorant of order 1, can be expressed via an integral over the bicircle 𝕋2 . p Theorem 9.20. Under the above conditions, f belongs to Bp,p ω,1 if and only if f ∈ H and p

∫ ∫( 𝕋 𝕋

|f (ζ ) − f (η)| |dζ ||dη| ) < ∞. ω(|ζ − η|) |ζ − η|

This can be obtained from the relation 1

∫ 0

π

‖Δ1t f ‖pp

dt|dζ | 󵄨p 󵄨 . dt ≍ ∫ ∫󵄨󵄨󵄨f (ζeit ) − f (ζ )󵄨󵄨󵄨 ω(t)p t [ω(|eit − 1|)]p |eit − 1| −π 𝕋

p

Let B (1 < p < ⬦) denote the space of f ∈ H(𝔻) such that p−2 󵄨p 󵄨 Bp (f ) := (− ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z))

1/p

1 < p < ⬦.

,

𝔻 p

The space B is called the invariant (or diagonal) Besov spaces. The seminorm Bp is Möbius invariant in the sense that Bp (f ∘ σ) = Bp (f ) for all σ ∈ Möb(𝔻) and f ∈ Bp . As a particular case of Theorem 9.20, we have the following: Theorem 9.21. Let 1 < p < ⬦. A function f ∈ H(𝔻) belongs to Bp if and only if f ∈ H p and Bp,1 (f ) := (− − ∫∫ 𝕋 𝕋

1/p

|f∗ (ζ ) − f∗ (η)|p |dζ ||dη|) |ζ − η|2

< ∞.

The seminorm Bp,1 is also Möbius invariant, and the Douglas formula B2 = B2,1 holds [160]. When p ≤ 1, we choose an integer N such that Np − 2 > −1 and define the space Bp by the requirement N−1

Np−2 󵄨p 󵄨 󵄨p 󵄨 dA(z) < ∞. ‖f ‖pBp = ∑ 󵄨󵄨󵄨f (j) (0)󵄨󵄨󵄨 + ∫󵄨󵄨󵄨f (N) (z)󵄨󵄨󵄨 (1 − |z|2 ) j=0

𝔻

(9.42)

9.6 Invariant Besov spaces | 321

Of course, instead of the ordinary derivatives, we can use one of the operators 𝒥 N , ℛN , and 𝒟[N] . It turns out that the norm of Bp (p ≤ 1) is Möbius invariant in the sense that ‖f ∘σ‖ ≤ C‖f ‖ for all σ ∈Möb(𝔻), where C is independent of f and σ; see [41] for p = 1 and [628] for p < 1. , and therefore it can ≃ HΛp,p The space Bp (p ∈ ℝ+ ) coincides with the space Bp,p 1/p 1/p be renormed in several ways. One of them is a particular case of Theorem 6.14. Theorem 9.22. Let p ∈ ℝ+ . A function f ∈ H(𝔻) belongs to Bp if and only if f ∈ H p and ∞

∑ En (f )pp < ∞.

n=0

Exercise 9.28. If 0 < p < q < ⬦, then Bp ⊂ Bq , and the inclusion is strict. We have B1 ⊂ H ∞ and hence Bp ⊂ H ∞ if (and only if) p ≤ 1. Exercise 9.29. According to the Coifman–Rochberg theorem on atomic decomposition of Bergman spaces (Theorem 3.16), a function f ∈ H(𝔻) belongs to Bp , p ≤ 1, if and only if D[N] f can be represented in the form ∞

D[N] f (z) = ∑ an n=1

(1 − |wn |2 )γ , (1 − w̄ n z)γ+N

where γ > 0, wn ∈ 𝔻, and ‖{an }‖ℓp ≤ C‖D[N] f ‖Ap

Np−2

f ∈ Bp if and only if

∞

f 󸀠 (z) = ∑ an n=1

≍ ‖f ‖Bp . Taking γ = 1, we obtain that

∞ 1 − |wn |2 = ∑ (−an )σw󸀠 n (z). 2 (1 − w̄ n z) n=1

From this and from the inclusion Bp ⊂ H ∞ we can deduce that ‖f ∘ σ‖Bp ≤ C‖f ‖Bp for all σ ∈Möb (𝔻). Further information If p = 2, then Bp it coincides with the Dirichlet space 𝒟. The norm in 𝒟 is given by ‖f ‖2𝒟 = |f (0)|2 + 𝒟(f ), where ∞

󵄨2 󵄨 󵄨 󵄨2 2 𝒟(f ) = ∫ − 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 dA = ∑ n󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 . 𝔻

n=1

Concerning deeper properties od 𝒟, see Carleson [113] (representation theorems), Nagel, Rudin, and Shapiro [398] (tangential limits). Krotov [328] extended the results of [398] to the Hardy–Sobolev spaces on the ball; see also Twomey [588] and Cima et al. [122, Section 2.6]. For further information and references, we refer to the recent papers [42] and [183]. Here we give Carleson’s representation formula in a simplified

322 | 9 Lipschitz spaces of higher order form. Let f = BSF, where F ∈ H 2 , B is the Blaschke product with the zeros zn , n ≥ 1, and S is the singular inner function generated by a singular measure dσ. Then 2

2

󵄨2

∞

1 − |z |2

dσ(ξ )

n 𝒟(f ) = 𝒟(F) + ∫ − 󵄨󵄨󵄨F(ζ )󵄨󵄨󵄨 ( ∑ + 2∫ )|dζ |. 2 |ζ − ξ |2 n=1 |ζ − zn |

󵄨

𝕋

(9.43)

𝕋

Richter and Sundberg [499] extended this formula to a class of weighted Dirichlet spaces; see also PhD thesis of Chacon Perez [117]. Is there something similar for invariant Besov spaces? The expression inside the brackets in (9.43) occurs in the problem of the nontangential derivative of an inner function. Generalizing and improving two old results of M. Riesz and Frostman, Ahern and Clark [13, 14] proved the following theorem. Theorem (Ahern–Clark). Let φ(z) = B(z) exp(− ∫ − 𝕋

ζ +z dν(ζ )), ζ −z

z ∈ 𝔻,

(9.44)

where B is a Blaschke product with zeros zn , n ≥ 1, and dν is a positive (not necessarily singular) measure on 𝕋. Then φ has a finite nontangential derivative at ζ ∈ 𝕋 if and only if 1 − |zn |2 dν(ξ ) + 2∫ < ∞. 2 |ζ − ξ |2 n=1 |ζ − zn | ∞

h(ζ ) := ∑

𝕋

and we have |φ (ζ )| = h(ζ ). 󸀠

A proof of this result along with many other results on inner functions can be found in Mashreghi’s book [379]. Note that every self-map φ of 𝔻 can be represented in the form (9.44). For further information and references, see [122, Ch. 1, § 1.7]. Exercise 9.30. The Möbius dual (see (6.44)) of Bp is isomorphic to B for p ≤ 1 and to 󸀠 Bp for 1 < p < ⬦. (Compare with (6.45).) Exercise 9.31. If an inner function belongs to Bp for some p > 0, then it is a finite Blaschke product. In the case p > 1, this was proved by Kim [307]. See Further notes 9.5 for the case of weak Besov spaces. Exercise 9.32. Applying the decomposition theorem for Besov spaces (Theorem 6.4), we can prove the following. Let f ∈ Bp . p ̂ – If p ≥ 2, then ∑∞ n=1 n|f (n)| < ∞. ∞ – If 1 < p < 2, then ∑n=1 supk≥n |f ̂(k)|p < ∞. p−1 supk≥n |f ̂(k)|p < ∞. – If p ≤ 1, then ∑∞ n=1 n This substantially improves the implication f ∈ Bp ⇒ f ̂(n) = o(n1/p ) proved by Zhu [626] for p ≥ 1. A proof of these implications and some other properties of Bp can be found in [419, (3)].

Further notes and results | 323

Further notes and results Theorem 9.1 solves a problem posed by Shields and Williams [533, Problem C] and provides more information. Theorem 9.5 is essentially due to them [534], although they proved it under the hypotheses that ψ is slow and Δ1 ψ(n) 󵄨 󵄨 ψ is convex, or ψ is concave, or 󵄨󵄨󵄨Δ2 ψ(n)󵄨󵄨󵄨 ≤ −c , n

n ≥ 1.

(𝒮𝒲 )

However, the proofs are similar. Section 9.2 follows the author’s paper [423]. Proposition 9.6 (for analytic functions) has recently been generalized by Kolomoitsev [319] to noninteger values of n, with an appropriate definition of the moduli of smoothness of fractional order. The most of Section 9.4, including Subsection 9.4.1, is contained in the paper [426]. Some particular cases of Theorem 9.9 were previously obtained by Janson [271] and Blasco–de Souza [87, Theorems 2.1, 2.2]. In these papers the majorant ω is assumed to be regular of order 1 or 2 (i. e., “Dini” and “b1 ” or “b2 ”); see Proposition 4.16. The proof in the text differs from the existing ones in that we use neither the Poisson nor the Cauchy kernel, and perhaps it can be used to prove more general results. In the case where ω(t) = t α (0 < α < 1), all the results of Section 9.5, except Theorems 9.14, 9.15, and 9.18, were proved by Böe [91]. Using some results of [174], Böe proved a variant of our Theorem 9.17 and then deduced the other results from this one. In his paper the reader will find further interesting results about inner functions and Blaschke products. Theorem 9.15 probably does not hold if it is assumed that ω is fast but not regular. On the other hand, having in mind the paper [440], where the case of q = ∞ and p > 0 was considered, it is reasonable to expect that Theorem 9.16 is valid under the hypothesis that ω is fast. The first results such as Theorem 9.19 were proved by Richter and Shields [498] (Dirichlet space), Aleman [26], and Dyakonov [175]; see also Walsch [594]. Ahern [8, Theorem 6] proved Theorem 9.14 when ω(t) = t α , 0 < α < 1, and then used it and his Theorem 2.25 to prove Corollary 6.31. 9.1. For a slow majorant ψ defined on [1, ∞), let t

̃ = 1 + ∫ ψ(x) dx, ψ(t) x 1

t ≥ 1.

Theorem 1 of [534], together with our proof of Theorem 9.5, gives the following fact, which was proved in [534] under hypothesis (𝒮𝒲 ). 󸀠

̃ Theorem. Let ψ be a slow majorant. (i) If u ∈ hp (ψ), where p ∈ {1, ∞}, then ũ ∈ hp (ψ), p and (ii) there is a function u ∈ h (ψ) such that ̃ Mp (r, u)̃ ≥ cψ(

1 ). 1−r

324 | 9 Lipschitz spaces of higher order This shows again (take ψ(t) ≡ 1) that the Riesz conjugate function theorem does not hold for p ∈ {1, ∞}. Statement (i) is easy to prove and extend to the case p < 1: Mpp (r, u) = 𝒪(ψ(

1 ̃ 1 )). )) 󳨐⇒ Mpp (r, u)̃ = 𝒪(ψ( 1−r 1−r

This generalizes the second inequality of Theorem 1.15. 9.2. Concerning Theorem 9.4, we note that in the case n = 1, there is a stronger result of Bary and Stechkin [59, Theorem 7]. Theorem (BS). Let ω be a majorant of order 1. Then the Hilbert operator maps Λω into itself if and only if ω is regular. p,q 9.3. In view of Theorem 9.19, it is a natural question when the space HΛp,q ω = Bω is contained in H ∞ . This can be translated to the language of coefficient multipliers3 : Is the function g(z) = 1/(1 − z) (i. e., the sequence (1, 1, . . . , 1, . . .)) a multiplier from Bp,q ω to H ∞ ? Since X = Bp,q is “homogeneous”, the set of these multipliers, denoted by X∗ = ω ∞ A (X, H ), is identical to X , provided that q ≤ ⬦. The proof of Theorem 6.10 can easily ∗ −n be modified to show that g ∈ (Bp,q ω ) if and only if the sequence ω(2 )‖Vn ∗ f ‖(H p )∗

belongs to ℓq . Since (see Lemma 6.15) 󸀠

2n(1−1/p ) = 2n/p , p ≥ 1, ‖Vn ‖(H p )∗ ≍ { n(1/p−1) 2 ‖Vn ‖∞ ≍ 2n/p , p < 1, 󸀠

∞ −n n/p we see that Bp,q , n ≥ 0, belongs to ℓq . The ω ⊂ H if and only if the sequence ω(2 )2 latter can be stated in the following form: the function ω(t)/t 1/p , 0 < t < 1, belongs 󸀠 ∞ (α > 0) if and only if α > 1/p, whilst to Lq+0 . In particular, if q > 1, then HΛp,q α ⊂ H if q ≤ 1, then the inclusion holds if and only if α ≥ 1/p. Note that if q ≠ ∞, then the equivalences remain true if H ∞ is replaced with A(𝔻). 󸀠

9.4. Concerning Remark 9.5, there is a rich theory concerning connections between the moduli of smoothness and the best approximation by polynomials. The first results in this area were obtained by Jackson in the doctoral thesis [269]. “Jackson’s theorem” usually means the validity of the inequality Ek (g)X ≤ CX,n ωn (g, 1/k)X ,

k ≥ 1,

(9.45)

in the case where X = Lp (𝕋) (1 ≤ p < ⬦) or X = C(𝕋), although he proved it only for n = 1, X = L1 (𝕋), and X = C(𝕋). The case n > 1 was treated by Stechkin [547]. Further historical and mathematical facts can be found in Bary and Stechkin’s paper [59] and 3 A theory of multipliers will be considered in Chapter 12.

Further notes and results | 325

the recent monographs DeVore and Lorents [145] and Stepanets [557]. Storozhenko [569, 571] proved that (9.45) continues to be true for X = H p , that is, Ek (f )p ≤ Cp,n ωn (f∗ , 1/k)p ,

n ≥ 1, k ≥ 1, p ∈ ℝ+ .

(9.46)

It is interesting that this inequality for n = 1, together with relations (9.23), only gives Ek (A1 )p ≤ Cp ξk,p (k ≥ 2), where ξk,p = k −1/2p for p > 1/2, ξk,p = (log2 k)/k for p = 1/2, and ξk,p = 1/k for p < 1/2. On the other hand, we proved that Ek (A1 )p ≤ Cp k −1/2p for all k and p, which can also be deduced from (9.46) and (9.23) by taking sufficiently large n for a fixed p. 9.5 (Inner functions and weak Besov spaces). Gröhn and Nicolau [218] proved the following: Theorem (GN). An inner function belongs to the weak Bp space (p > 1) if and only if it is an exponential Blaschke product. See Further notes 6.5. A function f ∈ H(𝔻) is in weak-Bp if its derivative belongs to the space Lp,⋆ (μp ), where dμp (z) = (1 − |z|)

p−2

dA(z).

̃1 . It consists If p = 1, then the space is trivial. Instead, the authors consider the space ℒ w −1 1,⋆ of f ∈ H(𝔻) such that (1 − |z|) |f (z)| belongs to L (𝔻, A). It is proved that the inclũ1 ⊂ H 1,⋆ hold, where H 1,⋆ is the weak H 1 space (see Section 2.9). Theorem sions H 1 ⊂ ℒ w ̃1 , then I is a 2 of [218] states that if the derivative of an inner function I belongs to ℒ w finite Blaschke product. 9.6. In a recent paper [497], Reijonen proves the following: ̂ Theorem (R). Let p, q be positive real numbers, and let φ ∈ L1 (0, 1) and φ(r) := 1 ∫r φ(x) dx be such that φ ∈ 𝒟, that is, ̂ − Bφ(1

1−r 1+r ̂ ≤ C φ( ̂ ) ≤ φ(r) ) K 2

(9.47)

for some constants C > 0, B > 1, and K > 1. Then the relation 1

‖1 − |Ir |‖p q 󵄩󵄩 󸀠 󵄩󵄩q p,q ) φ(r) dr 󵄩󵄩I 󵄩󵄩Aφ ≍ ∫( 1−r 0

holds for any inner function I if and only if r

∫ 0

̂ φ(s) φ(r) ds ≤ C , (1 − s)q (1 − r)q

0 < r < 1.

(9.48)

326 | 9 Lipschitz spaces of higher order (Concerning properties of 𝒟, see Proposition 4.17.) The set of weights satisfying ̂q . (9.48) is denoted by 𝒟 Condition (9.48) can be rewritten by using partial integration on the left side. It follows that (9.48) is equivalent to r

∫ 0

̂ ̂ φ(s) φ(r) , ds ≤ C (1 − r)q (1 − s)q+1

0 < r < 1.

(9.49)

This is equivalent to the condition ̂ φ(r)/(1 − r)q−ε is almost increasing in r ∈ (0, 1) for some ε > 0.

(9.50)

(This is a reformulation of Proposition 4.16, case (ii).) On the other hand, conditions (9.47) are equivalent to the condition that the funĉ − r), 0 < r < 1, is almost normal in our sense, which means that there are tion φ(1 ̂ ̂ constants a > 0 and b > 0 such that φ(r)/(1 − r)a (resp., φ(r)/(1 − r)b ) is almost increasing (resp., almost decreasing); see page 149. We can easily prove that this implies ‖f ‖qAp,q φ

1

≍ ∫ Mpq (r, f ) 0

̂ φ(r) dr, 1−r

p,q p,q ̂ − t)1/q is almost normal. that is, Ap,q φ = Hϕ = H[ϕ] , where ϕ(t) := φ(1 Condition (9.50) along with the hypothesis that ϕ is fast means that ϕ is regular in our sense (see p. 266). Then ω(t) = t/ϕ(t) is also regular, and I 󸀠 ∈ Ap,q φ if and only if p,q . Thus the “if” part of Theorem (R) follows from Theorem 9.14. = HΛ I ∈ Bp,q ω,1 ω,1 An example is in order. Let

φ(r) =

(1 − r)q . ω(1 − r)q (1 − r)

Here ω ∈ L1 (0, 1) is a fast majorant (of order 1). Take ω(t) = t logε (2/t), where ε > max(1/q, 1). Then φ ∈ L1 (0, 1), and r

̂ = ∫ log−εq φ(r) 0

2 dr 2 ≍ log1−εq . 1−r 1−r 1−r

This function is not normal, and therefore (“if” part of) Theorem (R) does not cover Theorem 9.14. In [497], the reader will find other results concerning, for example, Blaschke products (Theorem 11). We also recommend [474] for a discussion on the membership of the derivative of a Blaschke product in Bergman-type spaces.

Further notes and results | 327

Several variables The first results on Lipschitz (or Besov) spaces on the real sphere were obtained by Greenwald [214, 215], who considered some tangential moduli of continuity of first and second orders. Lizorkin and Nikol’skĭ [360, 406] used moduli of continuity of arbitrary order and characterized these spaces via best approximation by spherical polynomials. The theory of Lipschitz spaces on the complex ball is simpler; the first results were given by Kwon, Koo, and Cho [335]; further information and results can be found in [275]4 and [450].

4 The author of the papers [13]–[15] in the list of references is Pavlović, not Oswald.

10 Littlewood–Paley theory This chapter contains some properties of the Littlewood–Paley g-function. Using the Fefferman–Stein vector maximal theorem and the area theorem of Calderón, we prove that f ∈ H p , p ∈ ℝ+ , if and only if the Littlewood–Paley g-function, in one of three variants, belongs to Lp . Then we present Oswald’s results on a generalized g-function. In Section 10.5, we give two proofs of a generalized area theorem. A section is devoted to Littlewood–Paley inequalities for a subharmonic function with quasinearly subharmonic Laplacian. In Section 10.7, we give a modified Calderón’s proof of Calderón’s area theorem; as a result, we have the hyperbolic variant of Calderón’s theorem with application to composition operators from the Bloch space to H p . The next section contains the discussion of the action of an integration operator from H p to H q , where 0 < p, q < ∞. The last section is devoted to the Triebel–Lizorkin spaces. Throughout this chapter, we suppose, unless specified otherwise, that p and q belong to ℝ+ . The symbol ⬦ is not used.

10.1 Vector maximal theorems and Calderon’s area theorem The admissible approach regions are defined by SB (ζ ) = {z ∈ 𝔻: |z − ζ | < B(1 − |z|)} (B = const > 1, ζ ∈ 𝕋).

(10.1)

The sets SB (ζ ) satisfy ∫( ∫ h(z) dA(z))|dζ | ≍ ∫ h(z)(1 − |z|) dA(z), 𝕋 SB (ζ )

(10.2)

𝔻

where dA is the Lebesgue measure in ℂ, h ≥ 0 is a measurable function on 𝔻, and the equivalence constants are independent of h. We can take as before Uζ ,c = the convex hull of {|z| < c} ∪ {ζ } ,

(10.3)

where c ∈ (0, 1) is a constant. For our purposes, it is convenient to use the sets Tζ = {rζeiθ : 0 ≤ r < 1, |θ| ≤ 1 − r}.

(10.4)

The sets Uζ ,c and Tζ also satisfy the analogue of (10.2). Unlike Uζ ,c and SB (ζ ), the set Tζ contains no any disc centered at zero. We could use the sets Tζ ,c = {rζeiθ : 0 ≤ r < 1, |θ| ≤ c(1 − r)}, where c > 2π, but this is not important. https://doi.org/10.1515/9783110630855-010

330 | 10 Littlewood–Paley theory As in the scalar-valued case, using the vector maximal theorem (Theorem C.24), we obtain the following: Theorem 10.1. Let p > 1 and q > 1. Let un be a sequence of nonnegative functions subharmonic in 𝔻. Then ∞

q

p/q

∫( ∑ (M∗ un ) ) 𝕋

n=0

∞

p/q

q

dℓ ≤ Cp,q sup ∫( ∑ (un (rζ )) ) 0 0 and q > 0, then we choose γ > 0 such that p/γ > 1 and q/γ > 1 and apply γ Theorem 10.1 to the functions un and the indices p/γ and q/γ instead of p and q. If the functions un are continuous on the closed disc, then the right side of (10.5) can be replaced with ∞

∫( ∑ un q ) 𝕋

p/q

dℓ.

n=0

This will be used further. q Theorem 10.3. Let {fj }∞ 0 be a sequence in H . If

p > 0, q > 0, and m > max{0, 1/p − 1, 1/q − 1}, then p/q m q ∫( ∑ (σ∗ fj ) ) dℓ 𝕋 j=0 ∞

∞

q

≤ C ∫( ∑ |fj | ) 𝕋

j=0

p/q

dℓ.

Proof. The hypothesis m > max{0, 1/p − 1, 1/q − 1} implies that there exists s, 0 < s ≤ 1, such that m > 1/s − 1 and 1/s − 1 > max{0, 1/p − 1, 1/q − 1}. By (5.33) we have Σ :=

p/q q m ∫( ∑ (σ∗ fn ) ) dℓ 𝕋 n=0 ∞

∞

s

≤ C ∫( ∑ [ℳ(Mrad |fn | )] 𝕋

n=0

q/s

p/q

)

dℓ

10.2 The Littlewood–Paley g-theorem

∞

s

p0 /q0

q0

= C ∫( ∑ [ℳ(Mrad |fn | )] ) 𝕋

| 331

n=0

dℓ,

where p0 = p/s > 1 and q0 = q/s > 1. Using Theorem C.24 with the indices p0 and q0 , we obtain ∞

p0 /q0

s q0

Σ ≤ C ∫( ∑ (Mrad |fn | ) (ζ )) 𝕋

n=0

dℓ.

Now we can apply Theorem 10.1 or Theorem 10.2 to conclude the proof. Theorem 10.4 (Calderón). Let Dζ , ζ ∈ 𝕋, be one of the sets 𝒮B (ζ ), Uζ ,c , or Tζ . If f is analytic in the unit disc, f (0) = 0, p > 0, and q > 0, then p/q

󵄨 󵄨2 ‖f ‖pp ≍ ∫(∫ |f |q−2 󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 dA)

|dζ |.

𝕋 Dζ

Calderón stated this theorem for the upper half-plane. However, to apply his proof (postponed to Section 10.7, Theorems 10.16 and 10.18) to the unit disc and more general classes of functions, only one fact must be explained (Lemmas 10.18 and 10.26, respectively). The function 󵄨

󵄨2

𝒜f (ζ ) = ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 dA(z), 󸀠

Dζ

introduced in [365], is called the Luzin area function. If f is univalent, then 𝒜f (ζ ) coincides with the area of the image of Dζ under f . Concerning deeper properties of the area function, we refer to Zygmund [639, Ch. XIV]. As a particular case of Calderón’s theorem, we have the following: Theorem 10.5. Let p > 0, and f ∈ H(𝔻) with f (0) = 0. Then f ∈ H p if and only if 𝒜f ∈ Lp (𝕋), and we have ‖f ‖p ≍ ‖𝒜f ‖p . Problem 10.10. It follows from the theorem that if f , g ∈ H(𝔻) are such that |f 󸀠 (z)| ≤ |g 󸀠 (z)| for all z ∈ 𝔻, then ‖f ‖p ≤ Cp ‖g‖p . It would be interesting to find the optimal Cp .

10.2 The Littlewood–Paley g-theorem For f ∈ H(𝔻), the function 1

1/2

󵄨2 󵄨 󸀠 𝒢 [f ](ζ ) = (∫(1 − r)󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 dr) , 0

ζ ∈ 𝕋 := 𝜕𝔻,

332 | 10 Littlewood–Paley theory is called the (Littlewood–Paley) g-function associated with f . The corresponding “maximal function” is defined as 1/2

1

󵄨2 󵄨 󸀠 𝒢∗ [f ](ζ ) = (∫(1 − r) sup 󵄨󵄨󵄨f (ρζ )󵄨󵄨󵄨 dr) . 0 0), then 󵄨 󵄨 lim (1 − r) sup󵄨󵄨󵄨f 󸀠 (ρζ )󵄨󵄨󵄨 = 0

r→1−

for almost all ζ ∈ 𝕋.

ρ 2, where v is real valued and harmonic. By Theorem 3.5, Δu is QNS. If f is analytic, then we can take u = |f |2 in Theorems 10.13 and 10.14 getting the following variant of the Littlewood–Paley theorem and also of Flett’s theorem (p ≤ 1). Theorem 10.15. Let f ∈ H(𝔻), and let Kp (f ) = ∫(1 − |z|) 𝔻

󵄨p 󵄨󵄨f (z)󵄨󵄨󵄨 dA(z).

p−1 󵄨 󸀠 󵄨

348 | 10 Littlewood–Paley theory If Kp (f ) < ∞ and 0 < p ≤ 2, then f ∈ H p and ‖f ‖pp − |f (0)|p ≤ Cp Kp (f ). If p ≥ 2 and f ∈ H p , then Kp (f ) < ∞ and Kp (f ) ≤ Cp (‖f ‖pp − |f (0)|p ). See Further notes 10.8 for an improved variant of this theorem for p > 1. Local estimates for the Riesz measure In what follows, we suppose that u is a nonnegative subharmonic function defined in 𝔻 and denote by μ the Riesz measure of u. As we have seen, the formula I(r, u) − u(0) =

1 r dμ(z) ∫ log 2π |z|

(0 < r < 1)

(10.25)

r𝔻

holds (see Theorem 2.8). Lemma 10.15. We have 1 1 dμ(z). ∫ log 2π |z|

I(u) − u(0) =

𝔻

Proof. This follows from (10.25) and the monotone convergence theorem. Lemma 10.16. Let q ≥ 1, and let μq be the Riesz measure of uq . Then q

{μ(E)} ≤ Cq μq (5E)

(10.26)

for every disc E such that 6E ⊂ 𝔻. The constant Cq depends only of q. If E is a disc of radius R, then rE denotes the concentric disc of radius Rr. Proof. By translation the proof reduces to the case where E is centered at zero. Then, since μ(E) = ν((1/r)E), where ν is the Riesz measure of the function u(rz), we can assume that the radius of E is fixed, for example, E = ε𝔻, ε = 1/6. Using the simple inequalities q

q

(I(r, u) − u(0)) ≤ (I(r, u)) − u(0)q and (I(r, u))q ≤ I(r, uq ), which hold because q > 1, we see from (10.25) (applied to u and uq ) that q

(

1 1 r r dμ(z)) ≤ dμ (z). ∫ log ∫ log 2π |z| 2π |z| q r𝔻

r𝔻

Letting r = 4ε, we get q

{μ(2ε𝔻)} ≤ C ∫ |z|−1 dμq (z), 4ε𝔻

(10.27)

10.6 Littlewood–Paley-type inequalities | 349

where we have applied the estimate log(4ε/|z|) ≥ log 2

for |z| < 2ε,

and

log(4ε/|z|) ≤ 1/|z|.

Therefore, to prove (10.26), we have to remove |z|−1 . To do this, we translate the “center” of (10.27) to get q

{μ(2εDa )} ≤ C ∫ |z − a|−1 dμq (z) 4εDa

for a ∈ ε𝔻, where Da = {z : |z − a| < 1}. Since ε𝔻 ⊂ 2εDa and 4εDa ⊂ 5ε𝔻, we see that q

{μ(ε𝔻)} ≤ C ∫ |z − a|−1 dμq (z). 5ε𝔻

Now we integrate this inequality over the disc ε𝔻 with respect to dA(a) and apply Fubini’s theorem, which finishes the proof because supz∈𝔻 ∫5ε𝔻 |z − a|−1 dA(a) < ∞. Proof of Theorem 10.12. From (10.26) it follows that q

∫(1 − |z|) {μ(Eε (z))} dA(z) ≤ C ∫(1 − |z|) μq (E5ε (z)) dA(z). −1

−1

𝔻

(10.28)

𝔻

Further, from μq (E5ε (z)) = ∫ dμq (w) E5ε (z)

and Fubini’s theorem it follows that the right-hand side of (10.28) equals ∫ dμq (w) ∫ (1 − |z|) dA(z), −1

𝔻

󸀠 (w) E5ε

where Eε󸀠 (w) = {z : |z − w| < ε(1 − |z|)}.

(10.29)

󸀠 Since z ∈ E5ε (w) implies |z| − |w| < 5ε(1 − |z|) and thus 1 − |w| < (1 + 5ε)(1 − |z|), we see that −1 −1 󵄨 󵄨 󸀠 (w)󵄨󵄨󵄨(1 − |w|) . ∫ (1 − |z|) dA(z) ≤ (1 + 5ε)󵄨󵄨󵄨E5ε 󸀠 (w) E5ε

󸀠 Since (1 − 5ε)(1 − |z|) < 1 − |w| for z ∈ E 󸀠 (w), we have |E5ε (w)| ≤ C 󸀠 (1 − |w|)2 , where 󸀠 2 C = π(5ε/(1 − 5ε)) . Combining all these results, we get q

∫(1 − |z|) {μ(Eε (z))} dA ≤ Cq ∫(1 − |w|) dμq (w). −1

𝔻

𝔻

350 | 10 Littlewood–Paley theory This completes the proof of (10.21) because of Lemma 10.15 and the inequality 1 − |w| ≤ log(1/|w|). Proof of Theorem 10.14. Fix ε < 1/6. Applying Lemma 10.16 to the pair uq , (uq )1/q , we obtain, because 1/q > 1, q

μq (Eε (z)) ≤ Cq (μ(E5ε (z))) ,

(10.30)

where μq and μ are the Riesz measures of uq and u. On the other hand, since Δu is quasinearly subharmonic, we have q

q

(μ(E5ε (z))) = ( ∫ Δu dA) ≤ C1 (1 − |z|) E5ε (z)

≤ C2 ∫ (1 − |z|)

2q−2

2q

q

sup (Δu(w))

w∈E5ε (z)

q

(Δu(z)) dA(z).

6ε𝔻

It follows that ∫(1 − |z|) μq (Eε (z)) dA(z) ≤ C ∫(1 − |z|) −1

2q−3

dA(z) ∫ (Δu)q dA, E6ε (z)

𝔻

𝔻

where C depends only on q. Hence, as in the proof of Theorem 10.12, ∫(1 − |z|) dμq (z) ≤ C ∫(1 − |z|)

2q−1

(Δu)q dA.

𝔻

𝔻 q

This implies that I(u ) < ∞ because of Lemma 10.15 applied to uq . To deduce (10.22) from (10.31), we rewrite (10.24) as 1/q

(

1 r dμ (z)) ∫ log 2π |z| q r𝔻

≤

1 r dμ(z). ∫ log 2π |z| r𝔻

Hence ∫ log ε𝔻

ε dμ (z) ≤ C sup(Δu)q ≤ C ∫ (Δu)q dA ≤ CM, |z| q ε𝔻 2ε𝔻

where M = ∫(1 − |z|) 𝔻

and hence I(uq ) − u(0)q = ∫ log 𝔻

1 dμ (z) |z| q

2q−1

q

(Δu(z)) dA(z),

(10.31)

10.7 A proof of Calderón’s theorem and generalizations | 351

= ∫ log ε𝔻

ε 1 dμq (z) + log ∫ dμq (z) + |z| ε ε𝔻

∫ log ε≤|z| 0. If G ≥ 0, we apply this case to the functions G + ϵ or (G2 + ϵ)1/2 , ϵ > 0, and then let ϵ tend to 0. Observe that, in the case p = 2, Theorem 10.16 says that ‖G‖22 ≍ ∫ G2 dA + ∫ Δ(G2 )(z)(1 − |z|) dA(z). 𝔻/2

𝔻

(10.34)

10.7 A proof of Calderón’s theorem and generalizations | 353

Lemma 10.18. If G ∈ 𝔾, then (10.34) holds. This is a consequence of the following lemma due to Stoll [564, Proposition 4.3] (cf. [567, Lemma 2.6]), the relation − u(ζ )|dζ | = u(0) + ∫ 𝕋

1 1 dμ (z), ∫ log 2π |z| u 𝔻

and the inequalities log 1/|z| ≥ 1 − |z| and ∫ u dA ≤ C ∫ u dℓ, 𝕋

𝔻/2

where u ≥ 0 is subharmonic in a neighborhood of 𝔻. Lemma 10.19. Let u ≥ 0 be a function subharmonic in a neighborhood of 𝔻, then ∫ log 𝔻

1 dμ(z) ≤ C sup u + C ∫(1 − |z|) dμ(z), |z| 1 𝔻 𝔻

4

where dμ is the Riesz measure of u. Proof. We need to estimate J(ε) = ∫ε𝔻 log(1/|z|) dμ, where ε is small. We have, by elementary inequalities and the Riesz representation theorem, J(ε) = ∫ log ε𝔻

1 ε dμ + log ∫ dμ |z| ε ε𝔻

1 εe ε dμ + log ∫ log dμ ≤ ∫ log |z| ε |z| ε𝔻

ε𝔻

ε𝔻

εe𝔻

1 εe ε dμ + log ∫ log dμ ≤ ∫ log |z| ε |z| 1 = 2π(M1 (ε, u) − u(0)) + 2π log (M1 (εe, u) − u(0)). ε Now taking εe = 1/4, we obtain the desired result. Proof of the inequality ‖SG‖q ≤ C‖G‖q (q > 0) This inequality proves that ∫ Gq dA + ‖SG‖qq ≤ C‖G‖qq 𝔻/2

since obviously ∫ Gq dA ≤ C‖G‖qq . 𝔻/2

We begin with, essentially, a particular case of the desired inequality.

354 | 10 Littlewood–Paley theory Lemma 10.20. Let h ∈ Lp (𝕋), h ≥ 0, and let u be the Poisson integral of h. Then ‖Su‖p ≤ C‖h‖p ,

1 < p ≤ 2.

Proof. By Lemma 1.38 and the reverse Hölder’s inequality with exponents p/(p − 2) and p/2 we have ‖h‖pp ≥ c ∫ Δ(u2 )(z)u(z)p−2 (1 − |z|) dA(z) 𝔻

≥ c ∫ |dζ | ∫ Δ(u2 )(z)u(z)p−2 K(z, ζ ) dA(z) 𝕋

𝔻

≥ c ∫(M∗ u)p−2 (Su)2 dℓ 𝕋

2 p−2 2 ≥ c‖M∗ u‖p−2 p ‖Su‖p ≥ c‖h‖p ‖Su‖p .

The result follows. (Note that Δ(u2 ) = 2|∇u|2 .) We will use the following property of SG. Lemma 10.21. S(Gp ) ≤ Cp (M∗ G)p−1 SG, p > 1. Proof. We have 2

S(Gp ) (ζ ) = ∫ Δ(G2p ) dA ≍ ∫ G2p−2 Δ(G2 ) dA, Uζ

Uζ

from which we easily obtain the desired inequality. To prove the equivalence, observe that the function u = G2 is log-subharmonic, and therefore |∇u|2 ≤ uΔu.

(10.35)

Δ(uα ) = α(α − 1)uα−2 |∇u|2 + αuα−1 Δu.

(10.36)

On the other hand,

If α ≥ 1, then, in view of (10.35), we have the relation 2

S(Gα ) ≍ ∫ G2α−2 Δ(G2 ) dA.

(10.37)

Uζ

If 0 < α < 1, then by (10.35) Δ(uα ) ≥ α(α − 1)uα−1 Δu + αuα−1 Δu = α2 uα−1 Δu. Since obviously Δ(uα ) ≤ αuα−1 Δu, we see that (10.37) holds for all α. The result follows.

10.7 A proof of Calderón’s theorem and generalizations | 355

Lemma 10.22. If the inequality ‖SG‖s ≤ C‖G‖s

(all G ∈ 𝔾, C independent of G)

(10.38)

holds for some s > 0, then it holds for all q ∈ (0, s). Proof. Here and elsewhere, we assume that G = Fρ , where F ∈ 𝔾 and 0 < ρ < 1. We obtain the final conclusion by letting ρ → 1− and using Fatou’s lemma along with the fact that the constant C is independent of ρ. Let 0 < q < s and p = qs > 1. Then by

Lemma 10.21 applied to G1/p we have SG ≤ C(M∗ (G1/p ))

p−1

S[G1/p ] = C(M∗ G)(p−1)/p S[G1/p ].

Hence by Hölder’s inequality with indices p = s/q and p/(p − 1) = s/(s − q) we have q󵄩 󵄩 ‖SG‖qq ≤ C 󵄩󵄩󵄩(M∗ G)q(p−1)/p (S[G1/p ]) 󵄩󵄩󵄩1 q󵄩 󵄩 󵄩 󵄩 ≤ C 󵄩󵄩󵄩(M∗ G)q(p−1)/p 󵄩󵄩󵄩p/(p−1) 󵄩󵄩󵄩S[G1/p ] 󵄩󵄩󵄩p 󵄩󵄩 1/p 󵄩q = C‖M∗ G‖(p−1)q/p 󵄩󵄩S[G ]󵄩󵄩󵄩s . q

From this and from (10.38), via the maximal theorem, it follows that 󵄩󵄩 1/p 󵄩󵄩q q ‖SG‖qq ≤ C‖G‖(p−1)q/p 󵄩󵄩G 󵄩󵄩s = C‖G‖q . q This finishes the proof. In view of Lemma 10.22, the following fact completes the proof of the inequality ‖SG‖q ≤ C‖G‖q for all q > 0. Lemma 10.23. We have the inequality ‖SG‖q ≤ C‖G‖q , q > 4. Proof. Let q > 2. Let h(ζ ), ζ ∈ 𝕋, be any bounded positive measurable function. Then ∫(SG)2 h dℓ = ∫ h(ζ ) |dζ | ∫ K(z, ζ )Δ(G2 )(z) dA(z) 𝔻

𝕋

𝕋

2

= ∫ Δ(G )(z) dA(z) ∫ h(ζ )K(z, ζ )|dζ |. 𝔻

𝕋

Since K(z, ζ ) ≤ C(1 − |z|)P(z, ζ ), where P is the Poisson kernel, P(z, ζ ) = (1 − |z|2 )/|ζ − z|2 , we see that ∫(SG)2 dℓ ≤ C ∫ Δ(G2 )(z)(1 − |z|)u(z) dA(z), 𝕋

where u = 𝒫 [h].

𝔻

356 | 10 Littlewood–Paley theory We continue with the relation Δ(G2 u) = uΔ(G2 ) + 2∇(G2 )∇u ≥ uΔ(G2 ) − 2G|∇G| |∇u|, which implies uΔ(G2 ) ≤ Δ(G2 u) + 2G|∇G| |∇u|,

(10.39)

and hence ∫(SG)2 h dℓ ≤ C1 ∫ Δ(G2 u)(1 − |z|) dA(z) 𝔻

𝕋

+ C2 ∫ G|∇G| |∇u|(1 − |z|) dA 𝔻

≤ C1 ∫ Δ(G2 u) log 𝔻

1 dA + C2 ∫ G|∇G| |∇u|(1 − |z|) dA |z| 𝔻

2

= C1 ∫ G h dℓ + C2 ∫ G|∇G| |∇u|(1 − |z|) dA 𝔻

𝕋 2

≤ C1 ∫ G h dℓ 𝕋

󵄨 󵄨󵄨 󵄨 + C3 ∫ |dζ | ∫ G(z)󵄨󵄨󵄨∇G(z)󵄨󵄨󵄨 󵄨󵄨󵄨∇u(z)󵄨󵄨󵄨K(z, ζ ) dA 𝕋

𝔻 2

≤ C1 ∫ G h dℓ + C4 ∫(M∗ G)(SG)(Su) dℓ. 𝕋

𝕋

Here we have used the Cauchy–Schwarz inequality and then the log-subharmonicity of G to conclude that |∇G|2 ≤ (1/4)Δ(G2 ) and hence that 󵄨2 󵄨 ∫󵄨󵄨󵄨∇G(z)󵄨󵄨󵄨 K(z, ζ ) dA(z) ≤ (SG)2 (ζ ). 𝔻

Now let p = q/(q − 1) (< 2) and apply the three-term Hölder inequality with indices 2q, 2q, p: ∫ G2 h dℓ ≤ ‖G‖22q ‖h‖p , 𝕋

∫(M∗ G)(SG)(Su) dℓ ≤ ‖M∗ G‖2q ‖SG‖2q ‖Su‖p . 𝕋

It follows that ∫(SG)2 h dℓ ≤ C‖G‖22q ‖h‖p + C‖M∗ G‖2q ‖SG‖2q ‖Su‖p 𝕋

≤ C‖G‖22q ‖h‖p + C‖G‖2q ‖SG‖2q ‖Su‖p

≤ C‖G‖22q ‖h‖p + C‖G‖2q ‖SG‖2q ‖h‖p ,

10.7 A proof of Calderón’s theorem and generalizations | 357

where we have used Lemma 10.20 (valid because p < 2). Taking the supremum over all h with ‖h‖p = 1, we get 󵄩 󵄩 ‖SG‖22q = 󵄩󵄩󵄩(SG)2 󵄩󵄩󵄩q ≤ C‖G‖22q + C‖G‖2q ‖SG‖2q , which implies ‖SG‖2q ≤

C + √C 2 + 4C ‖G‖2q . 2

This completes the proof of (10.23) and therefore of the inequality ‖SG‖q ≤ C‖G‖q for all q > 0. q

q

Proof of the inequality ‖G‖q ≤ C ∫𝔻/2 Gq dA + C‖SG‖q , q > 0 In proving this, we use the already proved reverse inequality, Lemma 10.18, and the following lemma. Let S1 [G](ζ )2 = ∫ G2 dA + S[G]2 (ζ ). 𝔻/2

Lemma 10.24. If ασ + β(1 − σ) = 1, where 0 < σ < 1 and α, β > 0, then σ

1−σ

S1 [G] ≤ CS1 [Gα ] S1 [Gβ ]

,

where C is independent of G ∈ 𝔾. Proof. First, observe that (2α − 2)σ + (2β − 2)(1 − σ) = 0. Then we have (SG)2 (ζ ) = ∫ Δ(G2 ) dA = ∫ Δ(G2 )G(2α−2)σ G(2β−2)(1−σ) dA. Uζ

Uζ

Similarly, ∫ G2 dA = ∫ G2 G(2α−2)σ G(2β−2)(1−σ) dA. 𝔻/2

𝔻/2

It follows from these inequalities and Hölder’s inequality that σ

S1 [G]2 ≤ ( ∫ G2 G2α−2 dA + ∫ Δ(G2 )G2α−2 dA) 𝔻/2

Uζ 1−σ

× ( ∫ G2 G2β−2 dA + ∫ Δ(G2 )G2β−2 dA) 𝔻/2

Uζ

Now the desired conclusion follows from (10.37).

.

358 | 10 Littlewood–Paley theory Now we a ready to prove the inequality ‖G‖q ≤ C‖S1 G‖q . Assume first that G ∈ 𝔾, where G is continuous in a neighborhood of the closed disc. We have, by Lemma 10.18, 󵄩2 󵄩2 󵄩 󵄩 ‖G‖qq = 󵄩󵄩󵄩Gq/2 󵄩󵄩󵄩2 ≍ 󵄩󵄩󵄩S1 [Gq/2 ]󵄩󵄩󵄩2 2σ 2(1−σ) 󵄩 󵄩 󵄩󵄩 ≤ C 󵄩󵄩󵄩S1 [Gαq/2 ] S1 [Gβq/2 ] 󵄩1 2σ 󵄩󵄩 αq/2 2(1−σ) 󵄩 󵄩 ] (S1 G) = C 󵄩󵄩S1 [G 󵄩󵄩1 , where we have used Lemma 10.24 with 2q 2 q+2 α= , β= , σ= , q+2 q 2(q + 1)

1−σ =

q . 2(q + 1)

Hence, by Hölder’s inequality with the indices (q + 1)/q and q + 1, 2σ 󵄩 󵄩 󵄩 󵄩 ‖G‖qq ≤ C 󵄩󵄩󵄩S1 [Gαq/2 ] 󵄩󵄩󵄩(q+1)/q 󵄩󵄩󵄩(S1 G)2(1−σ) 󵄩󵄩󵄩q+1 .

Since 2σ(q + 1)/q = (q + 2)/q, we have 󵄩2σ 󵄩 󵄩󵄩 αq/2 2σ 󵄩 ] 󵄩󵄩󵄩(q+1)/q = 󵄩󵄩󵄩S1 [Gαq/2 ]󵄩󵄩󵄩(q+2)/q , 󵄩󵄩S1 [G and, since 2(1 − σ) = q/(q + 1), we also have 󵄩󵄩 2(1−σ) 󵄩 󵄩󵄩 = ‖S1 G‖2(1−σ) , 󵄩󵄩(S1 G) q 󵄩q+1 so 󵄩2σ 󵄩 . ‖G‖qq ≤ C 󵄩󵄩󵄩S1 [Gαq/2 ]󵄩󵄩󵄩(q+2)/q ‖S1 G‖2(1−σ) q On the other hand, we have proved that 󵄩 󵄩󵄩 󵄩 αq/2 󵄩 ]󵄩󵄩󵄩(q+2)/q ≤ C 󵄩󵄩󵄩Gαq/2 󵄩󵄩󵄩(q+2)/q = C‖G‖αq/2 , 󵄩󵄩S1 [G q whence 2σ

‖G‖qq ≤ C(‖G‖αq/2 ) ‖S1 G‖2(1−σ) , q q and hence , ‖G‖q(1−ασ) ≤ C‖S1 G‖2(1−σ) q q where we have used the fact that ‖G‖q < ∞. Since q(1 − ασ) = 2(1 − σ), we see that ‖G‖q ≤ C‖S1 G‖q , which completes the proof of in the case where G is defined in a neighborhood of 𝔻. If G is arbitrary, then we apply the result to the functions Gρ (z) = G(ρz), ρ < 1. Then a little work is needed to complete the proof of the theorem. Remark. Considering the family {Gδ : δ > 0} is of substantial importance for the validity of (the modified) Calderón’s proof even if we only want to prove the Lusin area theorem 10.5. Remark 10.25. In modifying the original Calderón’s proof the author was motivated by Stoll’s paper [567]. Concerning Stoll’s paper [567], see Further notes 10.9.

10.7 A proof of Calderón’s theorem and generalizations | 359

The case where the Laplacian is log-subharmonic If Δ(F 2 ) is log-subharmonic, for example, if F = ‖{fn }‖ℓ2 , fn ∈ H(𝔻), and F(0) = 0, then we can prove a theorem that says somewhat more than Theorem 10.16. Theorem 10.18. Let 0 < p, q < ∞, and let G be a log-subharmonic C 2 function on 𝔻 with G(0) = 0 such that Δ(G2 ) is log-subharmonic. Then p/q

‖G‖pp ≍ ∫(∫ Gq−2 Δ(G2 ) dA)

(10.40)

|dζ |,

𝕋 Uζ

p/q

‖G‖pp ≍ ∫(∫ Gq−1 ΔG dA)

(10.41)

|dζ |.

𝕋 Uζ

The proof of this theorem is obtained from the proof of Theorem 10.16 by removing the summand ∫𝔻/2 Gp dA from the definition of S1 , so that S1 = S, and using the following lemma instead of Lemma 10.18. Note that if G(0) ≠ 0, then we have to add G(0)p to the right-hand sides of (10.40) and (10.41). Lemma 10.26. If G = F p/2 , where p > 0 and F ∈ C 2 is log-subharmonic with F(0) = 0 such that Δ(F 2 ) is log-subharmonic, then we have the relation ‖G‖22 ≍ ∫ Δ(G2 )(z)(1 − |z|) dA(z).

(10.42)

𝔻

Proof. Let p < 2. Proceeding in a similar way as in the proof of Theorem 2.17, we conclude that the Riesz measure of G2 is absolutely continuous and that, as we have seen before, Δ(G2 ) ≍ F p−2 Δ(F 2 ). Then using Green’s formula as in the proof of Lemma 3.23, we get ‖G‖22 = ‖F‖pp ≍ ∫ F p dA + ∫ F(z)p−2 Δ(F 2 )(z)(1 − |z|) dA(z). 𝔻

𝔻

The proof of Lemma 3.23 shows that it is remains to prove that ∫[Δ(F 2 )(z)]

p/2

p

(1 − |z|) dA(z) ≥ c ∫ F p dA. 𝔻

𝔻

To do this, using Theorem 10.14 applied to ur , by integration in r we get 2q

q

− uq dA − u(0)q ≤ C ∫(1 − |z|) (Δu(z)) dA(z) ∫ 𝔻

(0 < q ≤ 1);

(10.43)

𝔻

see Exercise 10.17. Now taking u = F 2 and q = p/2, we conclude the proof for p < 2. If p > 2, then the function F p−2 Δ(F 2 ) is log-subharmonic, and an application of the maximum principle gives the result.

360 | 10 Littlewood–Paley theory 10.7.1 Hyperbolic Hardy classes p For p > 0, we denote by Hhyp the class of self-mappings φ of 𝔻 for which 2p

hp (φ) := sup ∫ − (κφ (rζ )) |dζ | < ∞, 0 0, and φ ∈ H(𝔻, 𝔻) with φ(0) = 0, then q−1

hp (φ) ≍ ∫(∫ (log 𝕋 Dζ

1 ) 1 − |φ|2

where φ[h] = the hyperbolic derivative of φ.

p/q

󵄨󵄨 [h] 󵄨󵄨2 󵄨󵄨φ 󵄨󵄨 dA)

φ󸀠 , 1 − |φ|2

|dζ |,

(10.45)

10.7 A proof of Calderón’s theorem and generalizations | 361

1 Relation (10.45) becomes clearer if we define κφ = log 1−|φ| 2 and then use formula (10.41). If φ is a univalent function, then the integral

󵄨

𝒜h (φ)(ζ ) := ∫ 󵄨󵄨󵄨φ

dA(z) 󵄨󵄨 dA = ∫ (1 − |z|2 )2

[h] 󵄨󵄨2

Dζ

φ(Dζ )

is the hyperbolic area of φ(Dζ ). Thus, as a particular case of Theorem 10.19, we have an area theorem for the hyperbolic Hardy spaces. Corollary 10.27. If 0 < p < ∞ and φ ∈ H(𝔻, 𝔻) with φ(0) = 0, then p

󵄨2 󵄨 hp (φ) ≍ ∫(∫ 󵄨󵄨󵄨φ[h] 󵄨󵄨󵄨 dA) |dζ |. 𝕋 Dζ

As an application of Theorem 10.19, we present a theorem of Kwon on the compactness of some composition operators. Theorem 10.20. Let φ ∈ H(𝔻, 𝔻) and p ∈ ℝ+ . Then the following conditions are equivalent: (a) 𝒞φ : B 󳨃→ H p is bounded; (b) 𝒞φ : B 󳨃→ H p is compact; and (c) φ belongs to p/2 . the hyperbolic Hardy class Hhyp

Proof. Assume, as we may, that φ(0) = 0. The implication (c) ⇒ (a) is a consequence of, for example, Theorems 10.5 and 10.19 (q = 1). To prove the converse, we use the bi-Bloch lemma to find two Bloch functions f1 and f2 such that 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 2 −1 󵄨󵄨f1 (z)󵄨󵄨 + 󵄨󵄨f2 (z)󵄨󵄨 ≥ (1 − |z| ) . From this we obtain p/2

󵄨 󵄨2 (∫ 󵄨󵄨󵄨(f1 ∘ φ)󸀠 󵄨󵄨󵄨 dA)

Dζ

Dζ

≥ cp (∫ Dζ

p/2

󵄨 󵄨2 + (∫ 󵄨󵄨󵄨(f2 ∘ φ)󸀠 󵄨󵄨󵄨 dA) p/2

|φ󸀠 |2 dA) (1 − |φ|2 )2

,

where Dζ is a Stoltz-type domain. Integrating this inequality in ζ ∈ 𝕋 and then using the boundedness of 𝒞φ together with Theorems 10.5 and 10.19, we conclude that (a) implies (c). It remains to prove that (c) implies (b). Let {fn } ∈ B be a sequence such that ‖fn ‖B ≤ 1 and fn 󴁂󴀱 0. By Theorem 7.17 it suffices to prove that ‖Cφ (fn )‖p → 0. This also follows from the two theorems. Namely, by (c) the function p/2

󵄨 󵄨2 A(ζ ) := (∫ 󵄨󵄨󵄨φ[h] 󵄨󵄨󵄨 dA) Dζ

362 | 10 Littlewood–Paley theory is finite for all ζ ∈ E ⊂ 𝕋 with |E| = |𝕋|. Let ζ ∈ E. Since |(fn ∘ φ)󸀠 | ≤ |φ[h] |, we have, by the dominated convergence theorem, limn hn (ζ ) = 0 for all ζ ∈ E, where p/2

󵄨2 󵄨 hn (ζ ) = (∫ 󵄨󵄨󵄨(fn ∘ φ)󸀠 󵄨󵄨󵄨 dA)

.

Dζ

On the other hand, {hn } has the integrable dominant A(ζ ), ζ ∈ E. So the dominated convergence theorem concludes the proof. Hyperbolic g-theorem From Theorem 10.18 we can deduce various generalizations of the g-theorem. We formulate one of them in the case of hyperbolic Hardy spaces. Theorem 10.21. If p > 0, q ≥ 1, and φ ∈ H(𝔻, 𝔻) with φ(0) = 0, then 1

q−1

1 ) hp (φ) ≍ ∫(∫(log 1 − |φ(rζ )|2 𝕋

0

p/q

󵄨2 󵄨󵄨 [h] 󵄨󵄨φ (rζ )󵄨󵄨󵄨 (1 − r) dr)

|dζ |.

(10.46)

z ∈ 𝔻,

(10.47)

In the case q = 1, this was proved by Kwon [331, Theorem 1].

10.8 Integration operators on Hardy spaces Here we add some information on the Volterra-type operator 1

z

0

0

f (tz)ℛg(tz) (Jg f )(z) = ∫ dt = ∫ f (w)g 󸀠 (w) dw, t

where g ∈ H(𝔻) and, recall, ℛg(z) = g 󸀠 (z)z. Necessary and sufficient conditions for g to be such that Jg maps H p into H q (p, q ∈ ℝ+ ) are known. In particular, Jg maps H p into H p if and only if g ∈ BMOA.

(10.48)

Pommerenke [485] observed that the relations ℛ(Jg f ) = f ℛg

and 󵄨2 󵄨2 󵄨 󵄨 ‖h‖22 ≍ 󵄨󵄨󵄨h(0)󵄨󵄨󵄨 + ∫󵄨󵄨󵄨h󸀠 (z)󵄨󵄨󵄨 (1 − |z|) dA(z) 𝔻

give 󵄨2 󵄨2 󵄨 󵄨 ‖ℛJg f ‖22 ≍ ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(z)󵄨󵄨󵄨 (1 − |z|) dA(z), 𝔻

(10.49)

10.8 Integration operators on Hardy spaces | 363

which via the Carleson measure characterization of BMOA means that (10.48) holds for p = 2. Aleman and Siskakis [32] extended this to the case p = q ≥ 1, whilst the remaining cases were treated by Aleman and Cima [30]. Theorem 10.22. Let p < q, g ∈ H(𝔻), and α = 1/p − 1/q. (i) If α ≤ 1, then Jg maps H p to H q if and only if g ∈ HΛα . (ii) If α > 1, then Jg maps H p to H q if and only if g is constant, that is, Jg = 0. Proof. The necessity in both cases is simple. As in [30] (see also Aleman [27]), assuming that ‖Jg f ‖q ≤ C‖f ‖p , we have −1/q−1 󵄨 󵄨󵄨 , 󵄨󵄨ℛ(Jg f )(z)󵄨󵄨󵄨 ≤ C‖f ‖p (1 − |z|)

whence, by (10.49), −1/q−1 󵄨 󵄨󵄨 󵄨󵄨 . 󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(z)󵄨󵄨󵄨 ≤ C‖f ‖p (1 − |z|)

Now take 1/p

f (z) = (1 − |z|2 )

̄ −2/p . (1 − wz)

Since ‖f ‖p ≤ 1, we have (1 − |z|2 )

󵄨󵄨ℛg(z)󵄨󵄨󵄨 ≤ C(1 − |z|)−1/q−1 , 󵄨 󵄨

−1/p 󵄨

which implies g ∈ HΛα . To prove the sufficiency (in case (i)), using the hypothesis that g ∈ HΛα in the form |ℛg(tz)| ≤ Ct(1 − |tz|)α−1 ≤ Ct(1 − t)α−1 , we get 1

󵄨 󵄨 󵄨 󵄨󵄨 α−1 󵄨󵄨Jg f (z)󵄨󵄨󵄨 ≤ C ∫󵄨󵄨󵄨f (tz)󵄨󵄨󵄨(1 − t) dt. 0

Then using Aleksandrov’s proof of the Hardy–Littlewood–Sobolev theorem 5.8, we finish the proof of Theorem 10.22. Theorem 10.23. Let p ∈ ℝ+ . Then Jg maps H p into H p if and only if g ∈ BMOA. Proof. We first consider the “if” part. Let g ∈ BMO and f ∈ H(𝔻). Let p > 2. As in [418], by the Hardy–Stein identity and (10.49) we have 󵄨2 󵄨2 󵄨 󵄨p−2 󵄨 󵄨 ‖Jg f ‖pp ≍ ∫󵄨󵄨󵄨Jg f (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛf (z)󵄨󵄨󵄨 (1 − |z|) dA(z) 𝔻

whence, by Hölder’s inequality, (p−2)/p

‖Jg f ‖pp ≤ C(∫ |Jg f |p dmg ) 𝔻

2/p

(∫ |f |p dmg ) 𝔻

,

(10.50)

364 | 10 Littlewood–Paley theory where dmg (z) = |ℛg(z)|2 (1 − |z|) dA(z). Now by the Carleson measure characterization of BMO (see Theorem 7.26) we conclude that 2 ‖Jg f ‖pp ≤ C‖g‖2(p−2)/p ‖Jg f ‖pp−2 ‖g‖4/p BMO BMO ‖f ‖p ,

which implies ‖Jg f ‖p ≤ C‖g‖BMO ‖f ‖p for f ∈ H(𝔻), where C is independent of f . In the general case, we apply this case to the functions fr , 0 < r < 1. Let 0 < p < 2. Using the Littlewood–Paley g-theorem 10.6, the radial maximal theorem, Hölder’s inequality, and the Carleson measure characterization of BMO (or Knese’s lemma 7.6), we obtain ‖Jg f ‖pp

p/2

1

󵄨2 󵄨2 󵄨 󵄨 ≍ ∫(∫󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(rζ )󵄨󵄨󵄨 (1 − r) dr) 0

𝕋

|dζ |

1

≤ C ∫(Mrad f (ζ ))

(2−p)p/2

𝕋 1

󵄨2 󵄨p 󵄨 󵄨 (∫󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(rζ )󵄨󵄨󵄨 (1 − r) dr) 0

p/2

|dζ |

p/2

󵄨2 󵄨p 󵄨 󵄨 (∫ ∫󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(rζ )󵄨󵄨󵄨 (1 − r) rdr |dζ |) ≤ C‖f ‖(2−p)p/2 p 𝕋 0

p/2

󵄨2 󵄨p 󵄨 󵄨 (∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(z)󵄨󵄨󵄨 (1 − |z|) dA(z)) = C‖f ‖(2−p)p/2 p 𝔻

≤

2 ‖g‖pBMO ‖f ‖pp /2 C‖f ‖(2−p)p/2 p

= C‖g‖pBMO ‖f ‖pp .

This finishes the proof of the “if” part of the theorem. In proving the “only if” part, we can assume that g and f are in H(𝔻). Then we proceed exactly as before but use the reverse Hölder inequality. For example, assuming that f ∈ H(𝔻) and 0 < p < 2, using (10.50) and the reverse Hölder inequality, we obtain (p−2)/p

‖Jg f ‖pp ≥ c(∫ |Jg f |p dmg )

2/p

(∫ |f |p dmg ) 𝔻

𝔻

2/p

2(p−2)/p (∫ |f |p dmg ) ≥ c‖Jg f ‖p−2 p ‖g‖BMO

,

𝔻

and hence 2/p

‖Jg f ‖2p ≥ c‖g‖2(p−2)/p (∫ |f |p dmg ) BMO

.

𝔻

Now taking f ∈ H p such that ‖f ‖p = 1 and ∫𝔻 |f |p dmg ≥ c‖g‖2BMO , we get ‖Jg ‖ ≥ c‖g‖BMO .

10.8 Integration operators on Hardy spaces | 365

Theorem 10.24. Let 0 < q < p. Then Jg maps H p into H q if and only if g ∈ H ϱ , where 1/ϱ = 1/q − 1/p. Proof [27, 418]. In proving the sufficiency, we use the g-theorem 10.6. We have 1

󵄨2 󵄨2 󵄨 󸀠 󵄨 𝒢 [Jg f ](ζ ) = (∫(1 − r)󵄨󵄨󵄨g(rζ )󵄨󵄨󵄨 󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 dr)

1/2

≤ Mrad g(ζ )𝒢 [f ](ζ ).

0

Hence, by Hölder’s inequality, the maximal theorem, and the g-theorem, we get 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝒢 [Jg f ]󵄩󵄩󵄩q ≤ ‖Mrad g‖ϱ 󵄩󵄩󵄩𝒢 [f ]󵄩󵄩󵄩p ≤ C‖f ‖p ‖g‖ϱ , as desired. In proving the necessity, the key is the following fact. (A) Let p > q > 0 be such that Jg maps H p into H q . Then Jg (H p1 ) ⊂ H q1 whenever p > p1 > q1 > 0 and 1 1 1 1 − . − = q p q1 p1 Moreover, ‖Jg ‖H p1 ,H q1 ≤ C‖Jg ‖H p ,H q . Before proving (A) (from [27]), we continue the proof of the theorem exactly as in [418]. As usual, we may suppose that g ∈ H(𝔻). Consider first the case where ϱ = mp for some positive integer m. Then g ∈ H ϱ if and only if g m ∈ H p , and since g m+1 = (m + 1)Jg (g m ), with the notation fm = g m , the Hardy–Stein identities, together with (10.49), give 󵄨2 󵄨ϱ−2 󵄨 󵄨 ϱ ‖g‖H ϱ ≍ ∫󵄨󵄨󵄨g(z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(z)󵄨󵄨󵄨 (1 − |z|) dA(z) 𝔻

󵄨2 󵄨2 󵄨 󵄨mp−2−2m 󵄨󵄨 󵄨 = ∫󵄨󵄨󵄨g(z)󵄨󵄨󵄨 󵄨󵄨fm (z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛg(z)󵄨󵄨󵄨 (1 − |z|) dA(z) 𝔻

󵄨 󵄨 = C ∫󵄨󵄨󵄨Jg fm (z)󵄨󵄨󵄨

mp−2−2m m+1

󵄨2 󵄨󵄨 󵄨󵄨ℛ(Jg fm )(z)󵄨󵄨󵄨 (1 − |z|) dA(z).

𝔻

Since mp mp − 2 − 2m = − 2 = q − 2, m+1 m+1 another use of the Hardy–Stein identities yields ϱ

ϱq/p

‖g‖H ϱ ≍ ‖Jg fm ‖qH q ≤ ‖Jg ‖qH p ,H q ⋅ ‖fm ‖qH p = ‖Jg ‖qH p ,H q ⋅ ‖g‖H ϱ . Since ϱ − ϱq/p = q, this implies the desired inequality ‖g‖H ϱ ≤ C‖Jg ‖H p ,H q .

366 | 10 Littlewood–Paley theory Let 0 < q < p < ∞ be arbitrary and take a positive integer m with p1 := ϱ/m < p. Then, by statement (A) and the case considered before, we get ‖g‖H ϱ ≤ C‖Jg ‖H p1 ,H q1 ≤ C‖Jg ‖H p ,H q . It remains to prove (A). Define s > 0 by 1/s + 1/p = 1/p1 . Then by the inner– outer factorization, if f ∈ H p1 , then we can write f = uv, where u ∈ H s , v ∈ H p , and ‖u‖s ‖v‖p = ‖f ‖p1 ; see Exercise 2.19. Since Jg (uv) = JJg v u and v ∈ H p , we have, by hypothesis, that Jg v ∈ H q . Hence, by the sufficiency of the theorem, Jg f = JJg v u ∈ H q1 because 1 1 1 1 1 1 1 − = − + − = , q1 s q1 p1 p1 s q that is, 1/s = 1/q1 − 1/q. The proof that ‖Jg ‖H p1 ,H q1 ≤ C‖Jg ‖H p ,H q is left to the reader. This completes the proof of the theorem. 10.8.1 A generalized integration operator Here we define a generalized version of Jg , namely Jgβ f (z)

1

β−1

1 1 = ℛ (f ℛ g(z)) = ∫(log ) Γ(β) t −β

β

0

f (tz)ℛβ g(tz) dt. t

(10.51)

We generalize Pommerenke’s theorem. β

Theorem 10.25. If g ∈ H(𝔻) and β > 0, then Jg maps H 2 into H 2 if and only if g ∈ BMOA. Proof. We start from the relation 2β−1

1 󵄨2 󵄨 ‖h‖22 ≍ ∫󵄨󵄨󵄨ℛβ h(z)󵄨󵄨󵄨 (log ) |z|

dA(z)

𝔻

2β−1 󵄨2 󵄨 ≍ ∫󵄨󵄨󵄨ℛβ h(z)󵄨󵄨󵄨 (1 − |z|) dA(z),

(10.52)

𝔻 β

β

where h ∈ h(𝔻) with h(0) = 0. Taking h = Jg f and using the formula ℛβ (Jg f ) = f ℛβ g, we obtain 2β−1

1 󵄩󵄩 β 󵄩󵄩 󵄨2 󵄨2 󵄨 β 󵄨 󵄩󵄩Jg f 󵄩󵄩2 ≍ ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛ g(z)󵄨󵄨󵄨 (log ) |z|

dA(z).

(10.53)

𝔻

Hence, by the Carleson measure characterization of BMOA (see Theorem 7.26) we see that our theorem is a consequence of the following:

10.8 Integration operators on Hardy spaces | 367

Theorem 10.26. Let β > 0 and g ∈ H(𝔻). Then g ∈ BMOA if and only if the measure 2β−1

1 󵄨2 󵄨 dmg,β (z) := 󵄨󵄨󵄨ℛβ g(z)󵄨󵄨󵄨 (log ) |z|

dA(z),

z ∈ 𝔻,

is a Carleson measure. The same holds if we replace ℛβ with 𝒥 β . Proof. We first prove: (A) If dmg,β is Carleson, then so is dmg,β+k for every k ∈ ℕ+ . To prove this, consider a slightly modified Carleson window: Wε (ζ ) = {z ∈ 𝔻: |z − ζ | < ε}, ζ ∈ 𝕋, 0 < ε < 1. A measure μ is Carleson if and only if μ(Wε (ζ )) ≤ Cε, where C is independent of ζ and ε. As before, we let Hε (z) = {w ∈ 𝔻: |σz (w)| < ε}, ε < 1. If h ∈ H(𝔻), then −2k−2 󵄨2 󵄨 󵄨2 󵄨󵄨 k ∫ 󵄨󵄨󵄨h(w)󵄨󵄨󵄨 dA(w); 󵄨󵄨𝒥 h(z)󵄨󵄨󵄨 ≤ C(1 − |z|) Hε (z)

see (10.18). Now taking h = 𝒥 β g, we obtain −2k−2 󵄨2 󵄨 󵄨2 󵄨󵄨 k+β ∫ 󵄨󵄨󵄨𝒥 β g(w)󵄨󵄨󵄨 dA(w). 󵄨󵄨𝒥 g(z)󵄨󵄨󵄨 ≤ C(1 − |z|) Hε (z)

Then choose a constant c > 1 such that Hε (z) ⊂ Wcε (ζ ) whenever z ∈ Wε (ζ ). Since 1 − |z| ≍ 1 − |w| for w ∈ Hε (z), we have 2β+2k−1 󵄨2 󵄨 dA(z) ∫ dmg,β+k ≤ C ∫ 󵄨󵄨󵄨𝒥 β+k g(z)󵄨󵄨󵄨 (1 − |z|)

Wε (ζ )

Wε (ζ )

≤C ∫

2β−3 󵄨2 󵄨 dA(w) dA(z) ∫ 󵄨󵄨󵄨𝒥 β g(w)󵄨󵄨󵄨 (1 − |w|)

Wε (ζ ) Hε (z)

2β−3 󵄨2 󵄨 dA(w) ∫ K(z, w) dA(z), ≤ C ∫ 󵄨󵄨󵄨𝒥 β g(w)󵄨󵄨󵄨 (1 − |w|) Wε (ζ )

Wε (cζ )

where z 󳨃→ K(z, w) is the characteristic function of Hε (w) (and w 󳨃→ K(z, w) is the characteristic function of Hε (z)). Since the absolute value of the inner integral is less than (or equal to) |Hε (w)| and |Hε (w)| ≍ (1 − |w|)2 , we see that 2β−1 󵄨2 󵄨 mg,β+k (Wε (ζ )) ≤ C ∫ 󵄨󵄨󵄨𝒥 β g(w)󵄨󵄨󵄨 (1 − |w|) dA(w) ≤ Cε. Wcε (ζ )

This finishes the proof of (A).

368 | 10 Littlewood–Paley theory It follows from (A) that it remains to prove that if dmg,β is Carleson for some 0 < β < 1, then so is for β = 1. We start from the inequality (1 − |w|2 )2−β 󵄨󵄨 1 󵄨 󵄨󵄨 β 󵄨 󵄨󵄨𝒥 g(z)󵄨󵄨󵄨 ≤ C ∫ 󵄨𝒥 g(w)󵄨󵄨󵄨 dA(w). ̄ 3 󵄨 |1 − wz|

(10.54)

𝔻

This can be deduced from the formula 𝒥

β−1

g(z) = cβ ∫ 𝔻

(log(1/|w|))2−β 1 𝒥 g(w) dA(w) ̄ 2 (1 − wz)

by an elementary but tedious computation (see also Exercise 10.33). By the Cauchy– Schwarz inequality from (10.54) we get |𝒥 1 g(w)|2 (1 − |w|)2−β (1 − |w|)2−β 󵄨2 󵄨󵄨 β dA(w) dA(w) ∫ 󵄨󵄨𝒥 g(z)󵄨󵄨󵄨 ≤ C ∫ ̄ 4 ̄ 2 |1 − wz| |1 − wz| 𝔻

≤ C(1 − |z|)

−β

∫ 𝔻

1

𝔻 2−β

2

|𝒥 g(w)| (1 − |w|) ̄ 2 |1 − wz|

dA(w).

Hence 2β−1 β−1 󵄨2 󵄨 dA(z) ≤ C ∫ (1 − |z|) (I1 (z) + I2 (z)) dA(z), ∫ 󵄨󵄨󵄨𝒥 β g(z)󵄨󵄨󵄨 (1 − |z|)

Wε (ζ )

Wε (ζ )

where I1 (z) = ∫ W2ε (ζ )

I2 (z) =

|𝒥 1 g(w)|2 (1 − |w|)2−β dA(w), ̄ 2 |1 − wz|

∫ 𝔻\W2ε (ζ )

|𝒥 1 g(w)|2 (1 − |w|)2−β dA(w). ̄ 2 |1 − wz|

By Fubini’s theorem, β−1 󵄨2 󵄨 ∫ 󵄨󵄨󵄨𝒥 β g(z)󵄨󵄨󵄨 (1 − |z|) I1 (z) dA(z)

Wε (ζ )

β−1

(1 − |z|) 2−β 󵄨2 󵄨 dA(z) ≤ ∫ 󵄨󵄨󵄨𝒥 1 g(w)󵄨󵄨󵄨 (1 − |w|) dA(w) ∫ ̄ 2 |1 − wz| W2ε (ζ )

𝔻

󵄨2

󵄨 ≤ C ∫ 󵄨󵄨󵄨𝒥 1 g(w)󵄨󵄨󵄨 (1 − |w|) W2ε (ζ )

2−β

β−1

(1 − |w|)

dA(w) ≤ Cε.

10.8 Integration operators on Hardy spaces | 369

Here we have used the inequality ∫ 𝔻

(1 − |z|)a a−b+2 dA(z) ≤ C(1 − |w|) , ̄ b |1 − wz|

w ∈ 𝔻,

(10.55)

valid under the conditions a > −1 and a − b + 2 < 0. On the other hand, if w ∈ 𝔻 \ W2ε (ζ ) and z ∈ Wε (ζ ), then ̄ | − |ζ − z| ≥ 2ε − ε = ε. ̄ + w(ζ ̄ − z)󵄨󵄨󵄨󵄨 ≥ |1 − wζ ̄ = 󵄨󵄨󵄨󵄨1 − wζ |1 − wz| Now choose s such that 0 < s < β. It follows that I2 (z) ≤ C

∫ 𝔻\W2ε (ζ )

≤ Cε−s ∫ 𝔻

̄ 1−β |𝒥 1 g(w)|2 (1 − |w|)|1 − wz| dA(w) s+(2−s) ̄ |1 − wz|

|𝒥 1 g(w)|2 (1 − |w|) dA(z). ̄ 1+β−s |1 − wz|

The last integral is ≤∫ 𝕋

|dη| s−β ≤ C(1 − |z|) ̄ 1+β−s |1 − ηz|

because dm1,g is a Carleson measure. Hence ∫ (1 − |z|)

β−1

I2 (z) dA(z) ≤ Cε−s ∫ (1 − |z|)

s−1

dA(z) ≤ Cε−s εs+1 ,

Wε (ζ )

Wε (ζ )

which completes the proof of the theorem. β

Problem 10.12. We conjecture that Jg maps H p into H p if and only if g ∈ BMOA. Using Theorem 10.26 and the generalized g-theorem (Theorem 10.9), we can prove that if β p < 2 and g ∈ BMOA, then Jg maps H p into H p . On the other hand, if 0 < p < 2 and β

Jg maps H p into H p , then g ∈ BMOA. The answer to the conjecture is affirmative if the following relation holds: 2β−1 󵄨2 󵄨p−2 󵄨 󵄨 󵄨p 󵄨 dA(z). ‖f ‖pp ≍ 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨󵄨ℛβ f (z)󵄨󵄨󵄨 (1 − |z|)

(?)

𝔻 β

Exercise 10.28. Let α = 1/p − 1/q > 0. If α = β, then Jg maps H p into H q if and only if β

g belongs to the Hardy–Sobolev space HSα∞ . If α < β, then Jg maps H p into H q if and only if g belongs to the Besov space B∞,∞ . α

370 | 10 Littlewood–Paley theory

10.9 Addendum: Tents and Triebel–Lizorkin spaces In this section, we suppose that 0 < p, q ≤ ∞ but p ≠ ∞ or q ≠ ∞. The tent spaces were introduced by Coifman, Meyer, and Stein [131] in the case of a half-space. Many authors consider these spaces without explaining what is a tent. In [131] and [130], it is said that the tent T(E) over a set E ⊂ 𝕋 is the complement of the set ⋃ζ ∈E̸ Uζ ,c . It is then clear that if I ⊂ 𝕋 is an arc such that |I| ≤ δ(c), then T(I) is a triangle “comparable” with the Carleson window W(I). For small |I|, this agrees with the Peláez’s definition of T(I) [459]. It is interesting that Cohn and Verbitsky [130] define a Carleson measure by ‖μ‖𝒞ℳ := sup I⊂𝕋

μ(T(I)) < ∞. |I|

Tents appear in a natural way in the proof of the Carleson measure theorem: see Garnett [198, p. 31]. In what follows, we write Uζ = Uζ ,c although proving that the definition of tent spaces is independent of a particular choice of c is very difficult; see [131]. In [130, 522]3 the tent space Tqp on the unit disc is defined in the following way. Let 󵄨q

󵄨

𝒜q f (ζ ) = (∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 dτ(z))

1/q

,

q < ∞,

Uζ

where dτ(z) = (1 − |z|2 )−2 dA(z), the Möbius invariant measure on 𝔻. The space Tqp , p < ∞, q < ∞, consists of measurable functions f on 𝔻 such that 𝒜q f ∈ Lp (𝕋). By Theorem 10.10 a function f ∈ H(𝔻) belongs to H p (p < ∞) if and only if the function 𝒥 β f (z)(1 − |z|)β belongs to T2p for some (for all) β > 0. But the space Tq∞ is not defined by “𝒜q f ∈ L∞ (𝕋)”, which surprised the author when he encountered tent spaces for the first time. It is defined as the space of measurable functions f with ‖f ‖Tq∞ = (sup I⊂𝕋

|f (z)|q 1 dA(z)) ∫ |I| 1 − |z|

1/q

< ∞,

T(I)

where T(I) is the tent over the arc I. In other words, f ∈ Tq∞ if and only if dμ(z) :=

|f (z)|q dA(z) 1 − |z|

is a Carleson measure on 𝔻. The norm on this space is comparable with ‖μ‖1/q 𝒞M , where ‖μ‖𝒞M is the Carleson norm of μ. 3 The Rydhe’s paper [522] is entirely devoted to the unit disc.

10.9 Addendum: Tents and Triebel–Lizorkin spaces | 371

p In [131] and [130], T∞ (p < ∞) is defined as the space of functions that are continuous in 𝔻, have nontangential limits a. e. on 𝕋, and satisfy ‖f ‖T∞p = ‖M∗ f ‖Lp (𝕋) < ∞, where M∗ is the non-tangential maximal function associated with Uζ . Some authors p (e. g., [362, 364, 459, 470]) define T∞ as the space of measurable functions f with

󵄨p 󵄨 ‖f ‖pT p = ∫(ess sup󵄨󵄨󵄨f (z)󵄨󵄨󵄨) |dζ | < ∞. ∞

z∈Uζ

𝕋

1 An advantage of the former definition is in that the dual of T∞ is equal to the space of Carleson measures [131, Proposition 1], and this duality leads to an atomic 1 decomposition of T∞ [131, Proposition 2], which was used by Luecking in proving the following magical statement [364, Proposition 1].

Proposition 10.29. If s > 0 and λ > max{1, 1/s}, then λ

∫(∫( 𝕋 𝔻

s

1 − |z| s ) dν(z)) |dζ | ≍ ∫(ν(Uζ )) |dζ | |z − ζ | 𝕋

for every positive locally finite measure ν on 𝔻, where the equivalence constants are independent of ν. Luecking attributes the case s > 1 to Harboure et al. [227]. As a matter of fact, Luecking worked in a half-space. This form of Proposition appears in [47, Lemma 4], where the same idea of the proof is used.4 A generalization to weighted Bergman spaces was proved in [462, Lemma 4]. Factorization of tent spaces p In [130], it was proved that Tqp = T∞ ⋅ Tq∞ for p ∈ ℝ+ and q ∈ ℝ+ . This means: (1) If p ∞ p f ∈ T∞ and g ∈ Tq , then fg ∈ Tq and ‖fg‖Tqp ≤ C‖f ‖T∞p ‖g‖Tq∞ , where C is independent of f and g; and (2) conversely, any function h ∈ Tqp can be factored as h = fg, where ‖f ‖T∞p ≤ C‖h‖Tqp and ‖g‖Tq∞ ≤ 1. There are other theorems of this type (“strong factorization theorems”), which includes Hardy, Hardy–Sobolev and Triebel–Lizorkin spaces. One of them reads: −1 −1 Theorem. Let α > 0, 0 < p, p1 , p2 < ∞, and p−1 1 + p2 = p . Then α

p

𝒥 H =H

p1

⋅ 𝒥 α H p2 .

Here 𝒥 α X = {f ∈ H(𝔻): 𝒥 −α f ∈ X}. As an application of factorization theorems, the authors gave a relatively simple proof of the following embedding theorem. 4 But without mentioning Luecking’s name.

372 | 10 Littlewood–Paley theory Theorem. Let 0 < p < 2, α > 0, and let μ be a positive Borel measure on 𝔻. Let F(z) = μ(Dz )/(1 − |z|)1+αp , where Dz is the disc of radius (1 − |z|)/2 centered at z. Then the inequality 󵄨p 󵄨 ∫󵄨󵄨󵄨𝒥 α f (z)󵄨󵄨󵄨 dμ(z) ≤ C‖f ‖pH p 𝔻 p

∞ holds for every f ∈ H if and only if F ∈ T2/(2−p) .

In the case where s is an integer, this theorem was proved by Luecking [364]. Tribel–Lizorkin spaces p The Tribel–Lizorkin space Fq,α (𝔻) is defined as the class of functions f ∈ H(𝔻) such s−α s that the function (1 − |z|) 𝒥 f (z) belongs to Tqp . Here α ∈ ℝ, s > α, 0 < p < ∞, and p 0 < q ≤ ∞. For p < ∞, we have F2,0 = H p. p “The definition [of Fq,α ] is independent of a particular choice of s > α”; this is stated without a proof in [130]. In the case q ≥ 1, a proof is given by Rydhe [522], who α,τ considers a more general class Fp,q . p Oswald [412] considered the (Triebel–Lizorkin) class Fq,α (𝕋), p ∈ ℝ+ , 0 < q ≤ ∞, α ∈ ℝ, defined by the requirement that the function 1

󵄨q 󵄨 h(ζ ) = (∫󵄨󵄨󵄨𝒥 s f (rζ )󵄨󵄨󵄨 (1 − r)q(s−α)−1 dr)

1/q

ζ ∈𝕋

,

0

(s > α), belongs to Lp (𝕋) and proved that the definition is independent of s. Oswald p (𝔻). Our modified approach can be used to prove the did not discussed the spaces Fq,α following (see Remark 10.13): p p (𝕋). Moreover, (𝔻) ≃ Fq,α Theorem 10.27. Let p ∈ ℝ+ , 0 < q ≤ ∞, and α ∈ ℝ. Then Fq,α p an equivalent (quasi)norm on Fq,α can be given by p/q

∞

󵄨q 󵄨 (∫( ∑ (2nα 󵄨󵄨󵄨Vn ∗ f (ζ )󵄨󵄨󵄨) ) 𝕋

1/p

|dζ |)

n=0

.

(10.56)

If q = ∞, then (10.56) should be read 1/p

󵄨 p 󵄨 (∫(sup 2nα 󵄨󵄨󵄨Vn ∗ f (ζ )󵄨󵄨󵄨) |dζ |) 𝕋

.

n≥0

Proof. In this situation, we have nothing with Hardy spaces. Assume that g is a logsubharmonic function and temporarily define 1

βq−1

gq (ζ ) = (∫(1 − r) 0

q

1/q

g(rζ ) dr)

,

10.9 Addendum: Tents and Triebel–Lizorkin spaces | 373

∞

gq,d (ζ ) = ( ∑ 2

−βqn

n=0

q

1/q

g(rn ζ ) )

ζ ∈ 𝕋,

,

where rn = 1 − 2−n and 0 < q < ∞. If q = ∞, then replace the inner integral and the sum with sup (1 − r)β g(rζ ) and

sup 2−βn g(rn ζ ),

0 0 if and only if so is for all β > 0. Proof. As in the case of Theorem 10.26, it suffices to prove two statements: (A) If dμβ is Carleson, then so is dμβ+k for every integer k > 0. (B) If dμ1 is Carleson, then so is dμβ for 0 < β < 1. The proof of (A) is the same as the proof of the corresponding statement in the proof of Theorem 10.26. In proving (B), we first consider the case q ≤ 1. We start from the inequalities (∫(1 − |w|)

q

2/q−2+δ/q 󵄨

󵄨󵄨g(w)󵄨󵄨󵄨 dA(w)) ≤ C ∫(1 − |w|)δ 󵄨󵄨󵄨g(w)󵄨󵄨󵄨q dA(w), 󵄨 󵄨 󵄨 󵄨

𝔻

(10.60)

𝔻

where δ > −1, and 2/q−1−β

(1 − |w|) 󵄨 󵄨󵄨 β 󵄨󵄨𝒥 f (z)󵄨󵄨󵄨 ≤ C ∫ ̄ 2/q |1 − wz|

󵄨 󵄨󵄨 1 󵄨󵄨𝒥 f (w)󵄨󵄨󵄨 dA(w).

(10.61)

𝔻

Hence (1−β)q

󵄨q 󵄨󵄨 β 󵄨q (1 − |w|) 󵄨 1 󵄨󵄨𝒥 f (z)󵄨󵄨󵄨 ≤ C ∫󵄨󵄨󵄨𝒥 f (w)󵄨󵄨󵄨 ̄ 2 |1 − wz|

dA(w).

𝔻

Then considering two integrals just as in the proof of Theorem 10.26, we obtain the desired conclusion. (Inequality (10.55) is to be used, and s is to be chosen so that 0 < s < βq.)

376 | 10 Littlewood–Paley theory Let q > 1. We start from the inequality 2q−1

(1 − |w|) 󵄨 󵄨󵄨 β 󵄨 󵄨󵄨 1 󵄨󵄨𝒥 f (z)󵄨󵄨󵄨 ≤ C ∫ 󵄨𝒥 f (w)󵄨󵄨󵄨 dA(w) ̄ 2q+β 󵄨 |1 − wz| 𝔻

= C∫ 𝔻

(1 − |w|)2q−1 󵄨󵄨 1 󵄨 󵄨𝒥 f (w)󵄨󵄨󵄨 dA(w). ̄ (2q+β+1)−1 󵄨 |1 − wz|

Applying Jensen’s inequality to the measure dν(w) =

(1 − |w|)2q−1 dA(w), |1 − w|̄ 2q+1+β

we get (1 − |w|)2q−1 |𝒥 1 f (w)|q 󵄨q 󵄨󵄨 β dA(w) 󵄨󵄨𝒥 f (z)󵄨󵄨󵄨 ≤ C ∫ ̄ (2q+1+β)−q |1 − wz| 𝔻

× (∫ 𝔻

(1 − |w|)2q−1 dA(w)) ̄ 2q+1+β |1 − wz|

≤ C((1 − |z|) )

−β (q−1)

∫ 𝔻

q−1

(1 − |w|)2q−1 |𝒥 1 f (w)|q dA(w). ̄ q+1+β |1 − wz|

The rest of the proof is left to the reader. Exercise 10.32. Inequality (10.60) is a simple consequence of the inequality −2−δ δ󵄨 󵄨q 󵄨󵄨 󵄨q ∫ (1 − |w|) 󵄨󵄨󵄨g(w)󵄨󵄨󵄨 dA(w), 󵄨󵄨g(z)󵄨󵄨󵄨 ≤ C(1 − |z|)

f ∈ H(𝔻),

Eε (z)

where Eε (a) = {w: |w − a| < ε(1 − |a|)}, 0 < ε < 1. Exercise 10.33. The inequality (1 − |w|2 )s−1 󵄨󵄨 1 󵄨 󵄨 󵄨󵄨 β 󵄨𝒥 f (w)󵄨󵄨󵄨 dA(w), 󵄨󵄨𝒥 f (z)󵄨󵄨󵄨 ≤ C ∫ ̄ s+β 󵄨 |1 − wz|

0 < β < 1, s > 0,

𝔻

seems rather complicated, but its proof is not so difficult. Namely, we have 1

𝒥 f (z) = cs ∫ 𝔻

(1 − |w|2 )s−1 1 𝒥 f (w) dA(w), ̄ s+1 (1 − wz)

which is the standard Bergman reproducing formula applied to 𝒥 1 f ; see Section 3.6. Hence 1

1 1 𝒥 f (z) = 𝒥1−β 𝒥 f (z) = ∫(log ) 𝒥 1 f (tz) dt, Γ(1 − β) t β

1

0

−β

Further notes and results | 377

and therefore it remains to prove that 1

̄ −s−1 dt ≤ C|1 − wz| ̄ −s−β . ∫(1 − t)−β |1 − t wz| 0

The main step is the inequality |1 − ta|−1 ≤ 2|1 − a|−1 , a ∈ 𝔻, 0 < t < 1. (See [424, Lemmas 1 and 2].) ∞ Exercise 10.34. If α ∈ ℝ and 0 < q < ∞, then B∞,q ⊂ Fq,α . This extends the result α stated in Exercise 10.30.

Further notes and results The material of the first four sections is based on the author’s paper [449]. Littlewood and Paley proved the implications (a) ⇔ (b) (p > 1) of Theorem 10.6 (see [358, Theorem 7] and, for the case p > 0, [639, Ch. XIV (3.5)]) and (b) ⇒ (a) (p > 1) (see [358, Theorem 7] and [639, Ch. XIV (3.19)]). The equivalence (a) ⇔ (c) ⇔ (d) is, maybe, new. As noted in [412] and [9], the case p > 0 can be treated by the methods of Fefferman and Stein’s paper [187].5 The validity of Theorem 10.3 were noted by Oswald [412]. In the case p > 1, Theorem 10.5 is due to Marcinkiewicz and Zygmund [375]. The equivalence f ∈ H p ⇔ Q2 (f ) < ∞ ⇔ Q∗2 (f ) < ∞ of Theorem 10.9 was proved by Oswald [412, pp. 417–421]; our proof is somewhat simpler in technical details. The validity of Corollary 10.8 was conjectured by MacGregor and Sterner [369], who proved that this statement does not hold in H ∞ . The hyperbolic Hardy classes were introduced by Yamashita [611, 613]. The importance of these classes and the “g-theorems” in the study of composition operators from the Bloch space to H p was recognized by Kwon [331, 333]. Theorem 10.20 was proved by Kwon [331, 332] and subsequently by Pérez-González and Xiao [476]. Theorem 10.26 is stated in Jevtić’s paper [272], but the author does not understand the last step of his proof. 10.1. It may be interesting to deduce some other classical, relatively simple inequalities from the equivalence (a) ⇔ (d) of Theorem 10.6. Proof of Theorem 6.13. Let p ≥ 2. We use the inequality (x + y)p/2 ≥ xp/2 + yp/2 (x ≥ 0, y ≥ 0) to conclude that if f ∈ H p , then ‖f ‖pp ≥

cp

2π

∞

󵄨p 󵄨 ∫ ∑ 2−np 󵄨󵄨󵄨f 󸀠 (rn ζ )󵄨󵄨󵄨 |dζ |

𝕋 n=0

5 Flett [190] proved this inequality in the case where f has no zeros in 𝔻.

378 | 10 Littlewood–Paley theory

∞

= cp ∑ 2

−np

n=0

Mpp (rn , f 󸀠 )

1

≥ cp ∫(1 − r)p−1 Mpp (r, f 󸀠 ) dr, 0

as claimed. The proof of (b) is similar: use the inequality (x + y)p/2 ≤ xp/2 + yp/2 (x, y ≥ 0). Proof of Theorem 2.21 (weakened variant). In the case p ≥ 2, using the Minkowski inequality in the normed space Lp/2 , we get ‖f ‖2p

∞

≤ Cp (∫( ∑ 2 𝕋

n=0

p/2

󵄨2 󵄨󵄨f (rn ζ )󵄨󵄨󵄨 )

−2n 󵄨󵄨 󸀠

2/p

|dζ |)

∞

󵄨2 p/2 󵄨 ≤ Cp ∑ (∫(2−2n 󵄨󵄨󵄨f 󸀠 (rn ζ )󵄨󵄨󵄨 ) |dζ |)

2/p

n=0 𝕋 ∞

1

n=0

0

= Cp ∑ 2−2n Mp2 (rn , f 󸀠 ) ≤ Cp ∫(1 − r)Mp2 (r, f 󸀠 ) dr. In the case p < 2 the proof is similar: we use the reverse Minkowski inequality, which is valid because the summands are nonnegative. 10.2. Concerning the case p > 1 of Theorem 10.8, it should be noted that Littlewood and Paley proved a theorem that is deeper than the equivalence f ∈ H p ⇔ Q1 (f ) < ∞ (p > 1). Theorem (LP). Let λ = {λn }∞ 0 (λ0 > 0) be a lacunary sequence of integers, and let λ0 −1

Δ0,λ f = ∑ f ̂(j)z j , j=0

λn −1

Δn,λ f = ∑ f ̂(j)z j . j=λn−1

Then f ∈ H p (p > 1) if and only if p/2

∞

󵄨2 󵄨 ∫( ∑ 󵄨󵄨󵄨Δn,λ f (ζ )󵄨󵄨󵄨 ) 𝕋

n=0

|dζ | < ∞.

For a proof, see [639, Ch. XIV (4.24)]. As an application we have the following: Theorem (LP1). We have ∞

C (∑ −1

n=0

‖Δn,λ f ‖pp )

1/p

∞

≤ ‖f ‖p ≤ C( ∑

The reverse inequalities hold for 1 < p ≤ 2.

n=0

1/2 2 ‖Δn,λ f ‖p ) ,

p ≥ 2.

Further notes and results | 379

10.3. The following result from [149] shows that Flett’s inequality is rather elementary. Theorem (A). Let u be a regularly oscillating (real-valued) function on 𝔻, u ∈ OCK1 . Then we have the inequality p−1 󵄨p 󵄨 󵄨p 󵄨 󵄨p 󵄨 − sup 󵄨󵄨󵄨u(tζ )󵄨󵄨󵄨 |dζ | ≤ 󵄨󵄨󵄨u(0)󵄨󵄨󵄨 + C ∫󵄨󵄨󵄨∇u(z)󵄨󵄨󵄨 (1 − |z|) dA(z), ∫ 𝕋

0 0. If Ip (f ) = sup ∫ f (ry)p dσ(y) < ∞ 0 1, we have ∫𝜕B Sαp f (y, f ) dσ(y) ≤ Cα Ip (f ), where Cα is independent of f . Here Sα (y, f ) = ∫ (1 − |x|) Γα (y)

and Γα (y) = {x ∈ B: |x − y| ≤ α(1 − |x|)}.

2−n

Δ(f 2 )(x) dx,

Further notes and results | 383

Stoll proved that the converse holds for p ≥ 2, whereas the case 1 < p < 2 is an open problem. In his paper the reader will find several other results concerning variations of the g-function, for example, gλ∗ (y, f ) = ∫(1 − |x|)Δ(f 2 )(x) B

(1 − |x|)(λ−1)(n−1) dx, |x − y|λ(n−1)

y ∈ 𝜕B.

11 One-to-one mappings In this chapter, we prove some fundamental theorems of the theory of univalent functions due to Prawitz, Bieberbach, and Köbe, with applications to the problem of membership of univalent functions to some function classes. In Section 11.2, we are concerned with statements of the type “if f is univalent, then f ∈ X if and only if f ∈ Y”, where X is and Y are different classes of analytic functions. One of most interesting cases is where X = H p and Y = Bp,p , p > 0—a recent paper of Astala and Koskela enables a proof considerably shorter and clearer that the original one. We also give a proof of a theorem of Holland, Twomey, and Spencer. Section 11.4 contains a characterization of the boundary behavior of a harmonic quasiconformal homeomorphism of the unit disc. In the last section, we consider H p -theory of quasiconformal harmonic mappings. We define the class UHk , 0 < k < 1, of those h ∈ H(𝔻) for which there is g ∈ H(𝔻) such that f = h + ḡ is quasiconformal. It turns out that if f (𝔻) is convex in one direction, for example, horizontal, then this class possesses almost all properties as the class 𝒰 of all univalent functions. The Lusin area theorem plays an important role. In this chapter, we do not use the symbol ⬦.

11.1 Integral means of univalent functions Recall that we denote by 𝒰 the set of all one-to-one analytic functions from 𝔻 into ℂ. Theorem 11.1 (Prawitz [489]). Let f ∈ 𝒰 with f (0) = 0. Then for every p > 0, the function 󵄨−p 󵄨 Jp (r) = Jp (r, f ) = ∫ − 󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 |dζ |,

0 < r < 1,

𝕋

is decreasing. Here it is interesting that the function u = |f |−p is subharmonic in the annulus 𝔻 \ {0} but not in 𝔻, because u(0) = +∞. Also, the function −u is not subharmonic in 𝔻, and therefore we cannot apply Theorem 2.3. Proof. We have 󵄨−p−2 󵄨 2πJp󸀠 (r) = −p ∫󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 Re{f (rζ )f 󸀠 (rζ )ζ }|dζ | 𝕋

󵄨−p−2 󵄨 f (ζ )f 󸀠 (ζ ) dζ = −(p/r) Im ∫ 󵄨󵄨󵄨f (ζ )󵄨󵄨󵄨 |ζ |=r

= −(p/r) Im ∫ |w|−p−2 w̄ dw, Γr https://doi.org/10.1515/9783110630855-011

386 | 11 One-to-one mappings where Γr is the image under f of the circle |ζ | = r; the curve Γr is oriented positively. Now we apply Green’s formula (1.3) to the domain Ωr,R bounded by Γr and the circle |w| = R, where R > max|z|=r |f (z)|. Since p 𝜕 ̄ = − |w|−p−2 , (|w|−p−2 w) 𝜕w̄ 2 we have Im ∫ |w|−p−2 w̄ dw − Im ∫ |w|−p−2 w̄ dw = −p ∬ |w|−p−2 dA(w). Γr

|w|=R

Ωr,R

The first integral is equal to 2πR−p , and therefore Jp󸀠 (r) = −(p/r)R−p − (p2 /2πr) ∬ |w|−p−2 dA(w). Ωr,R

Letting R tend to ∞, we get Jp󸀠 (r) = −(p2 /2πr) ∬ |w|−p−2 dA(w), Ωr

where Ωr is the “exterior” of the curve Γr . This concludes the proof. Exercise 11.1. If f is univalent in 𝔻 with f (0) = 0, then p p M(r)−p ≤ −Jp󸀠 (r) ≤ m(r)−p , r r where M(r) = max|z|=r |f (z)| and m(r) = min|z|=r |f (z)|. 11.1.1 Distortion theorems Theorem 11.2 (Bieberbach). If f ∈ 𝒰 , then |f 󸀠󸀠 (0)| ≤ 4|f 󸀠 (0)|. Proof [265]. We can assume that f (0) = 0 and f 󸀠 (0) = 1. Then 2

f (z)−1 = (f (z)−1/2 )

2

= z −1 (1 − f 󸀠󸀠 (0)z/4 + z 2 h(z)) , where h is analytic in 𝔻. By Theorem 11.1, case p = 1, the function 󵄨2 󵄨 J1 (r, f ) = r −1 (1 + 󵄨󵄨󵄨f 󸀠󸀠 (0)/4󵄨󵄨󵄨 r 2 + r 4 M22 (r, h)) is decreasing. Hence the function r −1 + |f 󸀠󸀠 (0)/4|2 r = J1 (r, f ) − r 3 M22 (r, h) is decreasing, that is, (d/dr)(r −1 + |f 󸀠󸀠 (0)/4|2 r) ≤ 0, and hence |f 󸀠󸀠 (0)/4| ≤ 1.

11.1 Integral means of univalent functions | 387

Theorem 11.3 (Köbe 1/4-theorem). If f ∈ 𝒰 , then f (𝔻) contains the disc of radius |f 󸀠 (0)|/4 centered at f (0). Proof [265]. Let f (0) = 0. If w is not in the range of f , then the function g(z) = 1/(f (z) − w) is univalent in 𝔻, and hence |g 󸀠󸀠 (0)| ≤ 4|g 󸀠 (0)|. It follows that 󵄨 󸀠 󵄨 󵄨 󵄨󵄨 󸀠󸀠 󸀠 2 󵄨󵄨f (0) + 2f (0) /w󵄨󵄨󵄨 ≤ 4󵄨󵄨󵄨f (0)󵄨󵄨󵄨, whence, by the Bieberbach theorem, 2|f 󸀠 (0)2 /w| ≤ 4|f 󸀠 (0)| + |f 󸀠󸀠 (0)| ≤ 8|f 󸀠 (0)|. Thus |w| ≥ |f 󸀠 (0)/4|, and this concludes the proof. Theorem 11.4. If f is a conformal mapping of D ⊂ ℂ onto G, then |f 󸀠 (z)| δG (f (z)) 󵄨 󵄨 ≤ ≤ 4󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨, 4 δD (z)

(11.1)

z ∈ D.

Proof. The first inequality in (11.1) is just a reformulation of Theorem 11.3: the set f (𝔻) contains the disc of radius |f 󸀠 (z)|δD (z)/4. Applying the first inequality to the inverse function, we get the second inequality in (11.1). Theorem 11.5. If f ∈ 𝒰 , then there exists a point w belonging to the disc of radius |f 󸀠 (0)| centered at f (0) and such that w ∈ ̸ f (𝔻). Proof. Let f (0) = 0 and f 󸀠 (0) = 1. Assume that f (𝔻) contains the closed unit disc. Then there is R > 1 such that f (𝔻) ⊃ 𝔻R . It follows that the inverse function g maps 𝔻R into 𝔻, and hence, by the Schwarz lemma, |g 󸀠 (0)| ≤ 1/R, that is, |f 󸀠 (0)| ≥ R > 1, a contradiction. Exercise 11.2. Theorems 11.3 and 11.5 can be used to prove that if f ∈ 𝒰 , then 󵄨 󵄨 󵄨 󵄨 (1 − |z|2 )󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨/4 ≤ δf (𝔻) (f (z)) ≤ (1 − |z|2 )󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨,

z ∈ 𝔻.

Theorem 11.6 (Köbe distortion theorem). If f ∈ 𝒰 , then 1−r |f 󸀠 (z)| 1+r ≤ , ≤ (1 + r)3 |f 󸀠 (0)| (1 − r)3

|z| = r.

(11.2)

Proof. Applying Bieberbach’s theorem to the function g(w) = f (σz (w)), where σz (w) = z−w ̄ , we get 1−zw 󵄨󵄨 f 󸀠󸀠 (z) 4r 2r 2 󵄨󵄨󵄨󵄨 󵄨󵄨 . − 󵄨󵄨z 󸀠 󵄨≤ 󵄨󵄨 f (z) 1 − r 2 󵄨󵄨󵄨 1 − r 2 Since r 𝜕r𝜕 log |f 󸀠 | = Re(z ff 󸀠 ), we see that 󸀠󸀠

2r − 4 𝜕 󵄨 󵄨 2r + 4 ≤ . log󵄨󵄨󵄨f 󸀠 󵄨󵄨󵄨 ≤ 𝜕r 1 − r2 1 − r2 Now the desired result is obtained by integration.

388 | 11 One-to-one mappings As a consequence of the proof, we have two statements. Corollary 11.3. If f ∈ 𝒰 , then 󵄨󵄨 f 󸀠󸀠 (z) 󵄨󵄨 󵄨 󵄨 (1 − |z|2 )󵄨󵄨󵄨 󸀠 󵄨󵄨󵄨 ≤ 6. 󵄨󵄨 f (z) 󵄨󵄨 Corollary 11.4. If f ∈ 𝒰 , then 󵄨󵄨 f (z) − f (0) 󵄨󵄨 |f 󸀠 (0)| |f 󸀠 (0)| 󵄨󵄨 󵄨󵄨 ≤ ≤ 󵄨 󵄨 󵄨󵄨 (1 − |z|)2 , z (1 + |z|)2 󵄨󵄨󵄨 󵄨

z ∈ 𝔻.

(11.3)

Proof. The right-hand side inequality is obtained from (11.2) by integration. To prove the left-hand side inequality, assume that f (0) = 0 and f 󸀠 (0) = 1. If |f (z)| ≥ 1/4, then the inequality holds because |z|/(1 + |z|)2 < 1/4 for |z| < 1. Let |f 󸀠 (z)| < 1/4. Then the segment ℓ joining 0 and f (z) lies in f (𝔻) by the Köbe one-quarter theorem, and hence |f (z)| = ∫ℓ |dw|. By the change w = f (ζ ) we get 󵄨 󸀠 󵄨 󵄨 󵄨󵄨 󵄨󵄨f (z)󵄨󵄨󵄨 = ∫󵄨󵄨󵄨f (ζ )󵄨󵄨󵄨|dζ | L

≥∫ L

(where L = f −1 (ℓ))

1 − |ζ | 1 − |ζ | |z| |dζ | ≥ ∫ d|ζ | = . (1 + |ζ |)3 (1 + |ζ |)3 (1 + |z|)2 L

This completes the proof. Corollary 11.5. If f ∈ 𝒰 and f (z) ≠ 0 for all z ∈ 𝔻, then 2

(1 − |z|) /5 ≤

|f (z)| −2 ≤ 5(1 − |z|) . |f (0)|

(11.4)

Proof. By the previous corollary we have |f (z)| ≤ |f (0)| + |f 󸀠 (0)|/(1 − |z|)2 . On the other hand, by the 1/4-theorem, f (𝔻) contains the disc D of radius |f 󸀠 (0)|/4 centered at f (0). Since 0 ∈ ̸ D, we have |f 󸀠 (0)|/4 ≤ |f (0)|, which along with the preceding inequality proves part of (11.4). Now applying this result to the function f ∘ σa , we get |f (σa (z))| ≤ 5|f (σa (0))|/(1 − |z|)2 . Hence, taking z = a, we get |f (0)| ≤ 5|f (a)|/(1 − |a|)2 , which completes the proof. Corollary 11.6. If f ∈ 𝒰 , then 1 − |z| 󵄨󵄨󵄨󵄨 f (z) − f (0) 󵄨󵄨󵄨󵄨 1 + |z| ≤󵄨 . 󵄨≤ 1 + |z| 󵄨󵄨󵄨 zf 󸀠 (z) 󵄨󵄨󵄨 1 − |z| Proof. Let g(w) = f (σz (w)), w, z ∈ 𝔻. Applying Corollary (11.4) to g, we obtain |f 󸀠 (z)|(1 − |z|2 ) |f (σz (w)) − f (σz (0))| |f 󸀠 (z)|(1 − |z|2 ) ≤ ≤ . |w| (1 + |w|)2 (1 − |w|)2 Now taking w = z, we conclude the proof.

(11.5)

11.2 Hardy–Prawitz theorem and applications |

389

Remark. Rewriting (11.5) as 1 − |z| 󵄨󵄨󵄨󵄨 zf 󸀠 (z) 󵄨󵄨󵄨󵄨 1 + |z| ≤󵄨 󵄨≤ 1 + |z| 󵄨󵄨󵄨 f (z) − f (0) 󵄨󵄨󵄨 1 − |z| and then multiplying this inequality by (11.3), we come back to (11.2).

11.2 Hardy–Prawitz theorem and applications If p > 0, then the function Ip (r) = Ip (r, f ) = J−p (r, f ) is increasing, and this fact does not depends on the hypothesis f ∈ 𝒰 (Theorem 2.3). However, the proof of Theorem 11.1 gives additional information on Ip (r): If f ∈ 𝒰 , f (0) = 0, and f 󸀠 (0) = 1, then Ip󸀠 (r) =

p p p2 R − ∬ |w|p−2 dA(w). r 2πr Ωr,R

This implies that Ip󸀠 (r) ≤ (p/r)M(r)p . Combining this with Corollary 11.4, we get the following theorem [489]. Theorem 11.7 (Prawitz). If f ∈ 𝒰 with f (0) = 0, then ρ

p Ip (ρ, f ) ≤ p ∫ r −1 M∞ (r, f ) dr. 0

A little work is needed to deduce the following result from this theorem and the 2 Hardy M∞ -theorem 2.16. 1

p Theorem 11.8 (Hardy–Prawitz). A function f ∈ 𝒰 is in H p if and only if ∫0 M∞ (r, f ) dr < ∞. Moreover,

‖f ‖pp

1

p ≍ ∫ M∞ (r, f ) dr,

f ∈ 𝒰,

0

where the equivalence constants are independent of f . As a consequence of this fact and Corollary 11.4, we have the following: Corollary 11.7. If f ∈ 𝒰 and 0 < p < 1/2, then f ∈ H p . Also, M1/2 (r, f ) ≤ C(log where C is independent of r.

2

2 ) 1−r

and

Mp (r, f ) ≤ C(1 − r)1/p−2 ,

p > 1/2,

390 | 11 One-to-one mappings We have proved that part of the following theorem holds for every analytic function. Theorem 11.9. Let 0 < p < q < ∞ and f ∈ 𝒰 . Then ‖f ‖pp

1

≍ ∫(1 − r)−p/q Mqp (r, f ) dr. 0

Proof. By Theorems 11.8 and 5.4 it suffices to prove that 1

‖f ‖pp ≤ C ∫(1 − r)−p/q Mqp (r, f ) dr. 0

This follows from the inequalities M∞ (r, f ) ≤ C(1 − r)−1/q Mq (√r, f ) and Theorem 11.7. Taking q = 2 in the preceding theorem, we obtain the case p ≤ 2 of the following result [261, 546]; see (6.27) and (6.28). Theorem 11.10 (Holland–Twomey–Spencer). For a function f ∈ 𝒰 with f (0) = 0, we have the relation 1

p/2

‖f ‖p ≍ (∫ A(r, f )

1/p

dr)

,

p ∈ ℝ+ ,

0

where the equivalence constants depend only on p. Proof. We have only to prove the inclusion 𝒰 ∩ H p ⊂ ARp for 2 < p < ∞. We come back to the proof of Prawitz’s theorem (with the notation used there). Since d(arg w) = Im(w̄ dw)/|w|2 = Im(dw/w), we see that 2πr

d I (r, f ) = p ∫ |w|p d(arg w) dr p Γr

p/2

≥ p(∫ |w|2 d(arg w)) Γr

(∫ d(arg w)) Γr

= p(2π)1−p/2 (∫ Im(w̄ dw))

p/2

1−p/2

= p(2π)1−p/2 2p/2 A(r, f )p/2 .

Γr

This proves the desired result, provided that d(arg w) > 0. In the general case, we can use ideas from Hörmander’s proof of Prawitz’s theorem [265, pp. 160–161]. It is shown there that if f (0) = 0, then, for a fixed r, D(r) := 2πr(d/dr)Ip (r, f )/p can be represented

391

11.2 Hardy–Prawitz theorem and applications |

as mj

n

∑ ∫(Rp1,j (α) + ∑ [Rp2k+1,j (α) − Rp2k,j (α)]) dα j=1 A

k=1

j

0

( ∑ = 0), k=1

(11.6)

where Aj ⊂ [0, 2π] are disjoint intervals such that |A1 | + ⋅ ⋅ ⋅ + |An | = 2π, and R1,j < R2,j < ⋅ ⋅ ⋅ < R2mj +1,j . Assuming that p > 2 and applying the reverse Hölder inequality with exponents 2/p and 2/(2 − p), we obtain n

(∑ ∫(R21,j (α) j=1 A

D(r) ≥

+

j

mj

∑ [Rp2k+1,j (α) k=1

−

2/p Rp2k,j (α)] ) dα)

p/2

1−p/2

n

× (∑ ∫ dα)

.

j=1 A

j

Since (a − b)γ ≥ aγ − bγ when a > b > 0 and 0 < γ < 1, we conclude that D(r) ≥

n

(∑ ∫(R21,j (α) j=1 A

+

j

mj

∑ [R22k+1,j (α) k=1 p/2

= (∫ |w|2 d(arg w))

−

p/2 2 R2k,j (α)]) dα) (2π)1−p/2

(2π)1−p/2 ,

Γr

where Γr is the image of the circle |z| = r under f . Since 󵄨 󵄨 ∫ |w|2 d(arg w) = ∫ Im(w̄ dw) = 2󵄨󵄨󵄨f (r𝔻)󵄨󵄨󵄨, Γr

Γr

we have d p p/2 I (r, f ) ≥ 2 A(r, f )p/2 (2π)1−p/2 dr p 2πr just in the case where d arg f > 0. According to (6.26) and Lemma 4.8, we can rewrite Theorem 11.10 as follows. Theorem 11.11. For a function f ∈ 𝒰 , we have the relation p/2

∞

󵄨2 󵄨 ‖f ‖pp ≍ ∑ 2n(p/2−1) ( ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 ) n=0

k∈In

,

p ∈ ℝ+ .

392 | 11 One-to-one mappings Combining this theorem with the Hardy–Prawitz theorem, we obtain another theorem from [261]. Let ∞

󵄨 󵄨 P(r, f ) = ∑ 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨r n . n=0

Theorem 11.12 (Holland–Twomey). For a function f ∈ 𝒰 , we have the following relation: ‖f ‖pp

1

≍ ∫ P(r, f )p dr,

p ∈ ℝ+ .

0

Proof. The inequality “‖f ‖pp ≤ C . . .” immediately follows from the Hardy–Prawitz theorem. On the other hand, we have 1

∞

0

n=0

p

󵄨 󵄨 ∫ P(r, f )p dr ≍ ∑ 2−n ( ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨) k∈In

p/2

∞

≤ ∑ 2−n ( ∑ 1) n=0

k∈In

p/2

󵄨2 󵄨 ( ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 ) k∈In

∞

p/2

󵄨2 󵄨 ≤ C ∑ 2−n 2np/2 ( ∑ 󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 ) n=0

k∈In

.

Now the result follows from Theorem 11.11. Exercise 11.8. Let 1 ≤ p ≤ 2. A function f ∈ 𝒰 belongs to H p if and only if ∑(n + 1)p−2 |f ̂(n)|p < ∞. (The equivalence does not hold for large p > 2; see [261]). Exercise 11.9. Let 0 < p < 1. Then a function f ∈ 𝒰 belongs to H p if and only if ∞

󵄨p 󵄨 ∑ 2n(p−1) sup󵄨󵄨󵄨f ̂(k)󵄨󵄨󵄨 < ∞.

n=0

k∈In

We know that Hαp,q ⊊ Hβs,q , where β = α + 1/p − 1/s and p < s (Theorem 3.12). However, we have the following: Theorem 11.13. Let α > 0. Then Hαp,q ∩ 𝒰 = Hβs,q ∩ 𝒰 . Proof. We have to prove that if f ∈ Hβs,q ∩ 𝒰 , then f ∈ Hαp,q . Let s = ∞. Applying Prawitz’s theorem to the function f − f (0) and using the maximum modulus principle, we get 1 p (ρ, f ) dρ, and then ‖f ‖pp ≤ C ∫0 M∞ Mpp (r, f )

1

p ≤ C ∫ M∞ (ρr, f ) dρ. 0

11.3 A Littlewood–Paley theorem for univalent functions | 393

Now the result in the case s = ∞ is obtained by the “fractional integration proposition” p (Proposition 3.30); the subharmonicity of the function z 󳨃→ M∞ (ρ|z|, f ) is to be used. If p < s < ∞, then the result follows by the preceding case and the inclusions Hαp,q ⊂ Hβs,q ⊂ Hγ∞,q , where β = α + 1/p − 1/s and γ = β + 1/s = α + 1/p.

11.3 A Littlewood–Paley theorem for univalent functions p = Bp,p for p ≥ 2 and that the reThe Littlewood–Paley theorem states that H p ⊂ 𝒟p−1 verse inclusion holds for p < 2. However, as was proved in [57], we have the following: p , and Theorem 11.14 (Baernstein–Girela–Peláez). If p ∈ ℝ+ , then 𝒰 ∩ H p = 𝒰 ∩ 𝒟p−1 moreover ‖f ‖p ≍ ‖f ‖𝒟p , where the equivalence constants depend only on p. p−1

The proof of Theorem 11.14 given in [57] is very complicated and depends on various techniques, which nevertheless are important because can be used in other situations, for example, in proving an extension of a theorem of Pommerenke; see Further notes 11.1. We present a surprisingly simple proof due to Astala and Koskela [50], which is based on a remarkable result of Jones (Lemma 11.10 below). p Proof of Theorem. Let p ≥ 2. The validity of the inclusion 𝒰 ∩ H p ⊂ 𝒰 ∩ 𝒟p−1 follows Theorem 10.15 and is independent of the univalence. In proving the converse, we may assume that f (0) = 0. Then first apply the Hardy–Littlewood inequality 1

1

0

0

p p (r, f ) dr ≤ C ∫(1 − r)p M∞ (r, f 󸀠 ) dr ∫ r −1 M∞

and then the inequality p M∞ (r, f 󸀠 ) ≤ C(1 − r)−1 Mpp ((1 + r)/2, f 󸀠 ) dr.

Combining these inequalities with Theorem 11.7, we conclude the proof for p ≥ 2. Let 0 < p < 2. Assume first that f (z) ≠ 0 for all z ∈ 𝔻. Then, by Lemma 11.10, the measure p−1 󵄨−p 󵄨p 󵄨 󵄨 dμ(z) = 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z)

(11.7)

is Carleson, and consequently p−1 󵄨p 󵄨 󵄨p 󵄨 ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|) dA(z) = ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 dμ(z) ≤ C‖f ‖pp . 𝔻

𝔻

Let f ∈ 𝒰 be arbitrary. Then a well-known consequence of Schwarz’s lemma states that there exists ζ0 ∈ 𝕋 such that f (z) ≠ f (0) + ζ0 f 󸀠 (0) =: w0 for all z ∈ 𝔻. Applying the

394 | 11 One-to-one mappings preceding result to the function f (z) − w0 , we find that p−1 󵄨p 󵄨p 󵄨 󵄨 󵄨p 󵄨 ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|) dA(z) ≤ Cp (‖f ‖pp + 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + 󵄨󵄨󵄨f 󸀠 (0)󵄨󵄨󵄨 ) ≤ Cp ‖f ‖pp , 𝔻

which proves the theorem. Jones’ lemma The following fact is a very particular case of Jones’ lemma [279]. Lemma 11.10. If 0 < p < 2, f ∈ 𝒰 , and f (z) ≠ 0 for all z ∈ 𝔻, then the measure dμ defined by (11.7) is Carleson. Proof [50, Lemma 5.6]. We have p−1 󵄨−p 󵄨p 󵄨 󵄨 ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z) 𝔻 2ε/p 󵄨−2 󵄨2 󵄨 󵄨 dA(z)) ≤ (∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 )

p/2

𝔻

× (∫(1 − |z|2 )

(p−1−ε)2/(2−p)

(2−p)/2

dA(z))

𝔻 2ε/p 󵄨−2 󵄨2 󵄨 󵄨 ≤ C(∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z))

p/2

=: CI(f )p/2 .

𝔻

Here ε > 0 has been chosen so that 2(p − 1 − ε)/(2 − p) > −1 (e. g., ε = p/4). Now write I(f ) = I1 (f ) + I2 (f ), where I1 (f ) is taken over D1 = 𝔻 ∩ f −1 (ρ𝔻), and I2 (f ) over D2 = 𝔻 \ f −1 (ρ𝔻), where ρ = |f (0)|. Then, by (11.4), 󵄨−ε/p 󵄨−2+ε/p 󵄨󵄨 󵄨2 󵄨 󵄨 dA(z) I1 (f ) ≤ C1 ∫ 󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 󵄨󵄨f (0)󵄨󵄨󵄨 D1

= C1 ρ−ε/p

∫ f (𝔻)∩ρ𝔻

|w|−2+ε/p dA(w) ≤ C1 ρ−ε/p ∫ |w|−2+ε/p dA(w) ≤ C2 , ρ𝔻

where C2 is independent of f . In a similar way, we obtain I2 (f ) ≤ C3 , where C3 is independent of f . Consequently, p−1 󵄨−p 󵄨p 󵄨 󵄨 ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z) ≤ Cp .

(11.8)

𝔻

We use this inequality to prove that dμ is a Carleson measure, that is, p−1 󵄨 󵄨 󵄨−p 󵄨p 󵄨 󵄨 ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) 󵄨󵄨󵄨σa󸀠 (z)󵄨󵄨󵄨 dA(z) ≤ Cp . 𝔻

(11.9)

11.3 A Littlewood–Paley theorem for univalent functions | 395

To do this, we apply (11.8) to the function f ∘ σa , and after substitution σa (z) = w, we get (11.9) with the same constant Cp as in (11.8). (It is convenient to use the formula 1 − |z|2 = (1 − |w|2 )|σa󸀠 (w)|.) Exercise 11.11. If p = 2 and f ≠ 0 in 𝔻, then the measure μ defined by (11.7) is Carleson. In other words, the measure dμ(z) = |(log f )󸀠 (z)|2 (1 − |z|) dA(z) is Carleson, and hence, if f ∈ 𝒰 and f (z) ≠ 0 in 𝔻, then log f ∈ BMOA. Univalent functions in the Bloch space and in BMOA As a consequence of Theorems 11.14 and 7.2, we have the following: Theorem 11.15. Let p ∈ ℝ+ . A function f ∈ 𝒰 belongs to BMOA if and only if p−1 󵄨 󵄨 󵄨p 󵄨 Kp (f ) := sup ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|2 ) 󵄨󵄨󵄨σa󸀠 (z)󵄨󵄨󵄨 dA(z) < ∞, a∈𝔻

(11.10)

𝔻

and we have Kp (f ) ≍ ‖f ‖1/p ∗2 . Note that f (σa (0)) − f (a) = 0. Relation (11.10) means that if f ∈ 𝒰 ∩ BMOA, then the measure dμp (z) = |f 󸀠 (z)|p (1 − |z|2 )p−1 dA(z) is a Carleson measure. From this we can obtain the following result [485]. Corollary 11.12 (Pommerenke). We have 𝒰 ∩ BMOA = 𝒰 ∩ B. Observe that if f ∈ 𝒰 , then f ∈ B if and only if supw∈f (𝔻) δf (𝔻) (w) < ∞; this follows from (11.1). For an extension to univalent harmonic functions, see Further notes 11.4. Proof. Assuming that f ∈ H(𝔻), we have 2󵄨 󵄨 󵄨3 󵄨 ‖f ‖3∗2 ≤ C sup ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|2 ) 󵄨󵄨󵄨σa󸀠 (z)󵄨󵄨󵄨 dA(z) a∈𝔻

𝔻

󵄨 󵄨 󵄨2 󵄨 ≤ CB(f ) sup ∫󵄨󵄨󵄨f 󸀠 (z)󵄨󵄨󵄨 (1 − |z|2 )󵄨󵄨󵄨σa󸀠 (z)󵄨󵄨󵄨 dA(z) = CB(f )‖f ‖2∗2 , a∈𝔻

𝔻

whence ‖f ‖∗2 ≤ CB(f ), where C is independent of f . If f is arbitrary, then we apply this inequality to fρ and let ρ → 1− . As a consequence of Theorem 11.15 (p < 2) and Jones’ lemma, we have the following result from [54] and [125]. (See Exercise 11.11.) Theorem 11.16 (Baernstein–Cima–Shober). If f ∈ 𝒰 and f (z) ≠ 0 for all z ∈ 𝔻, then log f ∈ BMOA. Exercise 11.13. Let 0 < q < ∞. A function f ∈ 𝒰 belongs to the Triebel–Lizorkin space ∞ Fq,0 if and only if it belongs to BMOA.

396 | 11 One-to-one mappings

11.4 Quasiconformal harmonic mappings of the disc A mapping f : 𝔻 󳨃→ ℂ is said to be quasiconformal (QC) if it is one-to-one, absolutely continuous on straight lines, and satisfies the condition 󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨f (z)󵄨󵄨 ≤ KJf (z),

z ∈ 𝔻,

(11.11)

where |f 󸀠 (z)| = ‖df (z)‖, and Jf is the Jacobian of f . The function f is then called K-quasiconformal (K-QC); we denote by Kf the smallest K satisfying (11.11). Quasiconformal mappings are differentiable almost everywhere. If f is in addition harmonic, then 󵄨2 󵄨2 󵄨 󵄨 and Jf (z) = 󵄨󵄨󵄨h󸀠 (z)󵄨󵄨󵄨 − 󵄨󵄨󵄨g 󸀠 (z)󵄨󵄨󵄨 ,

󵄨 󸀠 󵄨 󵄨 󸀠 󵄨2 󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨f (z)󵄨󵄨 = (󵄨󵄨󵄨h (z)󵄨󵄨󵄨 + 󵄨󵄨󵄨g (z)󵄨󵄨󵄨) , so Kf = sup z∈𝔻

|h󸀠 (z)| + |g 󸀠 (z)| , |h󸀠 (z)| − |g 󸀠 (z)|

and the condition Kf < ∞ can be rewritten as kf := sup z∈𝔻

|h󸀠 (z)| < 1, |g 󸀠 (z)|

where f = h + g.̄ We also have Kf =

1 + kf

1 − kf

and kf =

Kf − 1

Kf + 1

.

We denote the class of all quasiconformal harmonic mappings f : 𝔻 󳨃→ ℂ by QCH. A very nice introduction to planar quasiconformal mappings and univalent harmonic mappings (not necessarily quasiconformal) was written by Ponnusamy and Rasila [487] (67 pp.). 11.4.1 Boundary behavior of QCH homeomorphisms of the disc Before stating the results, we note that if f is a QC-mapping from 𝔻 onto 𝔻, then it extends to a continuous homeomorphism of 𝔻 onto 𝔻 so that the restriction to 𝕋 is a homeomorphism of 𝕋; this is a result of Ahlfors [16]. Thus we have f (eiθ ) = eiφ(t) ,

where φ is strictly increasing, continuous, and φ(t + 2π) − φ(t) ≡ 2π.

(11.12)

Subsequently, Ahlfors and Beurling [75] characterized the boundary mapping eiφ by the condition φ(θ + 2t) − φ(θ + t) 1 ≤ ≤ M, M φ(θ + t) − φ(θ)

t > 0,

11.4 Quasiconformal harmonic mappings of the disc | 397

where M is a constant independent of θ and t. A function φ satisfying this condition is called quasisymmetric. We need, however, a substantial improvement of Ahlfors’ theorem. Theorem 11.17 (Mori [394]). If Φ is a K-quasiconformal homeomorphism of 𝔻, then 󵄨 󵄨 (1/C)|z1 − z2 |K ≤ 󵄨󵄨󵄨Φ(z1 ) − Φ(z2 )󵄨󵄨󵄨 ≤ C|z1 − z2 |1/K

(z1 , z2 ∈ 𝔻),

(11.13)

where C depends on |Φ(0)|; if Φ(0) = 0, then we can take C = 16. A proof can be found in Ahlfors [18]. The mapping Φ(z) = |z|1/K (z/|z|) shows that the exponent 1/K is optimal in the class of all K-quasiconformal mappings. We will use Mori’s theorem to characterize the boundary function of a QCHmapping. As a consequence of the proof, we show that a QCH-homeomorphism of 𝔻 satisfies the ordinary Lipschitz condition (see (11.33)), which along with the Heinz inequality |f 󸀠 (z)| ≥ 1/π, valid when f (0) = 0 [252] (see Corollary 11.14 for a weaker variant), gives the following: Theorem 11.18 (Pavlović [435]). If f is a harmonic K-quasiconformal homeomorphism of 𝔻, then it is bi-Lipschitz, that is, there is a constant L < ∞, depending only on K and |f (0)|, such that 1 󵄨󵄨󵄨󵄨 f (z1 ) − f (z2 ) 󵄨󵄨󵄨󵄨 ≤󵄨 󵄨 ≤ L (z1 , z2 ∈ 𝔻), L 󵄨󵄨󵄨 z1 − z2 󵄨󵄨󵄨 and consequently 1 1 − |f (z)| ≤ ≤ L, L 1 − |z|

z ∈ 𝔻.

(11.14)

We leave the deduction to the reader. It is known that a bi-Lipschitz mapping is quasiconformal. For Lipschitzianity in a more general situation, see Further notes 11.7. The QCH-analogue of the Ahlfors–Beurling theorem reads as follows. Theorem 11.19 (Pavlović [435]). Let f be a orientation-preserving harmonic homeomorphism of 𝔻. Then the following two conditions are equivalent: (a) f is quasiconformal. (b) f = 𝒫 [eiφ ], where the function φ has the properties: (i) φ(θ + 2π) − φ(θ) ≡ 2π; (ii) φ is increasing and bi-Lipschitz, and (iii) the Hilbert transformation of φ󸀠 is bounded. Note that the Hilbert transformation of φ󸀠 is defined as π

φ󸀠 (θ + t) − φ󸀠 (θ − t) 1 dt, H(φ )(θ) = − ∫ π 2 tan(t/2) 󸀠

+0

θ ∈ ℝ,

398 | 11 One-to-one mappings and that φ is bi-Lipschitz if and only if φ is absolutely continuous and satisfies the first two of the following three conditions: ess inf φ󸀠 > 0,

(11.15)

󸀠

ess sup φ < ∞,

(11.16)

󵄨󵄨 󵄨 󵄨󵄨 φ󸀠 (θ + t) − φ󸀠 (θ − t) 󵄨󵄨󵄨 ess sup󵄨󵄨󵄨 ∫ dt 󵄨󵄨󵄨 < ∞. 󵄨󵄨 t θ∈ℝ 󵄨󵄨󵄨 󵄨 +0

(11.17)

π

The last condition is equivalent to the boundedness of H(φ󸀠 ) because φ󸀠 ∈ L∞ and 1 1 − = 𝒪(t 2 ), 2 tan(t/2) t

t → 0.

Kalaj [286] considered harmonic mappings of 𝔻 onto a smooth convex domain G. Under the hypotheses of the Radó–Kneser–Choquet theorem 1.20, he obtained the following result, which we state without proof. Theorem (Kalaj [286]). Let G be convex, and let Γ := 𝜕G ∈ C 1,α for some α ∈ (0, 1). Then the following conditions are equivalent: (a) f is quasiconformal. (b) f is bi-Lipschitz in the Euclidean metric. (c) γ is absolutely continuous, γ 󸀠 ∈ L∞ , | ess inf𝕋 γ 󸀠 | > 0, and H(γ 󸀠 ) ∈ L∞ . Although Kalaj also applies Mori’s theorem, his proof is not a complete imitation of that of Theorem 11.19. We note that Γ ∈ C 1,α means that the natural parameterization of Γ is of class C 1,α . It is not difficult to check that if G = 𝔻, and then the equivalence (a) ⇔ (c) reduces to Theorems 11.18 and 11.19. The author of Theorem 11.18 was motivated by the pioneering work of Martio [378]. Theorem (Martio [378]). The mapping f = 𝒫 [eiφ ] is quasiconformal if φ ∈ C 1 (ℝ), min φ󸀠 > 0, and π

∫ 0

ω(t) dt < ∞, t

(11.18)

where ω(t) = sup{|φ󸀠 (x) − φ󸀠 (y)| : |x − y| < t}. Condition (11.18), known as the Dini condition (applied to φ󸀠 ), is sufficient but not necessary for the Hilbert transformation of φ󸀠 to belong to L∞ ; see comments following Theorem 1.17. In the case of self-mappings of 𝔻, a lot of examples can be produced by using analytic functions that map 𝔻 into a relatively compact subset of the right half-plane. Having such F with Re F(0) = 1, we take φ󸀠 (θ) = Re F∗ (eiθ ), and since H(φ󸀠 )(θ) =

11.4 Quasiconformal harmonic mappings of the disc | 399

θ

Im F∗ (eiθ ), we see that the function φ(θ) = ∫0 φ󸀠 (t) dt satisfies condition (b) of Theorem 11.19. On the other hand, if F maps 𝔻 into a strip 0 < a < Re z < b and is unbounded, then a < φ󸀠 < b a. e., but H(φ󸀠 ) is not bounded. Before passing to the proof of Theorem 11.19, note that by a result of Lewy [346] (see Duren [168, Section 2.2]) the Jacobian of a univalent harmonic function is zerofree in 𝔻, that is, as we may assume that 󵄨2 󵄨2 󵄨 󵄨 Jf (z) = 󵄨󵄨󵄨𝜕f (z)󵄨󵄨󵄨 − 󵄨󵄨󵄨𝜕f̄ (z)󵄨󵄨󵄨 > 0

(z ∈ 𝔻).

(11.19)

Being harmonic, the mapping f can be represented as f (z) = h(z) + g(z), g(0) = 0, where h and g are analytic in 𝔻 and uniquely determined by f . We can rewrite (11.19) as 󵄨󵄨 g 󸀠 (z) 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨󵄨 < 1 (z ∈ 𝔻). 󵄨󵄨 h (z) 󵄨󵄨

(11.20)

Thus we consider those f for which (11.20) can be improved to 󵄨󵄨 g 󸀠 (z) 󵄨󵄨 󵄨 󵄨 k = sup󵄨󵄨󵄨 󸀠 󵄨󵄨󵄨 < 1. z∈𝔻 󵄨󵄨 h (z) 󵄨󵄨

(11.21)

This condition is very strong, as follows from Theorems 1.20 and 11.19.

Proof of Theorem 11.19 As mentioned after the formulation of the theorem, it suffices to consider conditions (11.15)–(11.17). The proof that the three conditions are sufficient is short; we simply compute the radial limits of the modulus of the bounded analytic function g 󸀠 /h󸀠 and apply the maximum modulus principle. The necessity proof is more involved and depends on Mori’s theorem. Boundary values of the derivatives According to Ahlfors’ theorem, we may assume that f = 𝒫 [γ], where γ = eiφ with φ having the properties described in (11.12). In calculating the boundary values of the analytic functions h󸀠 and g 󸀠 , it is useful to use the formulas Df (z) 1 ), h󸀠 (z) = 𝜕f (z) = e−iθ (R0 f (z) − i 2 r 1 Df (z) g 󸀠 (z) = 𝜕f̄ (z) = eiθ (R0 f (z) + i ), 2 r

(11.22) (11.23)

400 | 11 One-to-one mappings where Df = 𝜕f /𝜕θ and R0 f = 𝜕f /𝜕r. The derivatives R0 f and Df are connected by the simple but fundamental fact that the function rR0 f = ℛf is equal to the harmonic conjugate of Df . It is easy to check that Df equals the Poisson–Stieltjes integral of γ = eiφ : Df (reiθ ) =

π

1 ∫ P(r, θ − t) dγ(t). 2π

(11.24)

−π

Hence, by Fatou’s theorem, the radial limits of Df exist almost everywhere, and limr→1− Df (reiθ ) = γ0󸀠 (θ) a. e., where γ0 is the absolutely continuous part of γ. It turns out that if γ is absolutely continuous, then limr→1− R0 f (reiθ ) = H(γ 󸀠 )(θ) a. e. Absolute continuity. The function γ, of course, need not be absolutely continuous. However: If π

󵄨 󵄨 sup ∫ − 󵄨󵄨󵄨R0 f (ρeiθ )󵄨󵄨󵄨 dθ < ∞, ρ −1, which is possible because (α−1)(2−β) → α−1 > −1 as β → 1− , we get max φ󸀠 ≤ C2 , where C2 depends only on K and |f (0)|. From this and (11.30) we get A(θ) ≤ 2K 2 C2 , and hence, by (11.29) and (11.26), |h󸀠 (eiθ )| ≤ C3 . Since h󸀠 ∈ H 1 , using Smirnov’s maximum principle, we see that 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨h (z)󵄨󵄨 ≤ C3

(z ∈ 𝔻),

(11.33)

where the constant C3 depends only on K and |f (0)|. In the general case, consider the mappings fn of 𝔻 onto 𝔻 defined by fn (z) = f (wn (z))/rn = hn (z) + gn (z)

(rn = 1 − 1/n, n ≥ 2),

where wn is the conformal mapping of 𝔻 onto Gn = f −1 (rn 𝔻), wn (0) = 0, wn󸀠 (0) > 0. Since the boundary of Gn is an analytic Jordan curve, the mapping wn can be continued analytically across 𝕋, which implies that fn has a harmonic extension across 𝕋. Since also 󵄨󵄨 g 󸀠 󵄨󵄨 󵄨󵄨 (g 󸀠 ∘ w )w󸀠 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n n 󵄨󵄨 󵄨 ≤ k, 󵄨󵄨 󸀠 󵄨󵄨 = 󵄨󵄨 󸀠 󵄨󵄨 hn 󵄨󵄨 󵄨󵄨 (h ∘ wn )wn󸀠 󵄨󵄨󵄨 we can appeal to the preceding particular case and conclude that 󵄨󵄨 󸀠 󵄨 󵄨󵄨 󸀠 󵄨󵄨h (wn (z))󵄨󵄨󵄨󵄨󵄨󵄨wn (z)󵄨󵄨󵄨/rn ≤ C3 , where C3 is independent of n and z. Since Gn ⊂ Gn+1 and ∪Gn = 𝔻, we can apply Carathéodory’s convergence theorem (Theorem 11.21): wn (z) 󴁂󴀱 z, whence wn󸀠 (z) 󴁂󴀱 1 (n → ∞). Thus inequality (11.33) holds in the general case. Using this and (11.26), we get φ󸀠 (θ) + |B(θ)| ≤ C4 . Finally, it remains to apply (11.28).

11.4 Quasiconformal harmonic mappings of the disc | 403

Some consequences of the proof It was proved by Clunie and Sheil-Small [128, Theorem 5.7] and Kalaj [287] (see Further notes 11.12) that if f = h + ḡ maps 𝔻 onto a convex domain, then the function h belongs to 𝒰 . It follows from our proof that if f is quasiconformal and f (𝔻) = 𝔻, then 1/C ≤ |h󸀠 | ≤ C. This means that: Theorem 11.20. If f is QCH and f (𝔻) = 𝔻, then h is bi-Lipschitz. The converse is not true. To see this, let D = 1 + 𝔻 and F(z) = 1 + z. Then, as before, let (comments following Martio’s theorem, p. 398) φ󸀠 (θ) = Re F(eiθ ) = 1 + cos θ, which implies that H(φ󸀠 ) = Im F(eiθ ) = sin θ. Then we use formula (11.26) to show that |h󸀠 (eiθ )| is bounded, which implies that h󸀠 is bounded on 𝔻. On the other hand, from (11.26) and (11.31) it follows that |h󸀠 (eiθ )| ≥ A(θ)/2 ≥ (1 − |f (0)|)/4, and hence 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨h (z)󵄨󵄨 ≥ (1 − 󵄨󵄨󵄨f (0)󵄨󵄨󵄨)/4

(11.34)

for all z ∈ 𝔻. Thus h is bi-Lipschitz, whereas f is not quasiconformal because min𝕋 φ󸀠 = 0. Corollary 11.14. If f is a harmonic homeomorphism of 𝔻, then 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨f (z)󵄨󵄨 = 󵄨󵄨h (z)󵄨󵄨 + 󵄨󵄨g (z)󵄨󵄨 ≥ (1 − 󵄨󵄨󵄨f (0)󵄨󵄨󵄨)/4,

z ∈ 𝔻.

Proof. As in the proof of the characterization theorem, we can assume that f is smooth up to the boundary of 𝔻. Then (11.34) holds, and hence |f 󸀠 (z)| ≥ |h󸀠 (z)| ≥ (1 − |f (0)|)/4.

The Carathéodory convergence theorem Here we prove the simplest variant of Carathéodory’s theorem; for the general form, see [212, Ch. II, § 5] and [167]. Theorem 11.21. Let fn : 𝔻 󳨃→ 𝔻 be a sequence of univalent functions such that fn (0) = 0, fn󸀠 (0) > 0, fn (𝔻) ⊂ fn+1 (𝔻) for every n, and ⋃ fn (𝔻) = 𝔻. Then fn (z) → z uniformly on compact subsets. Proof. The set {fn } is relatively compact in H(𝔻), and hence it suffices to prove that every H(𝔻)-convergent subsequence of {fn } converges to the function φ(z) = z. Therefore we can assume that fn tends, uniformly on compact subsets, to some function f ∈ H(𝔻). Clearly, f (𝔻) ⊂ 𝔻 and fn󸀠 (0) → f 󸀠 (0). Let Dρ = {z: |z| ≤ ρ}, ρ < 1. Since ∪fn (𝔻) = 𝔻 and Dρ is compact, we see that fn (𝔻) contains Dρ for n > n0 , where n0 is large enough. The function g(w) = fn−1 (ρw) maps 𝔻 into 𝔻 and g(0) = 0, and hence g 󸀠 (0) = ρ/fn󸀠 (0) ≤ 1 for n > n0 . It follows that fn󸀠 (0) → 1, and hence f 󸀠 (0) = 1. Hence f (z) = z by Schwarz’ lemma, and the proof is complete.

404 | 11 One-to-one mappings Problem 11.13. Let QCH∗ = {f∗ : f is QCH}. Is QCH∗ a group with respect to composition? The set of all quasiconformal harmonic homeomorphisms of 𝔻 is not a group because the composition of two harmonic functions need not be harmonic. On the other hand, the set of all quasiconformal transformations of 𝔻 is a group (cf. [18]).

11.5 H p theory of QCH mappings Astala and Koskela [50] proved many results on quasiconformal mappings, some of which are probably new even when reduced to conformal mappings. As particular cases of their results, we can extend almost all the results from Section 11.2 to the class QCH. They consider QC mappings in the unit ball of ℝn , n ≥ 2. We will restrict ourselves to the case n = 2. Jones’ lemma holds for arbitrary QC mappings; the proof is almost the same as in the case of univalent functions and is based on the following lemma. Lemma 11.15 ([50, Lemma 2.2]). If f : 𝔻 󳨃→ ℂ is a quasiconformal mapping such that f (z) ≠ 0 for z ∈ 𝔻, then there are positive constants b and C such that b

(1 − |z|) /C ≤

|f (z)| −b ≤ C(1 − |z|) . |f (0)|

(11.35)

Proof. Let f be K-quasiconformal. It is well known (see Lehto and Virtanen [344, p. 241]) that f can be represented as f = Φ ∘ F, where F is K-QC homeomorphism of 𝔻 with F(0) = 0, and Φ : 𝔻 󳨃→ ℂ is a univalent function. The hypothesis f ≠ 0 in 𝔻 implies F ≠ 0 in 𝔻. By Corollary 11.5 we have 󵄨󵄨 Φ(F(z)) 󵄨󵄨 󵄨 −2 󵄨󵄨 󵄨 󵄨2 󵄨 󵄨 (1 − 󵄨󵄨󵄨F(z)󵄨󵄨󵄨) /5 ≤ 󵄨󵄨󵄨 󵄨 ≤ 5(1 − 󵄨󵄨󵄨F(z)󵄨󵄨󵄨) . 󵄨󵄨 Φ(F(0)) 󵄨󵄨󵄨 On the other hand, as a consequence of Mori’s theorem 11.17, we have 1/K K 󵄨 󵄨 (1 − |z|) /16 ≤ 1 − 󵄨󵄨󵄨F(z)󵄨󵄨󵄨 ≤ 16(1 − |z|) .

Combining these inequalities, we get (11.35) with b = 2K and C = 80, completing the proof. The author was not able to find a “QC-free” proof of this lemma. But we continue without appealing to the theory of QC-mappings. Proposition 11.16. Let G : 𝔻 󳨃→ ℝ be a nonnegative log-subharmonic function of class C 2 . Then r

d s M (r, G) ≍ ∫ G(z)s−1 ΔG(z) dA(z), dr s |z| 0, and Gϵ = (G + ϵ)s . By elementary calculation we get ∫ Gϵ Δφ dA = ∫ φΔGϵ dA = ∫ [(s − 1)(G + ϵ)s−2 |∇G|2 + (G + ϵ)s−1 ΔG]φ dA.

(∗)

𝔻

𝔻

𝔻

Hence, by Fatou’s lemma, ∫ [(s − 1)Gs−2 |∇G|2 + Gs−1 ΔG]φ dA < ∞, 𝔻

and hence, because |∇G|2 ≤ GΔG, ∫ sφGs−1 ΔG dA < ∞

∫ φ|s − 1|Gs−2 |∇G|2 dA < ∞,

and

𝔻

𝔻

see page 354. This shows that we can apply the dominated convergence theorem in (*) to conclude that the Riesz measure of Gs is absolutely continuous and equal to dμ := [(s − 1)Gs−2 |∇G|2 + Gs−1 ΔG] dA. Hence, as a consequence of the Riesz representation formula, r

d s 1 Ms (r, G) = ∫ dμ. dr 2π |z| 1, then the reverse of (∗) is a particular case (q = 1) of the dual of (†) and holds for all f ∈ H p , 1 < p < ∞. See (6.23).

412 | 11 One-to-one mappings 11.2. In view of Theorem 11.14 and the inclusions (6.20) and (6.19), we can ask whether 𝒰 ∩ Bp,2 = 𝒰 ∩ H p . The answer is affirmative if p < 1/2 because then H p ⊂ Bp,2 and 𝒰 ⊂ H p . However, if 1/2 ≤ p < 2, then the univalent function f (z) = ((1 − z) log

−1/p

2e ) 1−z

(11.45)

belongs to Bp,2 \ H p . If 2 < p < ∞, then we use the function f (z) = (1 − z)−2/p (log

−1

2ep/2 ) ; 1−z

see [208, Section 8]. The full proof that the function (11.45) is univalent, given in [57], is based particularly on the paper [182], where various important facts on the class 𝒰 can be found. 11.3 (Mappings of finite distortion and harmonic mappings). A mapping f : Ω 󳨃→ Ω󸀠 between subdomains of ℂ is said to have finite distortion if it is sufficiently regular1 in the Sobolev sense and there is a measurable function 1 ≤ K(z, f ) < ∞ such that |f 󸀠 (z)| ≤ K(z)Jf (z), where |f 󸀠 (z)| and Jf (z) are the norm of the derivative of f and the Jacobian, respectively. Let 1 1 𝕂(z, f ) = (K(z, f ) + ). 2 K(z, f ) Let ℱ (Ω, Ω󸀠 ) denote the class of homeomorphisms f : Ω 󳨃→ Ω󸀠 of finite distortion such that 𝕂(z, f ) is integrable on Ω. The following theorem was proved by Astala et al. [49]. Theorem (A). Let Ω be a convex domain, and let f0 ∈ ℱ (Ω, Ω󸀠 ). Then the minimization problem min ∫ 𝕂(z, f ) dA(z), f ∈ℱ

f = f0

on 𝜕Ω,

(11.46)

Ω

has a unique solution. This solution is a C ∞ -diffeomorphism whose inverse is harmonic in Ω󸀠 . See also [268], where the analogous minimization problem for weighted Lp -space is considered. Here we note that 𝕂(z, f ) =

|𝜕f (z)|2 + |𝜕f̄ (z)|2 . |𝜕f (z)|2 − |𝜕f̄ (z)|2

As a consequence of (A) and Theorem 11.18, we have the following [49, Theorem 11.27]: 1 See [49] for the details, which are not important here.

Further notes and results | 413

Corollary. Let Ω = Ω󸀠 = 𝔻 and f0 ∈ QC. Then the unique minimizer of problem 11.46 is quasiconformal if and only if f0 is bi-Lipschitz. 11.4. Abu-Muhanna [3] proved a few important results about the class UH. Let f = h + ḡ ∈ UH. Theorem (A-M). There is no upper bound on the valency of h: more precisely, h can be infinite-valent and such that h and f are convex in the real direction and the spherical area of h is finite. Further, the following relations hold, which extend Pommerenke’s theorem stated before as Corollary 11.12: h ∈ B ⇐⇒ h ∈ BMOA ⇐⇒ f ∈ B ⇐⇒ f ∈ BMO. Here B is the harmonic Bloch space. Analogous relations for the corresponding “little” spaces hold. If h ≠ 0 in 𝔻, then we have the equivalence log h ∈ B ⇐⇒ log h ∈ BMOA. Returning to Theorem (A-M), we recall that the spherical area of the covering surface of f is defined by As (f ) = ∫ 𝔻

Jf (z)

(1 + |f (z)|2 )2

dA(z),

where Jf is the Jacobian of f . In [3], the question is posed if the spherical area of h is always finite. Abu-Muhanna et al. [4, Theorem 4] proved that this is true under the additional hypotheses that f is quasiconformal and that the domain h(𝔻) is hyperbolic; the latter means that the complement of h(𝔻) contains at least two points. 11.5. The class SH0 ⊂ UH is defined by the requirements h(0) = g(0) = 0, h󸀠 (0) = 1, and ̂ g 󸀠 (0) = 0. It was conjectured that |a2 | := |h(2)| ≤ 5/2 for f ∈ SH0 . In [168], it is proved that |a2 | < 49. In [4, Theorem 1], this estimate is improved to |a2 | < 21. 11.6 (Area distortion). Abu-Muhanna and Louhichi [5] proved several interesting facts about area distortion under a harmonic univalent mapping of 𝔻 into 𝔻. In particular, for a subset E of 𝔻, they define ‖E‖ = inf{∑ rj2 : ⋃ H(aj , rj ) ⊃ E}, j

j

where H(aj , rj ) are pseudohyperbolic discs. If E = ⋃nj=1 H(aj , rj ) is a disjoint union, then ‖E‖ = ∑nj=1 rj2 . One of the results (Theorem 4) reads as follows. Theorem (A-ML). If f ∈ UH maps 𝔻 onto 𝔻, then m(f (E)) ≤ ‖E‖, where dm = dA/π.

414 | 11 One-to-one mappings Because m(r𝔻) = ‖r𝔻‖, this improves an earlier results of Koh and Kovalev [316], who proved that m(f (r𝔻)) ≤ m(r𝔻). 11.7. Let Ω be a Jordan domain with rectifiable boundary, and let γ(s) be the arc-length parameterization of 𝜕Ω. We say that 𝜕Ω is Dini smooth if γ is of class C 1 and the modulus of continuity ω of arg(γ 󸀠 ) satisfy the (Dini) condition L

∫ 0

ω(s) ds < ∞ (L = the length of 𝜕Ω). s

Kalaj [288] improved his own result from [286] by proving the following: Theorem (Ka). Let G and Ω be Jordan domains with Dini smooth boundaries. If f is a harmonic quasiconformal mapping from G onto Ω, then f is Lipschitz. 11.8 (A Schwarz–Pick inequality for harmonic mappings). The following intriguing result was proved by Kalaj [289]. Theorem (Ka2). If f is a harmonic orientation-preserving diffeomorphism of the unit disc 𝔻 onto a Jordan domain Ω with rectifiable boundary of length 2πR, then we have the sharp inequality R 󵄨 󵄨󵄨 , 󵄨󵄨𝜕f (z)󵄨󵄨󵄨 ≤ 1 − |z|2

z ∈ 𝔻.

(11.47)

If the equality in (11.47) is attained for some a, then Ω is convex, and there are a holomorphic function μ : 𝔻 → 𝔻 and a constant θ ∈ [0, 2π] such that F(z) := e

−iθ

z

z

0

0

μ(t)dt z+a dt +∫ ). f( ) = R(∫ 2 1 + z ā 1 + t μ(t) 1 + t 2 μ(t)

(11.48)

Moreover, every function f defined by (11.48) is a harmonic diffeomorphism and maps the unit disc to a Jordan domain bounded by a convex curve of length 2πR, and inequality (11.47) is attained for z = a. There are various interesting consequences of this theorem. One of them is a solution of the second-order Beltrami equation. Corollary. For every positive constant R and every holomorphic function μ of the unit disc into itself, there is a unique convex Jordan domain Ω = Ωμ,R with perimeter 2πR such that the initial boundary problem f (z) = μ(z)fz (z), { { z̄ f (0) = R, { {z f { (0) = 0 󳨀󳨀󳨀→ admits a unique univalent harmonic solution f = fμ,R : 𝔻 onto Ω.

Further notes and results | 415

11.9 (Intrinsic Hardy–Orlicz spaces). In Section 5.1, we gave a definition of Hardy– Orlicz spaces. If f is a univalent function, then denote by |f (z)|I the intrinsic path distance between f (z) and f (0) in Ω := f (𝔻): 󵄨 󵄨 dI (f (z), f (0)) = 󵄨󵄨󵄨f (z)󵄨󵄨󵄨I = inf length (γ), γ

where the infimum is taken over all curves γ ⊂ Ω with endpoints f (z) and f (0). Then f is said to belong to the intrinsic Hardy–Orlicz space HIΨ if 󵄨 󵄨 sup ∫ Ψ(δ󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨I )|dζ | < ∞

00 Ψ(2x) < Ψ(x)

∞, then HIΨ ∩ 𝒰 = H Ψ ∩ 𝒰 . As an application, they proved the following [324, Corollary 1.2]: Theorem (KB). If Ψ satisfies the Δ2 -condition, then a univalent function f belongs to H Ψ if and only if 1

󵄨 󵄨 ∫ Ψ(∫󵄨󵄨󵄨f 󸀠 (rζ )󵄨󵄨󵄨 dr)|dζ | < ∞. 𝕋

(11.49)

0

11.10. Kalaj and Saksman [292] substantially improved the main result of [291], where the solutions of Δf = g, g ∈ C(𝔻), were considered. One of their results reads as follows. Theorem (KS). Assume that g ∈ Lp (𝔻) and p > 2. If f is a K-quasiconformal solution of Δf = g that maps the unit disc onto a bounded Jordan domain Ω ⊂ ℂ with C 2 -boundary, then f is Lipschitz continuous. The result is sharp since it fails in general if p = 2. 11.11 (Convex harmonic mappings). A univalent harmonic function f = h + ḡ defined on 𝔻 is said to be convex if f (𝔻) is convex. A notable result of Clunie and Sheil-Small [128, Theorem 5.7] says that if f is convex, then h is univalent. (See Further notes 11.12.) It follows that h ∈ H p for p < 1/2, and consequently g ∈ H p for p < 1/2. Hence M∗ f ∈ Lp (𝕋) for p < 1/2. It was conjectured in Duren [168, Section 8.5] that the class of convex harmonic mappings is a subset of the harmonic Hardy class h1/2 . Aleman and Martín [31] constructed a counterexample to this conjecture. On the other hand, it is surprising that if f (𝔻) is convex and is not a half-plane, then f ∈ h1 ; this was proved by Abu-Muhanna and Schober [7] in the case where f (𝔻) is not a strip and by Grigoryan and Nowak [216] when f (𝔻) is a strip. An interesting phenomenon was observed by Maria Nowak [409]. The function f = h + g,̄ where 1 z z h(z) = ( ) + 2 1 − z (1 − z)2

1 z z and g(z) = ( ), − 2 1 − z (1 − z)2

416 | 11 One-to-one mappings is univalent and maps 𝔻 onto the half-plane Re z > −1/2. It belongs to h1/2 although neither h nor g belongs to h1/2 . This implies that the maximal function M∗ f does not belong to L1/2 (𝕋) since otherwise the conjugate function f ̃ = −i(h − g)̄ belongs to h1/2 and hence f + if ̃ = 2h ∈ h1/2 . See Theorem 2.27 by Burkholder et al. Such things cannot happen in the class of quasiconformal mappings because of a result of Zinsmeister [630] reproved in [50, Theorem 4.1]. Theorem (Z). If f : 𝔻 󳨃→ ℂ is a quasiconformal mapping and p ∈ ℝ+ , then the following conditions are equivalent: (a) sup0 0) are calculated by means of a characterization of the “solid hull” of H p . We use the decomposition method to discuss the p1 ,q1 multipliers from Bp,q . α to Bβ

12.1 Multipliers on abstract spaces Let A and B be two sets of sequences indexed either by nonnegative integers or by all integers. A sequence {μn } is said to be a multiplier from A to B if {μn an } ∈ B for all {an } ∈ A. The set of all multipliers from A to B is denoted by (A, B). This can be applied in the case of classical sequence spaces such as ℓp and also in the case of admissible spaces of analytic or harmonic functions. For instance, we can identify f ∈ H(𝔻) or f ∈ h(𝔻) with the sequence an = f ̂(n) and conversely: if a sequence {an } satisfies ∞ n n the condition lim supn→∞ √|a n | < ∞, then the function f (z) = ∑n=0 an z belongs to H(𝔻), so we can treat ℓp and other similar spaces as admissible spaces of analytic (or harmonic) functions. The most of results in harmonic analysis can be expressed in terms of coefficient multipliers. For instance, the Hardy–Littlewood–Sobolev theorem says that the sequence (n + 1)1/q−1/p is a multiplier from H p to H q , where 0 < p < q < ⬦. A less self-evident example is the following reformulation of a variant of the Littlewood–Paley theorem, that is, of the inclusion Bp,p ⊂ H p (p ≤ 2):

The sequence (n + 1)−1/p is a multiplier from the Bergman space Ap to H p for 0 < p ≤ 2.

We leave the proof as an exercise.1 The sequence σnα f can be viewed as a sequence of multiplier transforms. As a corollary of the corresponding Hardy–Littlewood theorem, we have that if p ≤ 1 and α > 1/p − 1, then this sequence is uniformly bounded in n on H p . To unify the language, we will formulate most statements only for analytic functions. If μ is a sequence and f ∈ H(𝔻), then we write ∞

(μ ∗ f ) = ∑ μn f ̂(n)z n . n=0

1 In the case 1 ≤ p ≤ 2, this was proved by MacGregor and Zhu [368] by using interpolation of linear operators. https://doi.org/10.1515/9783110630855-012

418 | 12 Multipliers on Hp , BMOA, and Besov spaces The following fact can be easily proved by using the closed graph theorem. Theorem 12.1. If μ is a multiplier from X to Y, where X and Y are admissible spaces, then the operator Tμ : f 󳨃→ μ ∗ f , f ∈ X, belongs to L(X, Y). In particular, if Y = X, then the sequence μ is bounded. As a consequence of the theorem, we have |μn | ≤ C‖en ‖Y /‖en ‖X , which along with n n (1.21) implies lim supn √|μ n | ≤ 1, so the function f (z) = ∑ μn z belongs to H(𝔻). Therefore the set of multipliers from X to Y can also be described via Hadamard product: (X, Y) = {g ∈ H(𝔻) : g ∗ f ∈ Y for all f ∈ X} with the quasinorm ‖g‖(X,Y) = sup‖f ‖X ≤1 ‖f ∗ g‖Y . Exercise 12.1. (H q , H p ) = ℓ∞ for ∞ ≥ q ≥ 2 ≥ p > 0. Proposition 12.2. If X and Y are admissible spaces, then so is (X, Y). If, in addition, one of X, Y is R-admissible,2 then so is (X, Y). Proof. Consider the case of analytic functions. Let 𝒞 (z) = (1 − z)−1 . Take g ∈ (X, Y) and observe that g ∗ 𝒞r = gr , and so M∞ (r 2 , g) ≤ A(r)‖gr ‖Y = A(r)‖g ∗ 𝒞r ‖Y ≤ A(r)‖g‖(X,Y) ‖𝒞r ‖X , where A(r) is independent of g. This shows that (X, Y) ⊂ H(𝔻), and the inclusion is continuous. The inclusion H(𝔻) ⊂ (X, Y) is not hard to prove. It remains to prove that (X, Y) is complete. Assume that ‖gm − gn ‖(X,Y) → 0 (m, n → ∞). This implies, by Theorem 12.1, that there exists an operator T ∈ L(X, Y) such that ‖gn ∗ f − Tf ‖Y → 0, which implies that gn ∗ f → Tf in H(𝔻). On the other hand, since the inclusion (X, Y) ⊂ H(𝔻) is continuous, we have gm − gn → 0 (m, n → ∞) in H(𝔻), and hence there exists g ∈ H(𝔻) such that gn ∗ f → g ∗ f in H(𝔻). Thus Tf = g ∗ f . The second statement is easy to prove. Many results of Chapter 6 can be extended to homogeneous spaces; for the definition, see Section 1.4.c. We call a space an HFP-space if it is H-admissible, homogeneous, and FP. The following proposition explains the term “Fatou property”. Proposition 12.3. A homogeneous space X is an FP-space if and only the following implication holds: (I) If fn is a sequence in X such that ‖fn ‖X ≤ 1 and fn 󴁂󴀱 f , then f ∈ X and ‖f ‖X ≤ 1. Proof. Let X be p-Banach. Condition (I) implies that X is an FP-space because if f ∈ H(𝔻) and supw∈𝔻 ‖fw ‖X ≤ 1, then frn 󴁂󴀱 f (rn ↑ 1), whence, by (I) and the homogeneity of X, we get f ∈ X and ‖f ‖X ≤ 1. 2 See Section 1.4.a.

12.1 Multipliers on abstract spaces | 419

To prove the converse, let fn ∈ X, ‖fn ‖X ≤ 1, and fn 󴁂󴀱 f . It follows that ‖(fn )w − fw ‖X → 0 (n → ∞) for all w ∈ 𝔻. Hence 󵄩p 󵄩p 󵄩 󵄩p 󵄩 󵄩 ‖fw ‖pX ≤ 󵄩󵄩󵄩(fn )w − fw 󵄩󵄩󵄩X + 󵄩󵄩󵄩(fn )w 󵄩󵄩󵄩X ≤󵄩󵄩󵄩(fn )w − fw 󵄩󵄩󵄩X + 1, and hence ‖fw ‖X ≤ 1 for all w ∈ 𝔻. Since X is homogeneous, we have ‖f ‖X ≤ 1. This completes the proof. The Abel dual as a space of multipliers The Abel dual X A was defined on page 189. We also use the notation X ∗ = (X, H ∞ ),

X # = (X, A(𝔻)).

(12.1)

Let 𝒜 denote the class of all Abel summable sequences, that is, of sequences {cn }∞ 0 such that the series s(r) := ∑n cn r n , 0 < r < 1, converges and the limit limr→1− s(r) is finite. The Abel dual X A of X coincides with (X, 𝒜). However, the following problem arises: The space 𝒜 is locally convex but is not normable; the topology of 𝒜 is the intersection of the topologies of H(𝔻) and C[0, 1] and can be given by the family of norms 󵄨 󵄨 󵄨 󵄨 pr (f ) = sup󵄨󵄨󵄨f (z)󵄨󵄨󵄨 + max 󵄨󵄨󵄨f (ρ)󵄨󵄨󵄨 |z| 0,

is bounded in the sense of the theory of linear topological spaces. This means that, for all r and ε, there are a sequence {fn } ⊂ Vr,ε and a seminorm pρ such that limn pρ (fn ) = ∞. We can take fn to be the partial sums of the Taylor series of f (z) = δ/ρ + z, where δ is small enough. In turns out, however, that for any admissible space X, the set X A becomes a Banach space when endowed with an appropriate norm. To confirm this, observe first that an standard application of the Banach–Steinhaus principle shows that there is a constant C independent of f such that ‖{cn }‖X A ≤ C‖f ‖X . This gives, in n particular, |cn | ≤ C‖en ‖X , and hence by (1.21) the function g(z) = ∑∞ n=0 cn z is analytic A in 𝔻. Thus we can define X as X A = {g ∈ H(𝔻): lim− f ∗ g(r) exists and is finite}. r→1

The norm is given by 󵄨 󵄨 ‖g‖X A = sup{󵄨󵄨󵄨f ∗ g(r)󵄨󵄨󵄨: ‖f ‖X ≤ 1, 0 < r < 1}. It is not too difficult to prove that then X A is complete, but we will not do this because this fact is irrelevant for our aims. Here we collect some important although simple properties.

420 | 12 Multipliers on Hp , BMOA, and Besov spaces Proposition 12.4. Let X and Y be H-admissible spaces. Then the following statement hold. (i) If X is minimal and homogeneous, then X A ≅ X # ≅ X ∗ , and X A is HFP. (ii) If X is minimal, then (X, Y) ≅ (X, YP ). (iii) If X is homogeneous and Y is HFP, then (X, Y) ≅ (XP , Y) ≅ (XP , YP ), and, in particular, X ∗ ≅ (XP )∗ . (iv) A homogeneous Banach space X has FP-property if and only if X ∗∗ = X. Furthermore, if X is HFP, then X ∗∗ ≅ X. (v) If X and Y are HFP-spaces, then (X, Y) ≅ (Y ∗ , X ∗ ). Statement (i) says in particular that if f ∈ X and g ∈ X A , then the function g ∗ f extends to a continuous function on 𝔻. This can be applied to X = H p (p < 1) and Y = B1/p−1 (by Theorem 6.12). Proof. Statements (i), (ii), and (iii) are almost trivial, whereas (iv) can be easily deduced from the relation ‖fw ‖X = ‖fw ‖X ∗∗ , |w| < 1, which is obtained by means of (i). Finally, we have (X, Y) ⊂ (Y ∗ , X ∗ ) ⊂ (X ∗∗ , Y ∗∗ ) = (X, Y). From this, (iv), and the fact that the first two inclusions are isometrical we get (v). In general, statement (i) does not hold for nonminimal spaces. For example, it is a delicate result of Piranian et al. [479] that (H ∞ )A = 𝒦a , where, we recall, 𝒦 is the space of Cauchy transforms, and Ka is the “absolutely continuous part” of 𝒦. On the other hand (H ∞ )∗ = (A(𝔻))∗ = 𝒦 by Proposition 12.4(iii). Preduals and the second duals It is known that a dual space can have different preduals; see, for example, [211]. We will not consider this question. Instead, we show how to identify a predual of a minimal space in some important cases. Theorem 12.2. If X is a minimal Banach HFP-space, then Y A ≅ X, where Y = (X A )P . Proof. Let f ∈ Y and g ∈ X. Since Y ⊂ X A , we see that ⟨f , g⟩ exists and is finite and that |⟨f , g⟩| ≤ ‖f ‖Y ‖g‖X , which implies ‖g‖X ≥ ‖g‖Y A . Let g ∈ Y A . In view of Theorem 6.9 and the fact that X is minimal, we have, for 0 < ρ < 1, 󵄨 󵄨 ‖gρ ‖X = sup{󵄨󵄨󵄨⟨f , gρ ⟩󵄨󵄨󵄨 : f ∈ X A , ‖f ‖X A ≤ 1} 󵄨 󵄨 = sup{󵄨󵄨󵄨⟨fρ , g⟩󵄨󵄨󵄨 : f ∈ X A , ‖f ‖X A ≤ 1} 󵄨 󵄨 ≤ sup{󵄨󵄨󵄨⟨fρ , g⟩󵄨󵄨󵄨 : f ∈ X A , ‖fρ ‖X A ≤ 1} ≤ ‖g‖Y a , where the relations fρ ∈ (X A )P and ‖fρ ‖X A ≤ ‖f ‖X A have been used. Now the F-property and the homogeneity of X give g ∈ X and ‖g‖X ≤ ‖g‖Y A . This concludes the proof.

12.1 Multipliers on abstract spaces | 421

Theorem 12.3. Let X be a Banach HFP-space. Suppose that the space (XP )A is minimal. Then the second dual of XP is isometrically isomorphic to X. More precisely, we have ((XP )A )A ≅ X. Proof. Let Z = Y A , where Y = (XP )A . Let f ∈ H(𝔻). Then fw ∈ X ∩ Z for |w| < 1, and 󵄨 󵄨 ‖fw ‖Z = sup{󵄨󵄨󵄨⟨fw , g⟩󵄨󵄨󵄨 : g ∈ Y, ‖g‖Y ≤ 1}. Hence, by the Hahn–Banach theorem and the analytic variant of Theorem 6.9, ‖fw ‖Z = ‖fw ‖XP = ‖fw ‖X . Since both Z and X are HFP-spaces, we have that f ∈ Z if and only if f ∈ X. Also, ‖f ‖Z = ‖f ‖X because X and Z are HFP. AA ≅ Bp,∞ Corollary 12.5. If p ≥ 1, then (Bp,⬦ α . α )

Abstract Besov spaces Let X be a homogeneous H-admissible space. We define the “X-Besov” space BqX,α = {f : . . .} by the requirement that the function (1 − r 2 )s−α ‖𝒥 s fr ‖X , 0 < r < 1, belongs to Lq−1 for some s > α. It follows from Lemmas B.7 and B.8 (see Remark B.9) that ‖Vn ∗ f ‖X ≤ CX ‖f ‖X and 󵄩 󵄩 cr A2 2ns ‖Vn ∗ f ‖X ≤ 󵄩󵄩󵄩Vn ∗ 𝒥 s fr 󵄩󵄩󵄩X ≤ Cr a2 2ns ‖Vn ∗ f ‖X , n

n

where c, C, a, A are positive constant independent of f , n ≥ 1, and r ∈ (0, 1). Then, arguing as in the case of Bp,q α , we obtain BqX

P ,α

= BqX,α = Vαq [X] = Vαq [XP ].

(12.2)

The first and last equalities hold because ‖fr ‖X = ‖fr ‖XP and ‖Vn ∗f ‖X = ‖Vn ∗f ‖XP . The space BqX,α is homogeneous for all q, p, and α and is minimal if and only if q ≤ ⬦; it has the FP-property if and only if q ≠ ⬦. By Theorem 6.9 the dual of BqX,α , q ≤ ⬦, is isomorphic to its Abel dual. According to Theorem 6.10, relations (12.2), and Proposition 12.4, the following theorem holds. Theorem 12.4. If X is homogeneous and q ≤ ⬦, then (BqX,α )A ≃ Bq(X 󸀠

(BqX,α )∗

≃

󸀠 Bq(X )∗ ,−α P

for all q ≤ ∞.

P)

A ,−α

, and also

This can be used to give a somewhat simpler proof of Theorem 6.11 in the case q p,⬦ where p ≤ 1 or p = ∞. Namely, we have Bp,q for p < 1, α = Vα [X], where X = B X = B1,1 for p = 1, and X = B∞,1 for p = ∞. It is relatively easy to determine the Abel dual of these space (see the proof of Theorem 7.14). 12.1.1 Compact multipliers A multiplier g ∈ (X, Y) is said to be compact if the multiplier transform g ∗ : X 󳨃→ Y, g ∗ f = g ∗ f , is compact, that is, if g ∗ maps B(X), the unit ball of X, into a totally

422 | 12 Multipliers on Hp , BMOA, and Besov spaces bounded subset of Y. We recall that a metric space (S, d) is totally bounded if for each ε > 0, there is a finite set F ⊂ S such that, for every g ∈ S, we have d(g, f ) < ε for some f ∈ F. Lemma 12.6. (a) A subset S of an R-admissible space Y is totally bounded if (1) S is bounded and (2) sup ‖g − gr ‖Y → 0 as r → 1− . g∈S

(12.3)

(b) If Y is minimal and S ⊂ Y is totally bounded, then (12.3) holds. Proof. (a) Using Montel’s theorem and the definition of admissible spaces, we prove that, for a fixed r < 1, the set {gr : g ∈ B(Y)} is totally bounded. From this and from (12.3) it follows that S is totally bounded. The proof of (b) is straightforward. We note that the hypothesis that Y is minimal cannot be dropped. For instance, take f ∈ H ∞ \ A(𝔻). Then S = {f } is totally bounded, but (12.3)(2) does not hold. Denote by κ(X, Y) the space of all compact multipliers from X to Y. The following theorem shows that describing κ(X, Y) is practically equivalent to describing (X, Y). Theorem 12.5. If X and Y are R-admissible spaces, then κ(X, Y) = (X, Y)𝒫 . In particular, (X, Y) is minimal if and only if κ(X, Y) = (X, Y). Proof. Assuming that g ∈ ZP , where Z = (X, Y), we have ‖g − gr ‖Z = sup ‖g ∗ f − g ∗ fr ‖Y → 0 (r → 1− ). f ∈B(X)

By Lemma 12.6 this means that the set g ∗ (B(X)) is totally bounded in Y. This proves the inclusion ZP ⊂ κ(X, Y). Assuming that g ∈ κ(X, Y) \ ZP , we find ε > 0, a sequence fn ∈ B(X), and a sequence rn → 1− such that (a) ‖fn ∗ (g − grn )‖Y ≥ ε for all n. Since the sequence hn := fn ∗ (g − grn ) belongs to the totally bounded set g ∗ (B(X)) − g ∗ (B(X)) ⊂ Y, there is a subsequence, denote it again by hn , that converges to some h ∈ Y in the norm ̂ as n → ∞ for all j. It folof Y. This implies that (1/2)j fn̂ (j)[g(j) − g(j)rnj ] → (1/2)j h(j) j ̂ lows that h = 0 because (1/2) |fn (j)| ≤ C‖fn ‖X ≤ C, where C is independent of n. This contradicts (a) and proves the theorem.

12.2 Multipliers for Hardy and Bergman spaces In this section, we apply the Coifman–Rochberg theorem (Theorem 3.16) to describe the set (Ap , Y), 0 < p ≤ 1, where Y is a q-Banach space with q ≥ p, and also the set (H p , Y) with q > p, p < 1.

12.2 Multipliers for Hardy and Bergman spaces | 423

Let p ∈ ℝ+ and β > −1. The (already defined) Bergman space Apβ as a set is equal to Hαp,p , where α = (β + 1)/p. However, it is useful to introduce a different quasinorm, namely ‖f ‖pp,β = ‖f ‖pAp = β

β + 1 󵄨󵄨 β 󵄨p ∫󵄨󵄨f (z)󵄨󵄨󵄨 (1 − |z|2 ) dA(z). π 𝔻

One of reasons is that limβ→−1 ‖f ‖p,β = ‖f ‖p for f ∈ H p . If β = 0, then we write Ap = Apβ . Proposition 12.7. Let 0 < p ≤ 1 and β > −1, and let Y be an R-admissible p-Banach space. Then (Apβ , Y) = B∞ Y,(β+2)/p−1

κ(Ap , Y) = B⬦ Y,(β+2)/p−1 .

and

We note that B∞ Y,(β+2)/p−1 can be described by ‖Vn ∗ f ‖Y ≤ C2n(1−(β+2)/p) and that Vn ∗ f can be replaced by Δn f if {en } is a Schauder basis (see p. 467) in YP and Y is a Banach space; for example, H q (1 < q < ⬦), Aq (1 < q < ⬦), and so on. Proof. First, recall that the fractional derivative D[s] h is defined by D[s] h(z) =

∞ Γ(n + s + 1) ̂ 1 h(n)z n ; ∑ Γ(s + 1) n=1 Γ(n + 1)

see Remark 6.29. Let g ∈ (Apβ , Y). Let f (z) = (1 − z)−s−1 , where s > 0 is sufficiently large. Then D[s] (gw ) = g ∗ fw , and therefore

󵄩󵄩 [s] 󵄩󵄩p p p 󵄩󵄩D gw 󵄩󵄩Y = ‖g ∗ fw ‖Y ≤ C‖fw ‖p,β = (β + 1)p ∫ − 𝔻

(1 − |z|2 )β β+2−(s+1)p dA(z) ≤ C(1 − |w|) , −(s+1)p |1 − wz|

where we have chosen s so that (s + 1)p > β + 2. This proves the inclusion (Apβ , Y) ⊂ B∞ Y,(β+2)/p−1 . To prove the reverse inclusion, we use Theorem 3.16. Let f ∈ Apβ and g ∈ B∞ Y,(β+2)/p−1 . Then f can be represented as in (3.30) with γ = 1/p. We have ∞

1/p [t]

(f ∗ g)(z) = ∑ an (1 − |wn |2 ) n=1

D g(w̄ n z),

z ∈ 𝔻,

where t = 1/p + (β + 2)/p − 1, and ‖{an }‖p ≤ C‖f ‖p . It follows that ∞

󵄩p 󵄩 ‖f ∗ g‖pY ≤ ∑ |an |p (1 − |wn |2 )󵄩󵄩󵄩D[t] gw̄ n 󵄩󵄩󵄩Y n=1

424 | 12 Multipliers on Hp , BMOA, and Besov spaces ∞

≤ C ∑ |an |p (1 − |wn |2 )(1 − |wn |2 )

(β+2)−p−tp

n=1

∞

= C ∑ |an |p . n=1

This completes the proof of the first relation. The second relation follows from the first and Theorem 12.5. Using Corollary 3.37, Proposition A.5, and Proposition 12.7, we obtain a characterization of (H p , Y). Proposition 12.8. If 0 < p < q ≤ 1 and Y is an R-admisssible q-Banach space, then (H p , Y) = B∞ Y,1/p−1

and

κ(H p , Y) = B⬦ Y,1/p−1 .

It is useful to observe that the proposition is a formal particular case (β = −1) of Proposition 12.7 and the formal equality H q = Aq−1 .

Remark 12.9. As we see, the proof of the inclusion (Apβ , Y) ⊂ B∞ Y,(β+2)/p−1 depends neither on Y nor on the atomic decomposition. Thus this inclusion holds for all p ∈ (0, ⬦), β ≥ −1, and all R-admissible space Y. A little more work is needed to prove that κ(Apβ , Y) ⊂ B⬦ Y,(β+2)/p−1 . As the first example, we prove the following theorem. and κ(Ap , Aq ) ≃ Theorem 12.6. If 0 < p ≤ 1 and q ≥ p, then (Ap , Aq ) ≃ Bq,∞ 2/p−1−1/q

q,⬦ . B2/p−1−1/q

p q ⬦ Proof. By Proposition 12.7 we have (Ap , Aq ) = B∞ Aq ,2/p−1 and κ(A , A ) = BAq ,2/p−1 . In

view of (12.2), this means that f ∈ (Ap , Aq ) if and only if ‖Vn ∗ f ‖Aq ≤ C2n(1−2/p) . Now we use Lemma 2.6 to show that ‖Vn ∗ f ‖Aq ≍ 2−n/q ‖Vn ∗ f ‖q . The result follows.

, ∗) is a unital p-Banach algebra. Corollary 12.10. If 0 < p ≤ 1, then (Bp,∞ 1/p−1 This follows from the previous theorem and the general fact that (X, X) is a p-Banach algebra (with the function 1/(1 − z) as a unit) for an arbitrary admissible space X. 2,∞ . = ℓ2/p−1/2 Corollary 12.11. If 0 < p ≤ 1, then (Ap , A2 ) = B2,∞ 2/p−1/2

As an application of Proposition 12.8 and the canonical isomorphism between q B∞ q H ,1/p−1 and the Lipschitz space HΛ1/p−1 , we have the following theorem. Theorem 12.7. Let 0 < p < 1 and p < q ≤ ∞. Then (H p , H q ) ≃ HΛq1/p−1 and κ(H p , H q ) = q Hλ1/p−1. Note that choosing q = ∞, we get the Duren–Romberg–Shields theorem (Theorem 6.12). Problem 12.15. What is (H q , H p ) for 0 < p < q < 2? This question is interesting because (H q , H p ) is independent of p < q and in fact by the Nikishin–Stein theorem C.17,

12.2 Multipliers for Hardy and Bergman spaces | 425

(H q , H p ) = (H q , H q,⋆ ), where H q,⋆ is the weak H q -space; H q,⋆ can be defined by the requirement suprn>0󵄩 󵄩󵄩k=n 󵄩∞ 2⊖s,q⊖s s Exercise 12.26. If 2 ≤ p ≤ ∞, then (Bp,q . α , ℓ ) = ℓ−α

12.4.1 Monotone multipliers p ,q

1 1 It is probably not true that the space (Bp,q ), where 1 < p ≤ p1 < ⬦, can be α , Bβ described (except in some particular cases) within the class of known spaces.

Theorem 12.18. Let 1 ≤ p < p1 < ⬦, and let {λn }∞ 0 be a decreasing sequence of posp1 ,q itive real numbers. (i) This sequence is a multiplier from Bp,q if and only if α into Bα λn = 𝒪(n1/p1 −1/p ) (n → ∞). (ii) The sequence is a compact multiplier if and only if λn = o(n1/p1 −1/p ) (n → ∞). p ,q

∞ p p1 1 Proof. By Theorem 6.4 and Proposition 12.19, (Bp,q α , Bα ) = V [(H , H )]. So we have to prove that, under the conditions of the theorem,

sup ‖Vn ∗ g‖(H p ,H p1 ) < ∞, n

̂ where g(n) = λn .

To do this, we use the Young inequality (Theorem C.4): ‖Vn ∗ g‖(H p ,H p1 ) = sup{‖Vn ∗ g ∗ f ‖p1 : ‖f ‖p ≤ 1} ≤ ‖Vn ∗ g‖ϱ ,

(12.7)

where 1 + 1/p1 = 1/ϱ + 1/p. Since λn decreases in n and {en } is a Schauder basis in H ϱ (observe that ϱ > 1), we see that ‖Vn ∗ g‖ϱ ≤ Cλ2n−1 ‖Vn ‖ϱ ≍ λ2n−1 2n(1−1/ϱ) = Cλ2n−1 2n(1/p−1/p1 ) ≤ C.

12.4 Multipliers between Besov spaces | 435

̂ In the other direction, assume that {λn } = g(n) is a multiplier. Then ≍ 2nα ‖Vn ∗ g‖p . 2nα ‖Vn ∗ g‖p1 ≍ ‖Vn ∗ g‖Bαp1 ,q ≤ C‖Vn ∗ g‖Bp,q α Hence λ2n+1 2n(1−1/p1 ) ≍ λ2n+1 ‖Vn ‖p1 ≤ C‖Vn ‖p ≍ 2n(1−1/p) . Statement (i) follows. To prove (ii), note first that {λn } is compact if and only if {λn } ∈ Z := V ⬦ [(H p , H p1 )]. Then the sufficiency proof of (i) shows that the “if” part of (ii) holds. The following chain of the inequalities shows that the “only if” part holds: ‖λ‖Z = sup ‖Vn ∗ λ‖(H p ,H p1 ) ≥ sup ‖Vn ∗ Pn ∗ λ‖p1 /‖Pn ‖p n

n(1/p−1)

n

≥ c sup ‖Vn ∗ λ‖p1 2 n

≥ cλ2n+1 ‖Vn ‖p1 2n(1/p−1)

≥ cλ2n+1 2n(1/p−1/p1 ) . Here Pn = Vn−1 + Vn + Vn+1 . Theorem 12.18(i) can be deduced from Theorem 6.1 and Lemma 2.7, but we gave the above proof intentionally. The theorem remains true if we assume that p ≤ 1 < p1 . This can be verified by means of the following lemma. Lemma 12.27 (Convolution lemma [422]). If f ∈ H p and g ∈ H q , where 0 < p ≤ 1 and p ≤ q ≤ ∞, then Mq (r, f ∗ g) ≤ (1 − r)1−1/p ‖f ‖p ‖g‖q ,

0 < r < 1.

Consequently (by Lemma 2.6), ‖Vn ∗ λ ∗ f ‖q = ‖Pn ∗ λ ∗ Vn ∗ f ‖q ≤ C2n(1/q−1) ‖Pn ∗ λ‖q ‖Vn ∗ f ‖p . Proof. We need a particular case of Corollary 1.32: 1−1/p

M1 (r, F) ≤ (1 − r 2 )

‖F‖p ,

F ∈ H p , 0 < p < 1.

(12.8)

Assuming, as we may, that f and g belong to H(𝔻), we have h(r 2 w) := f ∗ g(r 2 w) = ∫ − f (r ζ ̄ )g(rζw)|dζ |, 𝕋

̄ is analytic. From this and (12.8) and hence |h(r 2 w)| ≤ M1 (r, F), where F(z) = f (z)g(zw) we infer 󵄨p 󵄨 󵄨󵄨h(r 2 w)󵄨󵄨󵄨p ≤ ‖F‖p = ∫ − 󵄨󵄨󵄨f (ζ ̄ )g(ζw)󵄨󵄨󵄨 |dζ |. p 󵄨 󵄨

1−p 󵄨

(1 − r 2 )

𝕋

436 | 12 Multipliers on Hp , BMOA, and Besov spaces Integration of this inequality over w ∈ 𝕋 along with the Lq/p -Minkowski inequality gives (1 − r 2 )

1−p

Mpp (r 2 , h) ≤ ‖f ‖pp ‖g‖pq .

The result follows. Generalization of the Hardy–Littlewood–Sobolev theorem Theorem 12.19. Let 1 ≤ p < p1 < ⬦, and let λn be as in Theorem 12.18. Then (i) {λn } is a multiplier from H p to H p1 if and only if λn = 𝒪(n1/p1 −1/p ) (n → ∞); and (ii) {λn } is a compact multiplier from H p to H p1 if and only if λn = o(n1/p1 −1/p ) (n → ∞). Proof. (i) The “only if” part is proved as in the case of Theorem 12.18. Let {λn } be decreasing and positive. Consider first the case p < 2 ≤ p1 . From the inclusions Bp,2 ⊂ H p and Bp1 ,p1 ⊃ H p1 it follows that (H p , H p1 ) ⊃ (Bp,2 , Bp1 ,p1 ) = V ∞ [(H p , H p1 )]. Then, by the proof of Theorem 12.18, {λn } ∈ (H p , H p1 ). p,p Let 1 ≤ p < p1 ≤ 2. Then Bp1 ,p1 ⊂ H p1 and H p ⊂ Bα 1 , where α = 1/p1 − 1/p; p,p see (6.22). Hence (H p , H p1 ) ⊃ (Bα 1 , Bp1 ,p1 ). Now we use the decomposition theorem p,p (Theorem 6.1) together with inequality (12.7) to prove that {λn } is a multiplier from Bα 1 p1 ,p1 into B , which concludes the proof when p < p1 ≤ 2. If 2 < p < p1 , then the conclusion is obtained from the preceding case by a duality argument. (ii) The “if” part is proved as in the case of (i). In proving the “only if” part, we use Theorem 7.17. In a routine manner, we prove that the operator Tf = λ ∗ f satisfies condition (7.13). Take fn = Vn /‖Vn ‖p . Using Lemma 2.6, we prove that fn 󴁂󴀱 0. On the other hand, ‖Tfn ‖q1 ≥ cλ2n+1 ‖Vn ‖q1 /‖Vn ‖p ≍ 1. The result follows. Exercise 12.28 ([165]). If λn is an arbitrary complex sequence such that λn = 𝒪(n1/q−1/p ), n → ∞, where 0 < p < 2 ≤ q < ⬦, then {λn } is a multiplier from H p to H q . See (5.16) and (5.17). Exercise 12.29. If q < p < ⬦, then every multiplier from Ap to Aq is compact. This can be deduced from Proposition 12.19. In [103], this was deduced from the deep fact that every bounded operator from ℓp to ℓq is compact and that Ap is isomorphic to ℓp ; see Theorems A.3 and A.7.

12.5 Some applications to composition operators The problem of characterizing composition operators that act from an admissible space X into a weighted H ∞ -space or a weighted Bloch space can be reduced to the problem of characterizing a special kind of coefficient multipliers. To fix ideas, we first consider a composition operator 𝒞φ : X 󳨃→ B. This operator maps X into B if and

12.5 Some applications to composition operators | 437

only if 󵄨󵄨 󵄨󵄨 ∞ 󵄨 󵄨󵄨 󵄨󵄨 ∑ f ̂(n)nφ(z)n−1 φ󸀠 (z)󵄨󵄨󵄨 ≤ C‖f ‖X (1 − |z|2 )−1 , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨n=0

z ∈ 𝔻,

where C depends neither on f nor on z. Thus the family of multiplier transforms defined by Mφ,z (n) = nφ(z)n−1 φ󸀠 (z)(1 − |z|2 ),

z ∈ 𝔻,

is uniformly bounded in X A . If X is homogeneous, then we can say somewhat more. Proposition 12.30. If X is homogeneous, then Cφ maps X into B if and only if the family of multiplier transforms defined by 󵄨 󵄨n−1 󵄨 󵄨 M|φ|,z (n) = n󵄨󵄨󵄨φ(z)󵄨󵄨󵄨 󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨(1 − |z|2 ) is uniformly bounded in X ∗ . Moreover, we have ‖𝒞φ ‖ = supz∈𝔻 ‖M|φ|,z ‖X ∗ . For a fixed z, the sequence M|φ|,z is a product of an increasing and a decreasing sequence with special properties. Using this fact, we can characterize composition operators from H p to B for all p < ⬦. We will consider the case p = 1, leaving the remaining cases as an exercise. Theorem 12.20. The operator 𝒞φ maps H 1 into B if and only if 󵄨 󵄨n−1 󵄨 󵄨 sup n2 󵄨󵄨󵄨φ(z)󵄨󵄨󵄨 󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨(1 − |z|2 ) < ∞.

n≥1, z∈𝔻

Proof. By Proposition 12.30 we have to compute ‖M|φ|,z ‖BMO . Since 󵄨 ̂ 󵄨󵄨 ‖g‖BMO ≤ C sup (n + 1)󵄨󵄨󵄨g(n) 󵄨󵄨, n

we have 󵄨 󵄨n−1 󵄨 󵄨 ‖𝒞 φ‖ ≤ C‖M|φ|,z ‖BMO ≤ C sup n2 󵄨󵄨󵄨φ(z)󵄨󵄨󵄨 󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨(1 − |z|2 ). n

On the other hand, ‖M|φ|,z ‖BMO ≥ c‖M|φ|,z ‖B ≥ c sup ‖Vk ∗ M|φ|,z ‖∞ k≥0

󵄨 󵄨2k+1 󵄨 ≥ c sup 2 󵄨󵄨φ(z)󵄨󵄨󵄨 󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨(1 − |z|2 )‖Vk ‖∞ k

k 󵄨󵄨

󵄨 󵄨2k+1 󵄨 󵄨 ≥ c sup 22k 󵄨󵄨󵄨φ(z)󵄨󵄨󵄨 󵄨󵄨󵄨φ󸀠 (z)󵄨󵄨󵄨(1 − |z|2 ). k

The result follows.

(12.9)

438 | 12 Multipliers on Hp , BMOA, and Besov spaces Observe that condition (12.9) can be written as supn n‖φn ‖B < ∞. More generally, 𝒞φ maps H p into B (p ≤ 1) if and only if 󵄩 󵄩 sup n1/p 󵄩󵄩󵄩φn 󵄩󵄩󵄩B < ∞. n

(12.10)

In the case p > 1 the corresponding condition reads 󵄨 −1/p−1 󵄨󵄨 󸀠 󵄨󵄨 󵄨 2 −1 (1 − 󵄨󵄨󵄨φ(z)󵄨󵄨󵄨) 󵄨󵄨φ (z)󵄨󵄨 ≤ C(1 − |z| ) ,

(12.11)

which works for p ≤ 1 as well [476]. However, a simple analysis shows that the last two conditions are equivalent. Therefore, we have the following: Theorem 12.21. The operator 𝒞φ maps H p (p ∈ ℝ+ ) into B if and only if one of conditions (12.10) and (12.11) is satisfied. Exercise 12.31. It might be interesting to consider composition operators from Bp,q α into H ∞ , B, or HΛβ , 0 < β < 1.

Further notes and results Various conditions on spaces of analytic functions were considered by Taylor in [580]; a wide list can be found in [89]. Our requirements for a space to be homogeneous or FP are somewhat stronger than that in [89]. Proposition 12.8 is in fact a particular case of Kalton’s theorem (to be stated in Chapter B) on linear operators from H p (0 < p < 1) to an arbitrary q-Banach space p < q ≤ 1. Blasco [80, 81] was the first who used Kalton’s ideas in considering coefficient multipliers, although he considered the case of Banach spaces, when the atomic decomposition need not be used. The reader should also consult his papers [82] and [83] for various results on mixed norm spaces. The inclusion Λq1/p−1 ⊂ (H p , H q ) (of Theorem 12.7) when p < 1 ≤ q was stated without proof by Hardy and Littlewood [237, 239]. Duren and Shields [171] proved the reverse inclusion for q ≥ 1; the general case was discussed in [385]. Theorem 12.8 is due to Nowak [408]. All these results were proved in a way that differs from that used in the text. Theorem 12.10 is only a reformulation of the relation (H 1 , H p ) = Bp,∞ , 2 ≤ p < ⬦, due to Hardy and Littlewood [240] (the inclusion (H 1 , H p ) ⊃ Bp,∞ ) and Stein and Zygmund [555] (the reverse inclusion). It is trivial that (H q , H p ) = ℓ∞ for q ≥ 2 ≥ p. As far as the author knows, the set (H p , H q ) (0 < p, q < ⬦) in other case has not been determined yet. (Various sufficient conditions are known and are of great importance; see Stein [551, 552].) In particular, it is not known what is (H p , H q ) for 2 > p ≥ q > 0. The notion of a solid space and the notation s(X) and S(X) were introduced by Anderson and Shields [39]. Theorems 12.13 and 12.14 were proved in [274]; see also Mastyło and Mleczko [380] and Further notes 12.2. In the case q ≥ p, Theorem 12.15

Further notes and results | 439

was proved by Duren and Shields [171], and in the general case by Jevtić and Pavlović [273]. Theorem 12.17 is a reformulation of [421, Theorem 4.1]; in the case where p ≤ 1 ≤ p1 and q = q1 , it appears in [83, Theorem 5.4]. Theorem 12.19 is perhaps new. As far as the author knows, the only paper in which monotone multipliers are considered is that of Buckley et al. [102]. Also, there are only a few papers devoted to compact multipliers; we note Buckley et al. [103] and Mleczko [392]. Theorem 12.5 and some of its consequences are new. We recommend to the reader the book of Persson and Popa [477] for the theory of Schur multipliers, which is, among various generalizations of scalar multipliers, the closest to the content of this chapter. See also the paper [488], where the tensor product of Banach spaces of infinite matrices is considered. 12.1. Using Theorem 3.16, we prove that if 0 < p, q ≤ 1, α > 0, and max{p, q} ≤ ϱ ≤ 1, ϱ ϱ,ϱ then the ϱ-Banach envelope of Hαp,q is equal to Hα+1/p−1/ϱ = Aβ , where β = ϱα + ϱ/p − 2. Therefore, by Proposition 12.7, (Hαp,q , Y) = B∞ α+1/p−1 (Y) for a ρ-Banach space Y. This was proved by Blasco [83, Theorem 3.2] for ϱ = 1. This paper and Yue’s paper [622] contain theorems on multipliers from Hαp,q into H s or As . Blasco proved his results for s ≥ 1, whereas Yue required only that s ≥ max{p, q}. 12.2. Bonet and Taskinen [93] proved that the solid hull of the space X ⊂ H(𝔻) defined by 1 󵄨 󵄨󵄨 ) 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ C exp( 1 − |z| consists of the complex sequences {bm }∞ m=0 for which (n+1)4

sup exp(−2n ) ∑ |bm |2 (1 − 2

n≥1

m=n4 +1

2m

1 ) n2

< ∞.

This is a particular case of a more complicated result; see [93, Theorem 2.2]. See [94] and [92] for further investigation in this direction. 12.3. Nawrocki [400] proved that the solid hull of the Nevanlinna class consists of the sequences {bn } for which sup |bn | exp(−c√n) < ∞ n

for some c > 0,

(†)

whereas the solid hull of the Smirnov class is described by replacing “some” with “all” in (†). (See Subsection A.4.1 for the definition and some properties of this class). 12.4 (Hadamard product and Qp spaces). Consider the following statement: (A) If f ∈ X and g ∈ Y, then f ∗ g ∈ Qp . (For the definition of Qp , see Further notes 7.2.)

440 | 12 Multipliers on Hp , BMOA, and Besov spaces [437, 607] If 0 < p < 1, then (A) holds in the following cases: (i) X = Bq1/q and Y = B1,∞ 0

for q < 2/(1 − p); (ii) X = Bq1/q and Y = H 1 for q = 2/(1 − p); (iii) X = Qp and Y = Bs,∞ 0 for s = 2/(2 − p); and (iv) X = Qp and Y = B.

Observe that the function g(z) = (1 − z)−1 belongs to B1,∞ 0 , and therefore, by (i), ⊂ Qp (⊂ BMOA) for q < 2/(1 − p), which is well known (cf. [606, Theorem 4.2.1]).

Bq1/q

12.5 (Hardy–Lorentz spaces). For a measurable function g on 𝕋, the functional ‖g‖p,q , p, q ∈ ℝ+ , is defined by ∞

q/p q−1

‖g‖p,q = (p ∫ μ(g, λ)

λ

1/q

dλ)

,

0

where μ(g, λ) = |{ζ ∈ 𝕋: |g(ζ )| > λ}|/2π. The Lorentz space Lp,q consists of those g for which ‖g‖p,q < ∞. The Hardy–Lorentz space HLp,q is defined in Lengfield’s paper [345] as the intersection of Lp,q and the Smirnov class 𝒩+ . If p = q, then HLp,q = H p . In [345], various interesting facts (not only) about multipliers of HLp,q can be found. Furthermore, the list of references is very rich and contains, for example, papers where interpolation theorems for HLp,q and Bp,q were proved; these theorems play a substantial role in Lengfield’s paper. On the other hand, the interpolation theorems used in [345] are formulated for Hardy–Lorentz spaces, MLp,q , defined by the requirement M∗ f ∈ Lp,q . We need some work to show that MLp,q = HLp,q ; see Section 2.9. Here we cite the inclusions [345, Theorem 4.1] p ,q

0 B1/p

0 −1/p

s,q , = Bs,q ⊂ HLp,q ⊂ H1/p−1/s 1/s−1/p

where p0 < p < s ≤ ∞ and 0 < q ≤ ∞. By Theorem 12.14 we have p ,q

0 S(B1/p

0 −1/p

∞,q ∞,q = ℓ1−1/p ) = ℓ1/p −1/p+1−1/p 0

0

and, similarly, ∞,q ∞,q ) = ℓ1/s−1/p+1−1/s S(Bs,q = ℓ1−1/p , 1/s−1/p

where for a fixed 0 < p < 1, we chose p0 < p < s ≤ 1. It follows that ∞,q , S(HLp,q ) = ℓ1−1/p

0 < p < 1, 0 < q ≤ ∞,

which reduces to Theorem 12.13 when q = p. See [276, p. 185], where this fact is stated and proved in a different language. Note that the usual definition of Lp,q goes via the decreasing rearrangement g ∗ of ∗ g: g (t) = inf{s ≥ 0: μ(g, s) ≤ t}. Then the Lorentz functional and the space Lp,q are

Further notes and results | 441

defined by ∞

‖g‖p,q = ( ∫ [g (t)t ∗

1/p q dt

]

0

t

1/q

)

and Lp,q = {g: ‖g‖p,q < ∞}. p 12.6 (Multipliers of “large” spaces). The space Ha,γ consists of all f ∈ H(𝔻) for which

Mp (r, f ) = 𝒪((1 − r)−γ−1 exp

a ), 1−r

ϱ,q

p for all values of the indices. where a > 0 and γ ∈ ℝ. It is “large” because Bα ⊂ Ha,γ Such spaces were considered for the first time in [433], where it was shown that the p corresponding space of harmonic functions is self-conjugate and that f ∈ Ha,γ if and p 󸀠 only if f ∈ Ha,γ+2 . It seems that Dostanić [156] is the only author who considered coefficient multipliers on these spaces. He proved the following: 1 1 Theorem (Do). The class (Ha,γ , Ha,γ ) is independent of γ and is equal to 1 }, {g ∈ H(𝔻): ϕ ∗ g ∈ Ha,−1 2√na n z . where ϕ(z) = ∑∞ n=0 e p ,q

1 1 12.7 (Multipliers from Bp,q , p ≥ 2 ≥ p1 ). Wojtasczyk [600] described the α into Bβ p p1 p1 sets (A , A ), (A(𝔻), A ), and some others. His method is completely new and is based on a few deep theorems from Banach space theory. Although all of these results can be radically simplified (see, e. g., [276, Section 12.4]), the author feels that the proofs from [600] wait someone who will apply them to obtain essentially new results. Here we note that (if p ≥ 2 ≥ p1 ) we have

p ,q

q

q⊖q

1 1 (Bp,q ) = (Vαq [H p ], Vβ 1 [H p1 ]) = Vβ−α 1 [ℓ∞ ], α , Bβ

where we have used the easily provable fact that (H p , H p1 ) = ℓ∞ . Hence p ,q

1 1 (Bp,q ) = ℓβ−α α , Bβ

∞,q⊖q1

.

, we get as the particular case (Ap , Ap1 ) = ℓ1/p−1/q1 . Since Ap = Bp,p −1/p ∞,p⊖p

12.8. Yanigahara [615] described the sets (𝒩 + , H p ), 0 < p ≤ ∞.. In [600], multipliers into 𝒩 and ℒ0 (𝕋) we considered. Recall that ℒ0 (𝕋) is the space of all a. e. finite measurable functions g on 𝕋 such that N0 (g) := ∫ − 𝕋

|g| dℓ < ∞. 1 + |g|

442 | 12 Multipliers on Hp , BMOA, and Besov spaces The following can be proved by using Banach’s principle (Theorem C.19) and the Nikishin–Stein theorem (Theorem C.17): (H p , 𝒩 ) = (H p , 𝒩 + ) = (H p , ℒ0 (𝕋)) = ℓ∞ ,

2 ≤ p ≤ ∞.

(±)

It should be noted that a sequence {λn } is a multiplier from H p into ℒ0 (𝕋) if for all f ∈ H p , there exists a finite limit ∞

lim− ∑ λn f ̂(n)r n ζ n

r→1 n=0

for a. e. ζ ∈ 𝕋. In fact, it suffices to prove (±) the Banach principle. Namely, the inclusion ℓ∞ ⊂ p (H , ⋅) holds because ℓ∞ ⊂ (H p , H 2 ). If {λn } ∈ (H p , ℒ0 ), then we consider the sequence of operators Tj f (ζ ) = ∑ λn f ̂(n)(1 − 1/j)n ζ n . Since Tmax f (ζ ) < ∞ for a. e. ζ ∈ 𝕋, we can appeal to the Banach’s principle to conclude that Tmax : H p 󳨃→ ℒ0 is continuous. This implies that, for every ε > 0, there is a constant C > 0 such that N0 (en /C) < ε for all n. Taking ε = 1/2, we get that λn is bounded. 12.9 (Tensor products of Banach spaces of analytic functions). In [89] (see also [597], [189], and [585, Sections V.4, VI.3]), the authors defined the space X ⊗ Y, where X and Y are H-admissible Banach spaces, to be the set of all functions h ∈ H(𝔻) that can be represented in the form ∞

h = ∑ fn ∗ gn , n=1

fn ∈ X, gn ∈ Y,

so that the series converges in H(𝔻) and ∑∞ n=1 ‖fn ‖X ‖gn ‖Y < ∞. The norm in X ⊗ Y is ‖f ‖ ‖g ‖ , where the infimum is taken over all the above given by ‖h‖X⊗Y = inf ∑∞ n=1 n X n Y representations. It turns out that X ⊗ Y is an H-admissible Banach space. The most important property of X ⊗ Y is described by the relation (X ⊗ Y, Z) = (X, (Y, Z)) and, in particular, (X ⊗ Y, H ∞ ) = (X, (Y, H ∞ )) = (X, Y ∗ ). Using the decomposition theorem for Besov spaces along with Theorem 2.20 we prove that H 1 ⊗ H 1 = B1,1 , and hence, by the duality relation (B1,1 )∗ = B, B = (B1,1 , H ∞ ) = (H 1 , BMOA). A chapter of the monograph [276] is devoted to the results of [89], but the presentation is somewhat reduced, so the interested reader is referred to the original source.

13 Decompositions of spaces with subnormal weights and applications We consider the space hp,q and its analytic subspace, where ϕ is a subnormal function [ϕ]

such as ϕ(t) = (log(2/t))−γ , 0 < t < 1, and prove a decomposition theorem, which we use to detect the dual spaces and coefficient multipliers. New spaces arise, which we call Jackson spaces because they are defined via best approximation by polynomials. The duality theorems are reformulated by using Bergman-type duality pairing. This Ψ f (n) = Ψ (|n| + 1)f ̂(n), f ∈ forced us to introduce the fractional derivative 𝒥 Ψ f by 𝒥̂ h(𝔻), where Ψ is subnormal on [1, ∞) with the additional property that 1/Ψ is convex. Although the ideas are similar to those used in the previous chapter, the realization requires completely different techniques.

13.1 Decompositions Let λ = {λn } (n ≥ 0) be a lacunary sequence of positive real numbers. Let Ψ be an even C ∞ -function on ℝ such that supp(Ψ) ⊂ [−1 − δ/2, 1 + δ/2], where 1 + δ = infn λn+1 /λn , and Ψ(x) = 1 for |x| < 1

and

0 < Ψ(x) ≤ 1 for 1 < |x| < 1 + δ/2.

As before, consider the trigonometric polynomials Wn = WnΨ as ∞ k Wn (eiθ ) = ∑ Ψ( )eikθ . n k=−∞

We have ‖Wn ‖Lp (𝕋) ≍ n1−1/p ,

n≥1

(for all p > 0).

(13.1)

Indeed, the inequality ‖Wn ‖Lp (𝕋) ≤ Cn1−1/p follows from Lemma 6.2. In the other direction, let M > 1 + δ be an integer, and apply Lemma 2.7 (q = ∞) to the polynomials eiMnθ W(eiθ ). Define the polynomials 𝒱n,λ : iθ

∞

𝒱0,λ (e ) = ∑ Ψ(k/λ0 )e iθ

k=−∞ ∞

𝒱n,λ (e ) = ∑ (Ψ( k=−∞

We may and will assume that δ ≤ 1. https://doi.org/10.1515/9783110630855-013

ikθ

,

k k ) − Ψ( ))eikθ , λn λn−1

n ≥ 1.

444 | 13 Decompositions of spaces with subnormal weights and applications Proposition 13.1. The polynomials 𝒱n = 𝒱n,λ , n ≥ 0, possess the following properties: ̂n ) ⊂ [λn−1 , λn+1 ] ∪ [−λn+1 , −λn−1 ] (λ−1 := 0). (i) supp(𝒱 (ii) 𝒱n ∗ 𝒱j = 0 for |j − n| ≥ 2. (iii) f (z) = ∑∞ n=0 𝒱n ∗ f (z) (f ∈ h(𝔻)). (iv) ‖𝒱n ∗ f ‖p ≤ Cp ‖f ‖p (f ∈ H p , 0 < p ≤ ∞). (v) ‖𝒱n ∗ f ‖p ≤ Cp ‖f ‖p (f ∈ hp , 1 ≤ p ≤ ∞). ̂n (j) = 1 for (1 + δ/2)λn−1 ≤ j ≤ λn+1 . (vi) 𝒱 Proof. Item (iv) is a consequence of (much stronger) Theorem 6.3. To prove (v), we use the inequality ‖f ∗ g‖p ≤ ‖f ‖p ‖g‖1 (p ≥ 1) with g = 𝒱n and then (13.1) and the definition of 𝒱n . Recall that f ∈ hp,q if the function Mp (r, f )ϕ(1 − r), 0 < r < 1, belongs to the spa[ϕ] ϕ󸀠 (1−r)

ce Lq (dmϕ ), where dmϕ (r) = ϕ(1−r) (see Section 4.3.g). We need an analogue of Lemma 2.6. For a function g ∈ h(𝔻), let n

σn g(z) = ∑ (1 − j=0

j )g (z), n + 1 [j]

where g[j] is defined by g[0] (z) = g(0) and, for j ≥ 1, j ̂ ̂ g[j] (z) = g(j)z + g(−j) z̄ j .

Observe that g(z) = ∑∞ j=0 g[j] (z), z ∈ 𝔻. Lemma 13.2. If X ⊂ h(𝔻) is a homogeneous Banach space, then ‖σn g‖X ≤ ‖g‖X for g ∈ X and n ≥ 0. Proof. Since σn g is a polynomial, we have, by the Hahn–Banach theorem, ‖σn g‖X = sup{‖σn g ∗ f ‖∞ : f ∈ X ∗ , ‖f ‖X ∗ ≤ 1}, where X ∗ = (X, h∞ ). Now the result follows from the well-known Fejér theorem (‖σn g ∗ f ‖∞ ≤ ‖g ∗ f ‖∞ ) and the inequality ‖g ∗ f ‖∞ ≤ ‖f ‖X ∗ ‖g‖X . Lemma 13.3. Let g = ∑nm g[j] ∈ X (0 ≤ m < n), where X is a homogeneous Banach space. Then 3−1 r 2n ‖g‖X ≤ ‖gr ‖X ≤ 2r m/2 ‖g‖X ,

0 < r < 1.

Proof. Let m ≥ 2. After two summations by parts, we find ∞

∞

j=0

0

gr = ∑ r j g[j] = (1 − r)2 ∑ r j (j + 1)σj1 g.

13.1 Decompositions | 445

Taking into account that σj1 g = 0 for j < m and using the inequality ‖σj1 g‖ ≤ ‖g‖ (Lemma 13.2), we get ∞

‖gr ‖ ≤ (1 − r)2 ∑ r j (j + 1)‖g‖ = r m (1 + m(1 − r)). j=m

Using the elementary inequality m(1 − r) + 1 ≤ 2r −m/2 , we prove half of the lemma. To prove the rest, let R = 1/r > 1 and f = gr . Then two summations by parts give n

n−1

j=0

j=0

1 f + Rn f . g = ∑ Rj f[j] = ∑ (Rj + Rj+2 − 2Rj+1 )(j + 1)σj1 f + (Rn − Rn+1 )nσn−1

Hence, by Lemma 13.2, n−1

‖g‖ ≤ (R − 1)2 ∑ Rj (j + 1)‖f ‖ + (R − 1)Rn n‖f ‖ + Rn ‖f ‖. j=0

Finally, using the inequalities n(R − 1) ≤ Rn − 1 ≤ Rn and n−1

n−1

j=0

0

∑ Rj (j + 1) ≤ n ∑ Rj = n(Rn − 1)(R − 1)−1 nR ≤ nRn (R − 1)−1 ,

we obtain ‖g‖ ≤ 3R2n ‖f ‖, and this concludes the proof. 13.1.a. For an admissible space X, we define the space hqX,[ϕ] = {f ∈ h(𝔻) : . . .} (ϕ is

subnormal) by the requirement ‖fr ‖X ∈ Lq (dmϕ ). Thus hp,q = hqhp ,[ϕ] . This includes the [ϕ] case of spaces of analytic functions.

To state a decomposition theorem for the spaces hp,q , we introduce a class of gen[ϕ] eralized Vαp,q spaces. Let C = {cn }∞ be a sequence of positive real numbers such that 0 cn+1 c ≤ sup n+1 < 1 or cn cn n cn+1 cn+1 ≤ sup < ∞. 1 < inf n c cn n n

0 < inf n

(13.2)

We call such a sequence normal. If ϕ is normal on [0, 1], then {ϕ(2−n )} is normal, and if ψ is normal on [1, ∞), then {ψ(2n )} is normal. We will consider a more general class of sequences defined by 0 < inf n

c cn+1 ≤ sup n+1 < ∞. cn cn n

(13.3)

Let 𝒱n be a sequence of polynomials satisfying the properties listed in Proposition 13.1. The space 𝒱Cq [X], where X is an admissible space, consists of those f ∈ h(𝔻) or f ∈ H(𝔻) for which {cn ‖𝒱n ∗ f ‖X }0 ∈ ℓq , ∞

0 < q ≤ ∞.

446 | 13 Decompositions of spaces with subnormal weights and applications Remark 13.4. Condition (13.3) guarantees that if X is an h-admissible (or H-admissible) space, then we have the continuous inclusions h(𝔻) ⊂ 𝒱Cq [X] ⊂ h(𝔻). The point is that this condition ensures the convergence of the series ∞

∑ c n r λn

n=0

and

∞

λn ∑ c−1 n r

n=0

(0 < r < 1).

Then, using the inclusion 𝒵 q := VCq [X] ⊂ h(𝔻), we prove that 𝒵 q is complete. If in addition X is R-admissible, then so is 𝒵 q . Further, if X is R-admissible, then 𝒵 q is minimal if and only if q ≤ ⬦, whereas 𝒵 ⬦ is equal to the closure of the polynomials in 𝒵 ∞ . Note that 𝒵 ∞ (resp., 𝒵 ⬦ ) is defined by ‖𝒱n ∗ f ‖X = 𝒪(1/cn ) (resp., o(1/cn )). If X is homogeneous, then so is 𝒵 q , and 𝒵 q has the Fatou property if and only if q ≠ ⬦. By means of Lemmas 13.3, 2.6, and 4.13, we prove the following decomposition theorem. Theorem 13.1. Let X be a homogeneous Banach space or X = H p (p > 0), let ϕ be a subnormal function on [0, 1], and let λn (λ0 ≥ 1) be a sequence such that C = {ϕ(1/λn )} is normal. Then hqX,[ϕ] ≃ 𝒱Cq [X]. Example 13.5. Let ϕ(t) = (log(e/t))−γ , 0 < t < 1, γ > 0. Then dmϕ (r) =

γ dr, (1 − r) log(e/(1 − r))

and therefore hp,q (q < ⬦) coincides with the space defined by [ϕ] 1

∫ Mpq (r, f )(log 0

−γq−1

e ) 1−r

dr < ∞. 1−r

If q = ∞, then this should be interpreted as Mp (r, f ) = 𝒪(log

γ

e ) , 1−r

r → 1− .

If q = ⬦, then replace “𝒪” with “o”. n In this case, taking λn = 22 , we can conclude that if p ≥ 1 and f ∈ h(𝔻), then f ∈ hp,q if and only if [ϕ] ∞

∑ 2nq ‖𝒱n,λ ∗ f ‖qp < ∞.

n=0

If f is analytic, then this equivalence holds for all p > 0.

13.2 Duality | 447

Remark 13.6. If {en } (−∞ < n < ∞) is a Schauder basis in the closure of the harmonic polynomials in Y, for example, if Y = hp , 1 < p < ⬦, then the theorem remains true if we replace 𝒱n,λ with Δn,λ = Δn,λ (eiθ ) = ∑ eikθ , k∈In,λ

where I0,λ = (−λ1 , λ1 ), and In,λ = (−λn+1 , −λn ] ∪ [λn , λn+1 ) for n ≥ 1. This follows from [421, Theorem 4.1]; condition (0.5) in that paper should read “…|j| ∈ ̸ [λn−1 , λn+N ), where N ≥ 0 …”. This decomposition was recently rediscovered by Peláez and Rättyä [460, Theorem 4]. They proved an analogous result for the analytic Bergman-type spaces 1 Ap,q φ (1 < p < ⬦), where φ ∈ L (0, 1) is a continuous function satisfying lim

r→1−

(1 − r)φ(r) 1

∫r φ(x) dx

= 0.

(13.4)

1

p,q However, if we take ϕ(t)q = ∫1−t φ(x) dx, then Ap,q φ = H[ϕ] , and condition (13.4) is equivalent to

lim+

t→0

tϕ󸀠 (t) = 0. ϕ(t)

The set of such ϕ is a subset of the class of Karamata slowly varying functions [527]. Therefore Theorem 4 of [460] is a particular case of [421, Theorem 4.1].

13.2 Duality We continue describing the dual of a space with its Abel dual. Proposition 13.7. Let X be an R-admissible quasi-Banach space such that ‖𝒱n ∗ f ‖X ≤ C‖f ‖X . If 0 < q ≤ ⬦ and C is a sequence satisfying (13.3), then the space VCq [X] is minimal, and we have A

q (𝒱Cq [X]) ≃ 𝒱1/C [(XP )A ]. 󸀠

Proof. Define the space ℓCq (X) by the requirement {cn ‖fn ‖X } ∈ ℓq . The main step is provq q ing that the operator T({fn }) = ∑∞ n=0 𝒱n ∗ fn is bounded from ℓC (X) to 𝒱C [X]. We have 𝒱n ∗ 𝒱j = 0 for |j − n| ≥ 2, and hence n+1

𝒱n ∗ T({fj }) = ∑ 𝒱n ∗ 𝒱j ∗ fj , j=n−1

n ≥ 0,

where 𝒱j = fj = 0 for j < 0. It follows that n+1

󵄩 󵄩󵄩 󵄩󵄩𝒱n ∗ T({fj })󵄩󵄩󵄩X ≤ C ∑ ‖fj ‖X . j=n−1

448 | 13 Decompositions of spaces with subnormal weights and applications Hence, for q < ⬦, ∞

∞

n=0

n=0

󵄩 q 󵄩 ∑ (cn 󵄩󵄩󵄩𝒱n ∗ T({fj })󵄩󵄩󵄩X ) ≤ C ∑ cqn (‖fn ‖qX + ‖fn−1 ‖qX + ‖fn+1 ‖qX ).

Now we use condition (13.3) to prove the boundedness. The rest is similar to the proof q󸀠 of Theorem 6.10 and is based on the duality (ℓCq (ℝ))󸀠 ≅ ℓ1/C (ℝ). This theorem, along with some additional facts, allows us to describe the dual of in terms of such spaces. This will be done later on. See Theorem 13.7.

hpX,[ϕ]

13.2.1 Jackson spaces From the preceding two statements we can get a description of the dual of hp,q in terms [ϕ] of “Jackson” spaces. For a subnormal function ϕ on [0, 1] extended to [0, ∞) so that it remains subnormal and satisfies the condition ϕ(1/x) ≍ 1/ϕ(x), x > 0, we define the q Jackson space EX,[ϕ] = {f ∈ X: . . . } by the requirement ∞

∑ [ϕ(n + 1)q − ϕ(n)q ]En (f )qX < ∞,

q < ⬦.

n=1

(13.5)

We write p,q E[ϕ] = Ehqp ,[ϕ] .

The norm can be given, for instance, by “‖f ‖X + the sum”. If q ∈ {⬦, ∞}, then we q define EX,[ϕ] by 1 En (f )X = 𝒪(ϕ( )) (q = ∞), n

1 En (f )X = o(ϕ( )) (q = ⬦), n

n → ∞.

By Proposition 8.4 we can assume that ϕγ is concave for some δ > 0, and in this case, we have ϕ(n + 1)q − ϕ(n)q ≍ ϕ(n)q−1 ϕ󸀠 (n) = ϕ(n)q

ϕ󸀠 (n) , ϕ(n)

n ≥ 1,

which resembles the definition of hpX,[ϕ] . In the case of the function from Example 13.5, we have ϕ(x) = (1 + log x)γ , x ≥ 1, and therefore 1 ϕ(n + 1)q − ϕ(n)q ≍ (log n)γ(q−1)+γ−1 , n

n ≥ 2.

13.2 Duality | 449

p,q It follows that in this case the harmonic space E[ϕ] , p ≥ 1, can be described by the condition

(log n)γq−1 En (f )qp < ∞ n n=2 ∞

∑

for q < ⬦ and by 1 ), logγ n

En (f )p = 𝒪(

n → ∞,

for q = ∞. Theorem 13.2. If q ≤ ⬦, then, under the hypotheses of Theorem 13.1, we have A

q (hqX,[ϕ] ) ≃ E(X 󸀠

P)

A ,[ϕ]

.

Proof. Let Y = (XP )A . We choose a lacunary sequence λ = {λn } of positive integers such that ϕ(λn ) is normal and satisfies ϕ(λn ) ≍ 2n . Application of Proposition 13.7

p shows that (hqX,[ϕ] )A ≃ 𝒱1/C [Y], where 1/C = {ϕ(λn )}. Then, proceeding just as in the 󸀠

p proof of Theorem 6.14, we conclude that g ∈ 𝒱1/C [Y] if and only if 󸀠

∞

1/q󸀠

( ∑ ϕ(λn )q Eλn (g)qY ) 󸀠

󸀠

n=0

(13.6)

< ∞.

In the case q󸀠 = ∞, this immediately gives the conclusion. Let q󸀠 < ⬦. Then summa󸀠 󸀠 tion by parts, together with the inequality ∑nk=0 ϕ(λk )q ≤ Cϕ(λn )q , shows that (13.6) is equivalent to ∞

∑ [Eλn (g)qY − Eλn+1 (g)qY ]ϕ(λn )q < ∞. 󸀠

󸀠

󸀠

n=0

This sum is equal to ∞

λn+1 −1

∞

∑ ϕ(λn )q ∑ [Ek (g)qY − Ek+1 (g)qY ] ≍ ∑ ϕ(k)q [Ek (g)qY − Ek+1 (g)qY ].

n=0

󸀠

k=λn

󸀠

󸀠

󸀠

󸀠

󸀠

k=λ0

Another summation by parts leads to the desired result. (The proof shows that the q󸀠 norms in (hqX,[ϕ] )A and EY,[ϕ] are equivalent.) When applied to the space X = hp , Theorem 13.2 gives the following: p ,q Theorem 13.3. If 1 ≤ p ≤ ∞ and q ≤ ⬦, then (hp,q )A ≍ E[ϕ] . [ϕ] 󸀠

󸀠

It is easy to check that if ϕ is normal, then ϕ(n + 1)q − ϕ(n)q ≍ ϕ(n)q /n, n ≥ 1; see Theorem 6.17.

450 | 13 Decompositions of spaces with subnormal weights and applications

13.3 Reformulation of the duality theorems In the “subnormal situation”, the following theorem provides, maybe, a more natural description of (hp,q )󸀠 . [ϕ] Theorem 13.4. If p ≥ 1 and q ≤ ⬦, then (hp,q )󸀠 ≃ hp[ϕ],q with respect to the duality pairing [ϕ] 󸀠

(f , g)ϕ = ∫ f (z)g(z)̄ dνϕ (z),

󸀠

2

where dνϕ (reiθ ) = (ϕ(1 − r)) dmϕ dθ.

(13.7)

𝔻

Recall that dmϕ (r) = ϕ󸀠 (1 − r) dr/ϕ(1 − r). To pass from the sum of blocks to the integral, we need a technical lemma. For a subnormal function Ψ on [1, ∞), define the operator 𝒥 Ψ : h(𝔻) 󳨃→ h(𝔻) by ∞

Ψ

𝒥 g(z) = ∑ Ψ (n + 1)g[j] (z). n=0

j ̂ ̂ Recall that g[j] (z) = g(j)z + g(−j) z̄j for j ≥ 1 and g[0] (z) = g(0).

Lemma 13.8. Let Ψ be a subnormal function such that 1/Ψ is convex on [1, ∞), let {λn }∞ 1 (λ1 ≥ 1) be an increasing sequence of positive integers such that Ψ (λn ) ≍ Ψ (λn+1 ), and λn+1 let g = ∑j=λ g[j] . If X is a homogeneous h-admissible Banach space, then n−1

󵄩󵄩 Ψ 󵄩󵄩 󵄩󵄩𝒥 g 󵄩󵄩 ≍ Ψ (λn )‖g‖,

n ≥ 1,

where the equivalence constants are independent of g, n, and X. Proof. Let k = λn−1 and m = λn+1 . Then m

Ψ

𝒥 g = ∑ Aj g[j] ,

where Aj = Ψ (j + 1).

j=k

Applying the Abel identity, we get m

m

j=0

j=0

∑ αj βj = ∑ (Δ1 αj )sj β + αm+1 sm β,

j

where sj β = ∑ν=0 βν and Δ1 αj = αj − αj+1 . Another application of the Abel identity gives m

m

j=0

j=0

1 β + ∑ (Δ2 αj )(j + 1)σj1 β. ∑ αj βj = αm+1 sm β + (m + 1)Δ1 αm+1 σm

Taking αj = Aj and βj = g[j] , we obtain Ψ

m

m

j=k

j=k

2

1

𝒥 g = ∑ Aj g[j] = ∑(Δ Aj )(j + 1)σj g 1 + (Am+1 − Am+2 )(m + 1)σm g + Am+1 g,

(13.8)

13.3 Reformulation of the duality theorems | 451

where we have used the relations sj g = σj1 g = 0 for j < k and sm g = g. Hence, by Lemma 13.2, m

󵄨 2 󵄨 󵄩󵄩 Ψ 󵄩󵄩 󵄩󵄩𝒥 g 󵄩󵄩 ≤ ∑󵄨󵄨󵄨Δ Aj 󵄨󵄨󵄨(j + 1)‖g‖ + (Am+2 − Am+1 )(m + 1)‖g‖ + Am+1 ‖g‖. j=k

Letting aj = 1/Aj , we have Δ2 Aj = −Aj Aj+2 Δ2 aj + 2Aj (Aj+2 − Aj+1 )(aj − aj+1 ). Since 1/Ψ is convex, we have that, for fixed x > 1, the function 1 Ψ (t)

Fx (t) :=

−

1 Ψ (x)

x−t

,

1 ≤ t < x,

is decreasing in t. Therefore aj − aj+1 = Fj+2 (j + 1) ≤ Fj+2 (1 + j/2)

≤ 2/(j + 2)Ψ (1 + j/2) ≤ Caj /(j + 1).

(13.9)

Here we have used the relation Ψ (1 + j/2) ≍ Ψ (1 + j), which holds because Ψ is subnormal. On the other hand, we have Δ2 aj ≥ 0 because 1/Ψ is convex. From this, (13.9), and the identity m

∑(Δ2 aj )(j + 1) = (ak − ak+1 )k − (am+1 − am+2 )(m + 1) + ak − am+1

j=k

we obtain m

m

󵄨 󵄨 ∑󵄨󵄨󵄨Δ2 Aj 󵄨󵄨󵄨(j + 1) ≤ Am Am+2 ∑(Δ2 aj )(j + 1)

j=k

m

j=k

+ C ∑ Aj (Aj+2 − Aj+1 )aj j=k

≤ Am Am+2 [(ak − ak+1 )k + ak ] + C(Am+2 − Ak+1 ) ≤ Am Am+2 (Cak + ak ) + CAm+2 ≤ CΨ (λn ).

In the same way, we get (m + 1)(Am+2 − Am+1 ) = Am+1 Am+2 (am+1 − am+2 )(m + 1)

≤ CAm+1 Am+2 am+1 = CAm+2 ≤ CΨ (λn ).

From these inequalities and (13.8) we obtain 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 󵄩󵄩 Ψ 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩𝒥 g 󵄩󵄩 = 󵄩󵄩∑ Aj g[j] 󵄩󵄩󵄩 ≤ CΨ (λn )‖g‖. 󵄩󵄩󵄩 󵄩󵄩󵄩j=k

(13.10)

452 | 13 Decompositions of spaces with subnormal weights and applications m In the other direction, let h = ∑m j=k Aj g[j] . Then g = ∑j=k aj hj . Now we have m

‖g‖ ≤ ∑(aj + aj+2 − 2aj+1 )(j + 1)‖h‖ j=k

+ (am+1 − am+2 )(m + 1)‖h‖ + am+1 ‖h‖

= ((ak − ak+1 )k − (am − am+1 )m + ak − am )‖h‖ + (am+1 − am+2 )(m + 1)‖h‖ + am+1 ‖h‖

≤ C(ak + am )‖h‖ ≤ CΨ (λn )−1 ‖h‖, where we have used (13.10) and (13.9). This completes the proof. Remark. Identity (13.10) can be verified by induction on m. Proof of Theorem 13.4. Let 𝒳 = hp,q and 𝒴 = hp[ϕ],q . Applying Hölder’s inequality, we [ϕ] get 󸀠

󸀠

1

󵄨 󵄨󵄨 󵄨󵄨(f , g)ϕ 󵄨󵄨󵄨 ≤ C ∫[Mp (r, f )ϕ(1 − r)][Mp󸀠 (r, g)ϕ(1 − r)] dmϕ (r). 0

Another application of Hölder’s inequality (with the indices q and q󸀠 ) shows that |(f , g)ϕ | ≤ C‖f ‖𝒳 ‖g‖𝒴 . Incidentally, this shows that the integral in (13.7) converges absolutely. To prove the converse, let L ∈ 𝒳 󸀠 . We extend ϕ to (0, ∞) by 1

ϕ(t)−2 = c ∫ r 2(t−1) ϕ(1 − r)2 dmϕ (r) 0

1

= c ∫ r 2(t−1) ϕ(1 − r)ϕ󸀠 (1 − r) dr,

t > 1,

(13.11)

0

where c is chosen so that ϕ(1+) = ϕ(1). Applying Lemma 13.8 with Ψ = ϕ2 , we see that ϕ(1/λn ) ≍ 1/ϕ(λn ). Since the function 1/ϕ2 is convex, applying Proposition 13.1 2

and Lemma 13.8, we conclude that there exists a unique h such that 𝒥 ϕ h ∈ 𝒱Cq [hp ] = 2

hp[ϕ],q , C = {ϕ(1/λn )}, and Lf = (2π/c)⟨f , h⟩. Letting g = 𝒥 ϕ h/c, we have 󸀠

󸀠

1

̂ ̂ ∫ r 2|n| ϕ(1 − r)2 dmϕ (r), h(n) = g(n) 0

and hence ∞

1

̂ ∫ r 2|n| ϕ(1 − r)ϕ󸀠 (1 − r) dr Lf = 2π ∑ f ̂(n)g(n) n=−∞

0

󸀠

󸀠

13.3 Reformulation of the duality theorems | 453 2π 1

= ∫ ∫ f (reiθ )g(re−iθ )ϕ(1 − r)ϕ󸀠 (1 − r) dθ dr 0 0

= (f , g)ϕ for all harmonic polynomials f ∈ hp,q . Since such polynomials are dense in this space, [ϕ]

we have Lf = (f , g)ϕ for all f ∈ hp,q . This completes the proof. [ϕ]

A particular case deserves special attention. The space h∞ (ψ) = {f ∈ h(𝔻) : . . .} was defined on p. 289 by the requirement 1 󵄨 󵄨󵄨 )), 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ C(ψ( 1 − |z|

z ∈ 𝔻,

where ψ > 0 is a subnormal majorant on [1, ∞) (such that ψ(∞) = ∞). For a positive measure μ on 𝔻, let h1 (μ) = {g: ∫ |g| dμ < ∞}. 𝔻

Since h∞ (ψ) = h∞,∞ , where ϕ(t) = 1/ψ(1/t), 0 ≤ t ≤ 1, we have the following theorem. [ϕ] Theorem 13.5. The predual of h∞ (ψ) is isomorphic to the space h1 (μ), where dμ(z) = ϕ󸀠 (1 − |z|) dA(z) and 1/ϕ(t) = ψ(1/t), under the duality pairing (f , g)ϕ . (See Further notes 13.1.) Example 13.9. Let ψ(t) = (1 + log t)γ , t ≥ 1, where γ > 0. Then we can take dμ(z) = (log

−γ−1

e ) 1 − |z|

̄ (f , g)ϕ = ∫ f (z)g(z)(log 𝔻

dA(z) , 1 − |z|

−2γ−1

e ) 1 − |z|

dA(z) . 1 − |z|

13.3.1 Duality for Bergman spaces The following theorem is well known in the case of analytic Bergman spaces. 1

̂ Theorem 13.6. Let φ be a Bergman weight such that the function φ(r) = ∫r φ(t) dt satisfies ̂ ≤ C φ( ̂ φ(r)

1+r ). 2

If 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ⬦, then the dual of the Bergman space Aφp,q is isomorphic to ̄ dA(z). If 1 < p < ⬦, then the Aφp ,q under the duality pairing [f , g]φ = ∫𝔻 f (z)g(z)φ(z) 󸀠

󸀠

analogous fact for Ap,q φ holds.

454 | 13 Decompositions of spaces with subnormal weights and applications Proof. We sketch the idea. Define ϕ(t) by ϕ(1 − r)ϕ󸀠 (1 − r) = φ(r). Then ϕ is subnormal 󸀠 󸀠 because of the “doubling” property of φ. Then consider the spaces hp,q and hp[ϕ,q] , [ϕ ] 1

2

= Aφp,q and hp[ϕ,q] = Aφp ,q . where ϕ1 = ϕ2/q and ϕ2 = ϕ2/q . It turns out that hp,q [ϕ ] 󸀠

󸀠

1

󸀠

󸀠

󸀠

2

Then a slight modification of the proof of Theorem 13.4 shows that the dual of hp,q is [ϕ ] 1

isomorphic to hp[ϕ,q] under the pairing 󸀠

󸀠

2

̄ ̄ − |z|)ϕ󸀠 (1 − |z|) dA(z) = ∫ f (z)g(z)ω(|z|) dA(z). ∫ f (z)g(z)ϕ(1 𝔻

𝔻

It should be remarked that if p > 1, then the proofs become much shorter because p {en }∞ −∞ is a Schauder basis in h . See Remark 13.6. Further reformulations. We have defined the space hqX,[ϕ] in Section 13.1.a and proved a decomposition theorem (Theorem 13.1 and a duality theorem, Proposition 13.7). It follows that if X is a homogeneous Banach space or X = H p , p > 0, then q (hqX,[ϕ] ) ≃ 𝒱1/C [(XP )∗ ], ∗

󸀠

0 < q ≤ ⬦,

where C = {ϕ(1/λn )}. In fact, this also holds for q = ∞, but this is not so important. Assuming that ϕ is extended to ℝ+ by (13.11) and using the preceding method, we obtain the following: Theorem 13.7. Let X be a homogeneous Banach space or H p , p > 0. Then the operator 2 q q󸀠 𝒥 ϕ acts as an isomorphism of (hX,[ϕ] )∗ onto h(X )∗ ,[ϕ] , where 0 < q ≤ ⬦. P

p

This is Theorem 5.1 of [421]. If X = H , p ≤ 1, and ϕ is not normal, then this leads p,q to unpleasant expressions, but there is one case where the dual of H[ϕ] is isomorphic to the Bloch space even if ϕ is not normal. This was observed in [421, Theorem 7.2] and rediscovered very recently in [471] in a particular case; see Further notes 13.3. To state the theorem, let ϕγ (t) = t γ /ϕ(t), t ∈ ℝ+ , where γ > α with α such that ϕ(t)/t α is decreasing. Note that ϕγ is normal. Then note that the dual M p of H p , p < 1, can be described by ‖f ‖M p = sup (1 − r)M∞ (r, 𝒥 1/p f ) < ∞. 0 0; see Proposition 4.1. We can p,q identify Bω,n with hp,q via a multiplier transform, where [ϕ] x

1

0

0

t nq−1 t nq−1 nq dt = x dt. ϕ(x) = ∫ ∫ ω(t)q ω(xt)q q

13.4 Multipliers on spaces with subnormal weights | 457

p,q The function ϕ is subnormal because t α /ω(t) decreases in t. Hence, if Bω,n is selfp,q q conjugate, then so is h[ϕ] , and therefore, by Theorem 13.10, ϕ is almost normal. After

passing to an equivalent normal function, we have x(ϕ(x)q )󸀠 ≍ ϕ(x)q , that is, ϕ(x)q 1 ≍ , xnq ω(x)q

0 < x < 1.

Since ϕ(x)/xγ increases in x for some γ > 0, we see that ω(x)/xn−γ is almost decreasing. This proves the theorem in the case q < ∞. We omit the (similar and simpler) proof in the remaining cases. Exercise 13.12. Let X and Y be h-admissible spaces. If one of them is self-conjugate, then so is (X, Y). p,q

Multipliers of H[ϕ] The analytic analogue of Theorem 13.1 holds for any homogeneous quasi-Banach space the norm of which is plurisubharmonic; this means that the functions z 󳨃→ ‖f +zg‖X , z ∈ ℂ, f , g ∈ X, are subharmonic. In such a space we have the inequality 󵄩󵄩 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 r n 󵄩󵄩󵄩 ∑ ak ek 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ ak r k ek 󵄩󵄩󵄩 ≤ r m 󵄩󵄩󵄩 ∑ ak ek 󵄩󵄩󵄩, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩j=m 󵄩 󵄩j=m 󵄩j=m

0 < r < 1,

which, together with Lemmas B.6 and 4.13, can be used to prove the analogue of Theorem 13.1. If the quasinorm in an arbitrary quasi-Banach space is plurisubharmonic, then it is called a PL-convex space (see [140], where the term “PL-convex” was introduced, and [420, 425], where the term “c-convex” is used). Clearly, all Banach spaces are PLconvex, but there are other important spaces, such as Lp , which possess this property. Besides H p , the spaces Bp,q α are PL-convex for all p, q, α. If 1 < p < ⬦, then the theorems on hp,q extend to the analytic spaces in an obvious [ϕ] p,q A way. If p ∈ (0, 1] ∪ {⬦}, then the structure of (H[ϕ] ) is complicated. For instance, we

1,q A q have (H[ϕ] ) = EBMOA,[ϕ] , q ≤ ⬦. This is a consequence of Fefferman’s duality theorem and the general formula 󸀠

A

q q (HX,[ϕ] ) = EY,[ϕ] , 󸀠

where Y = (XP )A , and q ≤ ⬦,

where X is a PL-convex space, and in particular X = H p , p > 0. Exercise 13.13. Let ψ be a subnormal function on [1, ∞), and let X be an H-admissible PL-convex space. Then (X, H ∞ (ψ)) = X ∗ (ψ), where Y(ψ) = {f ∈ H(𝔻) : . . .} is defined by the requirement ‖fr ‖Y = 𝒪(ψ(

1 )). 1−r

In particular, (H p , H ∞ (ψ)) = H p (ψ) for 1 < p < ⬦. 󸀠

458 | 13 Decompositions of spaces with subnormal weights and applications

13.5 Harmonic Besov spaces. Case 0 < p < 1 In this section, we assume that 0 < p < 1 (and q ∈ ℝ+ ). In this case, we can hardly expect that an analogue of the decomposition theorem 13.1 holds for the space hp,q [ϕ]

when ϕ is not normal. If ϕ is normal, then hp,q = hp,q is self-conjugate, and we can [ϕ] ϕ p,q ̂n (k) = V ̂−n (−k), k < 0, and apply the decomposition of H . Define Vn for n < 0 by V [ϕ]

redefine V0 so that ∑∞ n=−∞ Vn ∗ u = u. Then we have the following:

Theorem 13.12. Let ϕ be normal. A function u ∈ h(𝔻) belongs to hp,q if and only if [ϕ] ∞

q

∑ [ϕ(2−|n| )‖Vn ∗ u‖p ] < ∞.

n=−∞

(13.14)

Observe that Vn ∗ u ∈ H(𝔻) for all n ∈ ℤ \ {0}, so we have ‖Vn ∗ u‖p = ‖Vn ∗ u‖Lp (𝕋) . The norm of V0 is irrelevant. On the other hand, we can choose λn = 2n in the definition of 𝒱n,λ = 𝒱n and ask whether the sum in (13.14) can be replaced with ∞

q

∑ [ϕ(2−n )‖𝒱n ∗ u‖Lp (𝕋) ] .

(13.15)

n=0

Theorem 13.13. Theorem 13.12 remains true if the sum in (13.14) is replaced with the sum (13.15). Proof. In view of Theorem 13.12, it suffices to prove that ‖Vn ∗ u‖Lp (𝕋) ≤ C‖𝒱n ∗ u‖Lp (𝕋) ,

n ≥ 1.

We need the following inequality, which will be proved later on (see Corollary B.5): ‖Q ∗ f ‖X ≤ C2n(1/p−1) ‖Q‖Lp (𝕋) ‖f ‖X .

(13.16)

Here X is a p-Banach H-admissible homogeneous space and Q is a polynomial such ̂ ⊂ [−M2n , M2n ], where M is a positive constant. We can adjust the definithat supp(Q) ̂n (k) = 𝒱 ̂n (k) for k ≥ 0. Then take Pn = Vn−1 + Vn + Vn+1 , and tions of Vn and 𝒱n so that V then X = H p,

Q(eiθ ) = (𝒱n ∗ u)(eiθ ),

and f = Pn .

By (13.16) we have ‖Vn ∗ u‖p = ‖Pn ∗ 𝒱n ∗ u‖p ≤ C2n(1/p−1) ‖Pn ‖p ‖𝒱n ∗ u‖Lp (𝕋)

≤ C2n(1/p−1) 2n(1−1/p) ‖𝒱n ∗ u‖Lp (𝕋) = C‖𝒱n ∗ u‖Lp (𝕋) ,

where we have used the inequality ‖Pn ‖p ≤ C2n(1−1/p) ; see (6.8). We are done.

Further notes and results | 459

The Besov space Bαp,q consists of u ∈ h(𝔻) for which 𝒥 s u ∈ hp,q s−α , where s > α. From the preceding discussion it is easy to deduce the following: Theorem 13.14. A function u ∈ h(𝔻) belongs to Bαp,q if and only if ∞

q

∑ [2nα ‖𝒱n ∗ u‖Lp (𝕋) ] < ∞.

n=0

If u ∈ Bαp,q and α > 0, then by the Hardy–Littlewood projection theorem the Riesz p projection R+ u belongs to Bp,q α ⊂ H , and therefore u∗ can be represented in a unique way as u∗ = f∗ + ḡ∗ , where f , g ∈ H p . Then using the theorem, we prove that if u ∈ Bαp,q , then ∞

∑ nqα−1 En (u∗ )qLp (𝕋) < ∞,

n=1

where En denotes the best approximation in Lp (𝕋) by trigonometric polynomials of order n. The following is not clear to the author: Question. Is it true that if g ∈ Lp (𝕋) and ∞

∑ nqα−1 En (g)qLp (𝕋) < ∞,

n=1

̄ then g can be represented as g = f∗ + h̄ ∗ , and the function f (z) + h(z), z ∈ 𝔻, belongs p,q to Bα ? The following old result of Storozhenko et al. [573, Theorem 3.4] belongs to the same set of results. Theorem. Let p > 0. A function g ∈ Lp (𝕋) belongs to Λpα (𝕋), 0 < α < 1, if and only if En (g)Lp (𝕋) = 𝒪(n−α ).

Further notes and results The decomposition method of finding multipliers between mixed-norm spaces with subnormal weights was introduced in [421], but polynomials more complicated than Vn were used. Besides, these papers contain many typos, which make them rather difficult to read; nevertheless, the results are formulated correctly. A recent preprint of Peláez and Rättyä [465] contains further development of the theory of Bergman spaces with arbitrary radial weights. The ideas of Section 13.1 are from [421], but the proofs (for example, of Theorem 13.10) are simplified. Theorems 13.2, 13.4, and 13.11 are new. 13.1. Theorem 13.5 is a generalization and improvement of a result of Shields and Williams [533], who used two rather strong additional hypotheses:

460 | 13 Decompositions of spaces with subnormal weights and applications –

There is a positive finite Borel measure dη on [0, 1) such that 1

1 = ∫ r 2n dη(r), ψ(n + 1)

n ≥ 0.

0

–

There is a constant C such that 󵄨 󵄨 (n + 1)󵄨󵄨󵄨ψ(n) − 2ψ(n + 1) + ψ(n + 2)󵄨󵄨󵄨 ≤ C(ψ(n + 1) − ψ(n)) for n ≥ 1.

13.2 ([421, Part II, Theorem 3.2]). If the spaces h∞ (ψ) and h∞ (ϕ), where ψ and ϕ are subnormal functions on [1, ∞), have the same set of multipliers, then they are isomorphic via a multiplier transform. This is a solution to Problem B of [533]. More precisely and more generally, if (X, X) = (Y, Y), where X = h∞,q , then the operator and Y = h∞,q [ψ] [ϕ] ψ(|n| + 1) |n| inθ ̂ r u(n)e ϕ(|n| + 1) n=−∞ ∞

Tu(reiθ ) = ∑

acts as an isomorphism from X onto Y. The same holds in the case where X = h1,q and [ϕ]

. Here, as before, ϕ and ψ are chosen so that the functions 1/ϕ(x) and 1/ψ(x) Y = h1,q [ψ] are convex in x ≥ 1. 13.3 (Duality of Bergman spaces with small exponents). Perälä and Rättyä [471] 1 ̂ proved that if ω is a radial Bergman weight such that the function ω(r) = ∫r ω(t) dt ̂ ̂ + r)/2), then the dual of Apω , p < 1, is isomorphic to the has the property ω(r) ≤ C ω((1 Bloch space under the pairing ⟨f , g⟩W = lim− ∫ f (rz)g(z)W(z) dA(z), r→1

𝔻

where ̂ 1/p (1 − |z|) W(z) = (1/p − 1)w(z)

1/p−2

+

ω(z) 1/p−1 ̂ 1/p−1 (1 − |z|) ω(z) . p

The authors say: “One could equally well consider any of the summands appearing in the definition of W only, …”. So their result is essentially a reformulation of a particular case (q = p) of Theorem 13.8.

A Quasi-Banach spaces In the class of quasi-Banach spaces, the “basic principles of functional analysis” hold. A concise discussion of these principles is contained in Section A.3. The rest of the chapter is devoted to definitions and properties of q-Banach envelopes, Schauder bases, and Lebesgue sequence spaces, and, in Section A.6, to Lp -integrability of lacunary series with vector-valued coefficients.

A.1 Quasi-Banach spaces Let X be a (complex) vector space. A functional ‖ ⋅ ‖: X 󳨃→ [0, ∞) is called a quasinorm if the following conditions are satisfied: ‖f + g‖ ≤ K(‖f ‖ + ‖g‖),

(A.1)

where K (≥ 1) is a constant independent of f , g ∈ X; and ‖f ‖ > 0

(f ≠ 0),

‖λf ‖ = |λ|‖f ‖

(λ ∈ ℂ).

(A.2)

The couple (X, ‖ ⋅ ‖) is then called a quasinormed space. The standard example are Lebesgue spaces Lp . When p < 1, the functional ‖ ⋅ ‖ = ‖ ⋅ ‖Lp is not a norm but satisfies (A.1) with K = 21/p−1 and, moreover, ‖f + g‖p ≤ ‖f ‖p + ‖g‖p .

(A.3)

A functional satisfying (A.3) and (A.2) is called a p-norm. From (A.3) it follows that ‖f1 + f2 + ⋅ ⋅ ⋅ + fn ‖p ≤ ‖f1 ‖p + ‖f2 ‖p + ⋅ ⋅ ⋅ + ‖fn ‖p . A similar inequality holds in the general case although a quasinorm need not be a p-norm for any p. The space X is endowed with the structure of a topological vector space by declaring “a neighborhood of zero” to mean “a set containing {f : ‖f ‖ < 1/n} for some n = 1, 2, . . ..” (The “ball” {f : ‖f ‖ < 1} need not be an open set. Therefore a quasinorm, in contrast to a p-norm, need not be continuous.) This topology is metrizable, according to the Aoki–Rolewicz theorem (Theorem A.1); namely, if a p-norm ⦀ ⋅ ⦀ is equivalent to the original quasinorm, then the formula d(f , g) = ⦀f − g⦀p defines a metric that induces the same topology. If X is complete in this topology, then it is called a quasiBanach space. A p-Banach space is a complete p-normed space. A linear operator T : X 󳨃→ Y, where X and Y are quasi-Banach spaces, is continuous if and only if it is bounded, which means that ‖Tf ‖Y ≤ M‖f ‖X , where M is independent of f . We denote by L(X, Y) the class of all bounded linear operators that act from X to Y. https://doi.org/10.1515/9783110630855-014

462 | A Quasi-Banach spaces Theorem A.1 (Aoki–Rolewicz [509]). If ‖ ⋅ ‖ is a quasinorm on X, then there are p > 0 and a p-norm ⦀ ⋅ ⦀ on X such that ‖f ‖/C ≤ ⦀f ⦀ ≤ ‖f ‖, f ∈ X, where C is independent of f . The p-norm is defined by n

p

1/p

n

: f = ∑ fj },

⦀f ⦀ = inf{(∑ ‖fj ‖ )

j=1

j=1

where the infimum is taken over all finite sequences {fj } ⊂ X. A.1.a. To avoid unnecessary complications, we assume that “quasinorm” means “p-norm for some p ∈ (0, 1]”. The following statements are important although their proofs are very simple. Lemma A.1. Let X and Y be quasi-Banach spaces, and let E be a dense subset of X. Let Tn ∈ L(X, Y) be a sequence such that supn ‖Tn ‖ < ∞. If the limit limn→∞ Tn f exists for all f ∈ E, then it exists for all f ∈ X, and the operator Tf := limn→∞ Tn f is linear and continuous. Lemma A.2. Let T be a continuous linear operator from a quasi-normed space X to a quasi-normed space Y, and let E be a subset of X such that the linear hull of E is dense in X. If Y0 is a closed subspace of Y such that T(E) ⊂ Y0 , then T(X) ⊂ Y0 . The proofs of the following assertions are left to the reader as exercises. Proposition A.3. Let X be p-normed. Then X is complete if and only if convergence of the series ∑ ‖fn ‖p implies convergence of ∑ fn . If X is complete and ∑ fn converges, then ∞ p p ‖ ∑∞ n=1 fn ‖ ≤ ∑n=1 ‖fn ‖ . Proposition A.4. Let {fjk } (j, k ≥ 1) be a double sequence in a p-Banach space X. If ∑j,k ‖fjk ‖p < ∞, then the iterated series ∞

∞

j=1

k=1

∑( ∑ fjk )

∞

∞

k=1

j=1

∑ (∑ fjk )

and

converge and have the same sum.

A.2 q-Banach envelops In the general case, a quasi-Banach space is embedded into many q-Banach spaces; the “smallest” of them is called the q-Banach envelope of X. To be more precise, define the functional Nq (0 < q ≤ 1) on X as follows: 1/q

Nq (f ) = inf{(∑ ‖fj ‖q ) j

: ∑ fj = f }, j

A.2 q-Banach envelops | 463

where the infimum is taken over the set of finite sequences {fj } ⊂ X. This functional is a “q-seminorm”, that is, satisfies the conditions q

q

q

{Nq (f + g)} ≤ {Nq (f )} + {Nq (g)} ,

Nq (λf ) = |λ|Nq (f ).

The set {f ∈ X : Nq (f ) = 0} =: Ker Nq is a closed subspace of X. If Ker Nq = {0}, that is, if Nq is a q-norm, then the completion of the space (X, Nq ) is a q-Banach space and is called the q-Banach envelope of X; denote it by [X]q . According to the Aoki–Rolewicz theorem, there always exists q such that X = [X]q with equivalent quasinorms. A simple but illustrative example is X = ℓp ; then [X]q ≅ ℓq (p < q ≤ 1), and the corresponding quasinorms are equal. It is much more difficult to identify the envelops of the Hardy space H p (see Corollary 3.37). The importance of the space [X]q lies in the fact that every operator from X to an arbitrary q-Banach space extends to an operator on [X]q ; more precisely: Proposition A.5. Let X possess the q-Banach envelope (i. e., let Nq be a q-norm), and let Y be an arbitrary q-Banach space. If T ∈ L(X, Y), then there exists a unique operator S ∈ L([X]q , Y) such that Sf = Tf for all f ∈ X. The following fact is useful in identifying the envelope. Proposition A.6. Let X be continuously embedded into a q-Banach space Y in such a way that every f ∈ Y can be represented as f = ∑∞ n=1 fn , fn ∈ X, with ∞

∑ ‖fn ‖qX ≤ C‖f ‖qY ,

n=1

where C does not depend of f . Then Y = [X]q (with equivalent quasinorms). Proof. The space X is a dense subset of Y. Since X is dense in [X]q , we see that it suffices to prove that the q-norms ‖ ⋅ ‖Y and Nq are equivalent on X. Let f = ∑ fj , where {fj } is a finite sequence in X. Then ‖f ‖qY ≤ ∑ ‖fj ‖qY ≤ C q ∑ ‖fj ‖qX . Taking the infimum over {fj } ⊂ X, we get ‖f ‖Y ≤ CNq (f ). (Incidentally, this shows that Nq is a q-norm.) To prove the reverse inequality, let f ∈ X. Then f = ∑∞ n=1 fn , where ∞

∑ ‖fn ‖qX ≤ C‖f ‖qY .

n=1

Since Nq (fn ) ≤ ‖fn ‖X , we get and

q ∑∞ n=1 Nq (fn )

≤ C‖f ‖qY . Hence ∑ fn converges in [X]q to f ,

Nq (f )q ≤ ∑ Nq (fn )q ≤ C‖f ‖qY , which completes the proof.

464 | A Quasi-Banach spaces Proposition A.7. If X is a quasi-Banach space such that Ker Nq = {0} (0 < q ≤ 1), then each f ∈ [X]q can be represented as ∞

f = ∑ fn , n=1

fn ∈ X (the series converges in [X]q ),

q q such that ∑∞ n=1 ‖fn ‖X ≤ CNq (f ), where C is independent of f .

Proof. Let f ∈ Y := [X]q , ‖f ‖Y = 1, and 0 < ε < 1. First, we choose f1 , . . . , fk1 ∈ X such that ‖f − g1 ‖Y < ε,

k1

∑ ‖fj ‖qX < (1 + ε),

and

j=1

where k1

g1 = ∑ fj . j=1

k

2 Then choose g2 = ∑j=k f , f ∈ X, so that +1 j j 1

‖f − g1 − g2 ‖Y < ε2

and

k2

∑ ‖fj ‖qX ≤ (1 + ε)‖f − g1 ‖Y < (1 + ε)ε.

j=k1 +1

Continuing in this way, we find a strictly increasing sequence {kj } (k0 = 0) and vectors {fj }∞ 1 ⊂ X such that 󵄩󵄩 kn 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩f − ∑ fj 󵄩󵄩󵄩 < εn 󵄩󵄩 󵄩󵄩 󵄩󵄩 j=1 󵄩 󵄩Y

and

kν+1

∑ ‖fj ‖qX ≤ (1 + ε)εν

j=kν +1

(n ≥ 1, ν ≥ 0).

The series ∑∞ j=1 fj is convergent in Y (and its sum is clearly equal to f ) because ∞

∞

j=1

j=1

∑ ‖fj ‖qY ≤ ∑ ‖fj ‖qX ≤

1+ε < ∞. 1−ε

This incidentally proves the desired inequality. Exercise A.8. The dual of a quasi-Banach space X is X 󸀠 = L(X, ℂ). If X 󸀠 separates points, then the Banach envelope of X is equal to the completion of the normed space (X, N), where N(f ) = sup{|Λf | : Λ ∈ X 󸀠 , ‖Λ‖ ≤ 1}. Proposition A.9. Let X be a p-Banach space, and let T ∈ L(X, X) be an operator such that ‖I − T‖ < 1, where I is the identity operator. Then T is invertible, and 󵄩󵄩 −1 󵄩󵄩p p −1 󵄩󵄩T 󵄩󵄩 ≤ (1 − ‖I − T‖ ) .

A.3 The closed graph theorem

| 465

k Proof. Consider the series ∑∞ k=0 (I − T) . We have

󵄩󵄩 n 󵄩󵄩p n 󵄩󵄩 󵄩 󵄩󵄩 ∑ (I − T)k 󵄩󵄩󵄩 ≤ ∑ ‖I − T‖pk . 󵄩󵄩 󵄩󵄩 󵄩󵄩k=m 󵄩󵄩 k=m Therefore the series converges; denote its sum by S. Then we have ST = TS = I and pk ‖S‖p ≤ ∑∞ k=0 ‖I − T‖ , which was to be proved. Exercise A.10. If X = Lp (0, 1), 0 < p < 1, and 1 ≥ q > p, then Nq (f ) = 0 for all f ∈ X. This is connected with the relation L(X, Y) = {0}, where Y is an arbitrary q-Banach space. In particular, the dual of Lp (0, 1) is trivial; this was proved by Day [141].

A.3 The closed graph theorem Let X, Y be a pair of complete spaces such that X is a dense subset of Y, which means that each point of Y can be approximated by points of X. This does not imply that points of a ball K1 ⊂ Y can be approximated by points of any fixed ball K2 ⊂ X, that is, K 2 ⊃ K1 . (K 2 = the closure of K2 in the topology of Y.) Namely, as the following theorem states, if K 2 ⊃ K1 , then X = Y. Theorem A.2. Let X and Y be quasi-Banach spaces. Let T ∈ L(X, Y) be such that the closure of T(B), where B = {f ∈ X : ‖f ‖ < 1}, contains a neighborhood of zero in Y. Then the mapping T: X 󳨃→ Y is open, and the operator T̂ : X/ Ker T 󳨃→ Y is invertible. A mapping is open if it maps open sets onto open sets. If T ∈ L(X, Y), then the ̂ + Ker T) = Tf . The quasinorm in X/Z is operator T̂ ∈ L(X/ Ker T, Y) is defined by T(f defined by ‖f + Z‖ = inf{‖f − g‖ : g ∈ Z}. Proof. Because of the Aoki–Rolewicz theorem, we can suppose that X and Y are p-normed for some p < 1. Let δ > 0 and U = {f ∈ X : ‖f ‖p < δ}. From the hypotheses of the theorem it follows that there are balls Un = {f ∈ X : ‖f ‖p < δn } and Vn = {g ∈ Y : ‖g‖p < εn }, n ≥ 1, limn εn = 0, such that (1) Vn ⊂ T(Un ),

and (2)

∞

∑ δn < δ.

n=1

(A.4)

We will prove that V1 ⊂ T(Δ2 ); then it will be easy to complete the proof. Let g ∈ V1 . It follows from (A.4)(1) that there exists f1 ∈ U1 such that g − Tf1 ∈ V2 . Similarly, there is f2 ∈ U2 such that (g − Tf1 ) − Tf2 ∈ V3 . Continuing in this way, we get the sequence of relations n

g − ∑ Tfk ∈ Vn+1 , k=1

fk ∈ Uk .

466 | A Quasi-Banach spaces p It follows that g = ∑∞ n=1 Tfn . Since ‖fk ‖ < δk , inequality (A.4)(2) implies that the series ∑k fk converges; denote its sum by f . Thus we have g = Tf and ∞

‖f ‖p ≤ ∑ ‖fn ‖p < δ, n=1

which was to be proved. Theorem (The open mapping theorem). Let X and Y be complete spaces, and let T ∈ L(X, Y). If T is onto, then T is open, and T̂ acts as an isomorphism of X/ Ker T onto Y. In particular, T is invertible if it is onto and one-to-one. Proof. Let U denote the unit ball of X. Choose a neighborhood W of 0 ∈ X so that W −W ⊂ U. By hypothesis, the space Y is the union of the sets T(nW), n ≥ 1. According to Baire’s cathegory theorem, the closure of one of them has the nonempty interior, which implies that T(W) contains an open set V ≠ 0. Then V − V is a neighborhood of zero and the inclusions V − V ⊂ T(W) − T(W) ⊂ T(W) − T(W) ⊂ T(U) hold. Now we appeal to Theorem A.2 to conclude the proof. Exercise A.11. A subspace E of a quasi-Banach space X is said to have the Hahn–Banach extension property (HBEP) if each λ ∈ E ∗ has an extension Λ ∈ X ∗ . If E has HBEP, then Λ can be chosen so that ‖Λ‖X ∗ ≤ C‖λ‖E ∗ , where C is independent of λ. As a particular case of the open mapping theorem we have the following: Theorem (The theorem on equivalent norms). Let ‖ ⋅ ‖1 and ‖ ⋅ ‖2 be quasinorms on a vector space X such that ‖f ‖1 ≤ ‖f ‖2 for every f ∈ X. If X is complete with respect to both quasinorms, then there exists a constant C < ∞ such that ‖f ‖2 ≤ C‖f ‖1 for all f ∈ X. The uniform boundedness principle Theorem (Banach–Steinhaus). Let X and Y be quasi-Banach spaces, and let {As } ⊂ L(X, Y) be a family of operators. If sups ‖As f ‖ < ∞ for all f ∈ X, then sups ‖As ‖ < ∞. In particular, the limit of an everywhere convergent sequence of bounded operators is a bounded operator. Proof. Let ‖f ‖2 = ‖f ‖X + sups ‖As f ‖Y (f ∈ X). From the hypotheses it follows that the functional ‖ ⋅ ‖2 is a quasinorm on X. It is not hard to prove that the space (X, ‖ ⋅ ‖2 ) is complete, and therefore the conclusion follows from Theorem A.3. The proof of the Banach–Steinhaus theorem depends on the Baire category theorem. An elementary proof, essentially due to Hausdorff [246], can be found in [253]. Corollary A.12. Let B : X × Y 󳨃→ Z be a separately continuous bilinear operator, where X, Y, Z are quasi-Banach spaces. Then there is a constant C < ∞ such that ‖B(f , g)‖Z ≤ C‖f ‖X ‖g‖Y for all f ∈ X, g ∈ Y.

A.3 The closed graph theorem

| 467

“Separately continuous” means that every operator of the form f 󳨃→ B(f , g) (f ∈ X) or g → 󳨃 B(f , g) (g ∈ Y) is continuous. Schauder bases A sequence {en : n ≥ 1} in a quasi-Banach space X is called a Schauder basis of X if to each f ∈ X there corresponds a unique scalar sequence {λn (f )} such that f = ∑∞ n=1 λn (f )en , the series converging in the topology of X. Proposition A.13. If {en : n ≥ 1} is a Schauder basis of X, then the functionals λn are continuous, and the linear operators sn : X 󳨃→ X defined by sn f = ∑nk=1 λk (f )ek are uniformly bounded. Proof. Let ⦀f ⦀ = supn ‖sn f ‖. Since ‖f − sn f ‖ → 0, we have ‖f ‖ ≤ K⦀f ⦀, where K is the constant from (A.1), and therefore it suffices to prove that X is complete with respect to the quasinorm ⦀ ⋅ ⦀. Let {fj }∞ j=1 be a Cauchy sequence in ⦀ ⋅ ⦀. This implies, because of the completeness of ‖ ⋅ ‖, that there is a sequence {gn } such that sup ‖sn fj − gn ‖ → 0 n≥1

as j → ∞,

(A.5)

and that, for every k, the sequence {λk (fj )}∞ j=1 converges; let γk = limj λk (fj ). Since the functional λk is linear and the space sn (X) is finite-dimensional, it follows that λk (gn ) = limj λk (sn fj ) = limj λk (fj ) = γk for k ≤ n and λk (gn ) = 0 for k > n. Hence gn = ∑nk=1 γk ek . On the other hand, (A.5) implies that {gn } converges in ‖ ⋅ ‖ to some g. Thus g = ∑∞ n=1 γn en , whence gn = sn g. Returning to (A.5), we see that ⦀fj − g⦀ → 0 as j → ∞, which was to be proved. Exercise A.14. Let {ej : j ≥ 1} be a Schauder basis in a Banach spaces X, let {aj : j ≥ 1} be a sequence of complex numbers, and let {λn } be a monotone sequence of positive real numbers. Then, for m < n, 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 λm /C ≤ 󵄩󵄩󵄩 ∑ λj aj ej 󵄩󵄩󵄩 ≤ Cλn if {λj } is increasing, 󵄩󵄩 󵄩󵄩 󵄩j=m 󵄩 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 λn /C ≤ 󵄩󵄩󵄩 ∑ λj aj ej 󵄩󵄩󵄩 ≤ Cλm if {λj } is decreasing, 󵄩󵄩󵄩 󵄩󵄩󵄩j=m where C is a constant independent of m, n, {aj }, and {λj }. The closed graph theorem Theorem (The closed graph theorem). Let T : X 󳨃→ Y be a linear operator, where X and Y are complete spaces. Then T is continuous if the following condition is satisfied: For every sequence {fn } ⊂ X such that fn tends to 0 ∈ X and Tfn tends to some g ∈ Y, we have g = 0.

468 | A Quasi-Banach spaces Proof. It follows from the hypotheses that X is complete with respect to the quasinorm ‖f ‖2 = ‖f ‖X + ‖Tf ‖Y , so we can apply the theorem on equivalent norms.

A.4 F -spaces The closed graph theorem remains valid in a wider class of spaces, the so-called F-spaces. By the term “F-norm” on a vector space X we mean a functional N: X 󳨃→ [0, ∞) satisfying: (a) N(f ) = 0 ⇒ f = 0; (b) N(f + g) ≤ N(f ) + N(g); and (c) N(λf ) ≤ N(f ) for |λ| ≤ 1, and lim N(λf ) = 0.

(A.6)

λ→0

The formula d(f , g) = N(f − g) defines an invariant metric on X, and the topology induced by this metric is vectorial, which means in particular that multiplication by scalars is continuous on ℂ × X. In the case where the metric d is complete, the space X is called an F-space. A p-Banach space can be treated as an F-space by introducing the F-norm N(f ) = ‖f ‖pX . Besides, if X is a locally convex space with topology given by a sequence of seminorms pn (n = 0, 1, 2, . . . ), then the formula 2−n pn (f ) 1 + pn (f ) n=0 ∞

N(f ) = ∑

defines an F-norm on X that induces the same topology. A.4.1 The Nevanlinna class The usual introduction into the theory of Hardy spaces goes across the Nevanlinna class. This class consists of the functions f ∈ H(𝔻) for which π

󵄨 󵄨 N(f ) := sup ∫ − log(1 + 󵄨󵄨󵄨f (reiθ )󵄨󵄨󵄨) dθ < ∞. 0 1. n≥0 λn inf

(A.9)

This sequence and also the series (A.8) are said to be lacunary. What is important in the Gurariy–Matsaev theorem 4.6 is the validity of the inequality 1

∞

0

n=0

󵄨p 󵄨 ∫󵄨󵄨󵄨ℒ(r)󵄨󵄨󵄨 dr ≥ cp ∑ |an |p /λn ,

cp = const > 0,

without the hypothesis an ≥ 0; if an ≥ 0, then the proof is almost trivial. The proof of the reverse inequality is also rather elementary (see, e. g., [383], where a more general result was proved). Even the case of positive coefficients is very useful because it leads to the so-called blocking technique (see [221]). It is possible to extend Gurariy–Matsaev theorem in two directions. Firstly, we can consider the case where an are members of a quasi-Banach space, and, secondly, we can consider the class of functions F(x, y), 0 ≤ x ≤ 1, y ≥ 0, for which λa μα F(x, y) ≤ F(λx, μy) ≤ λb μβ F(x, y)

(0 < λ, μ < 1).

(A.10)

This class will be denoted by Δ2 (a, b; α, β); F ∈ Δ2 means that F belongs to Δ2 (a, b; α, β) for some values of the parameters. Theorem A.8. Let X be a p-Banach space, and let F ∈ Δ2 (a, b; α, β). Define ℒ by (A.8) (λ0 ≥ 1), where an ∈ X, and the series converges for 0 < r < 1. Then the conditions 1

󵄩 󵄩 ∫ F(1 − r, 󵄩󵄩󵄩ℒ(r)󵄩󵄩󵄩)(1 − r)−1 dr < ∞ 0

(A.11)

472 | A Quasi-Banach spaces and ∞

∑ F(1/λn , ‖an ‖) < ∞

(A.12)

n=0

and the corresponding quantities are equivalent, the equivalence constants depending only on p, α, β, a, and b. The author decided to omit the proof in this edition and refers the reader to the paper [448]. The main tool in the proof is the following lemma. Lemma A.16. Let λk+1 /λk ≥ q > 1 for all k ≥ 0. Then there exists ε > 0 such that, for all δ > 0, there exists a polynomial P(r) of the form P(r) = p1 r N + ⋅ ⋅ ⋅ + pN+1 r 2N such that for 2−(1+ε) < r < 2−1 ,

P(r) ≥ 1 0 < P(r) ≤ δr

for 0 < r < 2

0 < P(r) ≤ δ(1 − r) for 2

−(1+ε)/q

−q(1−ε)

,

< r < 1.

(A.13) (A.14) (A.15)

Further notes and results Let hp,q (𝔹N ) denote the class of all complex-valued functions f harmonic in the unit ϕ ball 𝔹N of ℝN (abbreviated f ∈ h(𝔹N )) such that 1

∫ Mpq (r, f ) 0

ϕq (1 − r) dr < ∞, 1−r

(A.16)

where ϕ is a normal weight, and 󵄨p 󵄨 Mp (r, f ) = (∫ 󵄨󵄨󵄨f (rζ )󵄨󵄨󵄨 dσN (ζ ))

1/p

.

𝕊N

Each f ∈ h(𝔹N ) can be represented in a unique way as ∞

f (rζ ) = ∑ r n fn (ζ ), n=0

where fn is a spherical harmonic of degree n, that is, the restriction to SN of a homogeneous harmonic polynomial of degree n. If ∞

f (rζ ) = ∑ r λn fλn (ζ ), n=0

where {λn } is lacunary, then f is said to be a function with Hadamard gaps.

(A.17)

Further notes and results | 473

We can associate with f a function g on (0, 1) with values in Lp (𝕊N ) by g(r)(ζ ) = f (rζ ). Consequently, condition (A.16) can be rewritten as 1

q 󵄩q ϕ (1 − r) dr 󵄩 < ∞. ∫󵄩󵄩󵄩g(r)󵄩󵄩󵄩p 1−r 0

Here ‖ ⋅ ‖p = ‖ ⋅ ‖Lp (𝕊N ) . From this, by means of Theorem A.8 applied with F(x, y) = ϕ(x)q yq , we immediately obtain the following: Theorem A.9. Let 0 < p ≤ ∞, 0 < q < ⬦, and ϕ a normal weight. Let f be a function (𝔹N ) if and only if with Hadamard gaps given by (A.17). Then f belongs to hp,q ϕ ∞

∑ ϕ(1/λn )q ‖fλn ‖qp < ∞.

n=0

This theorem covers the case of analytic Bergman spaces on the ball of ℂn although then the things are simple, and there are several papers where this case is considered. The interested reader can take this as an exercise. Theorem A.9 holds for q = ∞ with an appropriate interpretation of hp,∞ . Two ϕ spaces arise: the space hp,∞ (𝔹N ) = {f ∈ h(𝔹N ) : sup ϕ(1 − r)Mp (r, f ) < ∞} ϕ 0 0.

The (Luxemburg) quasinorm in LF (X) (resp., ℓF,λ (X)) is defined as the infimum of those t > 0 such that 1

󵄩 󵄩 ∫ F(1 − r, 󵄩󵄩󵄩f (r)󵄩󵄩󵄩/t)(1 − r)−1 dr ≤ 1 0

(resp., ∞

∑ F(1/λn , ‖an ‖/t) ≤ 1);

n=0

see [397]. If f is nonzero on a set of positive measure, then due to the properties of F, there exists a unique t > 0 such that 1

󵄩 󵄩 ∫ F(1 − r, 󵄩󵄩󵄩f (r)󵄩󵄩󵄩/t)(1 − r)−1 dr = 1, 0

and we have t = ‖f ‖LF (X) ; a similar statement holds for ℓF,λ . From this and from Theorem A.8 we can deduce the following result. Theorem. Let X be a quasi-Banach space. Let ℒ(r) be given by (A.9), where λ = {λn }, λ0 ≥ 1, is a lacunary sequence of positive real numbers, and an ∈ X. Then ℒ belongs to LF (X) if and only if {an } belongs to ℓF,λ (X), and there is a constant C > 0 depending only on F and X such that 󵄩 󵄩 C −1 󵄩󵄩󵄩{an }󵄩󵄩󵄩ℓ

F,λ (X)

󵄩 󵄩 ≤ ‖ℒ‖LF (X) ≤ C 󵄩󵄩󵄩{an }󵄩󵄩󵄩ℓ (X) . F,λ

B Bounded vector-valued analytic functions In this chapter, we apply the simplest case of the Coifman–Rochberg theorem on the atomic decomposition of Bergman spaces (Theorem 3.17) and its applications to homogeneous admissible spaces and, more generally, to bounded vector-valued analytic functions. A theory of vector-valued Besov and H p -spaces (p < 1) is sketched. The Banach envelope and the dual of H p (X) are determined for p < 1. Some applications to composition operators are also presented.

B.1 Some properties of homogeneous spaces If X is a homogeneous H-admissible space and f ∈ X, then the vector-valued function ∞

F(z) = Ff (z) = ∑ f ̂(n)en z n , n=0

z ∈ 𝔻,

is analytic and bounded, that is, belongs to H ∞ (X), and 󵄩 󵄩 ‖F‖∞,X := sup󵄩󵄩󵄩F(z)󵄩󵄩󵄩X = ‖f ‖X . z∈𝔻

It turns out that most properties of X can be deduced from the corresponding properties of H ∞ (X), so it makes sense to consider H ∞ (X) in itself. Let X be a quasi-Banach space. A function F : Ω 󳨃→ X, where Ω is a domain in ℂ, is said to be analytic if every point in Ω admits a neighborhood in which f can be expanded into a power series with X-valued coefficients. In the case where Ω is the ̂ unit disc, it turns out that the analyticity implies the existence of vectors F(n) such that ∞

n ̂ F(z) = ∑ F(n)z n=0

for all z ∈ 𝔻,

(B.1)

with uniform convergence on compact subsets. However, the proof of this fact is rather ̂ difficult. To avoid this path, we can agree that F is analytic in 𝔻 if there are vectors F(n) such that (B.1) holds. These vectors are uniquely determined and are called the Taylor coefficients of F and, as it is expected, satisfy the condition 󵄩 ̂ 󵄩󵄩1/n lim sup󵄩󵄩󵄩F(n) 󵄩󵄩 ≤ 1. n→∞

On the other hand, if {fn } is a sequence of vectors in X, then the condition lim sup ‖fn ‖1/n ≤ 1 n→∞

https://doi.org/10.1515/9783110630855-015

(B.2)

476 | B Bounded vector-valued analytic functions n is necessary and sufficient for the series ∑∞ n=0 fn z to converge for every z ∈ 𝔻. In the case of convergence, the sum of that series is analytic in 𝔻. Therefore the set of the functions F : 𝔻 󳨃→ X analytic in 𝔻 can be identified with the set of the formal power series satisfying (B.2), that is, with the set of the power series converging in 𝔻. We will denote this set by H(𝔻, X). We endow H(𝔻, X) with the topology of uniform convergence on compact subsets of 𝔻. There is a substantial difference between functions with values in a Banach space and those with values in a quasi-Banach space, namely the failure of the maximum modulus principle: Let Jp,0 denote the closed linear span in Lp (𝕋) (0 < p < 1) of the Cauchy kernels φz (ζ ) = (1 − ζz)−1 , where |z| ≤ 1 and ζ ∈ 𝕋. Let Q : Lp (𝕋) 󳨃→ Lp (𝕋)/Jp,0 be the quotient map and define v(z) = Q(u(z)), |z| ≤ 1, where u(z)(ζ ) = (1 − ζz)−1 . As noted by Aleksandrov [21], the (nonconstant) function v is analytic in 𝔻, continuous on 𝔻, and vanishes on 𝕋. Another difference is the failure of Montel’s theorem. Namely, if X is infinitedimensional, then the unit sphere is not totally bounded, which implies the existence of a sequence {fn } on the unit sphere such that infm=n̸ ‖fm − fn ‖ > 0. Then every subsequence of the sequence of constant functions Fn (z) = fn diverges in H(𝔻, X).

Hadamard product The Hadamard product of a (scalar) function ψ ∈ H(𝔻) and a function F ∈ H(𝔻, X) is defined by ∞

n ̂ F(n)z ̂ . (ψ ∗ F)(z) = ∑ ψ(n) n=0

The proof that ψ ∗ F belongs to H(𝔻, X) is straightforward. Proposition B.1. Let F ∈ H(𝔻, X), where X is a p-Banach space, and let the series ∑∞ k=1 ψk (z) = ψ(z), where ψj ∈ H(𝔻), converge uniformly on compact subsets. Then ∞

(ψ ∗ F)(z) = ∑ (ψk ∗ F)(z) (|z| < 1), k=1

the series being uniformly convergent on compact subsets of 𝔻. Proof. Let |z| = r < 1, FN (z) = ∑Nj=1 (ψj ∗ F)(z), and RN (z) = ∑∞ j=N+1 ψj (z). Then ∞

󵄩p 󵄩p 󵄩 󵄩󵄩 󵄩󵄩FN (z) − ψ ∗ F(z)󵄩󵄩󵄩 = 󵄩󵄩󵄩RN ∗ F(z)󵄩󵄩󵄩 ≤ ∑ AN (j), j=0

(B.3)

̂ N (j)|p ‖F(j)‖ ̂ p |z|jp . The sequence RN (z) is uniformly bounded on comwhere AN (j) = |R ̂ N (j)| ≤ pact subsets and therefore, for every ρ < 1, there exists a constant Cρ such that |R j j ̂ Cρ /ρ for all N and j. From this and from the inequality ‖F(j)‖ ≤ Kρ /ρ we get AN (j) ≤

B.1 Some properties of homogeneous spaces | 477

M(ρ)(r/ρ2 )jp , where Mρ = Cϱp Kρp . Thus by taking ρ2 = √r we see that the sequence AN has a summable majorant. On the other hand, from the hypothesis RN (z) → 0, uniformly on compact subsets, it follows AN (j) → 0 (N → ∞) for every j. Therefore we can apply the dominated convergence theorem: ∞

∞

j=0

j=0

lim ∑ AN (j) = ∑ lim AN (j) = 0.

N→∞

N→∞

Now the desired statement follows from (B.3). The proof is independent of the completeness of H(𝔻, X); this property will be proved later on. Inequalities for the coefficients Let H ∞ (X) denote the set of all bounded functions F ∈ H(𝔻, X); the quasinorm is given by 󵄩 󵄩 ‖F‖∞,X = sup󵄩󵄩󵄩F(z)󵄩󵄩󵄩X . |z|λ

and H(λ) ≤ (2/λ)q ∫ |hλ |q dσ = (2/λ)q ∫ |f |q dσ. S

|f |≤λ

Now we use the formula ∞

∞

∫ |Tf |s dμ = s ∫ μ(Tf , λ)λs−1 dλ ≤ s ∫ (G(λ) + H(λ))λs−1 dλ. Ω

0

0

Multiplying inequality (C.8) by sλ

s−1

∞

∞

0

0

and then integrating over λ ∈ (0, ∞), we get

s ∫ G(λ)λs−1 dλ ≤ s2p ∫ (λs−p−1 ∫ |f |p dσ) dλ |f |>λ |f |

= s2p ∫(∫ λs−p−1 dλ)|f |p dσ = S

0

s2p ∫ |f |s dσ. s−p

The analogous inequality for H(λ) is proved in a similar way.

S

500 | C Lebesgue spaces: Interpolation and maximal functions Paley’s theorem The implication ∞

󵄨p󸀠 󵄨 f ∈ Lp (𝕋) ⇒ ∑ 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 < ∞ n=−∞

(1 < p < 2),

(C.9)

which is a weak form of the Hausdorff–Young theorem, was improved by Hardy and Littlewood [234]: p−2 ∗ p (cn ) < ∞, where {cn∗ } is the Theorem C.5. If f ∈ Lp (𝕋), 1 < p < 2, then ∑∞ n=0 (n + 1) ̂ decreasing rearrangement of the sequence {f (n)}.

An application of Marcinkiewicz’s theorem yields a more general result, due to Paley [414]: Theorem C.6. Let (Ω, μ) be a finite measure space, and let {φn }∞ 1 be an orthonormal sequence in L2 (Ω, μ) such that supn ‖φn ‖∞ < ∞. Then ∞

∑ np−2 |an |p ≤ C‖f ‖pp ,

n=1

f ∈ Lp (Ω, μ),

where 1 < p < 2 and an = ∫Ω f φn dμ. For the converse of this theorem, see [639, Theorem XII(5.1)]. Proof. Let μ(Ω) = 1 and supn ‖φn ‖∞ = K. Define the measure σ on ℕ+ , the set of positive integers, by σ({n}) = n−2 . Define the operator T: L1 (Ω, μ) by (Tf )(n) = nan . Bessel’s inequality implies that T is of strong type (2, 2). To prove that T is of weak type (1, 1) and therefore to conclude the proof (by Marcinkiewicz’s theorem), observe that |an | ≤ K‖f ‖1 . Hence, if ‖f ‖1 = 1, then we have 󵄨 󵄨 σ{n: 󵄨󵄨󵄨Tf (n)󵄨󵄨󵄨 > λ} ≤ σ{n: Kn > λ} ≤ ∑ n−2 ≤ CK min{1, 1/λ}, n>λ/K

which concludes the proof. Exercise C.3. Theorem C.5 improves the implication (C.9), which can be deduced from the implication ∞

p

∑ (n + 1)p−2 (cn∗ ) < ∞ 󳨐⇒ cn∗ = 𝒪((n + 1)1/p−1 )

n=0

and the equality ∞

∞

∞

n=−∞

n=0

n=0

p󸀠 p󸀠 −p ∗ p 󵄨p󸀠 󵄨 (cn ) . ∑ 󵄨󵄨󵄨f ̂(n)󵄨󵄨󵄨 = ∑ (cn∗ ) = ∑ (cn∗ )

C.3 Classical maximal functions | 501

Exercise C.4 (Hardy–Littlewood [230]). If {bn }∞ −∞ is a sequence such that, for some q > 2, ∞

∑ (|n| + 1)

q−2

n=−∞

|bn |q < ∞,

̂ then there exists a function g ∈ Lq (𝕋) such that g(n) = bn . Exercise C.5. If f is a function harmonic in the unit ball 𝔹n ⊂ ℝn , then 1

n−2

(∫(1 − r)

1/p p M∞ (r, f ) dr)

≤ Cp,n ‖f ‖p ,

1 < p ≤ ∞.

(C.10)

0

This can be proved by considering the measure dμ(r) = (1 − r)n−2 dr on (0, 1) and the operator T defined by Tf (r) = M∞ (r, f ), 0 < r < 1. The function f may take values in ℝn . If f is quasiconformal (not necessarily harmonic), then a result of Astala and Koskela [50, Theorem 3.3] asserts that inequality (C.10) and its reverse hold for all p > 0.

C.3 Classical maximal functions The maximal function of a 2π-periodic function ϕ ∈ L1 (−π, π) is the (2π-periodic) function ℳϕ defined as θ+h

1 󵄨 󵄨 (ℳϕ)(θ) = sup ∫ 󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 dt. 2h 0 1, then ℳϕ ∈ Lp (−π, π) and ‖ℳϕ‖p ≤ Cp ‖ϕ‖p , where Cp depends only of p.

502 | C Lebesgue spaces: Interpolation and maximal functions Proof. Statement (b) is obtained from (a) by Marcinkiewicz’s theorem. To prove (a), let ϕ ∈ L1 (−π, π), let E = {θ ∈ (−π, π) : ℳϕ(θ) > 1}, and let K be a compact subset of the (open) set E. It suffices to find an absolute constant C such that |K| ≤ C‖ϕ‖1 . By the definition of ℳϕ and the compactness of E there are intervals Ii (i = 1, . . . , n) such that Ii ⊂ (−2π, 2π), K ⊂ ⋃ Ii , and |Ii | ≤ ∫I |ϕ(t)| dt. Assume that the sequence |Ii | i is decreasing. Let J1 = I1 and J2 = Ik , where k is the smallest i for which Ii ∩ J1 = 0. (If such k does not exist, then clearly K ⊂ ⋃i Ii ⊂ 3I1 , where 3I1 is the interval concentric with I1 , and |3I1 | = 3|I1 |, and we take J1∗ = 3I1 , stopping the procedure.) Then let J3 = Im , where m is the smallest i > k such that Ii ∩ (J1 ∪ J2 ) = 0. Continuing in this way, we find a sequence Jj ⊂ (−2π, 2π) of pairwise disjoint intervals such that K ⊂ ⋃ Jj∗ , where, for each j, Jj∗ is the interval concentric with Jj , and |Jj∗ | = 3|Jj |. It follows that 2π

󵄨 󵄨 󵄨 󵄨 (1/3)|K| ≤ ∑ |Jj | ≤ ∑ ∫󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 dt ≤ ∫ 󵄨󵄨󵄨ϕ(θ)󵄨󵄨󵄨 dθ, j

j J j

−2π

which gives the desired inequality with C = 6. Lebesgue points The maximal theorem has many important applications. It is useful, for example, in proving almost everywhere convergence. Usually, we can easily prove a. e. convergence for a dense set of functions and then use the maximal theorem to interchange the limits. Here we consider the existence of Lebesgue points. The Lebesgue point of a measurable function ϕ : ℝ 󳨃→ ℂ is a point x ∈ ℝ such that h

1 󵄨󵄨 󵄨 lim ∫ 󵄨󵄨ϕ(t + x) − ϕ(x)󵄨󵄨󵄨 dt = 0. h→0 2h −h

The set of all Lebesgue points of ϕ is called the Lebesgue set of ϕ. Theorem C.8. If a 2π-periodic function ϕ is integrable on (−π, π), then almost every point in ℝ is a Lebesgue point for ϕ. Corollary C.6. The inequality |ϕ(θ)| ≤ (ℳϕ)(θ) holds almost everywhere. Proof of the theorem. The operator h

Tϕ(x) = lim sup h→0

1 󵄨󵄨 󵄨 ∫ 󵄨ϕ(t + x) − ϕ(x)󵄨󵄨󵄨 dt 2h 󵄨 −h

satisfies: (a) T(ϕ1 +ϕ2 ) ≤ Tϕ1 +Tϕ2 ; (b) Tϕ ≤ |ϕ|+Mϕ; and (c) Tg = 0 if g is continuous.

C.3 Classical maximal functions | 503

Let ϕ ∈ L1 (−π, π), λ > 0, and ε > 0. Choose a continuous function g such that ‖ϕ − g‖1 < ε. From (a) we get Tϕ ≤ Tg + T(ϕ − g) = T(ϕ − g), and then, from (b), by Theorem C.7 and Chebyshev’s inequality we get 󵄨 󵄨󵄨 󵄨󵄨{θ : T(ϕ − g)(θ) > λ}󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨{θ : |ϕ − g|(θ) > λ/2}󵄨󵄨󵄨 + 󵄨󵄨󵄨{θ : ℳ(ϕ − g)(θ) > λ/2}󵄨󵄨󵄨 4π 2C 2(2π + C)ε ≤ ‖ϕ − g‖1 + ‖ϕ − g‖1 ≤ . λ λ λ Thus |{θ : T(ϕ − g)(θ) > λ}| = 0 for every λ > 0, because ε is arbitrary. It follows that |{θ : T(ϕ − g)(θ) > 0}| = 0, which completes the proof. Radial maximal function Let u be a complex-valued function on 𝔻. The radial maximal function of u is the function Mrad u defined on 𝕋 by 󵄨 󵄨 (Mrad u)(ζ ) = sup 󵄨󵄨󵄨u(rζ )󵄨󵄨󵄨 (ζ ∈ 𝕋). 0≤r 1, the convergence is dominated. Theorem C.9 (radial maximal). The operator Mrad maps h1 (resp., hp , p > 1) into L1,⋆ (𝕋) (resp., Lp (𝕋)) and is continuous. Proof. If u ∈ hp , p > 1, then u = 𝒫 [ϕ], ϕ ∈ Lp (𝕋), and therefore the result follows from Theorem C.7 and Proposition C.7. The same holds in the case where p = 1 and u = 𝒫 [ϕ]; if u ∈ h1 is arbitrary, then, by Theorem C.7 and Proposition C.7, ‖uρ ‖1 ‖u‖1 󵄨 󵄨 |{ζ ∈ 𝕋 : sup 󵄨󵄨󵄨u(rζ )󵄨󵄨󵄨 > λ} ≤ C ≤C , λ λ 0 1, then ̂ (n) = m f ̂(n) Tf n for every integer n.

(C.21)

C.6 Nikishin–Stein’s theorem

| 513

Proof. Since Lp is of type p for 0 < p ≤ 2, the Nikishin–Stein theorem tells us that the operator T is of weak type (p, p) for p = 1 and p = 2. Hence, by Marcinkiewicz’s theorem, T is of strong type (p, p) for 1 < p < 2. To prove the rest, let g(w) = wn , w ∈ 𝕋, for a fixed integer n. By the hypothesis, for every ζ ∈ 𝕋, we have ζ n (Tg)(w) = (Tg)(ζw) for a. e. w ∈ 𝕋. The function Tg belongs to L1 because g ∈ L2 and T is of strong type (p, p) for p ∈ (1, 2). It follows that if ϕ ∈ L∞ , then − (Tg)(w)ϕ(w)|dw| = ∫ − ζ −n (Tg)(ζw)ϕ(w)|dw| ∫ 𝕋

𝕋

for every ζ ∈ 𝕋. Integrating this with respect to ζ and using Fubini’s theorem, we get − ζ −n (Tg)(ζw)|dζ | − ϕ(w)|dw| ∫ − (Tg)(w)ϕ(w)|dw| = ∫ ∫ 𝕋

𝕋

𝕋

=∫ − ϕ(w)wn |dw| ∫ − ζ −n (Tg)(ζ )|dζ |. 𝕋

𝕋

Hence (Tg)(w) = wn ∫ − ζ −n (Tg)(ζ )|dζ | =: mn wn

for a. e. w ∈ 𝕋,

𝕋

which proves formula (C.20). The validity of (C.21) can then be deduced from the Weierstrass theorem that the trigonometric polynomials are dense in Lp . It remains to prove that T is of strong type (q, q) for q ≥ 2. By Marcinkiewicz’s theorem (or by the Riesz–Thorin theorem) we can assume that q > 2. Let f , g be trigonometric polynomials. Then, in view of (C.20), we have π

iθ

− (Tf )(e )g(e ∫ −π

−iθ

π

) dθ = ∫ − f (eiθ )(Tg)(e−iθ ) dθ. −π

Using this and the fact that T is of strong type (p, p) for p = q/(q − 1), we conclude that ‖Tf ‖q ≤ C‖f ‖q ,

(C.22)

where C is independent of f . Now let f ∈ Lq be arbitrary, and let fn be a sequence of trigonometric polynomials such that ‖fn − f ‖q → 0. The validity of (C.22) for trigonometric polynomials implies ‖Tfn ‖q ≤ Cq ‖f ‖q .

(C.23)

Since ‖fn − f ‖1 ≤ ‖fn − f ‖q and T is continuous from L1 to ℒ0 , we see that Tfn → Tf in measure; after extracting a subsequence, we can assume that Tfn → Tf almost everywhere. Now Fatou’s lemma and (C.23) give ‖Tf ‖q ≤ Cq ‖f ‖q , which was to be proved.

514 | C Lebesgue spaces: Interpolation and maximal functions

C.7 Banach’s principle and the theorem on a. e. convergence The following fact, known as Banach’s principle, plays an important role in applications of Theorem C.17 to maximal operators. Theorem C.19 (Banach’s principle). Let X be a quasi-Banach space, let Tn (n ≥ 1) be a sequence of continuous linear operators from X to ℒ0 (Ω, μ), and let 󵄨 󵄨 Tmax f (ω) := sup󵄨󵄨󵄨Tn f (ω)󵄨󵄨󵄨 < ∞ n≥1

for almost all ω ∈ Ω

for every f ∈ X. Then the operator Tmax : X 󳨃→ ℒ0 (Ω, μ) is continuous. Proof (cf. [197]). Let ℒ0 (ℓ∞ ) denote the set of functions F = (f1 , f2 , . . .) : Ω 󳨃→ ℓ∞ with measurable coordinates. The following two facts imply the validity of the theorem. (i) With the F-norm ∫ Ω

‖F(ω)‖∞ dμ(ω), 1 + ‖F(ω)‖∞

the set ℒ0 (ℓ∞ ) is an F-space. (ii) The operator Tg = (T1 g, T2 g, . . .) maps X into ℒ0 (ℓ∞ ) and satisfies the condition of the closed graph theorem. In general, we do not have Tmax f (ω) < ∞ a. e., but we have a partition of Ω: Theorem C.20 (Sawier [526]). Let X be a quasi-Banach space, let Tn (n ≥ 1) be a sequence of continuous linear operators from X to ℒ0 (Ω, μ), and let Tmax f (ω) := supn≥1 |Tn f (ω)|. Then there exists a decomposition Ω = Ω0 ∪ Ω1 of the measure space Ω, depending only on the sequence {Tn }, such that (a) Tmax f (ω) < ∞ a. e. on Ω0 for every f ∈ X and (b) Tmax f (ω) = ∞ a. e. on Ω1 for every f from a subset of X of the second Baire category. Proof (cf. [526]). Assume that A ⊂ Ω is a measurable set and f1 a member of X such that Tmax f1 (ω) = ∞ for almost every ω ∈ A. For all N > 1, let KN = {f : 𝒩A (f ) ≤ (1 − 1/N)μ(A)}, where 𝒩A (f ) = ∫ A

Tmax f (ω) dμ(ω), 1 + Tmax f (ω)

μ(A) > 0, and, by definition, ∞/(1 + ∞) = 1. By Fatou’s lemma, for each N, the set KN is closed. We claim that KN is nowhere dense, that is, the interior of KN is empty. Otherwise, there exist f ∈ KN and δ > 0 such that 𝒩A (f + αf1 ) ≤ (1 − 1/N)μ(A)

for 0 < α < δ

C.7 Banach’s principle and the theorem on a. e. convergence |

515

and in particular δ

(†)

∫ 𝒩A (f + αf1 ) dα ≤ (1 − 1/N)μ(A)δ. 0

On the other hand, if Tmax f1 (ω) = ∞ for some ω ∈ A, then Tmax (f +αf1 )(ω) = ∞, except, maybe, for one α ∈ (0, δ), which implies that δ

δ

∫ 𝒩A (f + αf1 ) dα = ∫ dμ(ω) ∫ 0

A

0

Tmax (f + αf1 )(ω) dα = μ(A)δ. 1 + Tmax (f + αf1 )(ω)

This contradicts (†). Hence the set {f ∈ X : Tmax f = ∞ a. e. on A} = the complement of ⋃ KN N>1

is of the second category in X. Let 𝒜 be the family of all measurable sets A ⊂ Ω such that Tmax f = ∞ a. e. on A for some f ∈ X. It is easy to see that there exists Ω1 ∈ 𝒜 such that μ(Ω1 ) = supA∈𝒜 μ(A). Then the desired sets are Ω0 = Ω \ Ω1 and Ω1 . In the case of rotation-invariant operators, we have the following alternative (cf. [550, 526]). The proof is similar to that of Theorem C.17. Theorem C.21 (Sawier–Stein). Let X be as in Theorem C.17, and let Tn : X 󳨃→ ℒ0 (𝕋) be a sequence of linear operators that commute with rotations. Then either (a) Tmax (f )(ω) < ∞ a. e. on Ω for every f ∈ X, or (b) Tmax f (ω) = ∞ a. e. on Ω for every f from a subset of the second category in X. Theorem on a. e. convergence Theorem C.22. Suppose the conditions of Theorem C.19 are satisfied. If the limit lim T f (ω) n→∞ n

:= Tf (ω)

a. e.

exists and is finite for every f from a dense subset of X, then it exists for every f ∈ X, and T is continuous as an operator from X to ℒ0 . Proof. Let X0 denote the dense subset. Consider the following sublinear operator on X: 󵄨 󵄨 Sf (ω) = lim sup󵄨󵄨󵄨Tm f (ω) − Tn f (ω)󵄨󵄨󵄨 (ω ∈ Ω). m,n→∞

By Banach’s principle this operator is continuous because Sf ≤ 2Tmax f . By Lemma C.12 we have μ(Sf , ε) ≤ c(ε/‖f ‖) (ε > 0, f ∈ X),

(C.24)

516 | C Lebesgue spaces: Interpolation and maximal functions where c(λ) → 0 as λ → ∞. On the other hand, since Sg = 0 for g ∈ X0 , we have S(f ) = S(f − g) for all f ∈ X, g ∈ X0 . From this and from (C.24) it follows that μ(Sf , ε) ≤ c(ε/‖f − gk ‖)

(ε > 0),

where, for a fixed f ∈ X, we have chosen a sequence gk ∈ X0 so that ‖f − gk ‖ → 0 (k → ∞). Thus μ(Sf , ε) = 0 for every ε > 0. The result follows. As before, we have the following alternative (cf. [550, 526]). Theorem C.23. Let X be as in Theorem C.17, and let Tn : X 󳨃→ ℒ0 (𝕋) be a sequence of linear operators that commute with rotations. If the limit limn→∞ Tn f (ω) exists and is finite a. e. for every f from a dense subset of X, then either (a) this limit exists a. e. for every f ∈ X, or (b) lim supn |Tn f (ω)| = ∞ a. e. in ω ∈ Ω for every f from a set of the second category in X.

C.8 Vector-valued maximal theorem The following powerful theorem was proved by Fefferman and Stein [186]. Theorem C.24 (Fefferman–Stein). Let 1 < p < ⬦, 1 < q < ⬦, and let {fn }∞ n=0 be a sequence of measurable functions on the unit circle. Then there is a constant C such that π

∞

it

q

p/q

∫ ( ∑ (ℳfn (e )) ) −π

n=0

π

∞

−π

n=0

󵄨q 󵄨 dt ≤ C ∫ ( ∑ 󵄨󵄨󵄨fn (eit )󵄨󵄨󵄨 )

p/q

dt,

(C.25)

where ℳ denotes the Hardy–Littlewood maximal operator. A proof is also in [554], pp. 50–56. In fact, Fefferman and Stein proved it in the case of functions defined on ℝ. The periodic version can then be deduced by considering the functions φn defined as φn (θ) = fn (eiθ ) (|θ| < 2π) and φn (θ) = 0 (|θ| > 2π). This theorem can be expressed in terms of the space Lp (ℓq ). This space consists of the sequences {fn }∞ 0 defined on 𝕋 for which π

∞

−π

n=0

󵄨q 󵄨 󵄩󵄩 ∞ 󵄩󵄩 − ( ∑ 󵄨󵄨󵄨fn (eit )󵄨󵄨󵄨 ) 󵄩󵄩{fn }0 󵄩󵄩Lp (ℓq ) := ∫

p/q

dt < ⬦.

Hence (C.25) can be written as 󵄩 ∞󵄩 󵄩󵄩 ∞󵄩 󵄩󵄩{ℳfn }0 󵄩󵄩󵄩Lp (ℓq ) ≤ C 󵄩󵄩󵄩{fn }0 󵄩󵄩󵄩Lp (ℓq ) . In the case p = 1, Theorem C.24 does not hold, but we have: (i) If {fn } ∈ L1 (ℓq ), then ‖{ℳfj (ζ )}‖ℓq is finite for almost every ζ ∈ 𝕋. (ii) If {fn } ∈ L1 (ℓq ), then there is a constant C > 0 such that 󵄨 C󵄩 󵄩 󵄩 󵄩 󵄨󵄨 󵄨󵄨{ζ ∈ 𝕋 : 󵄩󵄩󵄩{ℳfj (ζ )}󵄩󵄩󵄩ℓq > λ}󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩{fn }󵄩󵄩󵄩L1 (ℓq ) , λ

λ > 0.

Further notes and results | 517

Further notes and results C.1. From Thorin’s proof of the Riesz–Thorin theorem (see [65] or [436]) it is seen that theorem holds for all p0 , p1 > 0. Calderón and Zygmund [108] proved that it holds for all pj > 0 and qj > 0 (j = 1, 2). C.2. Inequality (C.1) is optimal in the sense that ‖f ‖p cannot be replaced by C‖f ‖p with C < 1. On the other hand, if f ̂ is the Fourier transformation of f ∈ Lp (ℝ), where 1 < p < 2, then we have 1/p 1/2 ‖f ̂‖Lp󸀠 (ℝ) ≤ (p1/p /p󸀠 ) ‖f ‖Lp (ℝ) 󸀠

(see [63, 53]). C.3 (Marcinkiewicz’s L log+ L-theorem). The class of those σ-measurable functions f on S for which the integral on the right-hand side of (C.26) is finite is denoted by L log+ L(S). Theorem. Let μ and σ be finite measures on Ω and S, respectively, let 1 < q ≤ ∞, and let T be a quasilinear operator from L1 (σ) to the set of nonnegative μ-measurable functions. If T satisfies (C.3)(p = 1) and (C.4), then ∫ Tf dμ ≤ K1 + K2 ∫ |f | log+ |f | dσ, Ω

(C.26)

S

where K1 and K2 are independent of f . The proof is similar to that of Theorem C.2. For the general Marcinkiewicz theorem, see Zygmund [639, Ch. XII, § 4]; the proof is much more difficult. C.4. By using Marcinkiewicz’s L log+ L-theorem and the proof of Paley’s theorem, it can be proved that, under the hypotheses of Paley’s theorem, we have the implication ∞

|an | < ∞. n=1 n

f ∈ L log+ L(Ω) ⇒ ∑

(C.27)

In the case of Fourier series, this result, in a equivalent form, was proved by Zygmund [631] (see [639, Ch. VI, Theorem (3⋅ 9)]). Moreover, (C.27) shows the implication cn∗ < ∞. n+1 n=0 ∞

f ∈ L log+ L(𝕋) ⇒ ∑

C.5. It follows from (a) and C.3 that the operator ℳ maps L log+ L(𝕋) into L1 (𝕋). On the other hand, Stein [553] proved that if g ∈ L1 (𝕋) is such that ℳg ∈ L1 (𝕋), then g ∈ L log+ L(𝕋). (For a proof, see Torchinsky [585, Theorem 5.4].)

518 | C Lebesgue spaces: Interpolation and maximal functions C.6. In the proof of Khintchine’s inequality, it is essential that rn is defined by rn (t) = r(2n−1 t), n ≥ 1, where r ∈ L∞ (ℝ) is a 1-periodic function such that r(t + 1/2) ≡ −r(t). As an example, we can take r(t) = ei2πt to get (a particular case of) Paley’s inequality (see Theorems 2.22 and 2.23, p. 66). 2 < ∞, then the series C.7. It was proved by Rademacher [493] that if ∑∞ k=0 |ak | ∞ ∞ 2 ∑∞ k=0 ak rk (t) converges a. e. On the other hand, if ∑k=0 |ak | = ∞, then ∑k=0 ak rk (t) diverges a. e. (Khintchine and Kolmogorov [306]). The proof of these facts can be found in Duren [166] and, in a stronger form, in Zygmund [639, Ch. V, Sec. 8].

C.8. The best constants in (C.14) were found by Haagerup [225]: 1/2−1/p

2 , { { { 1/2 1+p cp = {2 (Γ( 2 )/√π)1/p , { { {1,

p ≤ p0 ,

p0 ≤ p ≤ 2,

p ≥ 2,

1, p ≤ 2, Cp = { 1/2 1+p 1/p 2 (Γ( 2 )/√π) , p ≥ 2, where p0 ≈ 1, 84742 is the solution of the equation Γ((1 + p)/2) = √π/2 in the interval (1, 2). In some particular cases, the best constants were found by other authors; we refer to [225] for information and references. C.9 (The case of σ-finite measure in Nikishin’s theorem). If (Ω, μ) is a σ-finite space, then Nikishin’s theorem states that there is a positive measurable function g on Ω such that μ{ω :

p

|Tf (w)| ‖f ‖ > λ} ≤ ( ) . g(w) λ

See [601, Ch. III.H] for explanation and further results. However, in this case, the space ℒ0 (Ω, μ) cannot be defined as in the case of μ(Ω) < ∞. Namely, the topology in ℒ0 is defined by the requirement that 󵄨 󵄨 fj → 0 (in ℒ0 (Ω, μ)) iff μ(E ∩ {ω : 󵄨󵄨󵄨fj (ω)󵄨󵄨󵄨 > λ}) → 0 for all λ > 0 and E with μ(E) < ∞. C.10 (The case of ℝn ). Let 0 < p ≤ 2. Every continuous sublinear operator T : Lp (ℝn ) 󳨃→ ℒ0 (ℝn ) that is invariant under translations and dilations is of weak type (p, p). The same holds if ℝn is replaced by the half-space {(x1 , . . . , xn−1 , xn ) : xn > 0}. For a proof, see [197, Corollary II.6.9].

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19]

[20]

[21]

[22]

E. Abakumov and E. Doubtsov, Reverse estimates in growth spaces, Math. Z. 271 (2012), no. 1–2, 399–413. E. Abakumov and E. Doubtsov, Moduli of holomorphic functions and logarithmically convex radial weights, Bull. Lond. Math. Soc. 47 (2015), no. 3, 519–532. Y. Abu-Muhanna, Bloch, BMO and harmonic univalent functions, Complex Var. Theory Appl. 31 (1996), no. 3, 271–279. Y. Abu-Muhanna, R. M. Ali, and S. Ponnusamy, The spherical metric and univalent harmonic mappings, Monatshefte Math. 188 (2019), no. 4, 703–716. Y. Abu-Muhanna and I. Louhichi, Area distortion under some orientation preserving harmonic maps, J. Anal. 26 (2018), no. 1, 135–150. Y. Abu-Muhanna and A. Lyzzaik, The boundary behaviour of harmonic univalent maps, Pac. J. Math. 141 (1990), no. 1, 1–20. Y. Abu-Muhanna and G. Schober, Harmonic mappings onto convex domains, Canad. J. Math. 39 (1987), no. 6, 1489–1530. P. Ahern, The Poisson integral of a singular measure, Can. J. Math. 35 (1983), 735–749. P. Ahern and J. Bruna, Maximal and area integral characterizations of Hardy–Sobolev spaces in the unit ball of Cn , Rev. Mat. Iberoam. 4 (1988), no. 1, 123–153. P. Ahern and M. Jevtić, Duality and multipliers for mixed norm spaces, Mich. Math. J. 30 (1983), no. 1, 53–64. P. Ahern and M. Jevtić, Mean modulus and the fractional derivative of an inner function, Complex Var. Theory Appl.. 3 (1984), 297–311. P. Ahern and W. Rudin, Zero sets of functions in harmonic Hardy spaces, Math. Scand. 73 (1993), no. 2, 209–214. P. R. Ahern and D. N. Clark, Radial N-th derivatives of Blaschke products, Math. Scand. 28 (1971), 189–201. P. R. Ahern and D. N. Clark, On inner functions with Hp -derivative, Mich. Math. J. 21 (1974), 115–127. P. R. Ahern and D. N. Clark, On inner functions with Bp derivative, Mich. Math. J. 23 (1976), 107–118. L. V. Ahlfors, On quasiconformal mappings, J. Anal. Math. 3 (1954), 1–58. L. V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, Minneapolis, Minnesota, 1981. L. V. Ahlfors, Lectures on quasiconformal mappings, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987. I. Akramov, T. Dębiec, J. W. D. Skipper, and E. Wiedemann, Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum, arXiv preprint arXiv:1808.05029 (2018). A. B. Aleksandrov, Approximation by rational functions, and an analogue of the M. Riesz theorem on conjugate functions for Lp -spaces with p ∈ (0, 1), Math. USSR Sb. 35 (1979), 301–316. A. B. Aleksandrov, Essays on nonlocally convex Hardy classes, Complex analysis and spectral theory (Leningrad, 1979/1980), Lecture Notes in Math., vol. 864, Springer, Berlin, 1981, pp. 1–89. A. B. Aleksandrov, A-integrability of the boundary values of harmonic functions, Math. Notes 30 (1981), no. 1, 515–523.

https://doi.org/10.1515/9783110630855-017

520 | Bibliography

[23] [24] [25] [26] [27]

[28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

[41] [42] [43] [44] [45]

A. B. Aleksandrov, On the boundary decay in the mean of harmonic functions, Algebra Anal. 7 (1995), no. 4, 1–49. A. B. Aleksandrov, E-mail correspondence, February 2019. A. B. Aleksandrov, J. M. Anderson, and A. Nicolau, Inner functions, Bloch spaces and symmetric measures, Proc. Lond. Math. Soc., III. Ser. 79 (1999), no. 2, 318–352. A. Aleman, Hilbert spaces of analytic functions between the Hardy and the Dirichlet space, Proc. Am. Math. Soc. 115 (1992), no. 1, 97–104. A. Aleman, A class of integral operators on spaces of analytic functions, Topics in complex analysis and operator theory. Proceedings of the winter school in complex analysis and operator theory, Malaga, Spain, February 5–9, 2006, Universidad de Malaga, Malaga, 2007, pp. 3–30. A. Aleman, M. Carlsson, and A-M. Persson, Preduals of Qp -spaces, Complex Var. Elliptic Equ. 52 (2007), no. 7, 605–628. A. Aleman, M. Carlsson, and A.-M. Persson, Preduals of Qp -spaces II. Carleson imbeddings and atomic decompositions, Complex Var. Elliptic Equ. 52 (2007), no. 7, 629–653. A. Aleman and J. A. Cima, An integral operator on Hp and Hardy’s inequality, J. Anal. Math. 85 (2001), no. 1, 157–176. A. Aleman and M. J. Martín, Convex harmonic mappings are not necessarily in h1/2 , Proc. Am. Math. Soc. 143 (2015), no. 2, 755–763. A. Aleman and A. G. Siskakis, An integral operator on Hp , Complex Var. Elliptic Equ. 28 (1995), no. 2, 149–158. A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. S. Aljančić, On the integral moduli of continuity in Lp (1 < p < ∞) of Fourier series with monotone coefficients, Proc. Am. Math. Soc. 17 (1966), 287–294. S. Aljančić and D. Aranđelović, O-regularly varying functions, Publ. Inst. Math. (Belgr.) 22(36) (1977), 5–22. J. M. Anderson, On division by inner factors, Comment. Math. Helv. 54 (1979), no. 2, 309–317. J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. J. M. Anderson and D. Girela, Inequalities of Littlewood–Paley type, multipliers and radial growth of the derivative of analytic functions, J. Reine Angew. Math. 465 (1995), 11–40. J. M. Anderson and A. L. Shields, Coefficient multipliers of Bloch functions, Trans. Am. Math. Soc. 224 (1976), no. 2, 255–265. V. V. Andreev and J. A. Cima, Weak H1 , the real and complex case, Invariant subspaces of the shift operator. CRM workshop, Centre de Recherches Mathématiques, Université de Montréal, Montréal, Canada, August 26–30, 2013, Providence, RI: American Mathematical Society, 2015, pp. 299–317. J. Arazy, S. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145. N. Arcozzi, R. Rochberg, E. T. Sawyer, and B. D. Wick, The Dirichlet space: a survey, N.Y. J. Math. 17a (2011), 45–86. N. Aronszajn, T. M. Creese, and L. J. Lipkin, Polyharmonic functions, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1983. J. L. Arregui and O. Blasco, Bergman and Bloch spaces of vector-valued functions, Math. Nachr. 261/262 (2003), 3–22. H. Arroussi, Function and operator theory on Large Bergman spaces, Ph.D. thesis, Universitat de Barcelona, 2016, pp. vii+109.

Bibliography | 521

[46] [47] [48] [49] [50] [51]

[52] [53] [54] [55]

[56] [57] [58] [59] [60] [61]

[62] [63] [64] [65] [66] [67] [68] [69]

H. Arroussi and J. Pau, Reproducing kernel estimates, bounded projections and duality on large weighted Bergman spaces, J. Geom. Anal. 25 (2015), no. 4, 2284–2312. M. Arsenović, Embedding derivatives of ℳ-harmonic functions into Lp spaces, Rocky Mt. J. Math. 29 (1999), no. 1, 61–76. M. Arsenović and M. Pavlović, On Dyakonov type theorems for harmonic quasiregular mappings, Czechoslov. Math. J. 67 (2017), no. 2, 289–296. K. Astala, T. Iwaniec, G. J. Martin, and J. Onninen, Extremal mappings of finite distortion, Proc. Lond. Math. Soc. (3) 91 (2005), no. 3, 655–702. K. Astala and P. Koskela, Hp -theory for quasiconformal mappings, Pure Appl. Math. Q. 7 (2011), no. 1, 19–50, Special Issue: In honor of Frederick W. Gehring. S. Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, vol. I, Pap. Lect. Spec. Year Oper. Theory, Indianapolis/Indiana 1985/86, Pitman Res. Notes Math. Ser., vol. 171, 1988, pp. 1–50. S. Axler, P. Bourdon, and W. Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001. K. I. Babenko, An inequality in the theory of Fourier integrals, Izv. Akad. Nauk SSSR, Ser. Mat. 25 (1961), 531–542 (Russian). A. Baernstein II, Univalence and bounded mean oscillation, Mich. Math. J. 23 (1976), no. 3, 217–223. A. Baernstein II, Some sharp inequalities for conjugate functions, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 409–416. A. Baernstein II, Analytic functions of bounded mean oscillation, Aspects of contemporary complex analysis, Academic Press, 1980, pp. 3–36. A. Baernstein II, D. Girela, and J. Á. Peláez, Univalent functions, Hardy spaces and spaces of Dirichlet type, Ill. J. Math. 48 (2004), no. 3, 837–859. G. Bao, H. Wulan, and K. Zhu, A Hardy–Littlewood theorem for Bergman spaces, Ann. Acad. Sci. Fenn., Math. 43 (2018), 807–821. N. K. Bari and S. B. Stechkin, Best approximations and differential properties of two conjugate functions, Tr. Mosk. Mat. Obŝ. 5 (1956), 483–522 (Russian). N. K. Bary, Trigonometric series, Nauka, Moscow, 1961 (Russian). F. Bayart, O. F. Brevig, A. Haimi, J. Ortega-Cerdà, and K-M. Perfekt, Contractive inequalities for Bergman spaces and multiplicative Hankel forms, Trans. Am. Math. Soc. (2018), https://doi.org/10.1090/tran/7290. F. Beatrous and J. Burbea, Characterizations of spaces of holomorphic functions in the ball, Kodai Math. J. 8 (1985), 36–51. W. Beckner, Inequalities in Fourier analysis, Ann. Math. (2) 102 (1975), no. 1, 159–182. A. S. Belov, On a problem of Salem and Zygmund concerning the smoothness of an analytic function that generates a Peano curve, Mat. Sb. 181 (1990), no. 8, 1048–1060 (Russian). C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988. G. Bennett, D. A. Stegenga, and R. M. Timoney, Coefficients of Bloch and Lipschitz functions, Ill. J. Math. 25 (1981), 520–531. J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin, 1976. S. Bergman, Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. I, II, J. f. M. 169, 1–42; 172, 89–128 (1932) (German). S. Bergman, The kernel function and conformal mapping, Mathematical Surveys, vol. 5, American Mathematical Society, Providence, R.I., 1950.

522 | Bibliography

[70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80]

[81] [82]

[83] [84]

[85] [86] [87] [88] [89] [90] [91] [92] [93]

S. N. Bernstein, Sur l’approximation des fonctions continues par des polynomes, C. R. Acad. Sci. Paris 152 (1911), 502–504. S. N. Bernstein, On the best approximation of continuous functions by polynomial of a given degree, Soobshch. Kharkov Mat. Obshch. 13 (1912), no. 2, 49–194 (Russian). S. N. Bernstein, On majorants of finite or quasi-finite growth, Dokl. Akad. Nauk SSSR 65 (1949), 117–120 (Russian). K. O. Besov, The boundary behavior of components of polyharmonic functions, Math. Notes 64 (1998), no. 4, 450–460. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239–255. A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. C. J. Bishop, Bounded functions in the little Bloch space, Pac. J. Math. 142 (1990), no. 2, 209–225. C. J. Bishop, An indestructible Blaschke product in the little Bloch space, Publ. Mat., Barc. 37 (1993), no. 1, 95–109. E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Am. Math. Soc. 67 (1961), no. 1, 1. W. Blaschke, Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, Leipz. Ber. 67 (1915), 194–200. O. Blasco, Spaces of vector valued analytic functions and applications, Geometry of Banach spaces, Proc. Conf., Strobl/Austria 1989, Lond. Math. Soc. Lect. Note Ser., vol. 158, London Math. Soc., 1991, pp. 33–48. O. Blasco, Operators on weighted Bergman spaces (0 < p ≤ 1) and applications, Duke Math. J. 66 (1992), no. 3, 443–467. O. Blasco, Multipliers on weighted Besov spaces of analytic functions, Banach spaces. Proceedings of an international workshop on Banach space theory, held at the Universidad de Los Andes, Merida, Venezuela, January 6–17, 1992, Contemporary Mathematics, American Mathematical Society, Providence, RI, 1993, pp. 23–33. O. Blasco, Multipliers on spaces of analytic functions, Can. J. Math. 47 (1995), no. 1, 44–64. O. Blasco, A remark on Carleson measures from Hp to Lq (μ) for 0 < p < q < ∞, Proceedings of the lecture notes of the seminar, Universities of Malaga and Seville, Spain, September 2003–June 2004, Univ. de Sevilla, Secretariado de Publicaciones, Sevilla, 2004, pp. 11–19. O. Blasco, Remarks on weighted mixed norm spaces, Contemp. Math. 561 (2012), 137–154. O. Blasco and J. L. Arregui, Multipliers on vector valued Bergman spaces, Can. J. Math. 54 (2002), no. 6, 1165–1186. O. Blasco and G. S. de Souza, Spaces of analytic functions on the disc where the growth of Mp (F , r) depends on a weight, J. Math. Anal. Appl. 147 (1990), no. 2, 580–598. O. Blasco and M. Pavlović, Complex convexity and vector-valued Littlewood-Paley inequalities, Bull. Lond. Math. Soc. 35 (2003), no. 6, 749–758. O. Blasco and M. Pavlović, Coefficient multipliers on Banach spaces of analytic functions, Rev. Mat. Iberoam. 27 (2011), no. 2, 415–447. R. P. Boas, Jr., Isomorphism between Hp and Lp , Am. J. Math. 77 (1955), 655–656. B. Böe, A norm on the holomorphic Besov space, Proc. Am. Math. Soc. 131 (2003), no. 1, 235–241 (electronic). J. Bonet, W. Lusky, and J. Taskinen, Solid hulls and cores of weighted H∞ spaces, Rev. Mat. Complut. 31 (2019), no. 3, 781–804. J. Bonet and J. Taskinen, Solid hulls of weighted Banach spaces of analytic functions on the unit disc with exponential weights, Ann. Acad. Sci. Fenn. 43 (2018), 521–530.

Bibliography | 523

[94] [95] [96] [97] [98]

[99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118]

J. Bonet and J. Taskinen, Solid hulls of weighted Banach spaces of entire functions, Rev. Mat. Iberoam. 34 (2018), no. 2, 593–608. J. Bonet and D. Vukotić, A note on completeness of weighted normed spaces of analytic functions, Results Math. 72 (2017), no. 1–2, 263–279. A. Borichev and H. Hedenmalm, Weighted integrability of polyharmonic functions, Adv. Math. 264 (2014), 464–505. N. Bourbaki, Elements of mathematics. Functions of a real variable. Elementary theory. Transl. from the French, Springer, Berlin, 2004. P. S. Bourdon, J. H. Shapiro, and W. Sledd, Fourier series, mean Lipschitz spaces, and bounded mean oscillation, Analysis at Urbana, Vol. I (Urbana, IL, 1986–1987), London Math. Soc. LNS, vol. 137, 1995, pp. 81–110. J. Bourgain, New Banach space properties of the disc algebra and H∞ , Acta Math. 152 (1984), no. 1–2, 1–48. V. Božin, B. Karapetrović, Failure of Korenblum’s maximum principle in Bergman spaces with small exponents, Proc. Am. Math. Soc. 146 (2018), no. 6, 2577–2584. O. F. Brevig, J. Ortega-Cerdà, K. Seip, and J. Zhao, Contractive inequalities for Hardy spaces, Funct. Approx. Comment. Math. (2018), 16 pages, Advance publication. S. M. Buckley, P. Koskela, and D. Vukotić, Fractional integration, differentiation, and weighted Bergman spaces, Math. Proc. Camb. Philos. Soc. 126 (1999), no. 2, 369–385. S. M. Buckley, M. S. Ramanujan, and D. Vukotić, Bounded and compact multipliers between Bergman and Hardy spaces, Integral Equ. Oper. Theory 35 (1999), no. 1, 1–19. S. M. Buckley, Relative solidity for spaces of holomorphic functions, Math. Proc. R. Ir. Acad. 104A (2004), no. 1, 83–97 (electronic). J. Burbea, Sharp inequalities for holomorphic functions, Ill. J. Math. 31 (1987), 248–264. D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class Hp , Trans. Am. Math. Soc. 157 (1971), 137–153. A. P. Calderón, Commutators of singular integral operators, Proc. Natl. Acad. Sci. USA 53 (1965), 1092–1099. A. P. Calderón and A. Zygmund, A note on the interpolation of linear operators, Stud. Math. 12 (1951), 194–204. S. Campanato, Proprietà di Hölderianità di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17 (1963), 175–188 (Italian). D. Campbell, J. Cima, and K. Stephenson, A Bloch function in all Hp classes, but not in BMOA, Proc. Am. Math. Soc. 78 (1980), 228–230. T. Carleman, Zur Theorie der Minimalflächen, Math. Z. 9 (1921), 154–160. L. Carleson, An interpolation problem for bounded analytic functions, Am. J. Math. 80 (1958), 921–930. L. Carleson, A representation formula for the Dirichlet integral, Math. Z. 73 (1960), 190–196. L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. Math. 75 (1962), 547–559. L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–137. C. Cascante, J. Fàbrega, and J. M. Ortega, The corona theorem for weighted Hardy and Morrey spaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 13 (2014), no. 3, 579–607. G. R. Chacon Perez, Carleson-type inequalities in harmonically weighted Dirichlet spaces, Ph.D. thesis, University of Tennessee – Knoxville, 2010. HuaiHui Chen, The Schwarz–Pick lemma and Julia lemma for real planar harmonic mappings, Sci. China Math. 56 (2013), no. 11, 2327–2334.

524 | Bibliography

[119] G. Choquet, Sur un type de transformation analytique généralisant la représentation conforme et definie au moyen de fonctions harmoniques, Bull. Sci. Math., II. Ser. 69 (1945), 156–165. [120] J. A. Cima, A type of Volterra operator, Complex Anal. Synergies. 2 (2016), 4 pp. [121] J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy transform, Operator Theory: Advances and Applications, vol. 156, Birkhäuser, Basel, 2005. Expanded version of talks and papers presented at a conference on the occasion of the 75th birthday of Harold S. Shapiro, Santa Barbara, CA, USA, March 2003. [122] J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society (AMS), Providence, RI, 2006. [123] J. A. Cima and A. Nicolau, Inner functions with derivatives in the weak Hardy space, Proc. Am. Math. Soc. 143 (2015), no. 2, 581–594. [124] J. A. Cima and W. T. Ross, The backward shift on the Hardy space, Mathematical Surveys and Monographs, vol. 79, American Mathematical Society (AMS), Providence, RI, 2000. [125] J. A. Cima and G. Schober, Analytic functions with bounded mean oscillation and logarithms of Hp functions, Math. Z. 151 (1976), no. 3, 295–300. [126] G. Cleanthous, A. G. Georgiadis, and M. Nielsen, Molecular decomposition and Fourier multipliers for holomorphic Besov and Triebel–Lizorkin spaces, Monatshefte Math. 188 (2019), no. 3, 467–493. [127] B. D. MacCluer, Compact composition operators on Hp (BN ), Mich. Math. J. 32 (1985), 237–248. [128] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn., Ser. A 1 Math. 9 (1984), 3–25. [129] J. G. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions, Comment. Math. Helv. 59 (1984), 362–375. [130] W. S. Cohn and I. E. Verbitsky, Factorization of tent spaces and Hankel operators, J. Funct. Anal. 175 (2000), no. 2, 308–329. [131] R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. [132] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in Lp , Astérisque 77 (1980), 11–66. [133] P. Colwell, Blaschke products, Bounded analytic functions, University of Michigan Press, Ann Arbor, MI, 1985. [134] L. Colzani, Hardy and Lipschitz spaces on unit spheres, Ph.D. thesis, Washington University, St. Louis, 1982. [135] L. Colzani, Hardy-spaces on unit spheres, Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 4 (1985), 219–244. [136] L. Colzani, M. H. Taibleson, and G. Weiss, Maximal estimates for Cesàro and Riesz means on spheres, Indiana Univ. Math. J. 33 (1984), 873–889. [137] O. Constantin and J. Á. Pelaéz, Boundedness of the Bergman projection on Lp -spaces with exponential weights, Bull. Sci. Math. 139 (2015), no. 3, 245–268. [138] C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. [139] B. Davis, On the weak type (1, 1) inequality for conjugate functions, Proc. Am. Math. Soc. 44 (1974), 307–311. [140] W. J. Davis, D. J. H. Garling, and N. Tomczak-Jaegermann, The complex convexity of quasi-normed linear spaces, J. Funct. Anal. 55 (1984), 110–150. [141] M. M. Day, The spaces Lp with 0 < p < 1, Bull. Am. Math. Soc. 46 (1940), 816–823.

Bibliography | 525

[142] K. de Leeuw, Y. Katznelson, and J.-P. Kahane, Sur les coefficients de Fourier des fonctions continues, C. R. Acad. Sci., Paris, Sér. A 285 (1977), 1001–1003. [143] J. A. de Reyna, Pointwise convergence of Fourier series, Lecture Notes in Mathematics, vol. 1785, Springer-Verlag, Berlin, 2002. [144] G. S. de Souza and G. Sampson, A real characterization of the pre-dual of Bloch functions, J. Lond. Math. Soc. 2 (1983), no. 2, 267–276. [145] R. A. DeVore and G. G. Lorentz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften, 303, Springer-Verlag, Berlin, 1993. [146] E. Diamantopoulos and A. G. Siskakis, Composition operators and the Hilbert matrix, Stud. Math. 140 (2000), no. 2, 191–198. [147] J. Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, vol. 485, Springer-Verlag, Berlin, 1975. [148] J. Diestel, H. Jarchow, A. Tonge, Absolutely summing operators, Camb. Stud. Adv. Math., vol. 43, Cambridge University Press, Cambridge, 2008. [149] O. Djordjević and M. Pavlović, On a Littlewood-Paley type inequality, Proc. Am. Math. Soc. 135 (2007), no. 11, 3607–3611 (electronic). p [150] A. E. Djrbashian and F. A. Shamoyan, Topics in the theory of Aα spaces, BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1988. [151] M. M. Djrbashian, On canonical representation of functions meromorphic in the unit disc, Dokl. Akad. Nauk Arm. SSR, vol. 3, 1945, pp. 3–9. [152] M. M. Djrbashian, On the problem of the representation of analytic functions, Soobshch. Inst. Mat. Mekh. Akad. Nauk Arm. SSR (1948), no. 2, 3–40. [153] E. P. Dolzhenko, On boundary properties of the components of polyharmonic functions, Math. Notes 63 (1998), no. 6, 724–735. [154] J. J. Donaire, D. Girela, and D. Vukotic, On univalent functions in some Möbius invariant spaces, J. Reine Angew. Math. 553 (2002), 43–72. [155] M. R. Dostanić, Unboundedness of the Bergman projections on Lp spaces with exponential weights, Proc. Edinb. Math. Soc., II. Ser. 47 (2004), no. 1, 111–117. [156] M. R. Dostanić, Multipliers in the space of analytic functions with exponential mean growth, Asymptot. Anal. 65 (2009), no. 3–4, 191–201. [157] M. R. Dostanić, M. Jevtić, and D. Vukotić, Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type, J. Funct. Anal. 254 (2008), no. 11, 2800–2815. [158] E. Doubtsov, Integral means of holomorphic functions as generic log-convex weights, Bull. Aust. Math. Soc. 97 (2018), no. 1, 88–93. [159] E. Doubtsov, Composition operators between Bloch and BMOA spaces on polydiscs, J. Math. Sci. (N.Y.) 229 (2018), no. 5, 518–527. [160] J. Douglas, Solution of the problem of Plateau, Trans. Am. Math. Soc. 33 (1931), 263–321. [161] O. Dovgoshey and J. Riihentaus, Bi-Lipschitz mappings and quasinearly subharmonic functions, Int. J. Math. Math. Sci. 2010 (2010), 8 p. [162] O. Dovgoshey and J. Riihentaus, On quasinearly subharmonic functions, Lobachevskii J. Math. 38 (2017), no. 2, 245–254. [163] P. Duren and A. Schuster, Bergman spaces, Mathematical Surveys, vol. 100, American Mathematical Society (AMS), Providence, RI, 2004. [164] P. L. Duren, Extension of a theorem of Carleson, Bull. Am. Math. Soc. 75 (1969), 143–146. [165] P. L. Duren, On the multipliers of Hp spaces, Proc. Am. Math. Soc. 22 (1969), 24–27. [166] P. L. Duren, Theory of Hp spaces, Pure and Applied Mathematics, vol. 38, Academic Press, New York, 1970. [167] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983.

526 | Bibliography

[168] P. L. Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge, 2004. [169] P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on Hp spaces with 0 < p < 1, J. Reine Angew. Math. 238 (1969), 32–60. [170] P. L. Duren and A. L. Shields, Properties of Hp (0 < p < 1) and its containing Banach space, Trans. Am. Math. Soc. 141 (1969), 255–262. [171] P. L. Duren and A. L. Shields, Coefficient multipliers of Hp and Bp spaces, Pac. J. Math. 32 (1970), 69–78. [172] K. M. Dyakonov, Smooth functions and co-invariant subspaces of the shift operator, Algebra Anal. 4 (1992), no. 5, 117–147 (Russian). [173] K. M. Dyakonov, Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math. 178 (1997), no. 2, 143–167. [174] K. M. Dyakonov, The moduli of holomorphic functions in Lipschitz spaces, Mich. Math. J. 44 (1997), no. 1, 139–147. [175] K. M. Dyakonov, Besov spaces and outer functions, Mich. Math. J. 45 (1998), no. 1, 143–157. [176] K. M. Dyakonov, Weighted Bloch spaces, Hp , and BMOA, J. Lond. Math. Soc., II. Ser. 65 (2002), no. 2, 411–417. [177] K. M. Dyakonov, Holomorphic functions and quasiconformal mappings with smooth moduli, Adv. Math. 187 (2004), no. 1, 146–172. [178] K. M. Dyakonov, Addendum to “Strong Hardy–Littlewood theorems for analytic functions and mappings of finite distortion”, Math. Z. 254 (2006), no. 2, 433–437. [179] K. M. Dyakonov, Functions in Bloch-type spaces and their moduli, Ann. Acad. Sci. Fenn., Math. 41 (2016), 705–712. [180] E. M. Dyn’kin, A constructive characterization of the Sobolev and Besov classes, Proc. Steklov Inst. Math. 155 (1983), 39–74. [181] E. M. Dyn’kin, The pseudoanalytic extension, J. Anal. Math. 60 (1993), 45–70. [182] P. J. Eenigenburg and F. R. Keogh, The Hardy class of some univalent functions and their derivatives, Mich. Math. J. 17 (1970), 335–346. [183] O. El-Fallah, K. Kellay, and T. Ransford, Invariant subspaces of the Dirichlet space, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51 (2010), pp. 133–141. [184] D. O. jun. Etter, Vector-valued analytic functions, Trans. Am. Math. Soc. 119 (1965), 352–366. [185] P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335–400. [186] C. Fefferman and E. M. Stein, Some maximal inequalities, Am. J. Math. 93 (1971), 107–115. [187] C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193. [188] L. Fejér and F. Riesz, Über einige funktionentheoretische Ungleichungen, Math. Z. 11 (1921), 305–314. [189] A. Figà-Talamanca, Translation invariant operators in Lp , Duke Math. J. 32 (1965), 495–501. [190] T. M. Flett, On some theorems of Littlewood and Paley, J. Lond. Math. Soc. 31 (1956), 336–344. [191] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. Lond. Math. Soc., III. Ser. 7 (1957), 113–141. [192] T. M. Flett, Some theorems on power series, Proc. Lond. Math. Soc., III. Ser. 7 (1957), 211–218. [193] T. M. Flett, On the rate of growth of mean values of holomorphic and harmonic functions, Proc. Lond. Math. Soc. 20 (1970), no. 3, 749–768. [194] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746–765. [195] T. M. Flett, Lipschitz spaces of functions on the circle and the disc, J. Math. Anal. Appl. 39 (1972), 125–158. [196] F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974), 593–602.

Bibliography | 527

[197] J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985, Notas de Matemática [Mathematical Notes], 104. [198] J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981. [199] J. B. Garnett and R. H. Latter, The atomic decomposition for Hardy spaces in several complex variables, Duke Math. J. 45 (1978), 815–845. [200] P. M. Gauthier and J. Xiao, BiBloch-type maps: existence and beyond, Complex Var. Theory Appl. 47 (2002), no. 8, 667–678. [201] F. W. Gehring, W. K. Hayman, and A. Hinkkanen, Analytic functions satisfying Hölder conditions on the boundary, J. Approx. Theory 35 (1982), 243–249. [202] S. Gergün, S. T. Kaptanoğlu, and A. E. Üreyen, Harmonic Besov spaces on the ball, Int. J. Math. 27 (2016), pp. 59. [203] D. Girela, Analytic functions of bounded mean oscillation, Complex function spaces, Univ. Joensuu Dept. Math. Rep. Ser., vol. 4, Univ. Joensuu, Joensuu, 2001, pp. 61–170. [204] D. Girela, C. González, and M. Jevtić, Inner functions in Lipschitz, Besov, and Sobolev spaces, Abstr. Appl. Anal. 2011 (2011), 26 p. [205] D. Girela, C. González, and J. Á. Peláez, Multiplication and division by inner functions in the space of Bloch functions, Proc. Am. Math. Soc. 134 (2006), no. 5, 1309–1314. [206] D. Girela, C. González, and J. Á Peláez, Toeplitz operators and division by inner functions, Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 85–103. [207] D. Girela and N. Merchán, A Hankel matrix acting on spaces of analytic functions, Integral Equ. Oper. Theory 89 (2017), no. 4, 581–594. [208] D. Girela, M. Pavlović, and J. Á. Peláez, Spaces of analytic functions of Hardy-Bloch type, J. Anal. Math. 100 (2006), 53–81. [209] D. Gnuschke and Ch. Pommerenke, On the growth of functions with Hadamard gaps, J. Lond. Math. Soc., II. Ser. 30 (1984), 441–450. [210] D. Gnuschke-Hauschild and Ch. Pommerenke, On Bloch functions and gap series, J. Reine Angew. Math. 367 (1986), 172–186. [211] G. Godefroy, Existence and uniqueness of isometric preduals: a survey, Banach space theory (Iowa City, IA, 1987), Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 131–193. [212] G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society, Providence, RI, 1969. [213] C. González and J. Á. Peláez, Univalent functions in Hardy spaces in terms of the growth of arc-length, J. Geom. Anal. 19 (2009), no. 4, 755–771. [214] H. C. Greenwald, Lipschitz spaces on the surface of the unit sphere of the euclidean n-space, Pac. J. Math. 50 (1974), no. 1, 63–80. [215] H. C. Greenwald, Lipschitz spaces of distributions on the surface of the unit sphere of the euclidean n-space, Pac. J. Math. 70 (1977), no. 1, 163–176. [216] A. Grigoryan and M. Nowak, Estimates of integral means of harmonic mappings, Complex Var. Theory Appl. 42 (2000), no. 2, 151–161. [217] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1956), 1–79. [218] J. Gröhn and A. Nicolau, Inner functions in weak Besov spaces, J. Funct. Anal. 266 (2014), no. 6, 3685–3700. [219] J. Gröhn and A. Nicolau, Inner functions in certain Hardy–Sobolev spaces, J. Funct. Anal. 272 (2017), no. 6, 2463–2486.

528 | Bibliography

[220] J. Gröhn, J. Á. Peláez, and J. Rättyä, Jointly maximal products in weighted growth spaces, Ann. Acad. Sci. Fenn., Ser. A 1 Math. 39 (2014), no. 1, 109–118. [221] K.-G. Groosse-Erdman, Blocking technique, weighted mean operators, and Hardy’s inequalities, Lecture Notes in Mathematics, vol. 1679, Springer-Verlag, Berlin, 1998. [222] V. I. Gurariy and V. I. Matsaev, Lacunary power sequences in the spaces C and Lp , Amer. Math. Soc., Transl., II. Ser. 72 (1968), 9–21. [223] V. P. Gurariy, Factorization of absolutely convergent Taylor series and Fourier integrals, J. Sov. Math. 4(1975) (1976), 313–328. [224] A. E. Gwilliam, On Lipschitz conditions, Proc. Lond. Math. Soc. 40 (1935), 353–364. [225] U. Haagerup, The best constants in the Khintchine inequality, Stud. Math. 70 (1981), no. 3, 231–283. [226] K. T. Hahn, Minimum problems of Plateau type in the Bergman metric space, Pac. J. Math. 14 (1964), 943–955. [227] E. Harboure, J. L. Torrea, and B. E. Viviani, A vector-valued approach to tent spaces, J. Anal. Math. 56 (1991), 125–140. [228] G. H. Hardy, The mean value of the modulus of an analytic function, Proc. Lond. Math. Soc. 14 (1915), no. 2, 269–277. [229] G. H. Hardy and J. E. Littlewood, A further note on the converse of Abel’s theorem, Proc. Lond. Math. Soc. (2) 25 (1926), 219–236. [230] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, Math. Ann. 97 (1927), no. 1, 159–209. [231] G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Z. (1928), no. 28, 612–634. [232] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1928), no. 1, 565–606. [233] G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), 81–116. [234] G. H. Hardy and J. E. Littlewood, Notes on the theory of series. XIII: Some new properties of Fourier constants, J. Lond. Math. Soc. 6 (1931), 3–9. [235] G. H. Hardy and J. E. Littlewood, Some new cases of Parseval’s theorem, Math. Z. 34 (1932), 620–633. [236] G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. Reine Angew. Math. 167 (1932), 405–423. [237] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no. 1, 403–439. [238] G. H. Hardy and J. E. Littlewood, Theorems concerning Cesàro means of power series, Proc. Lond. Math. Soc. (2) 36 (1934), 516–530. [239] G. H. Hardy and J. E. Littlewood, Notes on the theory of series. XXI: Generalizations of a theorem of Paley, Q. J. Math., Oxf. Ser. 8 (1937), 161–171. [240] G. H. Hardy and J. E. Littlewood, Theorems concerning mean values of analytic or harmonic functions, Q. J. Math., Oxf. Ser. 12 (1941), 221–256. [241] G. H. Hardy and W. W. Rogosinski, fourier series, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 38, Cambridge University Press, 1944. [242] A. Harnack, Existenzbeweise zur Theorie des Potentiales in der Ebene und im Raume, Leipz. Ber. (1886), 144–169. [243] A. Hartmann, X. Massaneda, A. Nicolau, and J. Ortega-Cerdà, Reverse Carleson measures in Hardy spaces, Collect. Math. 65 (2014), no. 3, 357–365. [244] W. W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Am. Math. Soc. 52 (1975), 237–241.

Bibliography | 529

[245] F. Hausdorff, Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen, Math. Z. 16 (1923), 163–169. [246] F. Hausdorff, Zur Theorie der linearen metrischen Räume, J. Reine Angew. Math. 167 (1932), 294–311. [247] V. P. Havin, The factorization of analytic functions that are smooth up to the boundary, Zap. Nauč. Semin. LOMI 22 (1971), 202–205. [248] V. P. Havin and F. A. Shamoyan, Analytic functions with a Lipschitzian modulus of the boundary values, Zap. Nauč. Semin. LOMI 19 (1970), 237–239. [249] W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, vol. 9, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976. [250] H. Hedenmalm, Bloch functions and asymptotic tail variance, Adv. Math. 313 (2017), 947–990. [251] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. [252] E. Heinz, On one-to-one harmonic mappings, Pac. J. Math. 9 (1959), 101–105. [253] J. Hennefeld, A nontopological proof of the uniform boundedness theorem, Am. Math. Mon. 87 (1980), no. 3, 217. [254] G. Herglotz, Über Potenzreihen mit positivem, reellem Teil im Einheitskreis, Leipz. Ber. 63 (1911), 501–511. [255] M. Hervé, Analytic and plurisubharmonic functions in finite and infinite dimensional spaces, Course given at the University of Maryland, College Park, Md., Spring 1970, Lecture Notes in Mathematics, vol. 198, Springer-Verlag, Berlin, 1971. [256] A. Hinkkanen, The sharp form of certain majorization theorems for analytic functions, Complex Var. Theory Appl. 12 (1989), no. 1–4, 39–66. [257] A. Hinkkanen, Majorization of the modulus of continuity of analytic functions, Comput. Methods Funct. Theory 8 (2008), no. 2, 303–325. [258] A. Hinkkanen, Growth conditions for analytic functions in unbounded open sets, Complex Var. Elliptic Equ. 56 (2011), no. 1–4, 59–80. [259] G. Hoepfner and J. Hounie, Atomic decomposition of holomorphic Hardy spaces in S1 and applications, Lecture Notes of Seminario Interdisciplinare di Matematica 7, Potenza: Università degli Studi della Basilicata, Dipartimento di Matematica e Informatica, 2008, pp. 189–206. [260] J. Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Stud. Math. 52 (1974), 159–186. [261] F. Holland and J. B. Twomey, On Hardy classes and the area function, J. Lond. Math. Soc., II. Ser. 17 (1978), 275–283. [262] F. Holland and J. B. Twomey, Explicit examples of Bloch functions in every Hp space, but not in BMOA, Proc. Am. Math. Soc. 95 (1985), 227–229. [263] F. Holland and D. Walsh, Criteria for membership of Bloch space and its subspace BMOA, Math. Ann. 273 (1986), 317–335. [264] B. Hollenbeck and I. E. Verbitsky, Best constants for the Riesz projection, J. Funct. Anal. 175 (2000), no. 2, 370–392. [265] L. Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston Inc., Boston, MA, 1994. [266] R. A. Hunt, On the convergence of Fourier series, Orthogonal expansions and their continuous analogues (Proc. Conf., Edwardsville, Ill., 1967), Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235–255. [267] K. Ivanov and P. Petrushev, Harmonic Besov and Triebel–Lizorkin spaces on the ball, J. Fourier Anal. Appl. 23 (2017), no. 5, 1062–1096.

530 | Bibliography

[268] T. Iwaniec, G. J. Martin, and J. Onninen, On minimisers of Lp -mean distortion, Comput. Methods Funct. Theory 14 (2014), no. 2–3, 399–416. [269] D. Jackson, Über die Genauigkeit der Annäheurung stetiger Funktionen durch ganze rationale Funktionen, Ph.D. thesis, Götingen, 1911. [270] D. Jackson, On approximation by trigonometric sums and polynomials, Trans. Am. Math. Soc. 13 (1912), 491–515. [271] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959–982. [272] M. Jevtić, On the Carleson measure characterization of BMO functions on the unit sphere, Proc. Am. Math. Soc. 114 (1992), no. 2, 379–386. [273] M. Jevtić and M. Pavlović, On multipliers from Hp to lq , 0 < q < p < 1, Arch. Math. (Basel) 56 (1991), no. 2, 174–180. [274] M. Jevtić and M. Pavlović, On the solid hull of the Hardy space Hp , 0 < p < 1, Mich. Math. J. 54 (2006), 439–446. [275] M. Jevtić and M. Pavlović, Besov–Lipschitz and mean Besov–Lipschitz spaces of holomorphic functions on the unit ball, Potential Anal. 38 (2013), no. 4, 1187–1206. [276] M. Jevtić, D. Vukotić, and M. Arsenović, Taylor coefficients and coefficient multipliers of Hardy and Bergman-type spaces, Springer, Cham, 2016. [277] D. R. Jocić, Equality of norms for a class of Bloch and symmetrically weighted Lipschitz spaces of vector valued functions and derivation inequalities for Pick functions, J. Funct. Anal. https://doi.org/10.1016/j.jfa.2018.12.005 (2018). [278] F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math. 14 (1961), 415–426. [279] P. W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), 41–66. [280] P. W. Jones, L∞ estimates for the 𝜕̄ problem in a half-plane, Acta Math. 150 (1983), no. 1–2, 137–152. [281] O. G. Jørsboe and L. Mejlbro, The Carleson–Hunt theorem on Fourier series, Lecture Notes in Mathematics, vol. 911, Springer-Verlag, Berlin, 1982. [282] J.-P. Kahane, Best approximation in L1 (T ), Bull. Am. Math. Soc. 80 (1974), 788–804. [283] J.-P. Kahane, Some random series of functions, second ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. [284] D. Kalaj, On Riesz type inequalities for harmonic mappings on the unit disk, Trans. Amer. Math. Soc., DOI: https://doi.org/10.1090/tran/7808. [285] D. Kalaj, Quasiconformal harmonic functions between convex domains, Publ. Inst. Math., Nouv. Sér. 76(90) (2004), 3–20. [286] D. Kalaj, Quasiconformal and harmonic mappings between Jordan domains, Math. Z. 260 (2008), no. 2, 237–252. [287] D. Kalaj, Quasiconformal harmonic mappings and close-to-convex domains, Filomat 24 (2010), no. 1, 63–68. [288] D. Kalaj, Quasiconformal harmonic mappings between Dini-smooth Jordan domains, Pac. J. Math. 276 (2015), no. 1, 213–228. [289] D. Kalaj, A sharp inequality for harmonic diffeomorphisms of the unit disk, J. Geom. Anal. 29 (2019), no. 1, 392–401. [290] D. Kalaj and M. Marković, Optimal estimates for harmonic functions in the unit ball, Positivity 16 (2012), no. 4, 771–782. [291] D. Kalaj and M. Pavlović, On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation, Trans. Am. Math. Soc. 363 (2011), no. 8, 4043–4061. [292] D. Kalaj and E. Saksman, Quasiconformal maps with controlled Laplacian, J. Anal. Math. 137 (2019), no. 1, 251–268.

Bibliography | 531

[293] D. Kalaj and M. Vuorinen, On harmonic functions and the Schwarz lemma, Proc. Am. Math. Soc. 140 (2012), no. 1, 161–165. [294] D. Kalaj and A. Zlatićanin, Quasiconformal maps with controlled Laplacian and Hölder continuity, Ann. Acad. Sci. Fenn., Math. (2019), no. 44, 797–803. [295] N. J. Kalton, Linear operators on Lp for 0 < p < 1, Trans. Am. Math. Soc. 259 (1980), 319–355. [296] N. J. Kalton, Analytic functions in non-locally convex spaces and applications, Stud. Math. 83 (1985), 275–303. [297] N. J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Stud. Math. 84 (1986), 297–324. [298] N. J. Kalton, Harmonic functions in non-locally convex spaces, Arch. Math. 50 (1989), no. 6, 538–549. [299] N. J. Kalton, N. T. Peck, and J. W. Roberts, An F -space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. [300] J. Karamata, Über die Hardy–Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930), 319–320. [301] J. Karamata, Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welche die Laplacesche und Stieltjessche Transformation betreffen, J. Reine Angew. Math. 164 (1931), 27–39. [302] Y. Katznelson, An introduction to harmonic analysis, Dover Publications Inc., New York, 1976. [303] I. R. Kayumov, S. Ponnusamy, and A. S. Kaliraj, Riesz–Fejér inequalities for harmonic functions, Potential Anal. (2018), 9 pp., accepted for publication. [304] C. N. Kellogg, An extension of the Hausdorff-Young theorem, Mich. Math. J. 18 (1971), 121–127. [305] A. Khintchine, Über dyadische Brüche, Math. Z. 18 (1923), 109–116. [306] J. Khintchine and A. N. Kolmogorov, Über Konvergenz von Reihen deren Glieder durch den Zufall bestimmt werden, Mat. Sb. 32 (1925), 123–138. [307] H. O. Kim, Derivatives of Blaschke products, Pac. J. Math. 114 (1984), no. 1, 175–190. [308] H. O. Kim and E. G. Kwon, Weighted subspaces of Hardy spaces, Can. J. Math. 40 (1988), no. 5, 1074–1083. [309] P. L. Kim, E. G. Kwon, and J. H. Park, On a Besov space and radial limits, Commun. Korean Math. Soc. 24 (2009), no. 4, 561–564. [310] S. V. Kislyakov, Fourier coefficients of boundary values of analytic functions on the disc and the bidisc, Tr. Mat. Inst. Steklova 155 (1981), 77–91. [311] S. V. Kislyakov, Interpolation of Hp -spaces: Some recent developments, Bar-Ilan University/distr. by the American Mathematical Society, Ramat Gan, 1999. [312] S. V. Kislyakov and Q. Xu, Interpolation of weighted and vector–valued Hardy spaces, Trans. Am. Math. Soc. 343 (1994), no. 1, 1–34. [313] S. V. Kislyakov and Q. Xu, Real interpolation and singular integrals, St. Petersburg Math. J. 8 (1996), no. 4, 593–615. [314] G. Knese, Uchiyama’s lemma and the John-Nirenberg inequality, Bull. Lond. Math. Soc. 45 (2013), no. 4, 683–692. [315] H. Kneser, Lösung der Aufgabe 41, Jahresber. Dtsch. Math.-Ver. 35 (1926), 123–124. [316] N.-T. Koh and L. V. Kovalev, Area contraction for harmonic automorphisms of the disk, Bull. Lond. Math. Soc. 43 (2011), no. 1, 91–96. [317] A. Kolmogoroff, Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fundam. Math. 7 (1925), 24–29 (French). [318] A. Kolmogoroff, Une série de Fourier-Lebesgue divergente partout, C.R. Acad. Sci. Paris 183 (1926), 1327–1328. [319] S. Y. Kolomoitsev, On moduli of smoothness and K-functionals of fractional order in the Hardy spaces, J. Math. Sci. 181 (2012), no. 1, 78–97.

532 | Bibliography

[320] S. V. Konyagin, On everywhere divergence of trigonometric Fourier series, Sb. Math. 191 (2000), no. 1, 97–120. [321] P. Koosis, Introduction to Hp spaces, Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. [322] B. Korenblum, A certain extremal property of outer functions, Mat. Zametki 10 (1971), no. 1, 53–56 (Russian). [323] B. Korenblum, BMO estimates and radial growth of Bloch functions, Bull., New Ser., Am. Math. Soc. 12 (1985), 99–102. [324] P. Koskela and S. Benedict, Intrinsic Hardy–Orlicz spaces of conformal mappings, Bull. Lond. Math. Soc. 47 (2015), no. 1, 75–84. [325] S. G. Krantz and S. Stević, On the iterated logarithmic Bloch space on the unit ball, Nonlinear Anal., Theory Methods Appl. 71 (2009), no. 5–6, A, 1772–1795. [326] M. A. Krasnosel’skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. [327] S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of linear operators, Nauka, Moscow, 1978 (Russian). [328] V. G. Krotov, An exact estimate of the boundary behavior of functions from Hardy–Sobolev classes in the critical case, Math. Notes 62 (1997), no. 4, 439–448. [329] Ü. Kuran, Subharmonic behaviour of |h|p (p > 0, h harmonic), J. Lond. Math. Soc., II. Ser. 8 (1974), 529–538. [330] E. G. Kwon, On the Hardy-Stein identity, Commun. Korean Math. Soc. 6 (1991), no. 1, 97–99. [331] E. G. Kwon, Composition of Blochs with bounded analytic functions, Proc. Am. Math. Soc. 124 (1996), no. 5, 1473–1480. [332] E. G. Kwon, Hyperbolic mean growth of bounded holomorphic functions in the ball, Trans. Am. Math. Soc. 355 (2003), no. 3, 1269–1294. [333] E. G. Kwon, Hyperbolic g-function and Bloch pullback operators, J. Math. Anal. Appl. 309 (2005), no. 2, 626–637. [334] E. G. Kwon, Quantities equivalent to the norm of a weighted Bergman space, J. Math. Anal. Appl. 338 (2008), no. 2, 758–770. [335] E. G. Kwon, H. W. Koo, and H. R. Cho, Holomorphic functions satisfying mean Lipschitz condition in the ball, J. Korean Math. Soc. 44 (2007), no. 4, 931–940. [336] E. G. Kwon and M. Pavlović, Bibloch functions an composition operators from Bloch type spaces to BMOA, J. Math. Anal. Appl. 382 (2011), no. 1, 303–313. [337] J. Laitila, Weighted composition operators on BMOA, Comput. Methods Funct. Theory 9 (2009), no. 1, 27–46. [338] J. Laitila, P. J. Nieminen, E. Saksman, and H. O. Tylli, Compact and weakly compact composition operators on BMOA, Complex Anal. Oper. Theory (2013), 163–181. [339] B. Łanucha, M. Nowak, and M. Pavlović, Hilbert matrix operator on spaces of analytic functions, Ann. Acad. Sci. Fenn., Math. 37 (2012), no. 1, 161–174. [340] P. Lefèvre, D. Li, H. Queffélec, and L. Rodríguez Piazza, Composition operators on Hardy–Orlicz spaces, Mem. Am. Math. Soc. 974 (2010), i–v +74. [341] P. Lefèvre, D. Li, H. Queffélec, and L. Rodríguez Piazza, Nevanlinna counting function and Carleson function of analytic maps, Math. Ann. 351 (2011), 305–326. [342] P. Lefèvre, D. Li, H. Queffélec, and L. Rodríguez Piazza, Some revisited results about composition operators on Hardy spaces, Rev. Mat. Iberoam. 28 (2012), no. 1, 57–76. [343] P. Lefèvre and L. Rodríguez Piazza, Absolutely summing Carleson embeddings on Hardy spaces, Adv. Math. 340 (2018), no. 1, 528–587. [344] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane. Translated from the German by K. W. Lucas. 2nd ed., Die Grundlehren der mathematischen

Bibliography | 533

[345] [346] [347] [348] [349] [350] [351] [352] [353]

[354]

[355] [356] [357] [358] [359] [360] [361] [362] [363] [364] [365] [366] [367] [368]

Wissenschaften, Band 126, Springer-Verlag, Berlin–Heidelberg–New York, VIII, 258 p. with 15 fig., 1973. M. Lengfield, A nested embedding theorem for Hardy–Lorentz spaces with applications to coefficient multiplier problems, Rocky Mt. J. Math. 38 (2008), no. 4, 1215–1251. H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Am. Math. Soc. 42 (1936), 689–692. H. Li and D. H. Luecking, BMO on strongly pseudoconvex domains: Hankel operators, duality ̄ and 𝜕-estimates, Trans. Am. Math. Soc. 346 (1994), no. 2, 661–691. J. Lindenstrauss, On complemented subspaces of m, Isr. J. Math. 5 (1967), 153–156. J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249. J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Isr. J. Math. 9 (1971), 263–269. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, vol. 338, Springer-Verlag, Berlin, 1973. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I, Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Springer-Verlag, Berlin, 1977. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin, 1979, Function spaces. M. Lindström and A. H. Sanatpour, Derivative-free characterizations of compact generalized composition operators between Zygmund type spaces, Bull. Aust. Math. Soc. 81 (2010), no. 3, 398–408. J. E. Littlewood, The converse on Abel’s theorem of power series, Proc. Lond. Math. Soc. 9 (1911), 434–448. J. E. Littlewood, On inequalities in the theory of functions, Proc. Lond. Math. Soc. (2) 23 (1925), v–ix + 481–519. J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. I, J. Lond. Math. Soc. 6 (1931), 230–233. J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. II, Proc. Lond. Math. Soc. 42 (1937), 52–89. C. Liu, A. Perälä, and J. Si, Weighted integrability of polyharmonic functions in the higher dimensional case, arXiv preprint arXiv:1807.03276 (2018). P. I. Lizorkin and S. M. Nikol’skiĭ, Approximation on a sphere in the metric of the continuous functions, Sov. Math. Dokl. 28 (1983), 407–410 (English). D. H. Luecking, Multipliers of Bergman spaces into Lebesgue spaces, Proc. Edinb. Math. Soc., II. Ser. 29 (1986), 125–131. D. H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), 345–368. D. H. Luecking, A new proof of an inequality of Littlewood and Paley, Proc. Am. Math. Soc. 103 (1988), no. 3, 887–893. D. H. Luecking, Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. Lond. Math. Soc. (3) 63 (1991), no. 3, 595–619. N. Lusin, Sur une propríeté des fonctions a carré sommble, Bull. Calcutta Math. Soc. 20 (1930), 139–154. W. Lusky, On generalized Bergman spaces, Stud. Math. 119 (1996), no. 1, 77–95. W. Lusky, On the isomorphic classification of weighted spaces of holomorphic functions, Acta Univ. Carol., Math. Phys. 41 (2000), no. 2, 51–60. T. MacGregor and K. Zhu, Coefficient multipliers between Bergman and Hardy spaces, Mathematika 42 (1995), no. 2, 413–426.

534 | Bibliography

[369] T. H. MacGregor and M. P. Sterner, Hadamard products with power functions and multipliers of Hardy spaces, J. Math. Anal. Appl. 282 (2003), no. 1, 163–176. [370] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Am. Math. Soc. 347 (1995), no. 7, 2679–2687. [371] K. M. Madigan, Composition operators on analytic Lipschitz spaces, Proc. Am. Math. Soc. 119 (1993), 465–473. [372] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. Lond. Math. Soc. (3) 51 (1985), 369–384. [373] N. G. Makarov, On the radial behavior of Bloch functions, Sov. Math. Dokl. 40 (1990), no. 3, 505–508 (English. Russian original). [374] J. Marcinkiewicz, Sur l’interpolation d’opérations, C. R. Acad. Sci., Paris 208 (1939), 1272–1273. [375] J. Marcinkiewicz and A. Zygmund, On a theorem of Lusin, Duke Math. J. 4 (1938), 473–485. [376] M. Marković, Differential-free characterisation of smooth mappings with controlled growth, Can. Math. Bull. 61 (2018), no. 3, 628–636. [377] M. Marković, Sharp inequalities over the unit polydisc, J. Funct. Anal. 268 (2015), no. 9, 2647–2671. [378] O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn., Ser. A 1 Math. 425 (1968), 3–10. [379] J. Mashreghi, Derivatives of inner functions, Fields Institute Monographs, vol. 31, Springer Science & Business Media, 2012. [380] M. Mastyło and P. Mleczko, Solid hulls of quasi-Banach spaces of analytic functions and interpolation, Nonlinear Anal. 73 (2010), no. 1, 84–98. [381] M. Mateljević and M. Pavlović, An extension of the Hardy–Littlewood inequality, Mat. Vesn. 6(19)(34) (1982), no. 1, 55–61. [382] M. Mateljević and M. Pavlović, On the integral means of derivatives of the atomic function, Proc. Am. Math. Soc. 86 (1982), 455–458. [383] M. Mateljević and M. Pavlović, Lp -behavior of power series with positive coefficients and Hardy spaces, Proc. Am. Math. Soc. 87 (1983), no. 2, 309–316. [384] M. Mateljević and M. Pavlović, Lp behaviour of the integral means of analytic functions, Stud. Math. 77 (1984), 219–237. [385] M. Mateljević and M. Pavlović, Multipliers of Hp and BMOA, Pac. J. Math. 146 (1990), no. 1, 71–84. [386] M. Mateljević and M. Pavlović, An extension of the Forelli–Rudin projection theorem, Proc. Edinb. Math. Soc., II. Ser. 36 (1993), no. 3, 375–389. [387] M. Mateljević and M. Pavlović, The best approximation and composition with inner functions, Mich. Math. J. 42 (1995), no. 2, 367–378. [388] B. Maurey, Théoremes de factorisation pour les operatèurs linéaires à valeurs dans les espaces Lp , vol. 11, Société Math. de France, 1974. [389] O. C. McGehee, L. Pigno, and B. Smith, Hardy’s inequality and the L1 norm of exponential sums, Ann. Math. 113 (1981), 613–618. [390] P. Melentijević, Invariant gradient in refinements of Schwarz and Harnack inequalities, Ann. Acad. Sci. Fenn., Math. 43 (2018), 391–399. [391] A. Michalak and M. Nawrocki, Banach envelopes of vector valued Hp spaces, Indag. Math. 13 (2002), no. 2, 185–195. [392] P. Mleczko, Compact multipliers on spaces of analytic functions, Arch. Math. 92 (2009), no. 4, 325–334. [393] A. Montes-Rodríguez, The essential norm of a composition operator on Bloch spaces, Pac. J. Math. 188 (1999), no. 2, 339–351.

Bibliography | 535

[394] A. Mori, On an absolute constant in the theory of quasiconformal mappings, J. Math. Soc. Jpn. 8 (1956), 156–166. [395] C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Am. Math. Soc. 43 (1938), no. 1, 126–166. [396] T. Muramatu, On the dual of Besov spaces, Publ. RIMS Kyoto Univ. 12 (1976), no. 1, 123–140. [397] J. Musielak, Orlicz spaces and modular spaces, Lect. Notes Math., vol. 1034, Springer-Verlag, Berlin, 1983. [398] A. Nagel, W. Rudin, and J. H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. Math. (2) 116 (1982), 331–360. [399] M. Nawrocki, Duals of vector-valued Hp -spaces for 0 < p < 1, Indag. Math. 2 (1991), no. 2, 233–241. [400] M. Nawrocki, On the solid hull of the Nevanlinna and Smirnov classes, Ann. Acad. Sci. Fenn., Math. 39 (2014), 897–904. [401] D. J. Newman, The nonexistence of projections from l1 to h1 , Proc. Am. Math. Soc. 12 (1961), 98–99. [402] A. F. Nikiforov and V. B. Uvarov, Special functions of mathematical physics. A unified introduction with applications, Transl. from the Russian by Ralph P. Boas, Birkhäuser, Basel–Boston, XVIII, 427 p., 1988. [403] E. M. Nikishin, Resonance theorems and superlinear operators, Russ. Math. Surv. 25 (1970), no. 6, 125–187. [404] E. M. Nikishin, A resonance theorem and series of eigenfunctions of the Laplacian, Math. USSR, Izv. 6 (1972), 788–806. [405] S. M. Nikol’skiĭ, Approximation of functions of several variables and imbedding theorems, Die Grundlehren der Mathematischen Wissenschaften, Band 205, Springer-Verlag, New York, 1975. Translated from the Russian by John M. Danskin, Jr. [406] S. M. Nikol’skiĭ and P. I. Lizorkin, Functional spaces on the sphere, related with approximation theory, Math. Notes 41 (1987), 286–291. [407] C. A. Nolder and D. M. Oberlin, Moduli of continuity and a Hardy–Littlewood theorem, Complex analysis, Joensuu 1987, Lecture Notes in Math., vol. 1351, Springer, Berlin, 1988, pp. 265–272. [408] M. Nowak, A note on coefficient multipliers (Hp , B) and (Hp , BMOA), Topics in complex analysis (Warsaw, 1992), Banach Center Publ., vol. 31, Polish Acad. Sci., Warsaw, 1995, pp. 299–302. [409] M. Nowak, Integral means of univalent harmonic maps, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 50 (1996), 155–162. [410] M. Nowak, E-mail correspondence, May 2019. [411] J. M. Ortega and J. Fàbrega, Pointwise multipliers and decomposition theorems in analytic Besov spaces, Math. Z. 235 (2000), no. 1, 53–81. [412] P. Oswald, On Besov–Hardy–Sobolev spaces of analytic functions in the unit disc, Czechoslov. Math. J. 33 (1983), 408–426. [413] C. Ouyang and J. Riihentaus, Characterizations of Hardy–Orlicz and Bergman–Orlicz spaces, Math. Scand. 80 (1997), 25–40. [414] R. E. A. C. Paley, Some theorems on orthogonal functions, Stud. Math. 3 (1931), 226–245. [415] R. E. A. C. Paley, On the lacunary coefficients of power series, Ann. Math. 34 (1933), 615–616. [416] D. Partyka and K. Sakan, Harmonic and quasiconformal mappings which agree on the boundary, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 49 (1995), 159–171. [417] D. Partyka and K. Sakan, A note on non-quasiconformal harmonic extensions, Bull. Soc. Sci. Lett. Łódź, Sér. Rech. Déform. 47(23) (1997), 51–63.

536 | Bibliography

[418] J. Pau, Integration operators between Hardy spaces on the unit ball of ℂn , J. Funct. Anal. 270 (2016), no. 1, 134–176. [419] M. Pavlović, A collection of unpublished articles, www.researchgate.net (1) Kelvin–Möbius transform of polyharmonic functions, (2) An invariant space of polyharmonic functions on R 2n , (3) Invariant Besov spaces. Taylor coefficients and applications, (4) Besov–Lipschitz spaces on the real ball (radial Lipschitz conditions), (5) Hardy–Stein identities for vector-valued functions. [420] M. Pavlović, Geometry of complex Banach spaces, Ph.D. thesis, Faculty of Mathematics, Belgrade, 1983, in Serbian. [421] M. Pavlović, Mixed norm spaces of analytic and harmonic functions. I, Publ. Inst. Math. (Belgr.) 40(54) (1986), 117–141, II, ibid. 41(55) (1987), 97–110. [422] M. Pavlović, An inequality for the integral means of a Hadamard product, Proc. Am. Math. Soc. 103 (1988), no. 2, 404–406. [423] M. Pavlović, Lipschitz spaces and spaces of harmonic functions in the unit disc, Mich. Math. J. 35 (1988), no. 2, 301–311. [424] M. Pavlović, Mean values of harmonic conjugates in the unit disc, Complex Var. Theory Appl. 10 (1988), no. 1, 53–65. [425] M. Pavlović, On the complex uniform convexity of quasi-normed spaces, Math. Balk. 5 (1991), no. 2, 92–98. [426] M. Pavlović, On the moduli of continuity of Hp -functions with 0 < p < 1, Proc. Edinb. Math. Soc. (2) 35 (1992), no. 1, 89–100. [427] M. Pavlović, Integral means of the Poisson integral of a discrete measure, J. Math. Anal. Appl. 184 (1994), no. 2, 229–242. [428] M. Pavlović, On subharmonic behaviour and oscillation of functions on balls in Rn , Publ. Inst. Math. (Belgr.) 55(69) (1994), 18–22. [429] M. Pavlović, A remark on the partial sums in Hardy spaces, Publ. Inst. Math. (Belgr.) 58(72) (1995), 149–152. [430] M. Pavlović, Subharmonic behaviour of smooth functions, Mat. Vesn. 48 (1996), no. 1–2, 15–21. [431] M. Pavlović, Decompositions of Lp and Hardy spaces of polyharmonic functions, J. Math. Anal. Appl. 216 (1997), no. 2, 499–509. [432] M. Pavlović, On Dyakonov’s paper: “Equivalent norms on Lipschitz-type spaces of holomorphic functions”, Acta Math. 183 (1999), no. 1, 141–143. [433] M. Pavlović, On harmonic conjugates with exponential mean growth, Czechoslov. Math. J. 49 (1999), no. 4, 733–742. [434] M. Pavlović, A Littlewood-Paley theorem for subharmonic functions, Publ. Inst. Math. (Belgr.) 68(82) (2000), 77–82. [435] M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk, Ann. Acad. Sci. Fenn., Math. 27 (2002), no. 2, 365–372. [436] M. Pavlović, Introduction to function spaces on the disk, Posebna Izdanja [Special Editions], vol. 20, Matematički Institut SANU, Belgrade, 2004. [437] M. Pavlović, Hadamard product in Qp spaces, J. Math. Anal. Appl. 305 (2005), no. 2, 589–598. [438] M. Pavlović, A short proof of an inequality of Littlewood and Paley, Proc. Am. Math. Soc. 134 (2006), no. 12, 3625–3627. [439] M. Pavlović, Lipschitz conditions on the modulus of a harmonic function, Rev. Mat. Iberoam. 23 (2007), no. 3, 831–845. [440] M. Pavlović, Remarks on Lp -oscillation of the modulus of a holomorphic function, J. Math. Anal. Appl. 326 (2007), no. 1, 1–11.

Bibliography | 537

[441] M. Pavlović, Derivative-free characterizations of bounded composition operators between Lipschitz spaces, Math. Z. 258 (2008), no. 1, 81–86. [442] M. Pavlović, Division by inner functions in a class of composition operators on Lipschitz spaces, Bull. Lond. Math. Soc. 40 (2008), no. 2, 199–209. [443] M. Pavlović, On the Holland–Walsh characterization of Bloch functions, Proc. Edinb. Math. Soc. 51 (2008), no. 2, 439–442. [444] M. Pavlović, Green’s formula and the Hardy–Stein identities, Filomat (Niš) 23 (2009), no. 3, 135–153. [445] M. Pavlović, Hardy–Stein type characterization of harmonic Bergman spaces, Potential Anal. 32 (2010), no. 1, 1–15. [446] M. Pavlović, Characterizations of the harmonic Hardy space h1 on the real ball, Filomat (Niš) 25 (2011), no. 3, 137–143. [447] M. Pavlović, Lacunary series in weighted spaces of analytic functions, Arch. Math. 97 (2011), no. 5, 467–473. [448] M. Pavlović, Integrability of vector-valued lacunary series with applications to function spaces, Ann. Mat. Pura Appl. 192 (2013), no. 5, 745–762. [449] M. Pavlović, On the Littlewood–Paley g-function and Calderon’s area function, Expo. Math.. 31 (2013), 169–195. [450] M. Pavlović, Addendum to “Besov–Lipschitz and mean Besov–Lipschitz spaces”: the Ahern–Schneider inequality, Potential Anal. 41 (2014), no. 1, 31–34. [451] M. Pavlović, Function classes on the unit disc. An introduction, De Gruyter Studies in Mathematics, vol. 52, De Gruyter, 2014. [452] M. Pavlović, Hardy–Orlicz spaces: Taylor coefficients, interpolations of operators, and univalent functions, J. Math. Sci. Adv. Appl. 28 (2014), 1–14. [453] M. Pavlović and M. R. Dostanić, On the inclusion H2 (U n ) ⊂ H2n (Bn ) and the isoperimetric inequality, J. Math. Anal. Appl. 226 (1998), no. 1, 143–149. [454] M. Pavlović and J. Á. Peláez, Weighted integrals of higher order derivatives of analytic functions, Acta Sci. Math. (Szeged) 72 (2006), no. 1–2, 73–93. [455] M. Pavlović and J. Á. Peláez, An equivalence for weighted integrals of an analytic function and its derivative, Math. Nachr. 281 (2008), no. 11, 1612–1623. [456] M. Pavlović and J. Riihentaus, Classes of quasi-nearly subharmonic functions, Potential Anal. 29 (2008), no. 1, 89–104. [457] M. Pavlović and J. Riihentaus, Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces, Positivity 15 (2011), no. 1, 1–10. [458] M. Pavlović and K. Zhu, New characterizations of Bergman spaces, Ann. Acad. Sci. Fenn., Math. 33 (2008), no. 1, 87–99, Addendum, ibid. 1(2009), 315–317. [459] J. Á. Peláez, Compact embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. Am. Math. Soc. 144 (2016), no. 3, 1095–1107. [460] J. À. Peláez and J. Rättyä, Generalized Hilbert operators on weighted Bergman spaces, Adv. Math. 240 (2013), no. 1, 227–267. [461] J. À. Peláez and J. Rättyä, Weighted Bergman spaces induced by rapidly increasing weights, Mem. Am. Math. Soc. 1066 (2014), iii–vi + 122. [462] J. Á. Peláez and J. Rättyä, Embedding theorems for Bergman spaces via harmonic analysis, Math. Ann. 362 (2015), no. 1–2, 205–239. [463] J. Á. Peláez and J. Rättyä, Two weight inequality for Bergman projection, J. Math. Pures Appl. (9) 105 (2016), no. 1, 102–130. [464] J. Á. Peláez and J. Rättyä, On the boundedness of Bergman projection, Proceedings of the 6th international school “Advanced courses of mathematical analysis VI”, Málaga, Spain, September 8–12, 2014, World Scientific, Hackensack, NJ, 2017, pp. 113–132.

538 | Bibliography

[465] J. Á. Peláez and J. Rättyä, Bergman projections induced by radial weights, arXiv preprint arXiv:1902.09837(2019). [466] A. Pełczyński, Projections in certain Banach spaces, Stud. Math. 19 (1960), 209–228. [467] A. Pełczyński, Banach spaces of analytic functions and absolutely summing operators, Expository lectures from the CBMS Regional Conference helf at Kent State University, July 11-16, 1976, Reg. Conf. Ser. Math., 30, American Mathematical Society (AMS), Providence, RI, 1977. [468] V. V. Peller, Hankel operators and their applications, Springer Monogr. Math., 2003. [469] A. Perälä, On the optimal constant for the Bergman projection onto the Bloch space, Ann. Acad. Sci. Fenn., Math. 37 (2012), no. 1, 245–249. [470] A. Perälä, Duality of holomorphic Hardy type tent spaces, arXiv preprint arXiv:1803.10584 (2018). [471] A. Perälä and J. Rättyä, Duality of weighted Bergman spaces with small exponents, Ann. Acad. Sci. Fenn., Math. 42 (2017), no. 2, 621–626. [472] F. Pérez-González and J. Rättyä, Univalent functions in Hardy, Bergman, Bloch and related spaces, J. Anal. Math. 105 (2008), no. 1, 125–148. [473] F. Pérez-González and J. Rättyä, Derivatives of inner functions in weighted Bergman spaces and the Schwarz–Pick lemma, Proc. Am. Math. Soc. 105 (2017), no. 5, 2155–2166. [474] F. Pérez-González, J. Rättyä, and A. Reijonen, Derivatives of inner functions in Bergman spaces induced by doubling weights, Ann. Acad. Sci. Fenn., Math. 42 (2017), no. 2, 735–753. [475] F. Pérez-González, J. Rättyä, and D. Vukotić, On composition operators acting between Hardy and weighted Bergman spaces, Expo. Math. 25 (2007), no. 4, 309–323. [476] F. Pérez-González and J. Xiao, Bloch–Hardy pullbacks, Acta Sci. Math. 67 (2001), no. 3–4, 709–718. [477] L-E. Persson and N. Popa, Matrix spaces and Schur multipliers. Matriceal harmonic analysis, Hackensack, NJ: World Scientific, 2014. [478] S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Stud. Math. 44 (1972), 165–179. (errata insert), Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, II. [479] G. Piranian, A. L. Shilds, and J. H. Wells, Bounded analytic functions and absolutely continuous measures, Proc. Am. Math. Soc. 18 (1967), 818–826. [480] H. R. Pitt, A note on bilinear forms, J. Lond. Math. Soc. 11 (1936), 174–180. [481] A. Plessner, Zur Theorie der konjugierten trigonometrischen Reihen, Mitt. Math. Sem. Giessen 10 (1923), 1–36. [482] Ch. Pommerenke, Über die Mittelwerte und Koeffizienten multivalenter Funktionen, Math. Ann. 145 (1962), 285–296. [483] Ch. Pommerenke, On Bloch functions, J. Lond. Math. Soc., II. Ser. 2 (1970), 689–695. [484] Ch. Pommerenke, Univalent functions, With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. [485] Ch. Pommerenke, Schlichte funktionen und analytische funktionen von beschränkter mittlerer oszillation, Comment. Math. Helv. 52 (1977), no. 1, 591–602. [486] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. [487] S. Ponnusamy and A. Rasila, Planar harmonic and quasiregular mappings, Topics in modern function theory. Based on mini-courses of the CMFT workshop on computational methods and function theory, Guwahati, India, January 3–10, 2008, Ramanujan Mathematical Society, Mysore, 2013, pp. 267–333.

Bibliography | 539

[488] N. Popa, Abel-Schur multipliers on Banach spaces of infinite matrices, Khayyam J. Math. 2 (2016), no. 1, 39–50. [489] H. Prawitz, Über Mittelwerte analytischer Funktionen, Ark. Mat. 20 (1927), no. 6, 1–12. [490] I. I. Privaloff, Sur le fonctions conjuguées, Bull. Soc. Math. Fr. 44 (1916), 100–103. [491] I. I. Privaloff, Das Cauchysche Integral, Sonderabdruck aus den Mitteilungen der physikalischen und mathematischen Fakultät der Universität Saratow, 1919 (Russian). [492] I. I. Privaloff, Sur certaines propriétés métriques des fonctions analytiques, J. Éc. Polytech. 24 (1924), 77–112. [493] H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 112–138. [494] T. Radó, Aufgabe 41, Jahresber. Dtsch. Math.-Ver. 35 (1926), 49. [495] W. Ramey and D. Ullrich, Bounded mean oscillation of Bloch pull-backs, Math. Ann. 291 (1991), no. 4, 591–606. [496] S. Ravisankar, Transversally Lipschitz harmonic functions are Lipschitz, Complex Var. Elliptic Equ. 58 (2013), no. 12, 1685–1700. [497] A. Reijonen, Derivatives of inner functions in weighted mixed norm spaces, J. Geom. Anal. (2018), https://doi.org/10.1007/s12220-018-0065-8. [498] S. Richter and A. Shields, Bounded analytic functions in the Dirichlet space, Math. Z. 198 (1988), no. 2, 151–159. [499] S. Richter and C. Sundberg, A formula for the local Dirichlet integral, Mich. Math. J. 38 (1991), no. 3, 355–379. [500] R. Riedl, Composition operators and geometric properties of analytic functions, Ph.D. thesis, Universität at Zurich, 1994. [501] F. Riesz, Über die Randwerte einer analytischen Funktion, Math. Z. 18 (1923), 87–95. [502] F. Riesz, Über eine Verallgemeinerung der Parsevalschen Formel, Math. Z. 18 (1923), 117–124. [503] F. Riesz, Sur une inégalité de M. Littlewood dans la théorie des fonctions, Proc. Lond. Math. Soc. 23 (1925), 36–39. [504] F. Riesz, Eine Ungleichung für harmonische Funktionen, Monatshefte Math. Phys. 43 (1936), 401–406. [505] F. Riesz and M. Riesz, Über die Randwerte einer analytischen Funktion, Quatriéme congrés des math. scand. 1916 (1916), 27–44. [506] M. Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1927), 218–244. [507] M. Riesz, Sur les maxima des formes bilinéaires et sur les fonctionelles linéaires, Acta Math. 49 ((1927)), 465–497. [508] J. Riihentaus, A generalized mean value inequality for subharmonic functions, Expo. Math. 19 (2001), no. 2, 187–190. [509] S. Rolewicz, Metric linear spaces, Monografie Matematyczne, Tom. 56 [Mathematical Monographs, vol. 56]. PWN-Polish Scientific Publishers, Warsaw, 1972. [510] M. Rosenblum and J. Rovnyak, Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. [511] G. Rozenblum, M. Ruzhansky, and D. Suragan, Isoperimetric inequalities for Schatten norms of Riesz potentials, J. Funct. Anal. 271 (2016), no. 1, 224–239. [512] G. Rozenblum and N. Vasilevski, Toeplitz operators in polyanalytic Bergman type spaces, arXiv preprint arXiv:1807.11216 (2018). [513] L. A. Rubel, A. L. Shields, and B. A. Taylor, Mergelyan sets and the modulus of continuity of analytic functions, J. Approx. Theory 15 (1975), no. 1, 23–40. [514] W. Rudin, The radial variation of analytic functions, Duke Math. J. 22 (1955), 235–242. [515] W. Rudin, Remarks on a theorem of Paley, J. Lond. Math. Soc. 32 (1957), 307–311.

540 | Bibliography

[516] W. Rudin, Projections on invariant subspaces, Proc. Am. Math. Soc. 13 (1962), no. 3, 429–432. [517] W. Rudin, Real and complex analysis, second ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1974. [518] W. Rudin, Function theory in the unit ball of Cn , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York, 1980. [519] W. Rudin, Composition with inner functions, Complex Var. Theory Appl.. 4 (1984), 7–19. [520] W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. [521] V. I. Ryazanov, The Cauchy–Stieltjes integrals in the theory of analytic functions, J. Math. Sci. (2018), 12 pp., https://doi.org/10.1007/s10958-018-3981-z. [522] E. Rydhe, On the characterization of Triebel–Lizorkin type spaces of analytic functions, J. Fourier Anal. Appl. 24 (2018), no. 6, 1491–1517. [523] E. Rydhe, Vectorial Hankel operators, Carleson embeddings, and notions of BMOA, Geom. Funct. Anal. 27 (2017), no. 2, 427–451. [524] R. Salem and A. Zygmund, A convexity theorem, Proc. Natl. Acad. Sci. USA 34 (1948), 443–447. [525] D. Sarason, Functions of vanishing mean oscillation, Trans. Am. Math. Soc. 207 (1975), 391–405. [526] S. Sawyer, Maximal inequalities of weak type, Ann. Math. (2) 84 (1966), 157–174. [527] E. Seneta, Regularly varying functions, Springer-Verlag, Berlin, 1976. [528] J. H. Shapiro, Linear topological properties of the harmonic Hardy spaces hp for 0 < p < 1, Ill. J. Math. 29 (1985), 311–339. [529] J. H. Shapiro, The essential norm of a composition operator, Ann. Math. (2) 125 (1987), 375–404. [530] J. H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. [531] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna class, Am. J. Math. 97 (1976), 915–936. [532] A. L. Shields and D. L. Williams, Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Am. Math. Soc. 162 (1971), 287–302. [533] A. L. Shields and D. L. Williams, Bounded projections, duality, and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 299/300 (1978), 256–279. [534] A. L. Shields and D. L. Williams, Bounded projections and the growth of harmonic conjugates in the unit disc, Mich. Math. J. 29 (1982), no. 1, 3–25. [535] N. A. Shirokov, Analytic functions smooth up to the boundary, Lecture Notes in Math., vol. 1312, Springer-Verlag, Berlin etc., 1988. [536] S. V. Shvedenko, Hardy classes and related spaces of analytic functions in the unit circle, polydisc, and the ball, J. Sov. Math. 39 (1987), no. 6, 3011–3087, Translation from Itogi Nauki Tekh., Ser. Mat. Anal. 23, 3–124. [537] A. G. Siskakis, Weighted integrals of analytic functions, Acta Sci. Math. 66 (2000), no. 3–4, 651–664. [538] W. T. Sledd, Some inequalities related to the Hausdorff–Young theorem, Proc. Am. Math. Soc. 42 (1974), 535–540. [539] W. T. Sledd, On multipliers of Hp spaces, Indiana Univ. Math. J. 27 (1978), 797–803. [540] W. T. Sledd and D. A. Stegenga, An H1 multiplier theorem, Ark. Mat. 19 (1981), 265–270. [541] V. Smirnoff, Sur les valeurs limites des fonctions, réguliéres à l’intérieur d’un cercle, J. Soc. Phys.-Math. Leningrad 2 (1929), 22–37.

Bibliography | 541

[542] B. Smith, A strong convergence theorem for H1 (T), Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981), Lecture Notes in Math., vol. 995, Springer, Berlin, 1983, pp. 169–173. [543] W. Smith, Composition operators between Bergman and Hardy spaces, Trans. Am. Math. Soc. 348 (1996), no. 6, 2331–2348. [544] W. Smith, Inner functions in the hyperbolic little Bloch class, Mich. Math. J. 45 (1998), 103–114. [545] S. Soboleff, Sur un théorème d’analyse fonctionnelle, Rec. Math., Moscow 4 (1938), 491–497 (Russian). [546] D. C. Spencer, On finitely mean valent functions, Proc. Lond. Math. Soc., II. Ser. 47 (1941), 201–211. [547] S. B. Stechkin, On the order of the best approximations of continuous functions, Izv. Akad. Nauk USSR. Ser. Mat. 15 (1951), no. 3, 219–242 (Russian). [548] D. A. Stegenga, Bounded Toeplitz operators on H1 and applications of the duality between H1 and the functions of bounded mean oscillation, Am. J. Math. 98 (1976), 573–589. [549] E. M. Stein, A maximal function with applications to Fourier series, Ann. Math. (2) 68 (1958), 584–603. [550] E. M. Stein, On limits of seqences of operators, Ann. Math. (2) 74 (1961), 140–170. [551] E. M. Stein, Classes Hp , multiplicateurs et fonctions de Littlewood–Paley, C. R. Acad. Sci. Paris 263 (1966), 716–719. [552] E. M. Stein, Classes Hp , multiplicateurs et fonctions de Littlewood–Paley. Applications de résultats anterieurs, C. R. Acad. Sci. Paris 263 (1966), 780–781. [553] E. M. Stein, Note on the class L log L, Stud. Math. 31 (1969), 305–310. [554] E. 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. [555] E. M. Stein and A. Zygmund, Boundedness of translation invariant operators on Hölder spaces and Lp -spaces, Ann. Math. (2) 85 (1967), 337–349. [556] P. Stein, On a theorem of M. Riesz, J. Lond. Math. Soc. 8 (1933), 242–247. [557] A. I. Stepanets, Methods of approximation theory, VSP, Utrecht, 2005. [558] K. Stephenson, Functions which follow inner functions, Ill. J. Math. 23 (1979), 259–266. [559] K. Stephenson, Construction of an inner function in the little Bloch space, Trans. Am. Math. Soc. 308 (1988), no. 2, 713–720. [560] W. J. Stiles, On properties of subspaces of ℓp , 0 < p < 1, Trans. Am. Math. Soc. 149 (1970), 405–415. [561] W. J. Stiles, Some properties of ℓp , 0 < p < 1, Stud. Math. 42 (1972), 109–119. [562] M. Stoll, A characterization of Hardy-Orlicz spaces on planar domains, Proc. Am. Math. Soc. 117 (1993), no. 4, 1031–1038. [563] M. Stoll, A characterization of Hardy spaces on the unit ball of Cn , J. Lond. Math. Soc. (2) 48 (1993), no. 1, 126–136. [564] M. Stoll, Harmonic majorants for eigenfunctions of the Laplacian with finite Dirichlet integrals, J. Math. Anal. Appl. 274 (2002), no. 2, 788–811. [565] M. Stoll, The Littlewood–Paley inequalities for Hardy–Orlicz spaces of harmonic functions on domains in ℝn , Potential theory in Matsue, Adv. Studia Pure Math., vol. 44, Math. Soc. Japan, Tokyo, 2006, pp. 363–376. [566] M. Stoll, Littlewood–Paley inequalities for subharmonic functions on domains in R n , Recent advances in harmonic analysis and applications, Springer Proceedings in Mathematics and Statistics, vol. 25, Springer, 2013, pp. 357–383.

542 | Bibliography

[567] M. Stoll, Littlewood–Paley theory for subharmonic functions on the unit ball in ℝN , J. Math. Anal. Appl. 420 (2014), no. 1, 483–514. [568] M. Stoll, Harmonic and subharmonic function theory on the hyperbolic ball, vol. 431, Cambridge University Press, Cambridge, 2016. [569] Eh. A. Storozhenko, Approximation of functions of class Hp , 0 < p ≤ 1, Math. USSR Sb. 34 (1978), 527–545. [570] Eh. A. Storozhenko, On the approximation of functions of the class Hp , 0 < p < 1, Soobshch. Akad. Nauk Gruz. SSR 88 (1978), 45–48 (Russian). [571] Eh. A. Storozhenko, Theorems of Jackson type in Hp , 0 < p < 1, Math. USSR, Izv. 17 (1981), 203–218. [572] Eh. A. Storozhenko, On a problem of Hardy and Littlewood, Mat. Sb. 119(161) (1982), 564–583 (Russian). [573] Eh. A. Storozhenko, V G. Krotov, and P. Owald, Direct and converse theorems of Jackson type in Lp spaces, 0 < p < 1, Mat. Sb. 98 (1975), 395–415 (Russian). [574] G. Sunouchi, On the summability of power series and Fourier series, Tohoku Math. J., II Ser. 7 (1955), 96–109. [575] O. Szász, Converse theorems of summability for Dirichlet series, Trans. Am. Math. Soc. 39 (1936), 117–130. [576] G. Szegö, Über die Randwerte einer analytischen Funktion, Math. Ann. 84 (1921), 232–244. [577] G. Szegö, Orthogonal polynomials. revised ed., Colloquium Publ., vol. 23, American Mathematical Society, New York City, 1959. [578] M. H. Taibleson, On the theory of Lipschitz spaces of distributions on euclidean n-spaces. II. Translation invariant operators, duality, and interpolation, J. Math. Mech. 14 (1965), 821–839. [579] P. M. Tamrazov, Contour and solid structural properties of holomorphic functions of a complex variable, Usp. Mat. Nauk 28 (1973), no. 1(169), 131–161. [580] A. E. Taylor, Banach spaces of functions analytic in the unit circle, I, Stud. Math. 11 (1950), no. 11, 145–170. [581] G. O. Thorin, An extension of a convexity theorem due to M. Riesz, Fysiograf. Sällsk. Lund Förhdl. 8 (1938), 166–170. [582] A. F. Timan and M. F. Timan, Generalized modulus of continuity and best approximation in the mean, Dokl. Akad. Nauk SSSR, n. Ser. 71 (1950), 17–20 (Russian). [583] M. F. Timan, Umkehrsätze der konstruktiven Funktionentheorie in Räumen Lp (1 ≤ p ≤ ∞), Mat. Sb., N. Ser. 46(88) (1958), 125–132 (Russian). [584] M. Tjani, Compact composition operators on some Möbius invariant Banach space, Ph.D. thesis, Michigan State University, 1996. [585] A. Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press Inc., Orlando, FL, 1986. [586] W. Trebels, Multipliers for (C, α)-bounded Fourier expansions in Banach spaces and approximation theory, Lecture Notes in Mathematics, vol. 329, Springer-Verlag, Berlin, 1973. [587] P. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Diss. Math. 131 (1976), 221. [588] J. B. Twomey, Tangential boundary behaviour of the Cauchy integral, J. Lond. Math. Soc., II. Ser. 37 (1988), no. 3, 447–454. [589] I. E. Verbitsky, An estimate of the norm of a function in a Hardy space in terms of the norms of its real and imaginary parts, Transl. Am. Math. Soc., Ser. 2 124 (1984), 11–15. [590] S. A. Vinogradov, Properties of multipliers of Cauchy–Stieltjes integrals and some factorization problems for analytic functions, Transl. Am. Math. Soc., Ser. 2 115 (1976), 1–32. [591] S. A. Vinogradov, Multiplication and division in the space of analytic functions with area integrable derivative, and in some related spaces, Zap. Nauč. Semin. POMI 222 (1995), 45–77.

Bibliography | 543

[592] V. S. Vladimirov, Generalized functions in mathematical physics. (obobshchennye funktsii v matematicheskoj fizike). 2nd ed., correct. and suppl., Sovremennye fiziko-tehnicheskie problemmy, Nauka, Moscow, 1979 (Russian). [593] M. Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. [594] D. Walsh, A property of univalent functions in Ap , Glasg. Math. J. 42 (2000), no. 01, 121–124. [595] J. Wang and J. Xiao, Analytic Campanato spaces by functionals and operators, J. Geom. Anal. 26 (2016), no. 4, 2996–3018. [596] F. B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Funct. Anal. 37 (1980), 218–234. [597] J. H. Wells, Some results concerning multipliers of Hp , J. Lond. Math. Soc. (2) 2 (1970), 549–556. [598] R. Wittmann, Coefficient inequalities for analytic functions in H1 , Math. Proc. Camb. Philos. Soc. 120 (1996), no. 2, 331–337. [599] P. Wojtaszczyk, Hp -spaces, p ≤ 1, and spline systems, Stud. Math. 77 (1984), 289–320. [600] P. Wojtaszczyk, On multipliers into Bergman spaces and Nevanlinna class, Can. Math. Bull. 33 (1990), no. 2, 151–161. [601] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. [602] H. Wulan, Compactness of composition operators on BMOA and VMOA, Sci. China Ser. A 50 (2007), no. 7, 997–1004. [603] H. Wulan, D. Zheng, and K. Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Am. Math. Soc. 137 (2009), no. 11, 3861–3868. [604] H. Wulan and K. Zhu, QK spaces via higher order derivatives, Rocky Mt. J. Math. 38 (2008), no. 1, 329–350. [605] H. Wulan and K. Zhu, Möbius invariant 𝒬K spaces, Springer, Cham, 2017. [606] J. Xiao, Holomorphic Q classes, Lecture Notes in Mathematics, vol. 1767, Springer-Verlag, Berlin, 2001. [607] J. Xiao, Geometric Qp functions, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. [608] J. Xiao and W. Xu, Composition operators between analytic Campanato spaces, J. Geom. Anal. 24 (2014), no. 2, 649–666. [609] J. Xiao and C. Yuan, Analytic Campanato spaces and their compositions, Indiana Univ. Math. J. 64 (2015), no. 4, 1001–1025. [610] Q. Xu, Littlewood-Paley theory for functions with values in uniformly convex spaces, J. Reine Angew. Math. 504 (1998), 195–226. [611] S. Yamashita, Hyperbolic Hardy class H1 , Math. Scand. 45 (1979), 261–266. [612] S. Yamashita, Gap series and α-Bloch functions, Yokohama Math. J. 28 (1980), no. 1–2, 31–36. [613] S. Yamashita, On hyperbolic Hardy classes, Comment. Math. Univ. St. Pauli 30 (1981), 65–69. [614] N. Yanagihara, The containing Fréchet space for the class N+ , Duke Math. J. 40 (1973), 93–103. [615] N. Yanagihara, Multipliers and linear functionals for the class N+ , Trans. Am. Math. Soc. 180 (1973), 449–461. [616] N. Yanagihara, The second dual space for the space N+ , Proc. Jpn. Acad. 49 (1973), 33–36. [617] C. Yang and W. Xu, Spaces with normal weights and Hadamard gap series, Arch. Math. 96 (2011), no. 2, 151–160. [618] S. Ye, Weighted composition operators from analytic Morrey spaces into Zygmund spaces, J. Comput. Anal. Appl. 25 (2018), no. 4, 770–784. [619] L. C. Young, On an inequality of Marcel Riesz, Ann. Math. (2) 40 (1939), 567–574.

544 | Bibliography

[620] W. H. Young, Sur la généralisation du théorème de Parseval, C. R. Acad. Sci. Paris Sér. A-B 155 (1912), 30–33. [621] W. H. Young, On the multiplication of successions of Fourier constants, Proc. R. Soc. Lond. 87 (1913), 331–339. [622] X. Yue, Coefficient multipliers on mixed norm spaces, Appl. Math. J. Chin. Univ. Ser. B 19 (2004), no. 3, 252–256. [623] M. Zafran, Multiplier transformations of weak type, Ann. Math. (2) 101 (1975), 34–44. [624] V. P. Zakharyuta and V. I. Yudovich, The general form of linear functionals on Hp , Usp. Mat. Nauk. 19 (1964), 139–142 (Russian). [625] R. Zhao and K. Zhu, Theory of Bergman spaces in the unit ball of Cn , Mém. Soc. Math. Fr.. 115 (2008), 1–103. [626] K. Zhu, Analytic Besov spaces, J. Math. Anal. Appl. 157 (1991), no. 2, 318–336. [627] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005. [628] K. Zhu, A class of Möbius invariant function spaces, Ill. J. Math. 51 (2007), no. 3, 977–1002. [629] K. Zhu, Operator theory in function spaces, 2nd ed., vol. 138, American Mathematical Society (AMS), Providence, RI, 2007. [630] M. Zinsmeister, A distortion theorem for quasiconformal mappings, Bull. Soc. Math. Fr. 114 (1986), 123–133. [631] A. Zygmund, Sur la convergence absolue des séries de Fourier, Proc. Lond. Math. Soc. 3 (1928), 194–196. [632] A. Zygmund, Sur les fonctions conjuguées, Fundam. Math. 13 (1929), 284–303. [633] A. Zygmund, Some points in the theory of trigonometric and power series, Trans. Am. Math. Soc. 36 (1934), 586–617. [634] A. Zygmund, On the convergence and summability of power series on the circle of convergence. II, Proc. Lond. Math. Soc. (2) 47 (1941), 326–350. [635] A. Zygmund, On certain integrals, Trans. Am. Math. Soc. 55 (1944), 170–204. [636] A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47–76. [637] A. Zygmund, A remark on the integral modulus of continuity, Rev., Ser. A, Univ. Nac. Tucuman 7 (1950), 259–269. [638] A. Zygmund, On a theorem of Marcinkiewicz concerning interpolation of operations, J. Math. Pures Appl. (9) 35 (1956), 223–248. [639] A. Zygmund, Trigonometric series: Vols. I, II, second edition, reprinted with corrections and some additions, Cambridge University Press, London, 1968.

Index ⬦ 93 − 1 ∫ Aλ (z) 206 Aαn 173 AK = (A, ℓ1 ) 193 p,q p,q Aφ , Aφ 137 A ≍ B 20 A(r, f ) 197 ARp 197 B6 (a, f ), B6 (f ) 223 Bj (a, f ) 222 Bp , Bp (f ) 320 BV[a, b] 9 Cc2 (D) 52 C ν,α (𝕋) 291 dh (a, z) 2 Dr (a) 17 D[s] f 204 D[s] f 204 dτ 2 Df 22 Eε (a) 106 en 15 En (f )X 15 p,q E[ϕ] 448 q

EX ,[ϕ] 448 (f , g)m 211 fn 󴁂󴀱 f 5 ∞ Fq,α 375 p Fq,α (𝔻) 372 p Fq,α (𝕋) 372 FP 16 f ̃ 21 ‖f ‖∗1 222 ‖f ‖∗2 222 ‖f ‖∗p 227 g[j] (z) 444 h∞ (ψ), h∞ (ψ)n 289 H∞ (ψ), H∞ (ψ)n 289 H∞ (X ) 477 hp 12 Hp 26 Hp,⋆ 82 p,q p,q hα , Hα 102 p,q p,q hϕ , Hϕ 135 p,q

p,q

h[ϕ] , H[ϕ] 140

p

H+ 81 p Hhyp 360

Hp (𝕋) 33 Hp (X ) 478 q hX ,[ϕ] 445 p,q HΛ̄ 305 ω,n

H(𝔻, 𝔻) 241 hΛω (𝔻) 271 HΛω (𝔻), HΛω (𝕋) 272 hΛω (𝕋) 271 p,q HΛω,n 304, 305 hA(𝔻) 13 HCK1 95 HFP 418 H(𝔻), h(𝔻) 4 p HS1/p 211 p

HSα 205 I(r, u) 48 β Jg f 366 Jg f 362 p,⬦ Lα 102 p,⋆ L (μ) 497 p,q Lα 102 q L+0 312 q L−1 102 L log+ L 37 M∗ u 506 Mrad u 503 M(𝕋) 6 Möb(𝔻) 2 Nφ (r, w) 72 Nφ (w) 72 NC 98 Of (z, r) 96 Op f (z, r) 96 OCK1 96 OCK2 127 ohp 19 P(a, ζ ) 3 P(r, θ) 5 ̃ θ) 22 P(r, PS[γ] 9 Qp 258 QNSK 95 R+ u 33 rj (t) 507

546 | Index

Rn f 200 RO 96 SB (ζ ) 329 sn f 37 S[G] = SG = S(G) 351 s(X ) 427 S(X ) 427 Tmax f 514 Tζ 329 Tq∞ 370 p T∞ 371 p Tq (p, q < ∞) 370 T (E) 370 Uζ ,ρ 12

u# 130 u& 130 u× 108 UHk , UH∗ 409 Vn 187 p,q Vα 188 q Vs [X ] 191 Wε (ζ ) 367 Wmax f 186 ψ Wn = Wn 184 W = WΨ 184 W (ζ , ε) 255 W (I) 126, 255 X 󸀠 15 (X , Y ) 418 XP 15 X ∗ = (X , H∞ ) 419 X # = (X , A(𝔻)) 419 X A 189 X ≃ Y 35 𝒜h (φ) 361 𝒜q f (ζ ) 370 𝒜f (ζ ) 331 ℬω 245 ℬ(ψ) 247 p,q Bα 181 𝒞φ , 𝒞(X , Y ) 241 𝒟 150 𝒟̌ 150 𝒟̂ 149 p 𝒟α 42 𝒢∗β [f ] 341 𝒢∗ [f ] 332 𝒢β [f ] 340 𝒢d [f ] 332

𝒢[f ] 332 (ℋ) 135 ℋf 87 𝒥s u 116 𝒥 Ψ g 450 𝒥 s u 118 𝒦 39 𝒦a , 𝒦s 39 ℒ0 509 ℳg 501 𝒫[ϕ] 6 ℛs f 182 ℛf 22 𝒰 385 𝒰k , 𝒰∗ 409 𝒱n,λ 443 q 𝒱C [X ] 445 B 183 b 236 p,q Bα 182 p,q Bϕ 203 p,q

Bω,n 304 p,q Bp,q = B0 183 p p Bα , bα 232 q BX ,α 421 B(f ) 236 hp (φ) 360 p,q ℓα 164 Δnt h(θ) 287 Δm λk 335 ΔNt g 120 Δ[α, β] 134 Δn f 182 p,q Δα 182 φ[h] 360 κφ (z) 360 κ(X , Y ) 422 Λ∗ , λ∗ 288 Λω,n = Λω,n (𝕋) 288 Λω (K) 265 p p Λω (𝕋), Λω (𝔻) 274 ωn (f , t)p 301 ωn (g, t) 287 ω(g, δ), Ω(g, δ) 265 ω(g, t)p , Ω(g, t)p 274 𝜕F , 𝜕F̄ 3 σnα f 173 σ∗α f 175 σa (z) 2

Index | 547

Abel dual 189, 419 Aleman lemma 319 approximation – by polynomials 15, 201 – of a singular inner function 206 – of the atomic function 206 – with inner functions 78 approximation by smooth functions 51 area function 197 – Luzin 331 Banach envelope 462 Banach principle 37, 177, 213, 514 Banach–Steinhauss principle 189 Bergman space – atomic decomposition 121, 122, 482 – Hardy–Stein characterization 131 – weighted harmonic 137 Bergman weight 136 Besov spaces – f-property 212 – abstract 421 – analytic 182 – best approximation 201 – decomposition 188, 203 – decomposition, p > 1 182 – duality 183, 195 – interpolation 214 – invariant 320 – isomorphism with Bp,q 204 – K-property 212 – lacunary series 189 – membership of inner functions 208 – normal 202 – singular inner functions 207 – vector-valued 485 Bi-Bloch lemma 247, 251 Blaschke condition 56 Blaschke product 56, 60, 63, 208, 211 – in Besov spaces 261 – in Hardy–Sobolev spaces 261 Bloch space 236 – compact composition operators 242 – Holland–Walsch characterization 260 – inner functions in b 258 – monotone coefficients 237 – Pavlović characterization 260 – predual – monotone coefficients 238

– weighted 245 – lacunary series 259 BMO 221 – B∞,2 ⊂ VMOA 233 p – B1/p ⊂ BMOA 232 – compact composition operators 243, 244 – f-property 227 – Fatou property 224 – Garsia norm 222 – homogeneity 224 – inner functions in VMOA 232 – invariance of seminorms 224 – lacunary series 236 – monotone coefficients 237 – Taylor coefficients 235 Bourgain’s lemma 161 Carleson window 255 (C, 1/p − 1)-convergence in Hp 174 Cesàro means of order α > −1 173 Chebyshev inequality 498 composition operators – from B to Hp 361 – from Hp to B 438 – on Lipschitz spaces 278 – derivative-free description 278 composition with inner functions 74, 76 convolution 4 convolution lemma 435 decomposition – of abstract Besov spaces 421 – of Besov spaces, 0 < p ≤ ∞ 188 – of Besov spaces, 1 < p < ⬦ 182 – of normal Besov spaces 203 – of polyharmonic spaces 128 – of spaces with subnormal weights 446 density of polynomials in Hp 28 diagonal spaces 147 distortion function 137 dual – of A(𝔻) 39 – of Bergman space with subnormal weights 453 – of Besov spaces, 0 < p ≤ ⬦ 195 – of Besov spaces, p > 1 183 – of H1 230 – of Hp , p < 1 195 – of Hp , p > 1 35 – of Hp (X ), p < 1 486

548 | Index

– of spaces with subnormal weights 449 q – of Vs [X ] 191 f-property 212, 317 Fejér kernel 300 formula – Carleson representation 322 – Cauchy integral 32 – Green 3, 54, 88, 190, 191, 225, 226, 229, 248 – Green–Poisson 3 – Jensen 54, 55 – Parseval 13, 36 – Riesz representation 53, 54 fractional derivative 204 fractional integral 116, 136, 204 fractional integration proposition 116 function – almost monotone 133 – atomic 57, 70, 206 – derivatives 207 – moduli of smoothness 304 – conjugate 21–23, 34 – convex of log r 48 – increasing 27 – inner 57, 59, 71, 74, 76, 208, 211, 232, 310, 317 – inner in Besov spaces 208 – Köbe 69 – Littlewood–Paley g-function 332 – log-subharmonic 46, 51, 61, 116 – logarithmically convex 50 – nearly convex 98–100 – normal 134, 137 – of bonded mean oscillation 221 – of class (ℋ) 197–199 – of vanishing mean oscillation 230 – outer 58, 59, 70, 319 – polyharmonic 101, 128 – positive harmonic 8, 71 – quasi-nearly subharmonic 95, 98, 103, 108, 113, 115, 116, 127, 136, 281, 347, 507 – Rademacher 507 – regularly oscillating 96, 119, 280, 379 – semicontinuous 45, 77 – singular inner 57, 206, 207 – strictly increasing 27 – subharmonic 45, 48, 50, 54, 55, 89, 95, 116, 346 – discontinuous 45 – subnormal 134

– superharmonic 47, 279 – univalent 69 – Weierstrass 288 Hadamard product 4, 476, 480 Hardy space – analytic 26 – and univalent functions 389 – atomic decomposition 90 – disc algebra 26 – equivalent ℓp (L2 )-norm 341 – harmonic 12 – harmonic, p < 1 14, 18, 19 – interpolation 160, 215 – on the sphere 89 – solid hull, p < 1 429 Hardy–Bloch spaces 183 Hardy–Littlewood decomposition lemma 26, 56, 178 Hardy–Sobolev space 205, 310 – membership of inner functions 211 Hardy–Stein identity 34, 64, 65, 88, 90, 226, 227 – asymptotic form 110 harmonic Schwarz lemma 86, 272 hyperbolic Hardy class 360 inequality – Ahern–Jevtić 208 – Bayart et al. 86 – Carleman 59 – Clunie–MacGregor 239 – Fejér–Riesz 61 – Flett 196, 197, 347, 348 – Girela–Pavlović–Peláez 240 – Hardy 41, 62, 172, 218 – Hardy Mp∞ 63 – Hardy–Littlewood (1/p − 1) 18, 29, 157, 175, 304 – Hardy–Littlewood Σ 21, 157, 162, 163, 173, 392 – Hardy–Littlewood M2p 65, 66, 89 – Hardy–Littlewood M2p 195, 240 p – Hardy–Littlewood Mq 163, 178, 196 – Harnack 8, 98 – Hausdorff–Young 165, 236, 496, 500 – Hedenmalm 262 – Heinz 397, 403 – Hilbert 61, 62 – Hollenbeck–Verbitsky 33

Index | 549

– isoperimetric 59 – John–Nirenberg 221, 228 – strong 221 – Kahane 216 – Kalaj 43 – Kalton (1/p − 1) 477 – Khintchine 508 – Khintchine–Haagerup 518 – Kolmogorov–Smirnov 69 – Korenblum 240 – Littlewood–Paley 196, 214, 240, 346–348, 381 – Luecking 381 – Makarov 239 – Mateljević–Pavlović 166 – Paley 518 – on lacunary series 66 – Pichorides 34 – Riesz–Zygmund 25, 61 – geometric interpretation 25 – Storozhenko 303, 304 – Verbitsky 43 – Wittman 179 inner factor 58 integrability of lacunary series 141, 142 – with complex coefficients 144 – with positive coefficients 142 – with vector coefficients 471 integrability of power series – with positive coefficients 141 integral means 12 – convexity 50 – log-convexity 50 – monotonicity 27 – monotonicity and convexity 48 – of polynomials 49 – of univalent functions 385 isometry Lp with hp 13 isomorphism Ap with ℓp 469 Jones lemma 394 K-property 212 Knese lemma 226 Lacunary series 66, 67, 89, 142, 251, 378 – and Peano curves 89 – in Besov spaces 189 – in BMOA 236 – in C[0, 1] 144

– in diagonal weighted spaces 147 – in quasi-Banach spaces 470 – in weighted L∞ -spaces 146 – in weighted mixed norm spaces 146 Laguerre polynomials 206 Laplacian 3 Lebesgue point 23, 24, 502 Lebesgue set 502 Lipschitz condition – for the modulus – of a harmonic function 284 – of an analytic function 275 – radial 307 Lipschitz space 265, 274 – analytic 272 – and spaces of harmonic functions 290 – BMO-type characterizations 312 – composition operators 278 – generalized 304 – harmonic on 𝔻 271 – of higher order 288 Littlewood’s conjecture 179 local-to-global estimates 106 log-subharmonicity lemma 46, 333 majorant 138, 265 – Δ2 -condition 289 – concave 267 – Dini 266, 268, 275 – fast 266, 267, 269, 276, 289, 310 – of order n 290 – on [1, ∞) 289 – regular 266, 276, 279, 307, 312 – regular of order n 290 – slow 266, 267, 271, 289 – slow of order n 290 Makarov low of iterated logarithms 239 maximal function 108, 113, 115, 116, 186 – main 501 – non-tangential 175, 333, 506 – radial 89, 503 maximal theorem – (C, α) 175 – complex 30, 167, 176, 208, 210, 505 – main 176, 501, 504 – radial 504 – subharmonic 504 – vector 516 – vector (C, α) 330, 338, 340

550 | Index

– vector log-subharmonic 330, 333 – vector subharmonic 330 – W - 187 maximum modulus principle 5, 14, 77 – Smirnov 31, 36 maximum principle 47, 223 mean value property 4 measure – Carleson 254, 381, 393 – Möbius invariant 2, 105 – Riesz 52, 346, 348 – local estimates 348 – of |f p | 63 mixed norm spaces – and Lipschitz spaces 120 – completeness 103 – harmonic 113, 115, 127 – Lebesgue 113, 115, 116 – minimal 104 – non-admissible 105 – weighted 136 – with normal weight 136 – with subnormal weights 139, 140 Möbius dual 211 Möbius group 2 moduli of continuity 265 moduli of smoothness 287 monotone coefficients in H1 170 monotone coefficients in Hp 169 multipliers 512 – between Besov spaces 433 – compact 421, 422 – decomposition method 431 – monotone – between Besov spaces 434 – between Hardy spaces 436 – of Kellogg spaces 428 – of spaces with subnormal weights 455 – preduals 420 – second dual 421 Nevanlinna class 468 non-tangential limits 12 – of an Hp -function 28 operator – composition 241 – compact 241 – from ℓp to ℓq 470

– Hilbert 23, 34, 298, 456 – Hilbert matrix 87 – invertible 464 – maximal 501 – of strong type 499 – of weak type 499 – quasilinear 160, 164, 498 – Riesz projection 33 – subadditive 498 – sublinear 501, 510 – Töplitz 212 Ortega–Fàbrega lemma 244 oscillation 96 outer factor 58 Poisson integral 76 – of a function 6 – semicontinuous 40, 78 – of a measure 7 – of f∗ , f ∈ H1 31 – of f∗ , f ∈ hp 13 – of |f∗ | 276 – of log |f∗ | 30 – of the conjugate function 36 – on the unit ball 44 Poisson kernel 3, 5, 19 – conjugate 22 Poisson–Stieltjes integral 9, 10, 32 predual – of H1 232 – of H∞ 39 – of the Bloch space 237 pseudohyperbolic metric 2 q-Banach envelope – of Hp 122 quasiconformal harmonic mappings 396 – bi-Lipschicity 397 – boundary correspondence 397 quasiconformal mappings 396 radial derivative 118, 181 radial limits 10–12 – and mean convergence 28 – of an hp -function 13 – of conjugate function 22 – of Hardy–Bloch function 213 representation – of outer functions 58

Index | 551

– of singular inner functions 58 Riesz polynomials 218 Schauder basis 35, 170, 175, 447, 467 – non-existence 218 Schwarz modulus lemma 98, 275 Smirnov class 468 space – admissible 14 – Campanato 257 – Dirichlet 321 – F -space 468 – Hardy–Lorentz 440 – HFP 418 – homogeneous 16 – Jackson 448 – Kellog 164 – minimal 15 – Morrey 257 – Musielak–Orlicz 474 – of Borel measures 6 – of Cauchy transforms 39 – of type p 510 – (p, q)-Zygmund 307 – p-normed 461 – p-Zygmund 307 – PL-convex 457 – quasinormed 461 – self-conjugate 112, 136, 139, 299, 456 – solid 427 – tent 370 – Triebel–Lizorkin 372 – weak Hardy 82 – weak Lebesgue 160, 214, 497 – with Fatou property 16 – Zygmund 288 Stoltz angle 12, 329 subharmonic behavior – of ‖F (z)‖X 482 – of |∇u|, u ∈ RO 96 – of |u|p |∇u|q 98 – of polyharmonic functions 101 subordination principle 69, 70, 88 – for BMOA 242 symmetric difference 287, 335 tangential derivative 119, 182 p Tauberian nature of B1/p 234

theorem – Abu-Muhanna 413 – Abu-Muhanna–Louhichi 413 – Ahern 70 – Ahern–Clark 322 – Ahern–Jevtić 208 – Aleksandrov 38 – on boundary decay 42, 44 – Almansi representation 101 – Aoki–Rolewicz 462 – Arroussi–Pau 155 – Astala–Iwaniec–Martin–Onninen 412 – Astala–Koskela 405 – Baernstein–Cima–Shober 395 – Baernstein–Girela–Peláez 393 – Banach–Steinhaus 466 – Bary–Stechkin 324 – Bieberbach 386, 387 – Bishop–Phelps 262 – Blasco–de Souza 275 – Bonet–Taskinen 439 – Bourdon–Shapiro–Sledd 232, 234 – Božin–Karapetrović 130 – Burkholder–Gundy–Silverstein 79 – Calderón 331, 352 – Calderón–Zygmund 517 – Carathéodory convergence 402, 403 – Carleson – on Carleson measures 255 – Carleson–Hunt 40, 174 – Carleson–Jacobs/Havin–Shamoyan 285 – Chen 41 – Cima–Nicolau 220 – closed graph 467 – Clunie–Sheil-Small 409 – Coifman–Rochberg 122 – Constantin–Peláez 153 – de Souza–Sampson 217 – Dostanić 441 – Duren–Romberg–Shields 195 – Dyakonov 261, 276 – Fatou 10, 36, 58 – Fefferman duality 230 – Fefferman multiplier 428 – Fefferman–Stein 506 – on subharmonic behavior 16, 18, 94 – Garsia 225, 227 – Gehring–Hayman–Hinkkanen 283 – Gnuschke–Pommerenke 155

552 | Index

– Gröhn–Nicolau 325 – Grothendieck 68 – Gurariy–Matsaev 144, 147 – Hardy–Littlewood 269 – fractional integration 118, 136 – harmonic conjugates 17, 105, 107, 109, 112, 126, 135 – on lacunary series 144 – projection 112 – Hardy–Littlewood fractional differentiation 188 – Hardy–Littlewood–Sobolev 166, 436 – Hardy–Littlewood–Sunouchi 175 – Hardy–Prawitz 389, 392 – Hardy–Rogosinski 174, 177 – Herglotz–Plessner 8 – Hinkkanen 283 – Holland–Twomey 392 – Holland–Twomey–Spencer 390 – Hurwitz 8, 27 – hyperbolic Calderón 360 – increasing inclusion 113, 189 – Kalaj 398, 414, 416 – Kalaj–Saksman 415 – Kalton 42 – Kalton maximum principle 482 – Karamata–Szász tauberian 235 – Kislyakov–Xu 160, 162, 164 – Köbe distortion 387 – Köbe one-quarter 387, 388 – Kolmogorov – on conjugate functions 37 – Kolmogorov–Smirnov 36 – Konyagin 179 – Koskela–Benedict 415 – Kwon 361 – Kwon–Pavlović 246 – Kwon–Pavlović–Zhu 131 – Laitila–Nieminen–Saksman – Tylli 244 – Lindelöf 40 – Lindström–Sanatpour 285 – Littlewood Tauberian 471 – Littlewood–Paley 378 – Littlewood–Paley g− 332, 335, 341 – local integrability 48 – Madigan 278 – Marcinkiewicz 160, 498 – Marcinkiewicz L log+ L 517

– Marković 260 – Martio 398 – McGehee–Pigno–Smith 179 – mixed embedding 115, 188 – Montel 5 – Mori 397, 402 – Nikishin 510 – Nikishin–Stein 37, 213, 424, 511–513 – on a.e. convergence 515 – on Banach envelope of Hp (X ) 485 – on equivalent norms 466 – on Lebesgue points 502 – on Riesz measure 52 – on strong convergence in H1 171 – open mapping 466 – Paley – on Fourier coefficients 500 – on lacunary series 66 – Pavlović 138, 290, 301, 305 – Pavlović–Peláez 137 – Pavlović–Pelaéz 139 – Peláez–Rättyä 153 – Pommerenke 395, 411 – Prawitz 385, 386, 389, 390, 392 – Privalov – absolute continuity of f∗ 33 – absolute continuity of f∗ 168 – conjugate functions 272, 298, 456 – uniqueness 30 – Privalov–Plessner 22, 24, 36, 58 – Radó–Kneser–Choquet 26 – Ramey–Ulrich 245 – Riesz – conjugate functions 34 – projection 33 – radial limits 28 – Riesz factorization 57 – Riesz projection 230 – Riesz representation 53 – Riesz the Brothers 31 – absolute continuity of f∗ 32 – absolutely continuous measures 32 – Riesz–Herglotz 7, 9, 32, 58 – Riesz–Szegö – on log |f∗ | 30 – uniqueness 30 – Riesz–Thorin 143, 202, 495 – Rogosinski 88 – Salem–Zygmund 89

Index | 553

– Sawier 514 – Sawier–Stein 177, 214, 515 – Shapiro 20, 38 – Shields–Williams 453 – harmonic conjugates 299 – Sidon 89 – Smirnov – on conformal mappings 41 – Smirnov factorization 58 – Stein 512 – Stephenson 76 – Stoll 382 – Storozhenko 275 – tangential differentiation 119

– Tauber 471 – Zinsmeister 416 – Zygmund 147, 292 – conjugate functions 37 – on (C, 1/p − 1)-convergence 174 transform(ation) – Cauchy 39, 212, 420 – of unimodular functions 262 – Hilbert 397, 398 – Kelvin 44 – Möbius 2, 44 Uchiyama lemma 225 uniform boundedness principle 466

De Gruyter Studies in Mathematics Volume 73 Carol Jacoby, Peter Loth Abelian Groups: Structures and Classifications, 2019 ISBN 978-3-11-043211-4, e-ISBN 978-3-11-042768-4, e-ISBN (ePUB) 978-3-11-042786-8 Volume 72 Francesco Aldo Costabile Modern Umbral Calculus: An Elementary Introduction with Applications to Linear Interpolation and Operator Approximation Theory, 2019 ISBN 978-3-11-064996-3, e-ISBN 978-3-11-065292-5, e-ISBN (ePUB) 978-3-11-065009-9 Volume 34/1 József Lörinczi, Fumio Hiroshima, Volker Betz Feynman-Kac-Type Theorems and Gibbs Measures: Volume 1, 2019 ISBN 978-3-11-033004-5, e-ISBN 978-3-11-033039-7, e-ISBN (ePUB) 978-3-11-038993-7 Volume 34/2 József Lörinczi, Fumio Hiroshima Applications in Rigorous Quantum Field Theory: Volume 2, 2019 ISBN 978-3-11-040350-3, e-ISBN 978-3-11-040354-1, e-ISBN (ePUB) 978-3-11-040360-2 Volume 71 Wolfgang Herfort, Karl H. Hofmann, Francesco G. Russo Periodic Locally Compact Groups: A Study of a Class of Totally Disconnected Topological Groups, 2018 ISBN 978-3-11-059847-6, e-ISBN 978-3-11-059919-0, e-ISBN (ePUB) 978-3-11-059908-4 Volume 70 Christian Remling Spectral Theory of Canonical Systems, 2018 ISBN 978-3-11-056202-6, e-ISBN 978-3-11-056323-8, e-ISBN (ePUB) 978-3-11-056228-6 Volume 69 Derek K. Thomas, Nikola Tuneski, Allu Vasudevarao Univalent Functions. A Primer ISBN 978-3-11-056009-1, e-ISBN 978-3-11-056096-1, e-ISBN (ePUB) 978-3-11-056012-1 Volume 68/1 Timofey V. Rodionov, Valeriy K. Zakharov Set, Functions, Measures. Volume 1: Fundamentals of Set and Number Theory, 2018 ISBN 978-3-11-055008-5, e-ISBN 978-3-11-055094-8 www.degruyter.com