Function Classes on the Unit Disc: An Introduction 9783110281903, 9783110281231

Table of contents :
Preface
1 The Poisson integral and Hardy spaces
1.1 The Poisson integral
1.1.1 Borel measures and the space h1
1.2 Spaces hp and Lp(T) (p > 1)
1.3 Space hp (p < 1)
1.4 Harmonic conjugates
1.4.1 Privalov–Plessner’s theorem and the Hilbert operator
1.5 Hardy spaces: basic properties
1.5.1 Radial limits and mean convergence
1.5.2 Space H1
1.6 Riesz projection theorem
1.6.1 Aleksandrov’s theorem
Further notes and results
2 Subharmonic functions and Hardy spaces
2.1 Basic properties of subharmonic functions
2.1.1 Maximum principle
2.2 Properties of the mean values
2.3 Riesz measure
2.3.1 Riesz’ representation formula
2.4 Factorization theorems
2.4.1 Inner–outer factorization
2.5 Some sharp inequalities
2.6 Hardy–Stein identities
2.6.1 Lacunary series
2.7 Subordination principle
2.7.1 Composition with inner functions
2.7.2 Approximation with inner functions
Further notes and results
3 Subharmonic behavior and mixed norm spaces
3.1 Quasi-nearly subharmonic functions
3.2 Regularly oscillating functions
3.3 Mixed norm spaces: definition and basic properties
3.4 Embedding theorems
3.5 Fractional integration
3.6 Weighted mixed norm spaces
3.6.1 Lacunary series in mixed norm spaces
3.6.2 Bergman spaces with rapidly decreasing weights
3.6.3 Mixed norm spaces with subnormal weights
3.7 Lq-integrability of lacunary power series
3.7.1 Lacunary series in C[0, 1]
Further notes and results
4 Taylor coefficients with applications
4.1 Using interpolation of operators on Hp
4.1.1 An embedding theorem
4.1.2 The case of monotone coefficients
4.2 Strong convergence in H1
4.2.1 Generalization to (C, a)-convergence
4.3 A (C, a)-maximal theorem
Further notes and results
5 Besov spaces
5.1 Decomposition of Besov spaces: case 1 < p < ⋄
5.2 Maximal function
5.3 Decomposition of Besov spaces: case 0 < p =8
5.3.1 Radial limits of Hardy–Bloch functions
5.4 Duality in the case 0 < p ≤∞
5.5 Embeddings between Hardy and Besov spaces
5.6 Best approximation by polynomials
5.7 Normal Besov spaces
5.8 Inner functions in Besov and Hardy–Sobolev spaces
5.8.1 Approximation of a singular inner function
5.8.2 Hardy–Sobolev space Sp 1/p
5.8.3 f-property and K-property
Further notes and results
6 The dual of H1 and some related spaces
6.1 Norms on BMOA
6.2 Garsia’s and Fefferman’s theorems
6.2.1 Fefferman’s duality theorem
6.3 Vanishing mean oscillation
6.4 BMOA and Bp 1/p
6.4.1 Tauberian nature of Bp 1/p
6.5 Coefficients of BMOA functions
6.6 Bloch space
6.7 Mean growth of Hp-Bloch functions
6.8 Composition operators on B and BMOA
6.8.1 Weighted Bloch spaces
6.9 Proof of the bi-Bloch lemma
Further notes and results
7 Littlewood–Paley theory
7.1 Vector maximal theorems and Calderon’s area theorem
7.2 Littlewood–Paley g-theorem
7.3 Applications of the (C,m)-maximal theorem
7.4 Generalization of the 𝑔-theorem
7.5 Proof of Calderón’s theorem
7.6 Littlewood–Paley inequalities
7.7 Hyperbolic Hardy classes
Further notes and results
8 Lipschitz spaces of first order
8.1 Definitions and basic properties
8.1.1 Lipschitz spaces of analytic functions
8.1.2 Mean Lipschitz spaces
8.2 Lipschitz condition for the modulus
8.3 Composition operators
8.4 Composition operators into HΛ pω
8.5 Inner functions
Further notes and results
9 Lipschitz spaces of higher order
9.1 Moduli of smoothness and related spaces
9.2 Lipschitz spaces and spaces of harmonic functions
9.3 Conjugate functions
9.4 Integrated mean Lipschitz spaces
9.4.1 Generalized Lipschitz spaces
9.5 Invariant Besov spaces
9.6 BMO-type characterizations of Lipschitz spaces
9.6.1 Division and multiplication by inner functions
Further notes and results
10 One-to-one mappings
10.1 Integral means of univalent functions
10.1.1 Distortion theorems
10.2 Membership of univalent functions in some function classes
10.3 Quasiconformal harmonic mappings
10.3.1 Boundary behavior of QCH homeomorphisms of the disk
10.4 Hp-classes of quasiconformal mappings
Further notes and results
11 Coefficients multipliers
11.1 Multipliers on abstract spaces
11.1.1 Compact multipliers
11.2 Multipliers for Hardy and Bergman spaces
11.2.1 Multipliers from H1 to BMOA
11.3 Solid spaces
11.3.1 Solid hull of Hardy spaces (0 < p < 1)
11.4 Multipliers between Besov spaces
11.4.1 Monotone multipliers
11.5 Multipliers of spaces with subnormal weights
11.6 Some applications to composition operators
Further notes and results
12 Toward a theory of vector-valued spaces
12.1 Some properties of admissible spaces
12.2 Subharmonic behavior of ||F(z)||x
12.2.1 Banach envelope of Hp(X), 0 < p < 1
12.3 Linear operators on Hardy and Bergman spaces
12.4 Proof of the Coifman–Rochberg theorem
Further notes and results
A Quasi-Banach spaces
A.1 Quasi-Banach spaces
A.2 q-Banach envelopes
A.3 Closed graph theorem
A.4 F-spaces
A.4.1 Nevanlinna class
A.5 Spaces lp
A.6 Lacunary series in quasi-Banach spaces
A.6.1 Lp-integrability of lacunary series on (0, 1)
Further notes and results
B Interpolation and maximal functions
B.1 Riesz–Thorin theorem
B.2 Weak Lp-spaces and Marcinkiewicz’s theorem
B.3 Classical maximal functions
B.4 Rademacher functions and Khintchin’s inequality
B.5 Nikishin’s theorem
B.6 Nikishin–Stein’s theorem
B.7 Banach’s principle and the theorem on a.e. convergence
B.8 Vector-valued maximal theorem
Further notes and results
Bibliography
Index

Citation preview

Miroslav Pavlović Function Classes on the Unit Disc

De Gruyter Studies in Mathematics

| Edited by Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, Missouri, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Volume 52

Miroslav Pavlović

Function Classes on the Unit Disc | An Introduction

Mathematics Subject Classification 2010 46E10, 46E15, 46E30, 30H10, 30H20, 30H25, 30H30, 30H35, 31A05, 31C45, 30J05, 30J15, 30C62, 30C55, 46A16, 47B33 Author Prof. Dr. Miroslav Pavlović University of Belgrade Faculty of Mathematics p.p. 550 11000 Belgrade Serbia [email protected]

ISBN 978-3-11-028123-1 e-ISBN 978-3-11-028190-3 Set-ISBN 978-3-11-028191-0 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ♾Printed on acid-free paper Printed in Germany www.degruyter.com

| To my family

Preface 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 through many books, some new properties of them have been described, whereas verifications of known proper­ ties 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 in­ tegral 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 min­ imum of the theory of Fourier series and Banach space techniques is also desirable. 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 B, e.g. the Fefferman–Stein vector maximal theorem and a theorem of Nik­ ishin, only should be understood and taken as granted. One more fact of such deep­ ness is used in Chapter 5, and concerns the real interpolation between Hardy spaces, but it arises because of the author’s ineffectiveness to find a simple proof, which cer­ tainly exists, of a theorem on radial limits of “Hardy–Bloch” functions. The author hopes that applications of these theorems in this text shows 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. Approximately 30 percent of the text already appeared in the author’s booklet In­ troduction to Function Spaces on the Disk [374],but “Classes” cannot be treated as an expanded version of “Spaces” because the latter is not a subset of the former, and the organization of text is significantly different.

Acknowledgment I am grateful to the following people: Marcin Marciniak (former Versita/De Gruyter), who invited and constantly encouraged me to write this book; Friederike Dittberner, Anja Moebius, and Britta Nagl (De Gruyter) for their kindness and patience in an­ swering all of my questions; Greg Knese (Washington University in St. Louis), Oscar Blasco (University of Valencia), Jie Xiao (Memorial University of Newfoundland), Dra­ gan Vukotić (Universidad Autónoma de Madrid), Wolfgang Lusky (University of Pader­ born), Jorge Hounie (Universidade Federal de São Carlos), John Garnett (University of California, LA), José Ángel Peláez (University of Malaga), Aimo Hinkkanen (University

1 in the author’s head

viii | Preface of Illinois at Urbana-Champaign), and Ern Gun Kwon (Andong National University), who provided me with texts that were unavailable in Belgrade; and, especially, an anonymous reviewer whose suggestions significantly affected the organization of the text. Belgrade, July 2013

Miroslav Pavlović

Contents Preface | vii 1 The Poisson integral and Hardy spaces | 1 1.1 The Poisson integral | 5 1.1.1 Borel measures and the space ℎ1 | 6 1.2 Spaces ℎ𝑝 and 𝐿𝑝 (𝕋) (𝑝 > 1) | 10 1.3 Space ℎ𝑝 (𝑝 < 1) | 12 1.4 Harmonic conjugates | 18 1.4.1 Privalov–Plessner’s theorem and the Hilbert operator | 19 1.5 Hardy spaces: basic properties | 22 1.5.1 Radial limits and mean convergence | 24 1.5.2 Space 𝐻1 | 27 1.6 Riesz projection theorem | 29 1.6.1 Aleksandrov’s theorem | 33 Further notes and results | 35 2 Subharmonic functions and Hardy spaces | 40 2.1 Basic properties of subharmonic functions | 40 2.1.1 Maximum principle | 42 2.2 Properties of the mean values | 42 2.3 Riesz measure | 45 2.3.1 Riesz’ representation formula | 47 2.4 Factorization theorems | 49 2.4.1 Inner–outer factorization | 50 2.5 Some sharp inequalities | 52 2.6 Hardy–Stein identities | 58 2.6.1 Lacunary series | 60 2.7 Subordination principle | 61 2.7.1 Composition with inner functions | 64 2.7.2 Approximation with inner functions | 68 Further notes and results | 69 3 3.1 3.2 3.3 3.4 3.5

Subharmonic behavior and mixed norm spaces | 74 Quasi-nearly subharmonic functions | 74 Regularly oscillating functions | 75 Mixed norm spaces: definition and basic properties | 83 Embedding theorems | 92 Fractional integration | 95

x | Contents 3.6 Weighted mixed norm spaces | 99 3.6.1 Lacunary series in mixed norm spaces | 102 3.6.2 Bergman spaces with rapidly decreasing weights | 102 3.6.3 Mixed norm spaces with subnormal weights | 105 3.7 𝐿𝑞 -integrability of lacunary power series | 109 3.7.1 Lacunary series in 𝐶[0, 1] | 112 Further notes and results | 114 4 Taylor coefficients with applications | 118 4.1 Using interpolation of operators on 𝐻𝑝 | 118 4.1.1 An embedding theorem | 121 4.1.2 The case of monotone coefficients | 126 4.2 Strong convergence in 𝐻1 | 129 4.2.1 Generalization to (𝐶, 𝛼)-convergence | 131 4.3 A (𝐶, 𝛼)-maximal theorem | 132 Further notes and results | 135 5 Besov spaces | 138 5.1 Decomposition of Besov spaces: case 1 < 𝑝 < ⬦ | 138 5.2 Maximal function | 140 5.3 Decomposition of Besov spaces: case 0 < 𝑝 ≤ ∞ | 143 5.3.1 Radial limits of Hardy–Bloch functions | 145 5.4 Duality in the case 0 < 𝑝 ≤ ∞ | 149 5.5 Embeddings between Hardy and Besov spaces | 155 5.6 Best approximation by polynomials | 160 5.7 Normal Besov spaces | 162 5.8 Inner functions in Besov and Hardy–Sobolev spaces | 164 5.8.1 Approximation of a singular inner function | 164 𝑝 5.8.2 Hardy–Sobolev space 𝑆1/𝑝 | 170 5.8.3 f-property and K-property | 171 Further notes and results | 172 6 6.1 6.2 6.2.1 6.3 6.4

The dual of 𝐻1 and some related spaces | 175 Norms on BMOA | 175 Garsia’s and Fefferman’s theorems | 179 Fefferman’s duality theorem | 183 Vanishing mean oscillation | 183 𝑝 BMOA and B1/𝑝 | 185

6.4.1 6.5 6.6

Tauberian nature of B1/𝑝 | 188 Coefficients of BMOA functions | 189 Bloch space | 189

𝑝

Contents | xi

6.7 Mean growth of 𝐻𝑝 -Bloch functions | 192 6.8 Composition operators on B and BMOA | 194 6.8.1 Weighted Bloch spaces | 197 6.9 Proof of the bi-Bloch lemma | 202 Further notes and results | 206 7 Littlewood–Paley theory | 211 7.1 Vector maximal theorems and Calderon’s area theorem | 211 7.2 Littlewood–Paley 𝑔-theorem | 213 7.3 Applications of the (𝐶, 𝑚)-maximal theorem | 217 7.4 Generalization of the 𝑔-theorem | 222 7.5 Proof of Calderón’s theorem | 224 7.6 Littlewood–Paley inequalities | 229 7.7 Hyperbolic Hardy classes | 235 Further notes and results | 238 8 Lipschitz spaces of first order | 241 8.1 Definitions and basic properties | 241 8.1.1 Lipschitz spaces of analytic functions | 246 8.1.2 Mean Lipschitz spaces | 247 8.2 Lipschitz condition for the modulus | 249 8.3 Composition operators | 251 8.4 Composition operators into 𝐻Λ𝑝𝜔 | 254 8.5 Inner functions | 260 Further notes and results | 261 9 Lipschitz spaces of higher order | 264 9.1 Moduli of smoothness and related spaces | 264 9.2 Lipschitz spaces and spaces of harmonic functions | 267 9.3 Conjugate functions | 275 9.4 Integrated mean Lipschitz spaces | 278 9.4.1 Generalized Lipschitz spaces | 280 9.5 Invariant Besov spaces | 284 9.6 BMO-type characterizations of Lipschitz spaces | 286 9.6.1 Division and multiplication by inner functions | 290 Further notes and results | 291 10 One-to-one mappings | 294 10.1 Integral means of univalent functions | 294 10.1.1 Distortion theorems | 295 10.2 Membership of univalent functions in some function classes | 298

xii | Contents 10.3 Quasiconformal harmonic mappings | 304 10.3.1 Boundary behavior of QCH homeomorphisms of the disk | 304 10.4 𝐻𝑝 -classes of quasiconformal mappings | 312 Further notes and results | 315 11 Coefficients multipliers | 318 11.1 Multipliers on abstract spaces | 318 11.1.1 Compact multipliers | 323 11.2 Multipliers for Hardy and Bergman spaces | 324 11.2.1 Multipliers from 𝐻1 to BMOA | 327 11.3 Solid spaces | 329 11.3.1 Solid hull of Hardy spaces (0 < 𝑝 < 1) | 331 11.4 Multipliers between Besov spaces | 332 11.4.1 Monotone multipliers | 335 11.5 Multipliers of spaces with subnormal weights | 337 11.6 Some applications to composition operators | 348 Further notes and results | 349 12 Toward a theory of vector-valued spaces | 352 12.1 Some properties of admissible spaces | 352 12.2 Subharmonic behavior of ‖𝐹(𝑧)‖𝑋 | 359 12.2.1 Banach envelope of 𝐻𝑝 (𝑋), 0 < 𝑝 < 1 | 362 12.3 Linear operators on Hardy and Bergman spaces | 364 12.4 Proof of the Coifman–Rochberg theorem | 369 Further notes and results | 374 A Quasi-Banach spaces | 375 A.1 Quasi-Banach spaces | 375 A.2 𝑞-Banach envelopes | 376 A.3 Closed graph theorem | 379 A.4 𝐹-spaces | 382 A.4.1 Nevanlinna class | 382 A.5 Spaces ℓ𝑝 | 383 A.6 Lacunary series in quasi-Banach spaces | 384 A.6.1 𝐿𝑝 -integrability of lacunary series on (0, 1) | 385 Further notes and results | 395 B B.1 B.2 B.3 B.4

Interpolation and maximal functions | 397 Riesz–Thorin theorem | 397 Weak 𝐿𝑝 -spaces and Marcinkiewicz’s theorem | 399 Classical maximal functions | 403 Rademacher functions and Khintchin’s inequality | 409

Contents | xiii

B.5 Nikishin’s theorem | 410 B.6 Nikishin–Stein’s theorem | 412 B.7 Banach’s principle and the theorem on a.e. convergence | 415 B.8 Vector-valued maximal theorem | 417 Further notes and results | 418 Bibliography | 421 Index | 443

1 The Poisson integral and Hardy spaces This chapter contains the 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 ra­ dial limits, the Privalov–Plessner on the 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, the well-known connection between the harmonic Hardy space ℎ𝑝 (1≤ 𝑝 ≤ ∞) and the Lebesgue space 𝐿𝑝 (𝕋) is presented without proof. A brief discussion of ℎ𝑝 for 𝑝 < 1 is in Section 1.3. In the last section we present a quick introduction to basic properties of (analytic) Hardy spaces. Our approach differs from that in other texts [129, 159, 273, 425, 430, 525] in which we first prove the Hardy–Littlewood decomposition lemma, and then deduce the radial limits 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 max­ imal functions in Appendix B, Section B.3. However, the reader can treat Section B.3 as a part of this chapter inserted before considering Hardy spaces.

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, we write ∫ − 𝑓 𝑑𝜇 = 𝑆

1 ∫ 𝑓 𝑑𝜇, 𝜇(𝑆) 𝑆

and in particular 2𝜋

2𝜋

1 ∫ ∫ 𝑓(𝑒𝑖𝜃 ) 𝑑𝜃, − 𝑓(𝑒 ) 𝑑𝜃 = 2𝜋 𝑖𝜃

0

∫ − 𝑓 𝑑𝐴 =

0

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

1 ∫ 𝑓(𝜁) |𝑑𝜁|, 2𝜋

where 𝕋 = 𝜕𝔻.

𝕋

The arc-length measure on 𝕋 will be denoted by 𝑑𝑙 and so ∫ 𝑓(𝜁) |𝑑𝜁| = ∫ 𝑓 𝑑𝑙. 𝕋

𝕋

2 | 1 The Poisson integral and Hardy spaces The two-dimensional measure of a measurable set 𝐺 ⊂ ℂ will be denoted by |𝐺|. Similarly, |𝑆| denotes the arc-length measure of 𝑆 ⊂ 𝕋. When dealing with spaces of analytic or harmonic functions it is convenient to use a new symbol, “⬦”, that has the following properties: 1 = 0 and 𝑥 < ⬦ < ∞ for all 𝑥 ∈ ℝ. ⬦ Let 𝑓 ∈ 𝐿𝑝 (𝕋), 0 < 𝑝 ≤ ∞, and 1/𝑝 𝑝

∫ |𝑓(𝜁)| |𝑑𝜁|) ‖𝑓‖𝐿𝑝 (𝕋) = ‖𝑓‖𝑝 = (−

.

𝕋

We write 𝐿⬦ (𝕋) = 𝐶(𝕋), and interpret the integral as in the case 𝑝 = ∞ : ‖𝑓‖⬦ = ‖𝑓‖∞ = ‖𝑓‖𝐿∞ (𝕋) . So we have 𝐿𝑝 (𝕋) ⊋ 𝐿𝑞 (𝕋) for 0 < 𝑝 < 𝑞 ≤ ∞.

Möbius transformations of the unit disk Every biholomorphic mapping (Möbius transformation) 𝜑 from 𝔻 onto 𝔻 can be rep­ resented 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. 1−|𝑎|2 – 𝜎𝑎󸀠 (𝑧) := (𝜎𝑎 )󸀠 (𝑧) = − |1− , |𝑎| < 1, |𝑧| ≤ 1. ̄ 2 𝑎𝑧| – We have (1 − |𝑎|2 )(1 − |𝑧|2 ) = |𝜎𝑎󸀠 (𝑧)|(1 − |𝑧|2 ) 1 − |𝜎𝑎 (𝑧)|2 = ̄ 2 |1 − 𝑎𝑧| and, more generally, 1 − 𝜎𝑎 (𝑧)𝜎𝑎 (𝑤) = –

–

(1 − |𝑎|2 )(1 − 𝑧𝑤)̄ . ̄ − 𝑎𝑤)̄ (1 − 𝑧𝑎)(1

The functional d(𝑎, 𝑧) = |𝜎𝑎 (𝑧)| (𝑎, 𝑧 ∈ 𝔻) is a metric on 𝔻, and is called the pseu­ dohyperbolic metric. It is Möbius invariant in sense that d(𝜎(𝑤), 𝜎(𝑧)) = d(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 par­ ticular that ∫ ℎ ∘ 𝜎𝑎 𝑑𝜏 = ∫ ℎ 𝑑𝜏, 𝔻

where ℎ ≥ 0 is a measurable function on 𝔻.

𝔻

1 The Poisson integral and Hardy spaces | 3

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 𝐶2 -function in 𝐷 = {𝑧 : |𝑧| < 𝑅}, then 𝜋

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

(1.1)

|𝑧| 1) For a Borel measurable function 𝑓 : 𝜌𝔻 󳨃→ ℂ, we define the integral means 𝑀𝑝 (𝑟, 𝑓), 0 < 𝑟 < 𝜌, by 1/𝑝

2𝜋 𝑖𝜃 𝑝

∫ |𝑓(𝑟𝑒 )| 𝑑𝜃) 𝑀𝑝 (𝑟, 𝑓) = ‖𝑓𝑟 ‖𝑝 = (− 0

where 𝑓𝑟 (𝑒𝑖𝜃 ) = 𝑓(𝑟𝑒𝑖𝜃 ).

,

1.2 Spaces ℎ𝑝 and 𝐿𝑝 (𝕋) (𝑝 > 1)

| 11

The harmonic Hardy space ℎ𝑝 (0 < 𝑝 ≤ ∞) is defined by ℎ𝑝 = { 𝑓 ∈ ℎ(𝔻) : ‖𝑓‖𝑝 = sup 𝑀𝑝 (𝑟, 𝑓) < ∞ }. 𝑟 1, then 𝑓 = P[𝑓∗ ], where 𝑓∗ (𝑒𝑖𝜃 ) = lim− 𝑓(𝑟𝑒𝑖𝜃 ). 𝑟→1

12 | 1 The Poisson integral and Hardy spaces The Poisson kernel also shows that boundedness of 𝑓∗ does not imply the bounded­ ness of 𝑓. However, if 𝑝 > 1, we have the following generalization of the maximum modulus principle. Corollary 1.7. If a function 𝑓 ∈ ℎ(𝔻) has radial limits 𝑓∗ (𝑒𝑖𝜃 ) almost everywhere, 𝑓∗ ∈ 𝐿∞ (𝕋), and 𝑓 ∈ ℎ𝑝 for some 𝑝 > 1, then 𝑓 ∈ ℎ∞ and ‖𝑓‖∞ = ‖𝑓∗ ‖∞ . Exercise 1.5. If 𝑓 ∈ ℎ𝑝 (𝑝 > 1) is real valued, then there are nonnegative functions 𝑝 𝑝 𝑓𝑗 ∈ ℎ𝑝 such that 𝑓 = 𝑓1 − 𝑓2 and ‖𝑓‖𝑝 = (‖𝑓1 ‖𝑝 + ‖𝑓1 ‖𝑝 )1/𝑝 . Exercise 1.6. 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

{ } ∫ |1 − 𝑟𝑒𝑖𝑡 |2𝑞−2 𝑑𝑡} 𝐾𝑝 (𝑟) = {− {−𝜋 }

(1/𝑝 + 1/𝑞 = 1).

Observe that 𝐾2 (𝑟) = (1 + 𝑟2 )1/2 . For the case of several variables, see [245], where the norm is expressed via the Euler gamma-function. Exercise 1.7. 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 1.8. 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.

1.3 Space ℎ𝑝 (𝑝 < 1) We have already defined the space ℎ𝑝 for 𝑝 < 1; see p. 11. 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 holds for ̂ 𝑝 ≥ 1 simply because |𝑓(𝑛)| ≤ ‖𝑓‖𝑝 . 1.1 (Admissible spaces). For the sake of simplicity, by the term “quasinormed space” we mean a vector space endowed with a 𝑝-norm (0 < 𝑝 ≤ 1); see Section A.1. A quasi­ normed space 𝑋 ⊂ ℎ(𝔻) will be called admissible (or ℎ-admissible) if it is complete, ℎ(𝔻) ⊂ 𝑋, and the inclusion 𝑋 ⊂ ℎ(𝔻) is continuous. These conditions imply lim

|𝑛|→+∞

|𝑛| √ ‖𝑒𝑛 ‖𝑋 = 1,

where 𝑒𝑛 (𝑟𝑒𝑖𝜃 ) = 𝑟|𝑛| 𝑒𝑖𝑛𝜃 .

(1.17)

An 𝐻-admissible space 𝑋 ⊂ 𝐻(𝔻) is defined in the same way. An important fact follows from the definition: for every 𝑧 ∈ 𝔻, the evaluation functional 𝑓 󳨃→ 𝑓(𝑧) is bounded. Thus the dual of 𝑋, denoted by 𝑋󸀠 , separates points.

1.3 Space ℎ𝑝 (𝑝 < 1) |

13

1.2 (Minimal spaces). An admissible space 𝑋 is said to be minimal if the set ℎ(𝔻) is dense in 𝑋. The space 𝑋 is minimal if and only if it satisfies one of the conditions: (a) For all 𝑓 ∈ 𝑋, we have lim𝑟→1− ‖𝑓 − 𝑓𝑟 ‖𝑋 = 0. (b) 𝑋 = 𝑋P , where 𝑋P is the closure in 𝑋 of the set P of all harmonic polynomials. (c) For all 𝑓 ∈ 𝑋 we have lim𝑛→∞ 𝐸𝑛(𝑓)𝑋 = 0, where 𝐸𝑛(𝑓)𝑋 = inf{‖𝑓 − 𝑄‖𝑋 : 𝑄 ∈ P𝑛 }, and P𝑛 is the set of all harmonic polynomials of degree ≤ 𝑛. A minimal 𝐻-admissible space 𝑋 is defined in a similar way. We only have to replace ℎ(𝔻), P𝑛 , P , with 𝐻(𝔻), P𝑛 , P , where P ⊂ 𝐻(𝔻) is now the set of all analytic polynomials, and P𝑛 the corresponding subset of P . If 𝑓 ∈ 𝑋, where 𝑋 is minimal, then sup0 0,

𝔻

we see that (1.19) implies (1.18). 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 3.10).

14 | 1 The Poisson integral and Hardy spaces Proof of Theorem 1.11. Assuming, as we may, that 𝑢 is real valued, we start from the inequality 𝐾 sup{|𝑢(𝑧)| : |𝑧 − 𝑎| < 𝑟}, (1.20) |∇𝑢(𝑎)| ≤ 𝑟 which is valid whenever the disk 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 so that (1 − |𝑎|)2 |𝑢(𝑎)|𝑝 ≥ (1 − |𝑧|)2 |𝑢(𝑧)|𝑝

for all 𝑧 ∈ 𝔻.

(1.21)

From Lagrange’s theorem and inequality (1.20) it follows that |𝑢(𝑎)| ≤ |𝑢(𝑧)| + 𝐾(𝑡/𝑟) sup |𝑢|

(𝑠 = 𝑡 + 𝑟)

𝐷𝑠 (𝑎)

provided 𝑧 ∈ 𝐷𝑡 (𝑎) and 𝐷𝑠 (𝑎) ⊂ 𝔻. Now choose 𝑡 and 𝑟 so 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 = 41/𝑝 . Next we choose 𝑡 and 𝑟 so 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 𝑡2 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. By translations and dilations, we obtain: Theorem 1.13. If 𝑓 is a function harmonic in a subdomain 𝐺 ⊂ ℂ and 𝐷𝑟 (𝑎) ⊂ 𝐺, then |𝑓(𝑎)|𝑝 ≤

𝐶𝑝 |𝐷𝑟 (𝑎)|

∫ |𝑓|𝑝 𝑑𝐴,

whenever 𝐷𝑟 (𝑎) ⊂ 𝐺.

𝐷𝑟 (𝑎)

If we apply (1.18) to the disk of radius 1 − |𝑧| centered at 𝑧 ∈ 𝔻, we get |𝑓(𝑧)|𝑝 ≤ 𝐶𝑝 (1 − |𝑧|)−2 ∫ |𝑓|𝑝 𝑑𝐴.

(1.22)

𝔻

This was the reason for choosing 𝑎 by (1.21). The following theorem shows that the inclusion ℎ𝑝 ⊂ ℎ(𝔻) is continuous for 𝑝 < 1.

1.3 Space ℎ𝑝 (𝑝 < 1) |

15

Theorem 1.14 (Hardy–Littlewood). For 𝑢 ∈ ℎ𝑝 (0 < 𝑝 < 1) we have |𝑢(𝑧)| ≤ 𝐶𝑝 (1 − |𝑧|)−1/𝑝 ‖𝑢‖𝑝 1/𝑝−1

̂ |𝑢(𝑛)| ≤ 𝐶𝑝 (|𝑛| + 1)

‖𝑢‖𝑝

(𝑧 ∈ 𝔻),

(1.23)

(𝑛 ∈ ℤ).

(1.24)

Proof. By (1.22) we have |𝑢(𝑟𝑒𝑖𝜃 )|𝑝 ≤ 𝐶𝑝 (1 − 𝑟)−2

|𝑢(𝑤)|𝑝 𝑑𝐴(𝑤)

∫ 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. This was to be proved. The following estimates are proved in a similar way as (1.23) and (1.24). 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

2 1/𝑝 ) ‖𝑢‖𝑝 1−𝑟

The first inequality holds for all 0 < 𝑝 ≤ ∞.

(0 < 𝑟 < 1).

16 | 1 The Poisson integral and Hardy spaces 1.3 (The space 𝑜ℎ𝑝 ). The space 𝑜ℎ𝑝 is the subspace of ℎ𝑝 consisting of 𝑢 ∈ ℎ(𝔻) such that lim𝑟→1− 𝑀𝑝 (𝑟, 𝑢) = 0. This is an infinite-dimensional space. Indeed, according to Exercise 1.4 (or to Proposition 1.2 below), 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 [438, Proposition 2.4]. Proposition 1.2. The Poisson kernel satisfies the following conditions: 𝑝 𝑀𝑝 (𝑟, 𝑃) = 𝑀𝑞𝑞 (𝑟, 𝑃) (𝑞 = 1 − 𝑝) and 𝑀𝑝 (𝑟, 𝑃) ≍ 𝛽𝑝 (𝑟), where for 0 < 𝑝 < 1/2;

𝛽𝑝 (𝑟) = (1 − 𝑟), 𝑒 2 = (1 − 𝑟) (log ) , 1−𝑟 = (1 − 𝑟)1/𝑝−1 ,

for 𝑝 = 1/2;

(1.25)

for 𝑝 > 1/2.

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

𝕋

where 𝜎𝑟 ∈ Möb(𝔻) is the involution 𝜎𝑟 (𝜁) = (𝑟 − 𝜁)/(1 − 𝑟𝜁). By the substitution 𝜁 = 󸀠 𝑝 󸀠 󸀠 − |𝜎 (𝜎𝑟 (𝜉))| |𝜎 (𝜉)| |𝑑𝜉|. From this, using the formula 𝜎 (𝜎𝑟 (𝜉)) = 𝜎𝑟 (𝜉) we get 𝐼𝑝 (𝑟) = ∫ 𝑟 𝑟 𝕋 𝑟 󸀠 1/𝜎𝑟 (𝜉), we obtain 𝐼𝑝 = 𝐼1−𝑝 . In order 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. 1.4. 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 param­ eters such that 𝑐𝐵 ≤ 𝐴 ≤ 𝐶𝐵; incidentally this means that 𝐴 is finite if and only if so is 𝐵. We call 𝑐 and 𝐶 the equivalence constants. Sentences of the following type occur of­ ten: “If 𝑃 holds, then 𝐴 ≍ 𝐵, where the equivalence constants are independent of . . . ” or “. . . where . . . depend only on . . . ” In most cases the sentence “where . . . ” may be omitted.

Isomorphic copy of ℓ∞ in ℎ𝑝 (0 < 𝑝 < 1) That the space ℎ𝑝 (𝑝 < 1) is not minimal is a consequence of the following assertion (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 (see [438]). Let ℎ denote the ℎ(𝔻)-topology. The space 𝑜ℎ𝑝 is closed in ℎ𝑝 and, by (1.23), 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.

1.3 Space ℎ𝑝 (𝑝 < 1)

| 17

Now we choose an increasing sequence 𝑟𝑛 ∈ (0, 1) and an increasing sequence 𝑘𝑛 ∈ ℕ+ in the following way. Since 𝑢𝑛 → 0 in ℎ, we can choose 𝑘1 so 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 so that 𝑝 𝑀𝑝 (𝑟, 𝑢𝑘1 ) < 𝜀/2 for 𝑟1 ≤ 𝑟 < 1. Next, we choose 𝑘2 so that |𝑢𝑘2 (𝑧)|𝑝 < 𝜀/22 and 𝑟2 > 𝑟1 𝑝 so that 𝑀𝑝 (𝑟, 𝑢𝑘2 ) < 𝜀/22 . Continuing in this way we find sequences 𝑟𝑛 ↑ 1 and 𝑣𝑛 such 𝑝 that 𝑀𝑝 (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, 𝑎𝑘 𝑣𝑘 ) = |𝑎𝑗 |𝑝 𝑀𝑝𝑝 (𝑟, 𝑣𝑗 ) + ∑ |𝑎𝑛 |𝑝 𝑀𝑝𝑝 (𝑟, 𝑢𝑛) 𝑛=1

𝑛=𝑗̸ 𝑝

𝑝

𝜀 𝑛 2 𝑛=𝑗̸

≤ |𝑎𝑗 | + sup |𝑎𝑛 | ∑ 𝑛=𝑗̸

≤ |𝑎𝑗 |𝑝 + sup |𝑎𝑛 |𝑝 𝜀 ≤ (1 + 𝜀)‖{𝑎𝑛 }‖𝑝∞ . 𝑛=𝑗̸

From this and (1.23) 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, 𝑝 denote it by 𝑢. Incidentally, the last inequality shows that ‖𝑢‖𝑝 ≤ (1 + 𝜀)‖{𝑎𝑛 }‖𝑝∞ . 𝑝 In the other direction, let 𝜀 < 1 and 𝜆 > 0 so that 𝜀 < 𝜆 < 1. Choose an index 𝑗 so that |𝑎𝑗 | > 𝜆‖{𝑎𝑛 }‖∞ . It is also clear that there exists 𝑟 ∈ [𝑟𝑗−1 , 𝑟𝑗 ) such that 𝑀𝑝 (𝑟, 𝑣𝑗 ) = 1. Then ‖𝑢‖𝑝𝑝 ≥ 𝑀𝑝𝑝 (𝑟, 𝑢) ≥ |𝑎𝑗 |𝑝 𝑀𝑝𝑝 (𝑟, 𝑣𝑗 ) − ∑ |𝑎𝑛 |𝑝 𝑀𝑝𝑝 (𝑟, 𝑣𝑛 ) 𝑛=𝑗̸

≥ |𝑎𝑗 |

𝑝

‖𝑢‖𝑝𝑝

−

𝜀‖{𝑎𝑛 }‖𝑝∞

𝑝

≥ (𝜆 − 𝜀)‖{𝑎𝑛 }‖𝑝∞ .

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. Whether there exists a function 𝑢 ∈ ℎ𝑝 (𝑝 < 1) such that ̂ |𝑢(𝑛)| ≥ 𝑐𝑝 (|𝑛| + 1)1/𝑝−1

(𝑛 ∈ ℤ) ?

(1.26)

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

2 1−𝑟

(0 < 𝑟 < 1) ?

(1.27)

Hardy and Littlewood [190] 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

18 | 1 The Poisson integral and Hardy spaces where 𝑃 is the Poisson kernel. It is clear that 𝑢 satisfies condition (1.26) for 𝑝 = 1/𝑁. That (1.27) is satisfied follows from the estimate (e.g. [438]) |𝐷𝑁 𝑃(𝑧)| ≤ 𝐶

1 − |𝑧|2 . |1 − 𝑧|𝑁+1

It should be noted that solving Problem 1.1 leads to solution of 1.2. Namely, (1.26) ̂ implies (1.27). Indeed, if (1.26) is satisfied, then |𝑓(𝑛)| ≥ 𝑐(𝑛 + 1)1/𝑝−1 , so the conclusion follows from the inequality ∞

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

which will be proved later on (see Theorem 4.1).

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

|𝑛| 𝑖𝑛𝜃 ̂ ̃ 𝑖𝜃 ) = −𝑖 ∑ (sign 𝑛) 𝑓(𝑛)𝑟 𝑒 . 𝑓(𝑟𝑒 𝑛=−∞

If 𝑓 is real valued, then 𝑓 ̃ is uniquely determined by the conditions: 𝑓 ̃ is real valued, ̃ = 0. 𝑓 + 𝑖𝑓 ̃ is analytic, and 𝑓(0) ̃ 𝑓= For an arbitrary 𝑓 ∈ ℎ(𝔻) we have 𝑓̃̃ = −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

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

0

where we write 𝜙(𝑥) instead of 𝜙(𝑒𝑖𝑥 ). Here 𝑃̃ denotes the conjugate Poisson kernel, 2𝑧 1+𝑧 ̃ 𝑃(𝑧) = Im = Im , i.e. 1−𝑧 1−𝑧 2𝑟 sin 𝜃 ̃ 𝜃) = 𝑃(𝑟𝑒 ̃ 𝑖𝜃 ) = . 𝑃(𝑟, 1 + 𝑟2 − 2𝑟 cos 𝜃 This kernel does not belong to ℎ1 . The kernels 𝑃 and 𝑃̃ are connected by the formula ̃ 𝜃) − 𝑃(𝑟,

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

(1.28)

̃ 𝜃). Note that 1/tan(𝜃/2) = 𝑃(1, Exercise 1.9. Let 𝑓(𝑟𝑒𝑖𝜃 ) be a function harmonic in 𝔻, and let R𝑓 = 𝑟 ̃ Then R𝑓 and 𝐷𝑓 are harmonic in 𝔻 and R𝑢 = 𝐷𝑢.

𝜕𝑓 𝜕𝑓 , and 𝐷𝑓 = . 𝜕𝑟 𝜕𝜃

1.4 Harmonic conjugates

| 19

1.4.1 Privalov–Plessner’s theorem and the Hilbert operator The following fundamental result is due to Privalov [406] and Plessner [398]. 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 )

𝑟→1

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

and

𝜀

𝑖𝜃 ̃ ) is contained in Corollary 1.5. This theorem guaran­ The existence of lim𝑟→1− 𝑃[𝑔](𝑟𝑒 tees the existence of the improper integral 𝜋

̃ 𝑖𝜃 ) = 𝑔(𝑒

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

There exists such a function 𝑔 ∈ 𝐶(𝕋) that this integral converges absolutely for no 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 (see [537, 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 operator². 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 ̃ ̃ = 𝑃[𝑔]. later on (see Theorem 1.39), if 𝑔̃ ∈ 𝐿1 , then P[𝑔] Proof of Theorem 1.17. It suffices to prove that 𝜋

{ 𝑔(𝜃 − 𝑡) − 𝑔(𝜃 + 𝑡) } 1 𝑖𝜃 ̃ 𝑑𝑡} = 0, lim− {𝑃[𝑔](𝑟𝑒 )− ∫ 𝑟→1 2𝜋 tan(𝑡/2) 1−𝑟 } {

(1.29)

under the hypothesis that 𝜃 is a Lebesgue point of 𝑔 (see p. 404 and Theorem B.8). We write the difference under lim𝑟→1− in (1.29) as 𝐼1 (𝑟) + 𝐼2 (𝑟), where 1−𝑟

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

2 Usually 2 tan(𝑡/2) is replaced by 𝑡.

20 | 1 The Poisson integral and Hardy spaces ̃ 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 the formula (1.28); it follows that 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 ̃ 󵄨󵄨 󵄨󵄨𝑃(𝑟, 𝑡) − 󵄨 ≤ const. 𝑃(𝑟, 𝑡) 󵄨󵄨 tan(𝑡/2) 󵄨󵄨󵄨 Thus

(1 − 𝑟 < |𝑡| < 𝜋).

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+

̃ almost everywhere. exists for all 𝜃 and is equal to 𝑔(𝜃) Proof. By partial integration, 𝜋

∫ 𝜀

𝛷(𝜃 + 𝜀) + 𝛷(𝜃 − 𝜀) − 2𝛷(𝜃) 𝛷(𝜃 + 𝑡) + 𝛷(𝜃 − 𝑡) − 2𝛷(𝜃) 𝑑𝑡 = 2 2 tan(𝜀/2) 4 sin (𝑡/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

whenever 𝛷󸀠 (𝜃) = 𝑔(𝜃).

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

1.4 Harmonic conjugates

| 21

Remark 1.1. If {𝑎𝑘 } is a convex sequence tending to 0, then the sum of the series 𝑎0 /2 + 1 ∑∞ 𝑛=1 𝑎𝑛 cos 𝑛𝜃 is positive for every 𝜃 ∈ (−𝜋, 𝜋) and belongs to 𝐿 (𝕋) (see [257, Theorem 4.1] or [537, Ch. V(1.5)]). In particular, the function ∞

𝑔(𝑒𝑖𝜃 ) = ∑ (log(𝑛 + 2))−1 cos 𝑛𝜃 𝑛=0 −1 is in 𝐿1 , while the function conjugate to 𝑔 is equal to ∑∞ 𝑛=1 (log(𝑛 + 2)) sin 𝑛𝜃 and is not 1 in 𝐿 (see [537, Ch. V(1.14)]).

Riesz–Zygmund inequality As we have seen, if 𝑔 ∈ ℎ1 , then the conjugate function 𝑔̃ need not belong to ℎ1 . A result of Riesz [419] and Zygmund [531] (cf. [537, Ch. IV (6.28)]) states that if 𝑔 ∈ ℎ1 , then 1

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

(1.30)

In other words: Theorem 1.19 (Riesz–Zygmund). If 𝑔 ∈ ℎ(𝔻) and 𝜕𝑔/𝜕𝜃 ∈ ℎ1 , then 𝜋 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕𝑔 󵄨󵄨 𝜕𝑔 1 󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨 (𝑟𝑒𝑖𝑡 )󵄨󵄨󵄨 𝑑𝑟 ≤ sup ∫ 󵄨󵄨󵄨 (𝑟𝑒𝑖𝜃 )󵄨󵄨󵄨 𝑑𝜃. 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕𝑟 2 0 1, this is true for all 𝑓 ∈ ℎ(𝔻), and the proof was left to the reader (Lemma 2.3). If 0 < 𝑝 ≤ 1, then the proof reduces to proving the inequality ∫ |𝑓(𝑟𝜁)|𝑝 |𝑑𝜁| ≤ ∫ |𝑓(𝜁)|𝑝 |𝑑𝜁|, 𝕋

𝑓 ∈ 𝐻(𝔻), 0 < 𝑟 < 1.

𝕋

Assume that 𝑓 has zeros in 𝔻 and let 𝐴(𝑧) be as in the proof of Lemma 1.1. Then we have 𝑓(𝑧) = 𝐴(𝑧)𝑔(𝑧)2/𝑝 , where 𝑔 is zero free in 𝔻. Since |𝐴(𝑧)| < 1 for 𝑧 ∈ 𝔻 and |𝐴(𝜁)| = 1 for 𝜁 ∈ 𝕋, we have ∫ |𝑓(𝑟𝜁)|𝑝 |𝑑𝜁| ≤ ∫ |𝑔(𝑟𝜁)|2 |𝑑𝜁| ≤ ∫ |𝑔(𝜁)|2 |𝑑𝜁| = ∫ |𝑓(𝜁)|𝑝 |𝑑𝜁|. 𝕋

𝕋

𝕋

𝕋

If 𝑓 is zero free in 𝔻, then we take 𝐴 = 1 and proceed as above. The rest of the lemma is not so important, and we leave the proof to the reader. (Note that the proof works for all 𝑝 ∈ ℝ+ .) Remark 1.2 (Monotone functions). We use the term “increasing” in the sense “𝑥 < 𝑦 ⇒ 𝜓(𝑥) ≤ 𝜓(𝑦)”, while the sentence “𝜓 is strictly increasing” means “𝑥 < 𝑦 ⇒ 𝜓(𝑥) < 𝜓(𝑦)”; the term “(strictly) decreasing” is interpreted in a simi­ lar way.

24 | 1 The Poisson integral and Hardy spaces Nontangential limits Theorem 1.21. Let 0 < 𝑝 ≤ ∞ and 𝑓 ∈ 𝐻𝑝 . Then 𝑓 has nontangential limits almost everywhere on 𝕋. Proof. Let 𝑓 = 𝑔 + ℎ, where 𝑔 and ℎ as in Lemma 1.1. Then the functions 𝑔1 = 𝑔𝑝/2 and ℎ1 = ℎ𝑝/2 a well defined and belong to 𝐻2 . Hence, by Theorem 1.8, they have 2/𝑝 2/𝑝 nontangential limits a.e. Since 𝑓 = 𝑔1 + ℎ1 we see that the same holds for 𝑓.

1.5.1 Radial limits and mean convergence Since 𝐻1 ⊂ ℎ1 , we see from the Riesz–Herglotz theorem and Fatou’s theorem that every function 𝑓 ∈ 𝐻1 has radial limits almost everywhere. However, the hypothesis that 𝑓 is analytic improves the properties of the boundary function substantially (see, e.g. Theorems 1.22 and 1.29). Theorem 1.22 (F. Riesz). Let 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 ≤ ∞. Then almost everywhere on 𝕋 there exist radial (and, moreover, nontangential, by Theorem 1.21) limits 𝑓∗ (𝑒𝑖𝜃 ) = lim− 𝑓(𝑟𝑒𝑖𝜃 ) , and the following relations hold: 𝑟→1

‖𝑓‖𝑝 = ‖𝑓∗ ‖𝑝

(𝑝 ≤ ∞),

(1.32)

𝜋

lim − ∫ |𝑓(𝑟𝑒𝑖𝜃 ) − 𝑓∗ (𝑒𝑖𝜃 )|𝑝 𝑑𝜃 = 0

𝑟→1−

(𝑝 < ∞).

(1.33)

−𝜋

Proof. When 𝑝 > 1, we may appeal to Theorem 1.10. Let 𝑓 ∈ 𝐻𝑝 and 1/2 < 𝑝 ≤ 1. Observe that (1.32) is implied by (1.33). Let 𝑝 > 1/2 and 𝑓𝑟 (𝑒𝑖𝜃 ) = 𝑓(𝑟𝑒𝑖𝜃 ). Then ‖𝑓𝑟 − 𝑓∗ ‖𝑝 = ‖(𝑔𝑟 − 𝑔∗ )(𝑔𝑟 + 𝑔∗ )‖𝑝 ≤ ‖𝑔𝑟 − 𝑔∗ ‖2𝑝 ‖𝑔𝑟 + 𝑔∗ ‖2𝑝 , where we have applied the Cauchy–Schwarz inequality. Since ‖𝑔𝑟 − 𝑔∗ ‖2𝑝 tends to 0 and ‖𝑔𝑟 + 𝑔∗ ‖2𝑝 is bounded, we can conclude that (1.33) holds for 𝑝 > 1/2. In the same way we reduce the case 𝑝 > 1/4 to the case 𝑝 > 1/2, etc. The set of all harmonic polynomials is not dense in ℎ𝑝 for 𝑝 ≤ 1. However: Theorem 1.23. If 0 < 𝑝 < ∞, then P is dense in 𝐻𝑝 ; in other words, 𝐻𝑝 is minimal for 𝑝 < ∞. Proof. It follows from (1.32) and (1.33) that ‖𝑓 − 𝑓𝜌 ‖𝑝 → 0 as 𝜌 → 1− . From this and Section 1.2 we obtain the result.

1.5 Hardy spaces: basic properties

| 25

Some simple inequalities Inequality (1.23) can be improved in the new situation. Lemma 1.3. If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 ≤ ∞, then |𝑓(𝑧)| ≤ (1 − |𝑧|2 )−1/𝑝 ‖𝑓‖𝑝 ,

(1.34)

and lim𝑟→1− 𝑀∞ (𝑟, 𝑓)(1 − 𝑟2 )1/𝑝 = 0. Corollary 1.10. If 𝑓 ∈ 𝐻𝑝 , then 𝑀𝑞 (𝑟, 𝑓) ≤ (1 − 𝑟2 )1/𝑞−1/𝑝 ‖𝑓‖𝑝

(1.35)

(𝑞 > 𝑝).

Proof. This follows from (1.34) and 𝜋

𝑞−𝑝

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

−𝜋

𝑀𝑝𝑝 (𝑟, 𝑓).

Taking 𝑞 = 1 > 𝑝 and 𝑟 = 1 − (1/(𝑛 + 1)) in (1.35), we get, via the inequality 𝑀1 (𝑟, 𝑓) ≥ 𝑛 ̂ |𝑓(𝑛)|𝑟 , one of many results of Hardy and Littlewood [191] (compare (1.24) and Prob­ lem 1.1): Theorem 1.24 (H–L (1/𝑝 − 1)-inequality). If 𝑓 ∈ 𝐻𝑝 (0 < 𝑝 < 1), then ̂ |𝑓(𝑛)| ≤ 𝐶𝑝 ‖𝑓‖𝑝 (𝑛 + 1)1/𝑝−1 , ̂ = 𝑜(𝑛1/𝑝−1 ), 𝑛 → ∞. and 𝑓(𝑛) 𝑝

𝑝 − |𝑓| 𝑑𝑙. Proof of Lemma 1.3. Assume that 𝑓 ∈ 𝐻(𝔻). Then, for 𝑝 < ⬦, we have ‖𝑓‖𝑝 = ∫ By the substitution 𝜁 = 𝜎𝑧 (𝜉) we get

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

𝕋

The function 𝑤 󳨃→ 𝑓(𝜎𝑧 (𝑤))(𝜎𝑧 )󸀠 (𝑤)1/𝑝 is analytic and therefore, by Lemma 1.2, ‖𝑓‖𝑝𝑝 ≥ |𝑓(𝜑(0))(𝜎𝑧 )󸀠 (0)1/𝑝 |𝑝 = |𝑓(𝑧)|𝑝 (1 − |𝑧|2 ), which proves the inequality. The rest holds because 𝐻𝑝 is minimal. Exercise 1.10. For a fixed 𝑧 ∈ 𝔻, equality occurs in (1.34) if and only if 1/𝑝

𝑓(𝑤) = 𝑐 ( where 𝑐 is a constant.

1 − |𝑧|2 ) ̄ 2 (1 − 𝑧𝑤)

,

26 | 1 The Poisson integral and Hardy spaces Exercise 1.11. If 𝑓 ∈ 𝐻𝑝 and 𝑝 ≥ 1, then 𝑀𝑝 (𝑟, 𝑓󸀠 ) ≤ (1 − 𝑟2 )−1 ‖𝑓‖𝑝 . For the case 𝑝 < 1, see Exercise B.6. Exercise 1.12. Considering the functions (1 − 𝑧)−𝛾 , 𝛾 > 0, one can prove that the ex­ ponent 1/𝑝 − 1 in Theorem 1.24 is optimal, i.e. that cannot be replaced by a smaller one.

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 ∈ 𝐻𝑝 [416, 482] but the proof is rather subtle. Theorem 1.25 (F. Riesz–Szegö). Let 𝑓 ≢ 0 belong to 𝐻𝑝 (𝑝 > 0). Then log |𝑓∗ | ∈ 𝐿1 (𝕋) and we have 𝜋 𝑖𝜃

∫ 𝑃(𝑟, 𝜃 − 𝑡) log |𝑓∗ (𝑒𝑖𝑡 )| 𝑑𝑡, log |𝑓(𝑟𝑒 )| ≤ −

(1.36)

0 ≤ 𝑟 < 1.

−𝜋

Before the proof we note an important consequence. Theorem 1.26 (Riesz–Szegö uniqueness theorem). If 𝑓 ∈ 𝐻𝑝 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 [406]. 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 [537, 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 [95, p. 12]. Proof of Theorem 1.25. Assume that |𝑓(0)| ≠ 0. Then, by the subharmonicity of 𝑓, 2𝜋

log |𝑓(0)| ≤ ∫ − log |𝑓(𝑟𝑒𝑖𝜃 )| 𝑑𝜃,

0 < 𝑟 < 1.

0

From the inequality 𝑝

log |𝑓(𝜌𝑒𝑖𝜃 )| ≤ (𝑀rad 𝑓(𝑒𝑖𝜃 )) /𝑝,

where 𝑀rad 𝑓(𝑒𝑖𝜃 ) = sup |𝑓(𝑟𝑒𝑖𝜃 )|, 0 1 this theorem is contained in Corollary 1.7, while in the general case it is a consequence of (1.36), or of the weaker inequality 𝜋 𝑖𝜃 𝑝

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

(0 ≤ 𝑟 < 1)

(1.37)

−𝜋

and (1.32). It is worthwhile to note that (1.37) can be deduced immediately from (1.33) and the inequality |𝑓(𝜌𝑧)|𝑝 ≤ P[ |𝑓𝜌 |𝑝 ](𝑧). In fact, inequality (1.36), together with Jensen’s inequality for the function 𝑥 󳨃→ 𝑒𝑥 , implies a more general fact, namely: Theorem 1.28 (Smirnov). If 𝑓 ∈ 𝐻𝑝 and 𝑓∗ ∈ 𝐿𝑞 for some 𝑞 > 𝑝, then 𝑓 ∈ 𝐻𝑞 .

1.5.2 Space 𝐻1 The results of this section, due to F. Riesz, M. Riesz, Privalov, and Smirnov, show how much 𝐻1 differs from ℎ1 . A function belonging to ℎ1 need not be equal to the Poisson integral of the bound­ ary function. However, as was proved by F. and M. Riesz: Theorem 1.29 (Riesz brothers). If 𝑓 ∈ 𝐻1 , then 𝑓∗ ∈ 𝐿1 and 𝑓 = P[𝑓∗ ]. This is easily deduced from the relation 𝑓(𝑟𝑧) = P[𝑓𝑟 ](𝑧) (𝑟 < 1), by means of (1.33). Exercise 1.13 (Cauchy integral formula). If 𝑓 ∈ 𝐻1 , then 𝑓(𝑧) =

𝑓 (𝜁) 1 ∫ ∗ 𝑑𝜁 2𝜋𝑖 𝜁 − 𝑧

(𝑧 ∈ 𝔻),

𝕋

and moreover

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

(𝑧 ∈ 𝔻, 𝑛 ≥ 1).

28 | 1 The Poisson integral and Hardy spaces 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 ∫ 𝜁𝑛 𝑑𝜇(𝜁) = 0 𝕋

for every 𝑛 = 1, 2, . . . . Then 𝜇 is absolutely continuous. ̂ ̂ for every 𝑘 ∈ ℤ (Proposition 1.1). = 𝜇(𝑘) Proof. Let 𝑓 = P[𝜇]. Then 𝑓 ∈ ℎ1 and 𝑓(𝑘) Therefore the condition of the theorem implies that 𝑓 is analytic. Thus 𝑓 ∈ 𝐻1 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 (Riesz brothers). If 𝑓 ∈ 𝐻1 and if the boundary function is almost ev­ erywhere 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 (1.11) and (1.13) ). Then, by the Riesz–Herglotz theorem, 𝑔 ∈ 𝐻1 and, by Theorem 1.29, 𝑔 = P[𝑔∗ ] and therefore 𝑔 = 𝑃𝑆[𝐺], where 𝜃

𝐺(𝜃) = ∫ 𝑔∗ (𝑒𝑖𝑡 )

(𝜃 ∈ ℝ).

0

Applying (1.13) again, with the obvious change of notation, and taking into account that the function 𝐺 is 2𝜋-periodic because 𝑔(0) = 0, we get 𝜋

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

It follows that 𝜋 𝑖𝜃

𝑓(𝑟𝑒 ) = 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 .

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

1.6 Riesz projection theorem |

29

In a similar way one proves the following [406, 407]: Theorem 1.32 (Privalov). The derivative of a function 𝑓 ∈ 𝐻(𝔻) belongs to 𝐻1 if and only if 𝑓 has absolutely continuous extension to 𝔻. The boundary function of the function (𝜕/𝜕𝜃)𝑓(𝑟𝑒𝑖𝜃 ) = 𝑖𝑟𝑒𝑖𝜃 𝑓󸀠 (𝑟𝑒𝑖𝜃 ), if 𝑓󸀠 ∈ 𝐻1 , is equal to (𝑑/𝑑𝜃)𝑓∗ (𝑒𝑖𝜃 ).

1.6 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 pro­ jection from ℎ2 onto 𝐻2 . This fact was generalized by M. Riesz [421]: Theorem 1.33 (Riesz projection theorem). If 1 < 𝑝 < ⬦, then 𝑅+ acts as a bounded projection from ℎ𝑝 onto 𝐻𝑝 . Remark 1.3. Hollenbeck and Verbitsky [218] proved that the best constant in the in­ equality ‖𝑅+ 𝑢‖𝑝 ≤ 𝐶𝑝 ‖𝑢‖𝑝 is given by 𝐶𝑝 = 1/ sin(𝜋/𝑝). From the projection theorem and Theorem 1.10 it follows that for every 𝜙 ∈ 𝐿𝑝 (𝕋) ̂ for 𝑛 ≥ 0 ̂ (1 < 𝑝 < ⬦) there exists a unique function 𝜓 ∈ 𝐿𝑝 (𝕋) such that 𝜓(𝑛) = 𝜙(𝑛) 𝑝 ̂ and 𝜓(𝑛) = 0 for 𝑛 < 0. This enables us to treat 𝑅+ as an operator from 𝐿 (𝕋) to 𝐻𝑝 (𝕋), 1 < 𝑝 < ⬦, where ̂ = 0 for 𝑛 < 0} 𝐻𝑝 (𝕋) = {𝜙 ∈ 𝐿𝑝 (𝕋) : 𝜙(𝑛)

(𝑝 ≥ 1).

(1.38)

However, the Riesz projection does not map 𝐿1 into 𝐻1 (𝕋) (see Remark 1.1); further­ more, 𝐻1 is not complemented in 𝐿1 , i.e. there is no bounded projection from 𝐿1 to 𝐻1 (see [433, Example 5.19], where it is shown that 𝐴(𝔻) is not complemented in 𝐶(𝕋))⁴. This was first proved by Newman [341] and later generalized by Rudin [429]. 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 .

4 If every subspace of a Banach space 𝑋 is complemented in 𝑋, then 𝑋 is isomorphic to a Hilbert space [296].

30 | 1 The Poisson integral and Hardy spaces Remark 1.4. It is a result of Pichorides [395] that the best possible constant in the in­ equality ‖𝑢‖̃ 𝑝 ≤ 𝐶𝑝 ‖𝑢‖𝑝 , with 𝑢 real valued, is {tan(𝜋/2𝑝), 𝐶𝑝 = { cot(𝜋/2𝑝), {

if 1 < 𝑝 ≤ 2, if 𝑝 ≥ 2.

In view of the connection between conjugate functions and the Hilbert operator (Pri­ valov–Plessner theorem (Theorem 1.17)), we have 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‖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 [466], based on the Hardy–Stein identities. As a consequence of Green’s formula and the identity Δ(𝑢𝑝 ) = 𝑝(𝑝 − 1)𝑢𝑝−2 |∇𝑢|2

(𝑢 > 0)

we have Lemma 1.4. Let 𝑢 > 0 belong to ℎ𝑝 , 1 < 𝑝 < ⬦. Then ‖𝑢‖𝑝𝑝 = |𝑢(0)|𝑝 +

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

Proof of Theorem 1.34. As mentioned above, it is sufficient to consider the case 1 < 𝑝 < 2. We may assume that 𝑢 is real valued and positive (see Exercise 1.5). Let 𝑢 ∈ ℎ(𝔻) and let 𝑓 ∈ 𝐻(𝔻) be such that 𝑢 = Re 𝑓. The function |𝑓|𝑝 is of class 𝐶2 (𝔻) because 𝑓 has no zeros in 𝔻 so we can apply Green’s theorem to get ‖𝑓‖𝑝𝑝 = |𝑓(0)|𝑝 +

𝑝2 1 ∫ − |𝑓|𝑝−2 |𝑓󸀠 (𝑧)|2 log 𝑑𝐴(𝑧). 2 |𝑧| 𝔻

From this, Lemma 1.4, and the inequality 𝑢𝑝−2 ≥ |𝑓|𝑝−2 , it follows that ‖𝑓‖𝑝𝑝 − |𝑓(0)|𝑝 ≤

𝑝 (‖𝑢‖𝑝𝑝 − |𝑢(0)|𝑝 ). 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. Remark 1.5. We have proved above 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 let 𝜌 → 1. This approach will be used very often and will be explained only when necessary.

1.6 Riesz projection theorem |

31

Remark 1.6. If 𝑓 is a conformal mapping of the disk 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 1.1. 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: 𝑛

𝑖𝑘𝜃 ̂ , where 𝑚 and 𝑛 are Theorem 1.36. Let 𝜙 ∈ 𝐿𝑝 (𝕋), 1 < 𝑝 < ⬦, and 𝜙𝑚,𝑛 (𝑒𝑖𝜃 ) = ∑ 𝜙(𝑘)𝑒 𝑘=𝑚 integers, 𝑚 < 𝑛. Then

(1.39)

‖𝜙𝑚,𝑛 ‖𝑝 ≤ 𝐶𝑝 ‖𝜙‖𝑝 ‖𝜙 − 𝜙𝑚,𝑛 ‖𝑝 → 0 as

𝑛 → ∞, 𝑚 → −∞.

(1.40)

Proof. Let 𝑒𝑘 (𝑒𝑖𝜃 ) = 𝑒𝑖𝑘𝜃 . Then 𝜙𝑚,𝑛 = 𝑒𝑚 𝑅+ (𝑒−𝑚 𝜙) − 𝑒𝑛+1 𝑅+ (𝑒−𝑛−1 𝜙). From this and The­ orem 1.33 we obtain (1.39), and from (1.39) and the Weierstrass approximation theorem we obtain (1.40). Now we can determine the dual of 𝐻𝑝 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 (𝐻𝑝 )󸀠 ≃ 𝐻𝑝 under the duality pairing ∞

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

(1.41)

𝑛=0

𝕋

the series being convergent in the ordinary sense. 1.5 (Convention). We write 𝑋 ≃ 𝑌, where 𝑋 and 𝑌 are quasi-Banach spaces, to denote that 𝑋 and 𝑌 coincide as sets and that their quasinorms are equivalent. If 𝑋 = 𝑌 as sets and ‖𝑓‖𝑋 = ‖𝑓‖𝑌 for all 𝑓, then we write 𝑋 ≅ 𝑌. Exercise 1.14 (Isomorphism 𝐿𝑝 with 𝐻𝑝 [71]). If 1 < 𝑝 < ⬦, then the formula ∞

∞

𝑛=0

𝑛=1

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

defines an isomorphism of ℎ𝑝 onto 𝐻𝑝 .

5 However, there are spaces, e.g. Bergman, in which this system is not a basis although the analog of the projection theorem holds. See Note 5.1.

32 | 1 The Poisson integral and Hardy spaces 󸀠

Exercise 1.15 (Parseval’s formula). If 𝑓 ∈ 𝐿𝑝 (𝕋) and 𝑔 ∈ 𝐿𝑝 (𝕋), and 1 < 𝑝 < ⬦, then ̂ ̂ 𝑟|𝑛| 𝑒𝑖𝑛𝜃 converges uniformly in 𝔻, and the Parseval’s formula the series ∑∞ 𝑛=−∞ 𝑓(𝑛)𝑔(𝑛) holds: 𝜋

∞

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

−𝜋

Kolmogorov–Smirnov theorem The Riesz projection theorem does not hold for 𝑝 = 1. Instead, the following weaker version holds [269, 451]. Theorem 1.38 (Kolmogorov–Smirnov). If 𝑓 ∈ 𝐻(𝔻) and Re 𝑓 ∈ ℎ1 , then 𝑓 ∈ 𝐻𝑝 for all 0 < 𝑝 < 1. Proof. Assume, as in the proof of Theorem 1.34, that 𝑢 > 0 and 𝑢 ∈ ℎ(𝔻), and addi­ tionally that 𝑓(0) = 𝑢(0). Then Lemma 1.4 is still valid so we have 𝑀𝑝𝑝 (1, 𝑢) = 𝑢(0)𝑝 −

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

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

𝔻

and hence

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

i.e. ‖𝑓‖𝑝𝑝 = |𝑓(0)|𝑝 +

𝑝2 𝑝 1 1 𝑝 ∫ − |𝑓|𝑝−2 |𝑓󸀠 |2 log 𝑑𝐴 ≤ 𝑢(0)𝑝 + ‖𝑢‖𝑝 = ‖𝑢‖ , 2 |𝑧| 1−𝑝 𝑝 1−𝑝 1 𝔻

as desired (see Remark 1.5).

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 func­ ̃ tion 𝑓 = P[𝑔] + 𝑖𝑃[𝑔] belongs to 𝐻𝑝 for 𝑝 < 1. Now Smirnov’s theorem (Theorem 1.28) 1 tells us that 𝑓 ∈ 𝐻 , 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[𝑔], and the result follows.

1.6 Riesz projection theorem | 33

It follows from the Theorem 1.38 and the complex maximal theorem (Theo­ rem B.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 [269]. Theorem 1.40 (Kolmogorov). If 𝑓 ∈ 𝐻(𝔻) and Re 𝑓 ∈ ℎ1 , then |{𝜁 ∈ 𝕋 : 𝑀rad 𝑓(𝜁) > 𝜆}| ≤

𝐶 ‖ Re 𝑓‖1 , 𝜆

𝜆 > 0,

where 𝐶 is an absolute constant. Equivalently, the operator 𝑢 󳨃→ 𝑀rad 𝑢̃ acts as a contin­ uous operator from ℎ1 into the weak 𝐿1 -space 𝐿1,⋆ (𝕋). For the definition and properties of weak 𝐿𝑝 -spaces, see Section B.2. Remark 1.7. For the best constant in the inequality ‖𝑢̃ ‖1,⋆ ≤ 𝐶‖𝑢‖1 , see [39, 104]. Proof. Let {𝑟𝑛 } be the sequence of all rational numbers in (0, 1), and let 𝑇𝑛 𝑢(𝑒𝑖𝜃 ) = ̃ 𝑖𝜃 ) < ∞ for almost all 𝜃. Now ̃ 𝑛 𝑒𝑖𝜃 ). By Corollary 1.5, we have 𝑇max 𝑢(𝑒𝑖𝜃 ) = 𝑀rad 𝑢(𝑒 𝑢(𝑟 Banach’s principle (Theorem B.19) tells us that 𝑇max maps ℎ1 into L0 (𝕋), continuously. (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 (Theorem B.17) to conclude the proof. Corollary 1.11. If 𝑓 ∈ 𝐻1 , then ‖𝑠𝑛 𝑓‖1,⋆ ≤ 𝐶‖𝑓‖1 , for 𝑛 ≥ 1, where 𝐶 is an absolute constant and 𝑛 𝑘 ̂ 𝑠𝑛𝑓(𝑧) = ∑ 𝑓(𝑘)𝑧 . (1.42) 𝑘=0 1

Consequently, if 𝑔 ∈ 𝐻 (𝕋), then the Fourier series of 𝑔 converges in measure to 𝑔. 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 𝐻1 (𝕋), this implies that ‖𝑠𝑛 𝑔−𝑔‖1,⋆ → 0, as 𝑛 → ∞, which implies that 𝑠𝑛 𝑔 tends to 𝑔 in measure. Exercise 1.16 (Zygmund’s theorem [530]). 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.

1.6.1 Aleksandrov’s theorem: 𝐿𝑝 (𝕋) = 𝐻𝑝 (𝕋) + 𝐻𝑝 (𝕋) Relation (1.32) shows that 𝐿𝑝 (𝕋) contains an isometric copy of 𝐻𝑝 (𝑝 > 0); denote this subspace by 𝐻𝑝 (𝕋). Thus 𝐻𝑝 (𝕋) = {𝑓∗ : 𝑓 ∈ 𝐻𝑝 }. If 𝑝 ≥ 1, then 𝐻𝑝 (𝕋) can

34 | 1 The Poisson integral and Hardy spaces be described by (1.38). In the case 𝑝 < 1, this cannot be applied, simply because the Fourier coefficients are not defined; then 𝐻𝑝 (𝕋) is equal to the 𝐿𝑝 -closure of ̂ = 0 for 𝑛 ≥ 1}, T+ = {𝜙 ∈ T : 𝜙(−𝑛) where T is the set of all trigonometric polynomials. Let 𝐻𝑝 (𝕋) = {𝜙 : 𝜙 ∈ 𝐻𝑝 (𝕋)}. One of consequences of the projection theorem is that 𝐿𝑝 (𝕋) = 𝐻𝑝 (𝕋)+𝐻𝑝 (𝕋), 1 < 𝑝 < ⬦. This fact was extended by Aleksandrov [14, 15] to the case 𝑝 < 1. However, in that case, the decomposition is not unique (up to an additive constant) because the intersection 𝐻𝑝 ∩ 𝐻𝑝 is equal to the linear span of the set of the functions 𝑔𝑎 (𝜁) = 1/(1 − 𝑎𝜁) (𝑎 ∈ 𝕋, 𝜁 ∈ 𝕋) (see [14, 15]). Theorem 1.41 (Aleksandrov). If 𝑓 ∈ 𝐿𝑝 (𝕋), 𝑝 < 1, then there are functions 𝑓1 ∈ 𝐻𝑝 (𝕋), 𝑓2 ∈ 𝐻𝑝 (𝕋), such that 𝑓 = 𝑓1 + 𝑓2 and ‖𝑓1 ‖𝑝 + ‖𝑓2 ‖𝑝 ≤ 𝐶𝑝 ‖𝑓‖𝑝 . Proof. Let 𝑋 denote the direct sum of the spaces 𝐻𝑝 and 𝐻𝑝 . Consider the operator 𝑇 : 𝑋 󳨃→ 𝐿𝑝 , 𝑇(𝑓1 , 𝑓2 ) = 𝑓1 + 𝑓2 . For every trigonometric polynomial 𝑓, ‖𝑓‖𝑝 ≤ 1, we will find (𝑓1 , 𝑓2 ) so that 𝑓 = 𝑇(𝑓1 , 𝑓2 ), where ‖(𝑓1 , 𝑓2 )‖ ≤ 𝐶𝑝 (and 𝐶𝑝 depends only on 𝑝) and then the result will follow from Theorem A.2. Let 𝑓 = ∑|𝑘|≤𝑛 𝑎𝑘 𝑒𝑖𝑘𝜃 and 𝛾(𝜁) = 𝜁𝑛 . Then 𝛾𝑓 and 𝛾𝑓 ̄ belong to 𝐻𝑝 . Put 𝜑 = 𝛾/(𝛾 − 1). ̄ . Then 𝜑 ∈ 𝐻𝑝 ∩ 𝐻𝑝 and 𝜑 + 𝜑̄ = 1. Now let 𝜑𝑡 (𝜁) = 𝜑(𝜁𝑒𝑖𝑡 ), 𝑡 ∈ ℝ, and 𝑔𝑡 = 𝑓𝜑𝑡 , ℎ𝑡 = 𝑓𝜑 𝑡 𝑝 ̄ Then 𝑔𝑡 and ℎ𝑡 are 𝐻 and 𝑓 = 𝑔𝑡 + ℎ𝑡 . 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 𝑝 𝑝 𝑝 𝑝 ‖ℎ𝑡 ‖𝑝 ≤ ‖𝑓‖𝑝 + ‖𝑔𝑡 ‖𝑝 ≤ 1 + ‖𝜑‖𝑝 . Finally, we take 𝑓1 = 𝑔𝑡 , 𝑓2 = ℎ̄ 𝑡 , and this completes the proof. As an application of Aleksandrov’s theorem, we have the following result of Shapiro [438]. 𝑝

Corollary 1.12. Let ℎP (0 < 𝑝 < 1) denote the closure in ℎ𝑝 of the set of all harmonic 𝑝 polynomials. Then ℎP /𝑜ℎ𝑝 ≃ 𝐿𝑝 (𝕋). 𝑝

𝑝

Proof. If 𝑢 ∈ ℎP , then there exists 𝑢 ∈ ℎP such that ‖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 𝑝 𝐿𝑝 (𝕋). We define the operator 𝑆 : ℎP 󳨃→ 𝐿𝑝 (𝕋) by 𝑆𝑢 = 𝑔. We have ‖𝑆𝑢‖𝐿𝑝 ≤ ‖𝑢‖ℎ𝑝 , by 𝑝 Fatou’s lemma, and the kernel of 𝑆 is 𝑜ℎ𝑝 . Since obviously 𝑆(ℎP ) ⊃ 𝐻𝑝 ∪𝐻𝑝 = 𝐿𝑝 (𝕋), we 𝑝 find, by means of the open mapping theorem, that ℎP /𝑜ℎ𝑝 is isomorphic to 𝐿𝑝 (𝕋). Exercise. Aleksandrov’s theorem does not hold for 𝑝 = 1.

1.6 Riesz projection theorem | 35

Further notes and results A lot of information and references (529) before 1985 can be found in the survey paper of Schvedenko [446]. The proof of Theorem 1.11 was given by Fefferman and Stein [149], and 2 years later by Kuran [280]. Kuran’s proof is very complicated and cannot be used in the subharmonic context. Fefferman and Stein’s proof can be found in Koosis [273] and Garnett [159]. As far as the author knows, Hardy and Littlewood nowhere formulated Theorem 1.11. For slightly more general results we refer to [366, 368]. Theorem 1.12 is formulated, without proof, in [190, Theorem 5]. In the same paper (Theorem 1) Theorem 1.14 was proved. Beside Theorem 1.16, Shapiro in [438] proved that 𝑜ℎ𝑝 contains an isomorphic copy of c0 = ℓ⬦ , the space of sequences tending to zero. Theorem 1.22, as well as some other fundamental theorems on Hardy spaces, is due to Riesz [416]; he introduced the term “Hardy spaces”. Theorems 1.29–1.31, due to Riesz brothers, were proved in their amazing pa­ per [420]. “This is a paper every analyst should read!” writes Koosis [273, Ch. II, Sec. A] – 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 𝑑𝜇(𝜁) (𝐾𝜇)(𝑧) = ∫ − , 𝑧 ∈ 𝔻. 1 − 𝜁𝑧̄ 𝕋

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–Ross [94, 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 , resp. 𝑑𝜇𝑠 , is the absolutely continuous, resp. singular, part of 𝑑𝜇; moreover, ‖𝜇‖ = ‖𝜇𝑎 ‖ + ‖𝜇𝑠 ‖. This implies that K = K𝑎 ⊕ K𝑠 , where K𝑎 , resp. K𝑠 , is defined via the absolutely, resp. singular, part of 𝑑𝜇. It is interesting that if 𝑑𝜈 is singular, then ‖𝐾𝜈‖K = ‖𝜈‖. The space K𝑎 is separable (K𝑠 is not) and is equal KP [94, Proposition 4.1.21]. Besides, the dual of K𝑎 can be identified with 𝐻∞ [94, Theorem 4.1.22]. The theory of the Cauchy integral is so beautiful and deep that anyone should read at least the expository paper [93] of the aforementioned authors.

36 | 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: lim𝑟→1 sup𝛿 −1, denote the (Dirichlet) space of functions 𝑓 ∈ 𝐻(𝔻) such that 1/𝑝 󸀠

𝑝

2 𝛼

(∫ |𝑓 (𝑧)| (1 − |𝑧| ) 𝑑𝐴(𝑧))

< ∞,

𝔻 𝑝

normed in one of obvious ways. We shall prove later on that D𝑝−1 ⊂ 𝐻𝑝 (0 < 𝑝 < 2) (see Theorem 7.14). Moreover, the inclusion is proper, which is proved by using lacu­ nary series, and therefore the following result of Kalton [250, Theorem 8.1] improves Theorem 1.41. 𝑝 𝑝 If 𝑓 ∈ 𝐿𝑝 (𝕋), 𝑝 < 1, then there are functions 𝑔 ∈ D𝑝−1 , ℎ ∈ D𝑝−1 , such that 𝑓 = 𝑔∗ + ℎ∗ and ‖𝑔‖D𝑝 + ‖ℎ‖D𝑝 ≤ 𝐶𝑝 ‖𝑓‖𝑝 . 𝑝−1 𝑝−1 The proof is omitted because it is based on the integration theory for functions with values in a quasi-Banach space, a complicated and extensive theme (see [250] and references therein). 1.9 (Boundary decay of harmonic functions). In [16], Aleksandrov 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).

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

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 exis­ tence 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. We also mention that if 𝐾 ⊂ 𝐷 is a compact set, then 𝑢 attains its maximum, 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. Here is a list of well-known properties of subharmonic functions: – In the case of 𝐶2 -functions there is a simple criterion of subharmonicity deduced from Green’s formula: A function 𝑢 ∈ 𝐶2 (𝐷) is subharmonic if and only if Δ𝑢 ≥ 0 in 𝐷.

2.1 Basic properties of subharmonic functions |

–

–

– –

–

–

41

Let 𝑢 ≢ −∞ be subharmonic in a domain 𝐷. Then there exists an increasing se­ quence of open sets 𝐷𝑛 , whose union is 𝐷, and a decreasing sequence of subhar­ monic functions 𝑢𝑛 ∈ 𝐶∞ (𝐷𝑛 ) tending to 𝑢 (see the proof of Theorem 2.5). From the formula Δ(𝑢 ∘ 𝜑)(𝑧) = (Δ𝑢)(𝜑(𝑧)) |𝜑󸀠 (𝑧)|2 , where 𝜑 is an analytic function and 𝑢 is 𝐶2 , we get: The composition 𝑢 ∘ 𝜑 is subharmonic if 𝑢 is subharmonic and 𝜑 is analytic. In a general case this assertion can be reduced to the “smooth” case by approximating an arbitrary subharmonic function by smooth ones. Every subharmonic function is locally integrable. An important example of a subharmonic function taking the value −∞ is the func­ tion 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 sub­ harmonic functions. The same holds for the limit of a decreasing sequence of sub­ harmonic functions. Let 𝜙 be an increasing convex function that is defined and continuous on an inter­ val 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 subhar­ monic 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 a segment) be a continuous function such that 𝐹(𝑧, 𝑡) is log-subharmonic in 𝑧 ∈ 𝔻 for 𝑏 every 𝑡 ∈ [𝑎, 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 is enough to prove that it is a uniform 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 is enough to prove that a finite sum of log-subharmonic functions is log-subharmonic, or what is the same, 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 and increasing in each variable separately (see [219, 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 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 +∞.

42 | 2 Subharmonic functions and Hardy spaces A function 𝑢 is called superharmonic if −𝑢 is subharmonic. Exercise 2.1. Let ℎ ≢ 0 be a real-valued harmonic function and 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.

2.1.1 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. Let us prove that 𝑀 is open as well, which will imply 𝑀 = 𝐷 or 𝑀 = 0, so the proof will be fin­ ished. 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 (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 2.1. If 𝑢 is upper semicontinuous on 𝐷 and subharmonic in 𝐷, then max{𝑢(𝑧) : 𝑧 ∈ 𝐷} = max{𝑢(𝜁) : 𝜁 ∈ 𝜕𝐷}. Corollary 2.2. 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 con­ tinuous on 𝐷. If 𝑢 ≤ ℎ on 𝜕𝐷, then 𝑢 ≤ ℎ in 𝐷; besides, 𝑢 < ℎ if 𝑢 is not harmonic.

2.2 Properties of the mean values Convexity and monotonicity 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 the inequality 𝜑(𝑟11−𝜆 𝑟2𝜆 ) ≤ (1 − 𝜆)𝜑(𝑟1 ) + 𝜆𝜑(𝑟2 ),

0 < 𝜆 < 1,

holds. If 𝜑 is of class 𝐶2 , then it is convex of log 𝑟 if and only if 𝜑󸀠󸀠 (𝑟) + 𝜑󸀠 (𝑟)/𝑟 ≥ 0.

2.2 Properties of the mean values | 43

Theorem 2.2. Let 𝑢 be subharmonic in the disk |𝑧| < 𝑅. Then the function 𝜋

∫ 𝑢(𝑟𝑒𝑖𝜃 ) 𝑑𝜃 𝐼(𝑟, 𝑢) := −

(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

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

For fixed 0 < 𝑟1 < 𝑟2 define the harmonic function ℎ(𝑧) = 𝑎 log |𝑧| + 𝑏 by ℎ(𝑟𝑗 ) = 𝐼(𝑟𝑗 ). 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 it follows that 𝐼(𝑟, 𝑢) is convex of log 𝑟. That 𝐼(𝑟, 𝑢) increases with 𝑟 follows from the fact that the function 𝜑(𝑥) = 𝐼(𝑟, 𝑒𝑥 ) is convex and bounded for −∞ < 𝑥 < log 𝑅, and this completes the proof in case of 𝐼(𝑟, 𝑢). In the 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 a useful fact [327, 449]: Lemma 2.2. If 𝑝 > 0 and 𝑓(𝑧) = ∑𝑛𝑘=𝑚 𝑎𝑘 𝑧𝑘 , then for 0 < 𝑝 ≤ ∞ we have 𝑟𝑛𝑀𝑝 (1, 𝑓) ≤ 𝑀𝑝 (𝑟, 𝑓) ≤ 𝑟𝑚 𝑀𝑝 (1, 𝑓),

0 < 𝑟 < 1.

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

𝑔 = ∑ 𝑎𝑘̄ 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) = 𝑟𝑚 󵄩󵄩󵄩 ∑ 𝑟𝑘−𝑚 𝑎𝑘 𝑒𝑘 󵄩󵄩󵄩 = 𝑟𝑚 󵄩󵄩󵄩 ∑ 𝑎𝑚+𝑘 𝑟𝑘 𝑒𝑘 󵄩󵄩󵄩 ≤ 𝑟𝑚 𝑀𝑝 (1, 𝑓), 󵄩󵄩𝑘=𝑚 󵄩󵄩 󵄩󵄩 𝑘=0 󵄩󵄩 󵄩 󵄩𝑝 󵄩 󵄩𝑝 which completes the proof. Lemma 2.3. If 𝑓 is a polynomial of degree ≤ 𝑛, then ‖𝑓‖𝑞 ≤ 𝐶(𝑛 + 1)1/𝑝−1/𝑞 ‖𝑓‖𝑝 , where 𝐶 depends only on 𝑝 and 𝑞.

0 < 𝑝 < 𝑞 ≤ ∞,

44 | 2 Subharmonic functions and Hardy spaces Proof. This can be deduced from Lemma 2.2 and Corollary 1.10 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}. Theorem 2.3. Let 𝑢 be subharmonic in the annulus 𝜌 < |𝑧| < 𝑅. Then the function 𝜋

∫ 𝑢(𝑟𝑒𝑖𝜃 ) 𝑑𝜃 𝐼(𝑟, 𝑢) = −

(𝜌 < 𝑟 < 𝑅)

−𝜋

is finite and convex of log 𝑟. The same holds for the function 𝐼∞ (𝑟, 𝑢). Proof. Since 𝑢 is locally integrable and 𝑟1

∫ 𝑟0 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. In order to prove that (b) implies (a) we start from the inequality 𝑥 + 𝑥2 ) ≤ 𝑠2 𝜑(𝑥1 ) + 𝑡2 𝜑(𝑥2 ), 2𝑠𝑡 𝜑 ( 1 2 where 𝑠 = 𝑒𝑐𝑥1 /2 𝜑(𝑥1 ), 𝑡 = 𝑒𝑐𝑥2 /2 𝜑(𝑥2 ), which implies that the discriminant of a cer­ tain quadratic form is nonpositive. Also an interesting proof can be given in the case where 𝜑 is 𝐶2 . Exercise 2.3. 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 assertion is useful in reducing some proofs for log-subharmonic func­ tions to the case of the moduli of analytic functions. Lemma 2.4. 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. Proof. 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

2.3 Riesz measure Let 𝐶20 (𝐷) be the set of all 𝐶2 -functions with compact support in 𝐷. Theorem 2.5 (Riesz measure theorem). Let 𝑢 ≢ −∞ be a function subharmonic in 𝐷𝑅 = 𝑅𝔻. Then there exists a unique positive measure 𝜇 on 𝐷𝑅 such that ∫ 𝜑𝑑𝜇 = ∫ 𝑢Δ𝜑 𝑑𝜇 𝐷𝑅

𝐷𝑅

for all 𝜑 ∈ 𝐶20 (𝐷𝑅 ). The measure 𝜇 = 𝜇𝑢 is called the Riesz measure of 𝑢. If 𝑢 ∈ 𝐶2 , then 𝑑𝜇𝑢 = Δ𝑢 𝑑𝐴. A sketch of the proof. Let 0 < 𝑟 < 𝑅, and define the functional Λ on 𝐶20 (𝐷𝑟 ) by Λ𝜑 = ∫ 𝑢Δ𝜑 𝑑𝐴. 𝐷𝑟

(2.3)

46 | 2 Subharmonic functions and Hardy spaces Let 𝜔(𝑧) = 𝜔(|𝑧|) be a radial function with compact support in 𝐷𝑅 , such that ∫𝐷 𝜔(𝑤) 𝑑𝐴(𝑤) = 1, and define the “sequence” of functions 𝑢𝜀 (0 < 𝜀 < 𝑅 − 𝑟) by 𝑅

𝑢𝜀 (𝑧) = ∫ 𝜔(𝑤)𝑢(𝑧 + 𝜀𝑤) 𝑑𝐴(𝑤) = 𝜀−2 ∫ 𝜔((𝑧 − 𝑤)/𝜀)𝑢(𝑤) 𝑑𝐴(𝑤). 𝐷𝑅

𝐷𝑅

These functions are defined in a disk 𝐷𝜌󸀠 , 𝑟 < 𝜌󸀠 < 𝑅, they are of class 𝐶∞ and subhar­ monic in 𝐷𝜌󸀠 , and 𝑢𝜀 ↓ 𝑢 as 𝜀 ↓ 0. From the formula 2𝜋

∫ − 𝑢𝑛 (𝜌𝑒𝑖𝜃 ) 𝑑𝜃 − 𝑢𝑛 (0) = 0

𝜌 1 ∫ Δ𝑢𝑛 (𝑧) log 𝑑𝐴(𝑧), 2𝜋 |𝑧| 𝐷𝜌

where 𝑟 < 𝜌 < 𝜌󸀠 , and 𝑢𝑛 = 𝑢1/𝑛 for 𝑛 ≥ 𝑚, where 𝑚 is fixed and such that 1/𝑚 < 𝑅 − 𝑟, we obtain log

2𝜋

2𝜋

0

0

𝜌 1 ∫ Δ𝑢𝑛 (𝑧) 𝑑𝐴(𝑧) ≤ ∫ − 𝑢𝑛(𝜌𝑒𝑖𝜃 ) 𝑑𝜃 ≤ ∫ − 𝑢𝑚 (𝜌𝑒𝑖𝜃 ) 𝑑𝜃. 𝑟 2𝜋 𝐷𝑟

Hence ∫𝐷 |Δ𝑢𝑛(𝑧)| 𝑑𝐴 ≤ 𝐶, where 𝐶 is a constant independent of 𝑛 and so 𝑟

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ 𝑢𝑛 Δ𝜑 𝑑𝐴󵄨󵄨󵄨 ≤ 𝐶 sup |𝜑(𝑧)|, 󵄨󵄨 󵄨󵄨 |𝑧|≤𝑟 󵄨󵄨𝐷 󵄨󵄨 󵄨 𝑟 󵄨

𝜑 ∈ 𝐶20 (𝐷𝑟 ).

Letting 𝑛 tend to ∞ we obtain |Λ𝜑| ≤ 𝐶‖𝜑‖∞ := 𝐶 sup|𝑧|≤𝑟 |𝜑(𝑧)|. Since 𝐶20 (𝐷𝑟 ) is dense in 𝐶0 (𝐷𝑟 ), we can apply the Riesz–Radon representation theorem to conclude that there exists a unique measure 𝑑𝜇𝑟 on 𝔻𝑟 such that Λ𝜑 = ∫ 𝜑 𝑑𝜇𝑟 ,

𝜑 ∈ 𝐶20 (𝐷𝑟 ).

𝐷𝑟

Since the functional Λ is positive (Λ𝜑 ≥ 0 for 𝜑 ≥ 0) as a limit of positive functionals, we see that the measure 𝜇𝑟 is positive (see [497, p. 31]). Thus to the sequence 𝑟𝑗 = 1−1/𝑗 there corresponds a unique sequence of positive measures 𝜇𝑗 such that 𝜇𝑗+1 = 𝜇𝑗 on 𝐷𝑟𝑗 , and therefore there exists a unique measure 𝜇 satisfying (2.3). The Riesz measure of the function log |𝑧 − 𝑎| is equal to 2𝜋𝛿𝑎 , where 𝛿𝑎 denotes the Dirac measure concentrated at the point 𝑎. For more information on the Riesz measure we refer to Hörmander [219] and Hayman–Kennedy [203].

2.3 Riesz measure |

47

2.3.1 Riesz’ representation formula Theorem 2.6 (Riesz representation theorem). Let 𝑢 be subharmonic in 𝐷𝑅 = {𝑧 : |𝑧| < 𝑅}, and 𝑢(0) > −∞. Then 𝜋

∫ 𝑢(𝑟𝑒𝑖𝜃 ) 𝑑𝜃 − 𝑢(0) = − −𝜋

1 𝑟 ∫ log 𝑑𝜇(𝑧) 2𝜋 |𝑧|

(2.4)

|𝑧| 𝑟. Choose two (bounded) sequences of 𝐶2 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 = ∫ 𝑢𝑛Δ𝑔𝑚 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.

48 | 2 Subharmonic functions and Hardy spaces If 𝐹 is a 𝐶2 -function in a neighborhood 𝔻, then an application of formula (1.2) to the function 𝑓 ∘ 𝜎𝑎 , 𝑎 ∈ 𝔻, leads to formula (1.4). The analogous formula for subharmonic functions reads 1 1 ∫ log 𝑑𝜇 (𝑤), 𝑧 ∈ 𝔻, (2.5) ∫ 𝑢(𝜁)𝑃(𝑧, 𝜁) |𝑑𝜁| − 𝑢(𝑧) = 2𝜋 |𝜎𝑤 (𝑧)| 𝑢 𝔻

𝕋

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, but which are not important in this text). The well-known Jensen’s formula is one of special cases of (2.4). Namely, if a func­ tion 𝑓 is analytic in 𝐷𝑅 and 𝑓(0) ≠ 0, then the Riesz measure of the function 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 (2.4) give Jensen’s formula, 𝜋

∫ log |𝑓(𝑟𝑒𝑖𝜃 )| 𝑑𝜃 = log |𝑓(0)| + ∑ log − |𝑎𝑘 | 𝑝 > 0? In order to prove the next theorem we need a simple but important inequality. Lemma 2.6 (Chebyshev). Let 𝑑𝜇(𝑟) = 𝜉(𝑟) 𝑑𝑟, where 𝜉(𝑟) > 0 for 𝑟 ∈ (0, 1), is a prob­ ability measure on (0, 1), and let 𝜓1 and 𝜓2 be real functions on (0, 1) such that 𝜓1 is

54 | 2 Subharmonic functions and Hardy spaces increasing and 𝜓2 is decreasing, then 1

1

1

∫ 𝜓1 (𝑟)𝜓2 (𝑟) 𝑑𝜇(𝑟) ≤ ∫ 𝜓1 (𝑟) 𝑑𝜇(𝑟) ∫ 𝜓2 (𝑟) 𝑑𝜇(𝑟). 0

0

0

Equality is attained if and only if 𝜓1 = const or 𝜓2 = const. Proof. This inequality is an immediate consequence of the relations (𝜓1 (𝑥) − 𝜓1 (𝑦)) × (𝜓2 (𝑥) − 𝜓2 (𝑦)) ≤ 0 and ∫(𝜓1 (𝑥) − 𝜓1 (𝑦))(𝜓2 (𝑥) − 𝜓2 (𝑦)) 𝑑𝜇(𝑥) 𝑑𝜇(𝑦) ≤ 0, Π

where Π = [0, 1] × [0, 1]. Theorem 2.13 ([330]). Let 𝑓 ∈ 𝐻𝑝 , let 𝑁 ≥ 2 be an integer, 𝑁𝛼 < 2, where 𝛼 ≥ 0. Then ∫ |𝑓(𝑧)|𝑁𝑝 |𝑧|−𝑁𝛼 (1 − |𝑧|2 )𝑁−2 𝑑𝐴(𝑧) ≤ 𝜋B(1 − 𝑏, 𝑁 − 1)‖𝑓‖𝑁𝑝 𝑝 ,

(2.11)

𝔻

where 𝑏 = 𝑁𝛼/2. Equality holds if and only if either 𝑓(𝑧) = 𝑐(1 − 𝑎𝑧)−2/𝑝 and 𝛼 = 0 or 𝑓(𝑧) = const and 𝛼 > 0. Proof. Denoting the integral in (2.11) by 𝐼 we have 1

𝐼 = ∫ 𝑟−2𝑏 𝐽(𝑟)(1 − 𝑟2 )𝑁−2 𝑟 𝑑𝑟, 0 2𝜋

where 𝐽(𝑟) = ∫0 |𝑓(𝑟𝑒𝑖𝜃 )|𝑁𝑝 𝑑𝜃. Applying Chebyshev’s inequality with respect to the probability measure 𝑑𝜇(𝑟) = 2(𝑁 − 1)(1 − 𝑟2 )𝑁−2 𝑟 𝑑𝑟 on [0, 1), we obtain 1

∫𝑟 0

1 −2𝑏

𝐽(𝑟) 𝑑𝜇(𝑟) ≤ (∫ 𝑟

1 −2𝑏

0

𝑑𝜇(𝑟)) (∫ 𝐽(𝑟) 𝑑𝜇(𝑟)) . 0

Hence 𝐼 ≤ (𝑁 − 1)B(1 − 𝑏, 𝑁 − 1) ∫ |𝑓(𝑧)|𝑁𝑝 (1 − |𝑧|2 )𝑁−2 𝑑𝐴(𝑧). 𝔻

Now the inequality (2.11) follows from Theorem 2.12. If 𝛼 = 0 and 𝑓(𝑧) = 𝑐(1 − 𝑎𝑧)−2 , then equality in (2.11) holds because of Theorem 2.12. If 𝛼 > 0 and 𝑓(𝑧) = const, then a direct computation shows that equality holds. Conversely, if equality holds, then 𝐽(𝑥) or 𝜓(𝑥) = 𝑥−2𝑏 is a constant. Thus 𝑓(𝑧) = 𝑐 for some constant 𝑐 or 𝑏 = 0. If 𝑏 = 0, then 𝑓(𝑧) = 𝑐(1 − 𝑎𝑧)−2/𝑝 , by Theorem 2.12. This was to be proved.

2.5 Some sharp inequalities | 55

Theorem 2.14 (Generalized Huber’s inequality). Let 𝑢 = 𝑢1 /𝑢2 , where 𝑢1 and 𝑢2 are functions log-subharmonic in 𝔻 and upper semicontinuous on 𝔻. Let 𝜇 = 𝜇2 be the Riesz measure of log 𝑢2 and 𝛼 := 𝜇(𝔻) < 2/𝑁, 𝑁 = 2, 3, . . . . Then 𝑁 𝑁

2 𝑁−2

∫ 𝑢(𝑧) (1 − |𝑧| )

𝑑𝐴(𝑧) ≤ 𝜋B(1 − 𝑁𝛼/2, 𝑁 − 1) (− ∫ 𝑢(𝜁) |𝑑𝜁|) .

𝔻

(2.12)

𝕋 󸀠

Equality holds if 𝑢(𝑧) = 𝑐|𝜎 (𝑧)||𝜎(𝑧)|

−𝛼

for some 𝜎 ∈ Möb(𝔻) and some constant 𝑐 ≥ 0.

Proof. We have log 𝑢𝑗 (𝑧) = ℎ𝑗 (𝑧) − ∫ log 𝔻

1 𝑑𝜇 (𝜁), |𝜎𝜁 (𝑧)| 𝑗

(2.13)

where ℎ𝑗 is the Poisson integral of log 𝑢𝑗 |𝕋 . Assume that 𝛼 > 0 and let ℎ = ℎ1 − ℎ2 . By (2.13) we have 𝑢(𝑧)𝑁 ≤ exp (𝑁ℎ(𝑧) + ∫ log 𝔻

𝑑𝜇(𝜁) 1 ). 𝑁𝛼 |𝜎𝜁 (𝑧)| 𝛼

(2.14)

̃ Since |𝑓| = 𝑒ℎ we have, by Let ℎ̃ be the harmonic conjugate of ℎ and let 𝑓 = exp(ℎ + 𝑖ℎ). (2.14) and Jensen’s inequality, 𝑢(𝑧)𝑁 ≤ |𝑓(𝑧)|𝑁 ∫ 𝔻

𝑑𝜇(𝜁) 1 , |𝜎𝜁 (𝑧)|𝑁𝛼 𝛼

which, via Fubini’s theorem, implies ∫ 𝑢(𝑧)𝑁 (1 − |𝑧|2 )𝑁−2 𝑑𝐴(𝑧) ≤ ∫ 𝐽(𝜁) 𝔻

𝔻

where 𝐽(𝜁) = ∫ |𝑓(𝑧)|𝑁 (1 − |𝑧|2 )𝑁−2 𝔻

𝑑𝜇(𝜁) , 𝛼

1 𝑑𝐴(𝑧). |𝜎𝜁 (𝑧)|𝑁𝛼

Now we use the substitution 𝑧 = 𝜎𝜁 (𝑤), i.e. 𝑤 = 𝜎𝜁 (𝑧) and the relation (1 − |𝑧|2 ) = (1 − |𝑤|2 )|𝜎𝜁󸀠 (𝑤)|, to obtain 𝐽(𝜁) = ∫ 𝔻

|𝑓(𝜎𝜁 (𝑤)|𝑁 |𝑤|𝑁𝛼

(1 − |𝑤|2 )𝑁−2 |𝜎𝜁󸀠 (𝑤)|𝑁 𝑑𝐴(𝑤).

Let 𝐹(𝑤) = 𝑓(𝜎𝜁 (𝑤))𝜎𝜁󸀠 (𝑤). By Theorem 2.13 we have 𝐽(𝜁) ≤ 𝜋B(1 − 𝑏, 𝑁 − 1)‖𝐹‖𝑁 1 ,

(2.15)

where 𝑏 = 𝑁𝛼/2. Since ∫ − |𝐹(𝑤)| |𝑑𝑤| = ∫ − |𝑓(𝜎𝜁 (𝑤))| |𝜎𝜁󸀠 (𝑤)| |𝑑𝑤| = ∫ − |𝑓(𝑧)| |𝑑𝑧| 𝕋

𝕋

𝕋

56 | 2 Subharmonic functions and Hardy spaces and 𝑓(𝑧) = 𝑢(𝑧) for 𝑧 ∈ 𝕋, the desired inequality, in the case 𝛼 > 0 follows from (2.15). If 𝛼 = 0, then 𝑢(𝑧)𝑁 ≤ |𝑓(𝑧)|𝑁 and the inequality follows from Theorem 2.12. That the function 𝑓(𝑧) = 𝑐|𝜎󸀠 (𝑧)| |𝜎(𝑧)|−𝛼 gives equality in (2.12) can be checked by simple calculation. Remark 2.2. Inequality 2.12 was proved in [330, Theorem 3], but the analysis of the equality case was not correct. Therefore we do not know whether “only if” can be added in the last sentence of Theorem 2.14.

Inequalities of Fejér–Riesz and Hilbert If 𝑔 is a function analytic in 𝔻, then a special 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 inequal­ ity [150]: 1

∫ |𝑓(𝑟)|𝑝 𝑑𝑟 ≤ 𝜋‖𝑓‖𝑝𝑝

(𝑓 ∈ 𝐻𝑝 , 𝑝 > 0).

(2.16)

−1 2𝑛 In particular, if 𝑝 = 2 and 𝑓(𝑧) = ∑∞ 𝑛=0 𝑎𝑛 𝑧 (𝑎𝑛 ≥ 0), then (2.16) yields

∑ 𝑚,𝑛≥0

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

(2.17)

This inequality, known as Hilbert’s inequality, can be deduced immediately from the equality 1

𝜋 2

∫ 𝑓(𝑟) 𝑑𝑟 = 𝑖 ∫ 𝑓(𝑒𝑖𝜃 )2 𝑒𝑖𝜃 𝑑𝜃, −1

0

a consequence of Cauchy’s integral theorem. From (2.17) it follows that ∑ 𝑚,𝑛≥0

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

1/2

∞

( ∑ |𝑏𝑛 |2 )

1/2

.

(2.18)

𝑛=0

Hardy’s inequality From Hilbert’s inequality we can obtain a slightly improved version of Hardy’s in­ equality (1.44). Namely: Theorem 2.15 (Hardy’s inequality). If 𝑓 ∈ 𝐻1 , then ̂ |𝑓(𝑛)| ≤ 𝜋‖𝑓‖1 . 𝑛=0 𝑛 + (1/2) ∞

∑

(2.19)

2.5 Some sharp inequalities |

57

Proof. Let 𝑓 ∈ 𝐻1 and let 𝑓 = 𝐵𝑔 be the Riesz’ factorization of 𝑓. Then the functions ̂ 𝐹 = 𝐵𝑔1/2 and 𝐺 = 𝑔1/2 belong to 𝐻2 and ‖𝑓‖1 = ‖𝐹‖22 = ‖𝐺‖22 . Let 𝑎𝑘 = |𝐹(𝑘)| and ̂ 𝑏𝑘 = |𝐺(𝑘)|. Then we have ∞ 𝑛 ̂ |𝑓(𝑛)| 𝑎𝑚 𝑏𝑛 1 ≤∑ . ∑ 𝑎𝑘 𝑏𝑛−𝑘 = ∑ 𝑛 + (1/2) 𝑛 + (1/2) 𝑚 + 𝑛 + (1/2) 𝑛=0 𝑛=0 𝑚,𝑛≥0 𝑘=0 ∞

∑

Now we use (2.18) to get ̂ |𝑓(𝑛)| ≤ 𝜋‖𝐹‖2 ‖𝐺‖2 = 𝜋‖𝑓‖1 𝑛=0 𝑛 + (1/2) ∞

∑

as desired. Remark 2.3. In the case 𝑝 = 2 the isoperimetric inequality (2.12) can be written as ̂ 2 |𝑓(𝑛)| ≤ ‖𝑓‖21 . 𝑛 + 1 𝑛=0 ∞

(2.20)

∑

It is interesting to compare this inequality with (2.19). In general, convergence of the ̂ ̂ 2 /(𝑛 + 1). series ∑ |𝑓(𝑛)|/(𝑛 + 1), with 𝑓 ∈ 𝐻(𝔻), does not imply convergence of ∑ |𝑓(𝑛)| 1 ̂ However, if 𝑓 ∈ 𝐻 , then |𝑓(𝑛)| ≤ ‖𝑓‖1 , and therefore (2.19) implies a weak form of (2.20), namely ∞ ̂ 2 |𝑓(𝑛)| ≤ 𝜋‖𝑓‖21 . ∑ 𝑛=0 𝑛 + 1 On the other hand, (2.20) implies ‖𝑓‖21 − |𝑓(0)|2 ≥ (1/2)|𝑓󸀠 (0)|2 , which cannot be de­ duced from (2.19). 𝑝 -inequality). If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 < ∞, then Theorem 2.16 (Hardy 𝑀∞ 1 𝑝 ∫ 𝑟−1/2 𝑀∞ (𝑟, 𝑓) 𝑑𝑟 ≤ 𝜋‖𝑓‖𝑝𝑝 . 0

Proof. Let 𝑓 = 𝐵𝑔2/𝑝 , where 𝐵 is a Blaschke product. Then 1

∫𝑟 0

1 −1/2

𝑝 𝑀∞ (𝑟, 𝑓) 𝑑𝑟

≤ ∫𝑟

1 −1/2

0 ∞ 𝑚,𝑛=0

∞

2

𝑛 ̂ ≤ ∫ 𝑟−1/2 ( ∑ |𝑔(𝑛)|𝑟 ) 𝑑𝑟 0

= ∑ as desired.

2 𝑀∞ (𝑟, 𝑔) 𝑑𝑟

𝑛=0

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

58 | 2 Subharmonic functions and Hardy spaces

2.6 Hardy–Stein identities By Theorem 2.5, the Riesz measure of a function 𝑣 subharmonic in 𝔻 is uniquely de­ termined by the requirement that ∫ 𝜑 𝑑𝜇𝑣 = ∫ 𝑣Δ𝜑 𝑑𝐴 𝔻

𝔻

for all 𝜑 ∈ 𝐶20 (𝔻), 𝜑 ≥ 0. Theorem 2.17. (i) If 𝑣 = |𝑓|𝑝 , 𝑝 > 0, 𝑓 ∈ 𝐻(𝔻), then 𝜇𝑣 is absolutely continuous and 𝑑𝜇𝑣 (𝑧) = 𝑝2 |𝑓(𝑧)|𝑝−2 |𝑓󸀠 (𝑧)|2 𝑑𝐴(𝑧). (ii) If 𝑣 = |𝑢|𝑝 , 𝑝 > 1, where 𝑢 ∈ ℎ(𝔻) and 𝑢 is real valued, then 𝜇𝑣 is absolutely contin­ uous and 𝑑𝜇𝑣 (𝑧) = 𝑝(𝑝 − 1)|𝑢(𝑧)|𝑝−2 |∇𝑢(𝑧)|2 𝑑𝐴(𝑧). Proof. (i) Let 𝑓 ∈ 𝐻(𝔻) and 𝜑 ∈ 𝐶20 (𝔻). We start from the relation ∫ |𝑓|𝑝 Δ𝜑 𝑑𝐴 = lim+ ∫ 𝑔𝜀 Δ𝜑 𝑑𝐴, 𝜀→0

𝔻

(2.21)

𝔻

where 𝑔𝜀 = (|𝑓|2 + 𝜀)𝑝/2 . Since 𝑔𝜀 ∈ 𝐶2 , we have ∫ 𝑔𝜀 Δ𝜑 𝑑𝐴 = ∫ 𝜑Δ𝑔𝜀 𝑑𝐴 = 𝑝 ∫ 𝜑 (|𝑓|2 + 𝜀)𝑝/2−2 [𝑝|𝑓|2 + 2𝜀]|𝑓󸀠 |2 𝑑𝐴. 𝔻

𝐷

𝔻

The function 𝜑 is integrable on 𝔻 so we can apply Fatou’s lemma together with (2.21) to obtain ∫ |𝑓|𝑝 Δ𝜑 𝑑𝐴 ≥ 𝑝2 ∫ 𝜑|𝑓|𝑝−2 |𝑓󸀠 |2 𝑑𝐴. 𝔻

𝔻 𝑝−2

󸀠 2

This shows that the function 𝜑|𝑓| |𝑓 | is integrable on 𝔻. Now use the dominated convergence theorem to conclude the proof in the case 𝑝 < 2. If 𝑝 ≥ 2, then the function |𝑓|𝑝 is of class 𝐶2 so we have ∫ |𝑓|𝑝 Δ𝜑 𝑑𝐴 = ∫ 𝜑Δ(|𝑓|𝑝 ) 𝑑𝐴, 𝔻

which immediately gives the result (take 𝜀 = 0 in 𝑔𝜀 ). The proof of the assertion (ii) is similar and we omit it.

2.6 Hardy–Stein identities

| 59

Combining Theorems 2.6 and 2.17 we get (see also Note 2.6): Theorem 2.18 (Hardy–Stein identities [182, 466]). If 𝑓 ∈ 𝐻(𝔻), then 𝑀𝑝𝑝 (𝑟, 𝑓) = |𝑓(0)|𝑝 +

𝑝2 𝑟 ∫ |𝑓(𝑧)|𝑝−2 |𝑓󸀠 (𝑧)|2 log 𝑑𝐴(𝑧) 2𝜋 |𝑧| |𝑧| 0 is a constant independent of 𝑓. This theorem does not extend to the case of 𝐿1 (𝕋), which can be seen from Remark 1.1. For a converse to Paley’s theorem, see Rudin [428].

2.7 Subordination principle | 61

Proof. Let 𝜆 = inf 𝑛≥1 follows that

𝑘𝑛+1 . 𝑘𝑛

From the Hardy–Littlewood 𝑀𝑝2 -theorem (Theorem 2.21) it ∞

𝑟𝑚+1

‖𝑓‖1 ≥ 𝑐1 ∑ ∫ 𝑀12 (𝑟, 𝑓󸀠 )(1 − 𝑟) 𝑑𝑟, 𝑚=1 𝑟 𝑚 𝑛−1 ̂ , we can apply Lemma where 𝑟𝑚 = 1 − 𝜆−𝑚 , 𝑐1 =const> 0. Since 𝑀1 (𝑟, 𝑓󸀠 ) ≥ 𝑛|𝑓(𝑛)|𝑟 3.11 below to prove the result. 𝑘𝑛 Theorem 2.23 (Paley). Let the series 𝑓(𝑧) = ∑∞ 𝑛=1 𝑎𝑛 𝑧 , where {𝑘𝑛 } is a lacunary se­ 2 quence, converge in 𝔻. Then 𝑓 ∈ 𝐻𝑝 (0 < 𝑝 < ∞) 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 .

(2.22)

Proof. In the case 1 ≤ 𝑝 < 2, the result is an immediate consequence of Paley’s the­ orem. Let 0 < 𝑝 < 1. Let 𝑓 be analytic in a neighborhood of the closed disk. Then, by means of the Cauchy–Schwarz inequality, we get (2−𝑝)/2

‖𝑓‖1 = ∫ |𝑓|𝑝/2 |𝑓|1−𝑝/2 ≤ ‖𝑓‖𝑝/2 𝑝 ‖𝑓‖2−𝑝 𝑝/2

(2−𝑝)/2

≤ ‖𝑓‖𝑝/2 𝑝 ‖𝑓‖2

.

𝑝/2

Since ‖𝑓‖1 ≥ 𝑐‖𝑓‖2 , we see that 𝑐 ‖𝑓‖2 ≤ ‖𝑓‖𝑝 . If 𝑓 is arbitrary, then we apply this in­ equality 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 opera­ ̂ )𝑧𝑘𝑛 , is bounded from 𝐻𝑞 to 𝐻2 . The adjoint 𝑃⋆ is formally equal tor 𝑃, (𝑃𝑓)(𝑧) = ∑ 𝑓(𝑘 𝑛 𝑞 󸀠 to 𝑃, and since (𝐻 ) = 𝐻𝑝 (Theorem 1.37), we have ‖𝑃𝑓‖𝑝 ≤ 𝐶𝑝 ‖𝑓‖2 for 𝑓 ∈ 𝐻2 . Hence ‖𝑓‖𝑝 ≤ 𝐶𝑝 ‖𝑓‖2 if 𝑃𝑓 = 𝑓. As an application of Theorems 2.23, 2.20, and Lemma 3.11 we get Proposition 2.1. If 0 < 𝑝 ≤ 2, and 𝑓(𝑧) = ∑ 𝑎𝑛 𝑧𝑘𝑛 , where {𝑘𝑛 } is lacunary, then ∞ 𝑑 2 𝑀𝑝 (𝑟, 𝑓) ≤ 𝐶 ∑ 𝑘𝑛 |𝑎𝑛 |2 𝑟2𝑘𝑛−1 . 𝑑𝑟 𝑛=1

If 2 ≤ 𝑝 < ∞, then the reverse holds.

2.7 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

62 | 2 Subharmonic functions and Hardy spaces |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 [302]. Theorem 2.24 (Subordination principle). If a function 𝑢 is subordinate to a subhar­ monic function 𝑈, then 𝜋

𝜋

− ∫ 𝑢(𝑟𝑒𝑖𝜃 ) 𝑑𝜃 ≤ − ∫ 𝑈(𝑟𝑒𝑖𝜃 ) 𝑑𝜃 −𝜋

(0 < 𝑟 < 1).

(2.23)

−𝜋

In the simplest case 𝜔(𝑧) = 𝜌𝑧, 0 < 𝜌 < 1, this theorem reduces to Theorem 2.2. Proof¹. 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

𝑖𝜃

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

𝑖𝜃

−𝜋

−𝜋

This concludes the proof because ℎ(𝑟𝑒𝑖𝜃 ) = 𝑈(𝑟𝑒𝑖𝜃 ). Example 2.2 (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 󵄨 󵄨󵄨 󵄨𝑝 𝑖𝜃 󵄨󵄨𝑝 󵄨 󵄨󵄨 1 + 𝑟𝑒𝑖𝜃 󵄨󵄨󵄨 𝑝 󵄨󵄨 1 + 𝑒 󵄨󵄨 󵄨󵄨 𝑑𝜃 ≤ 𝑐 − 󵄨󵄨 󵄨 ∫ |𝑓(𝑟𝑒 )| 𝑑𝜃 ≤ 𝑐 − ∫ 󵄨󵄨󵄨 ∫ − 󵄨󵄨 1 − 𝑒𝑖𝜃 󵄨󵄨󵄨 𝑑𝜃 󵄨 1 − 𝑟𝑒𝑖𝜃 󵄨󵄨󵄨 󵄨 −𝜋 −𝜋 󵄨 −𝜋 󵄨 𝑖𝜃 𝑝

𝑝

𝜋/2

𝑢(0)𝑝 2 =𝑐 ∫ (cot 𝜃)𝑝 𝑑𝜃 = 𝜋 cos(𝑝𝜋/2) 𝑝

(0 < 𝑝 < 1).

0

If 𝑓 is arbitrary, then we can use Theorem 1.4(ii) to reduce the proof to the preceding case. Exercise 2.10. It follows from 2.5 that if 𝑓 ∈ 𝐻𝑝 and 1/𝑓 ∈ 𝐻𝑝 for some 𝑝 > 0, then 𝑓 is outer. By the subordination principle, 𝑓 is in 𝐻𝑝 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 [4].

1 This proof was invented by Riesz [418].

2.7 Subordination principle |

63

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. With the hypotheses of the previous theorem, we have 𝜋 𝑝

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

where 1/2 {(1 − 𝑟) , { { { 2 𝛾𝑝 (𝑟) = {(1 − 𝑟)1/2 log , { 1 − 𝑟 { { 𝑝 {(1 − 𝑟) ,

1/2 < 𝑝 ≤ 1, 𝑝 = 1/2, 0 < 𝑝 < 1/2.

Proof. The analytic function 𝑎(𝑧) = − log 𝑓(𝑧) maps 𝔻 into the right half-plane. Re­ placing 𝑓 by 𝜁𝑓 for a suitable 𝜁 ∈ 𝕋, we may assume that 𝑎(0) > 0. It follows that 𝑓(𝑧) is subordinate to 1+𝑧 𝐴 𝜆 (𝑧) = exp (−𝜆 ), 1−𝑧 where 𝜆 > 0. The function −(1 − |𝑧|)𝑝 is subharmonic for 𝑝 ≤ 1, and therefore, by the subordination principle, 𝜋

𝜋 𝑝

𝑝

∫ (1 − |𝑓(𝑟𝑒𝑖𝜃 )|) 𝑑𝜃 ≥ − ∫ (1 − |𝐴 𝜆 (𝑟𝑒𝑖𝜃 )|) 𝑑𝜃. − −𝜋

−𝜋

In order 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

0

𝑝 (1 − 𝑟)𝑝 𝜆𝑃(𝑟, 𝜃) ) 𝑑𝜃 ≍ ∫ − ∫ (1 − |𝐴 𝜆 (𝑟𝑒 )|) 𝑑𝜃 ≍ ∫ ( 𝑑𝜃. 1 + 𝜆𝑃(𝑟, 𝜃) (1 − 𝑟 + 𝜃2 )𝑝 𝑖𝜃

−𝜋

𝑝

Introducing the change 𝜃 = 𝑡√1 − 𝑟 and computing the resulting integral we conclude the proof.

64 | 2 Subharmonic functions and Hardy spaces Corollary 2.4. If 0 < 𝑝 ≤ 1, and 𝑓 is an inner function with nonconstant singular factor, then 2𝜋

∫ (1 − |𝐼(𝑒𝑖𝜃 )|)𝑝 𝑑𝜃 ≥ 𝑐𝛾𝑝 (𝑟). 0

Exercise 2.11. If 𝑢 is a positive harmonic function, then 𝑀𝑝 (𝑟, 𝑢) ≥ 𝑐𝑝 𝛽𝑝 (𝑟), where 𝛽𝑝 has been defined in the formulation of Proposition 1.2. Problem 2.3. Concerning this exercise, it seems that it is not known if the above es­ timates hold for all harmonic functions. In [16], Aleksandrov proved that they hold if 𝑀𝑝 (𝑟, 𝑢) is replaced with 1/𝑝

1

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 Composition with inner functions Throughout this section we consider nonconstant inner functions. If 𝜔 is such a func­ tion, 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 , ‖𝑓 ∘ 𝜔‖𝑝 = ‖𝑓‖𝑝

and

if 𝜔(0) = 0,

due to Stephenson [468] (see Theorems 2.29 and 2.30). These relations as well as all other assertions in this section become obvious when specialized to the case 𝜔(𝑧) = 𝑧𝑛, 𝑛 ≥ 1. In general, the composition of two Lebesgue measurable functions need not be measurable. Proposition 2.2. Let 𝜔 be an inner function, |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 Subordination principle |

65

(iv) If 𝜙𝑛 are Lebesgue measurable functions on 𝕋 such that lim𝑛 𝜙𝑛 = 𝑓 a.e., then lim𝑛 𝜙𝑛 ∘ 𝜔∗ = 𝑓 ∘ 𝜔∗ a.e. ̄ and 𝜑 ∘ 𝜔(0) = 0. Proof. (i) If 𝜔(0) = 𝑎, then 𝜔 = 𝜑 ∘ (𝜑 ∘ 𝜔), where 𝜑(𝑧) = (𝑎 − 𝑧)/(1 − 𝑎𝑧) 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

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

(2.24)

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 (2.24), 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

2𝜋

∫ 𝜙(𝜔∗ (𝑒𝑖𝜃 )) 𝑑𝜃 = ∫ 𝜙(𝑒𝑖𝜃 ) 𝑑𝜃, 0

𝜙 ∈ 𝐶(𝕋);

0

recall 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 (𝑔 ∘ 𝜔∗ )−1 (𝐺) = 𝜔∗−1 (𝑔−1 (𝐺)). The set 𝑔−1 (𝐺) is Lebesgue measurable and therefore there exists a Borel set 𝐹 and a set 𝐸 of measure zero such that 𝑔−1 (𝐺) = 𝐹∪𝐸. Now 𝜔∗−1 (𝑔−1 (𝐺)) = 𝜔∗−1 (𝐹)∪ 𝜔∗−1 (𝐸). The set 𝜔∗−1 (𝐹) is Lebesgue measurable and 𝜔∗−1 (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)) provided that 𝜔∗ (𝜁) ∈ 𝕋 \ 𝐸, i.e. 𝜁 ∈ 𝜔∗−1 (𝕋) \ 𝜔∗−1 (𝐸), which proves the result because |𝜔∗−1 (𝐸)| = 0 and |𝜔∗−1 (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.

66 | 2 Subharmonic functions and Hardy spaces Theorem 2.27 ([432, 468]). If 𝜙 ∈ 𝐿1 (𝕋) and 𝜔 is an inner function with 𝜔(0) = 0, then 𝜙 ∘ 𝜔∗ ∈ 𝐿1 (𝕋) and 2𝜋

2𝜋 𝑖𝜃

∫ 𝜙(𝜔∗ (𝑒 )) 𝑑𝜃 = ∫ 𝜙(𝑒𝑖𝜃 ) 𝑑𝜃. 0

0

Proof. In order 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 2.2, we have 𝜙𝑛 ∘ 𝜔∗ → 𝜙 ∘ 𝜔∗ a.e. The result follows. Theorem 2.28. If 𝜙 ∈ 𝐿1 (𝕋) and 𝜔 is an inner function, |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 disk, 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 2.5. 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 above results one easily proves the following. Theorem 2.29 (Stephenson [468]). If 𝑓 ∈ 𝐻𝑝 , 𝑝 > 0, and 𝜔 is inner with 𝜔(0) = 0, then 𝑓 ∘ 𝜔 ∈ 𝐻𝑝 , (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 ∈ 𝐻𝑝 ⇒ 𝑓 ∈ 𝐻𝑝 . Theorem 2.30 (Stephenson [468]). If 𝑓 ∈ 𝐻(𝔻) and 𝜔 is an inner function, then 𝑓 ∘ 𝜔 ∈ 𝐻𝑝 implies 𝑓 ∈ 𝐻𝑝 . Proof. Assume that 𝜔(0) = 0. Let 𝑓 ∘ 𝜔 ∈ 𝐻𝑝 ,

𝑝 > 0,

𝑢 = |𝑓|𝑝 ,

𝑣 = |𝑓 ∘ 𝜔|𝑝 ,

and ℎ = P[𝑣∗ ].

2.7 Subordination principle | 67

We know that 𝑣 ≤ ℎ; see (1.37). Let 𝐷𝑟 = 𝑟𝔻, where 𝑟, 0 < 𝑟 < 1, is fixed. Then 𝑢 ≤ 𝑀 on 𝐷𝑟 for some constant 𝑀 < ∞. Let Ω𝑟 = 𝜔−1 (𝐷𝑟 ) = {𝑧 : |𝜔(𝑧)| < 𝑟} 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. conver­ gence 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, 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(𝐷𝑟 ) harmonic in 𝐷𝑟 such that 𝑢 ≤ 𝑢1 ≤ 𝑀 on 𝐷𝑟 and 𝑢1 (𝑧) = 𝑢(𝑧) for |𝑧| = 𝑟. By the mean value property, 𝜋

𝜋

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

(2.25)

−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) ∈ 𝐷𝑟 . 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 principle (Corollary 2.2) to conclude that 𝑈(0) − 𝜑(0) = 𝑢1 (0) − ℎ(0) − 𝜑(0) ≤ 0, which implies, via (2.25) 𝜋

∫ |𝑓(𝑟𝑒𝑖𝜃 )|𝑝 𝑑𝜃 ≤ ℎ(0) + 1. − −𝜋

68 | 2 Subharmonic functions and Hardy spaces Since ℎ(0) + 1 is a constant independent of 𝑟, we see that 𝑓 ∈ 𝐻𝑝 . 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 proof of which is 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 func­ tion on 𝕋 is lower semicontinuous on 𝔻. For (i), see assertion (a) after Definition 2.8 in [430]; for (ii), see Note 1.2.

2.7.2 Approximation with inner functions Let 𝜔 be an inner function with 𝜔(0) = 0. If 𝑓 ∈ 𝐻2 , then, by Rogosinski’s theorem (see Note 2.3) and Stephenson’s theorem, we have ∞

∞

𝑘=𝑛

𝑘=𝑛

̂ 2 ≥ ∑ |𝐹(𝑘)| ̂ 2, ∑ |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 ∈ 𝐻𝑝 , let 𝐸𝑛(𝑓)𝑝 = 𝐸𝑛 (𝑓)𝐻𝑝 = inf 𝑔∈P𝑛 ‖𝑓 − 𝑔‖𝑝 . Then the above inequality can be stated as 𝐸𝑛(𝑓 ∘ 𝜔)2 ≥ 𝐸𝑛 (𝑓)2 , 𝑓 ∈ 𝐻2 . It is interesting that this extends to the case 1 ≤ 𝑝 ≤ ∞. Theorem 2.31 (approximation with inner functions [332]). Let 1 ≤ 𝑝 ≤ ∞, 𝑓 ∈ 𝐻𝑝 , and let 𝜔 be an inner function with 𝜔(0) = 0. Then 𝐸𝑛(𝑓 ∘ 𝜔)𝑝 ≥ 𝐸𝑛 (𝑓)𝑝 . Proof. Assume first that 𝑝 is finite. Let 𝜋

∫ 𝑓(𝑒𝑖𝜃 )ℎ(𝑒𝑖𝜃 ) 𝑑𝜃. (𝑓, ℎ) = − −𝜋

We identify 𝐻𝑝 with 𝐻𝑝 (𝕋). From the general theory of best approximation in Banach spaces 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, while the proof of (ii) is straightforward – it is enough to observe that (ℎ, P𝑛 ) = 0 if and only ̂ if ℎ(𝑗) = 0 for 0 ≤ 𝑗 ≤ 𝑛. We get 𝐸𝑛 (𝑓 ∘ 𝜔)𝑝 ≥ sup{ |(𝑓 ∘ 𝜔, ℎ ∘ 𝜔)| : ℎ ∈ 𝐿𝑞 (𝕋), (P𝑛 , ℎ) = 0, ‖ℎ‖𝑞 ≤ 1}.

2.7 Subordination principle | 69

This concludes the proof in the case where 𝑝 is finite because |(𝑓 ∘ 𝜔, ℎ ∘ 𝜔)| = |(𝑓, ℎ)|, by Theorem 2.27. In the remaining case we have 𝐸𝑛 (𝑓 ∘ 𝜔)∞ ≥ 𝐸𝑛 (𝑓 ∘ 𝜔)𝑝 ≥ 𝐸𝑛(𝑓)𝑝 for all finite 𝑝. Since 𝐸𝑛 (𝑓)∞ = lim𝑝→∞ 𝐸𝑛(𝑓)𝑝 , we have 𝐸𝑛(𝑓 ∘ 𝜔)𝑝 ≥ 𝐸𝑛 (𝑓)𝑝 for all 𝑝, as claimed. Problem 2.4. If we change the notation and denote by 𝐸𝑛(𝑓)𝑝 the best 𝐿𝑝 approxi­ mation 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 the­ orem and cannot be applied to the case 𝑝 < 1. It would be interesting to study this case.

Further notes and results In the case 𝑝 = ∞, Corollary 2.3 is known as Hadamard’s three circles theorem. The case 𝑝 < ∞ was studied by Hardy [182] in the first paper from “the theory of Hardy spaces”. Carleman [85] proved Theorem 2.12 in the case when 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 products had not been invented yet. Observe that they, however, can be avoided at least in proving inequality (2.9). The author does not know who first proved (2.9) (𝑁 = 2) for an arbitrary 𝑓 ∈ 𝐻𝑝 . Theorem 2.12 for 𝑁 > 2 was proved by Burbea [80] although, 2 years earlier, a proof was sketched in [328]. It should be pointed out that all the proofs are essentially the same as that of Carleman. For the case 𝑁 = 2 of Theorem 2.14, see Huber’s paper [220], where the theorem was used in proving an isoperimetric inequality for surfaces in ℝ3 . One other paper of Huber [221] contains an extension of the Fejér–Riesz inequality to the quotient of two log-subharmonic functions. 2.1 (Harmonic Schwarz lemma [206]). If 𝑢 ∈ ℎ(𝔻) is real valued, |𝑢| ≤ 1, and 𝑢(0) = 0, then 𝑢 is subordinate to the function 𝑈(𝑧) =

1+𝑧 2 arg . 𝜋 1−𝑧

Hence |𝑢(𝑧)| ≤ 𝜋4 arctan |𝑧|, which holds also for complex-valued functions [206], 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 and 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 one can deduce that (A) if ‖𝑢‖𝑝 ≤ 1, with 𝑝 ≥ 1 and 𝑢(0) = 0, then 𝑀𝑝 (𝑟, 𝑢) ≤ (4/𝜋) arctan 𝑟.

70 | 2 Subharmonic functions and Hardy spaces The analogous statement for analytic functions, proved by using the subordina­ tion principle, reads: (B) If 𝑓 ∈ 𝐻𝑝 , 𝑝 ≥ 0, and 𝑓(0) = 0, then 𝑀𝑝 (𝑟, 𝑓) ≤ 𝑟‖𝑓‖𝑝 . 2.2 (Hilbert matrix transform). The Hilbert matrix transform of a function 𝑓 ∈ 𝐻(𝔻) is defined by 1 ∞ ∞ ̂ 𝑓(𝑟) 𝑓(𝑗) 𝑑𝑟 = ∑ 𝑧𝑛 ∑ , H𝑓(𝑧) = ∫ 1 − 𝑟𝑧 𝑛=0 𝑗=0 𝑗 + 𝑛 + 1 0

where the integral (or the series) is somehow defined. It is easy to see that the adjoint 󸀠 of the restriction 𝑅 : 𝐻𝑝 󳨃→ 𝐿𝑝 (0, 1), 1 ≤ 𝑝 < ∞, is equal to 𝑅⋆ : 𝐿𝑝 (0, 1) 󳨃→ (𝐻𝑝 )󸀠 , where 1 𝑔(𝑟) ⋆ 𝑅 𝑔(𝑧) = ∫ 𝑑𝑟. 1 − 𝑟𝑧 0

From this and (10.23) we find that 𝑅 maps 𝐿𝑞 (0, 1) into 𝐻𝑞 for 1 < 𝑞 < ∞, and that H maps 𝐻𝑞 into 𝐻𝑞 ; the latter is a result from [113]. Dostanić et al. [123], proved that ⋆

‖H𝑓‖𝑞 ≤ 𝐶𝑝 ‖𝑓‖𝑞 ,

1 < 𝑞 < ∞,

(†)

with the best constant 𝐶𝑝 = 𝜋/sin(𝜋/𝑞). In proving the inequality, they used the Hol­ lenbeck–Verbitsky result (see Remark 1.3). The latter and the Fejér–Riesz inequality 󸀠 give first ‖𝑅⋆ 𝑔‖𝑞 ≤ 𝜋1/𝑞 / sin(𝜋/𝑞)‖𝑔‖𝐿𝑞 (0,1) , and then (†). However, the proof that the constant 𝐶𝑝 is optimal is complicated. 2.3 (Rogosinski’s theorem [130, 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 prin­ ciple and Parseval’s formula: If 𝑓 is subordinate to 𝐹 ∈ 𝐻(𝔻), then 𝑛

𝑛

𝑘=0

𝑘=0

̂ 2 𝑟2𝑘 ≤ ∑ |𝐹(𝑘)| ̂ 2 𝑟2𝑘 , ∑ |𝑓(𝑘)|

0 < 𝑟 < 1, 𝑛 ≥ 0.

2.4 ([130]). If 𝑈 = |𝑓|𝑝 , 0 < 𝑝 < ∞, then strict equality holds for 0 < 𝑟 < 1 in (2.23) unless 𝑓 is constant or 𝜔(𝑧) = 𝛼𝑧, |𝛼| = 1. 2.5. Let 𝑅 > 0 and 𝜌 > 0. Let 𝑈 be a nonnegative function subharmonic in a neighbor­ hood 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

2.7 Subordination principle | 71

2.6 (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 𝐶2 so we can apply Green’s formula to |𝑓|𝑝 . In the case 𝑝 < 2, we apply this formula to the functions 𝑔 = (|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 𝐶2 (𝔻), we get Δ𝑔 =

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

= 𝑝(𝑝 − 2)(|𝑓|2 + 𝜀)𝑝/2−2 |𝑓|2 |𝑓󸀠 |2 + 2𝑝(|𝑓|2 + 𝜀)𝑝/2−1 |𝑓󸀠 |2 = 𝑝(|𝑓|2 + 𝜀)𝑝/2−2 [(𝑝 − 2)|𝑓|2 + 2|𝑓|2 + 2𝜀]|𝑓󸀠 |2 , whence Δ𝑔 = 𝑝(|𝑓|2 + 𝜀)𝑝/2−2 [𝑝|𝑓|2 + 2𝜀]|𝑓󸀠 |2 . Now proceed in a similar way as in proving Theorem 2.17 to finish the proof. 2.7. Hardy and Littlewood proved the inequality 1

‖𝑓‖2𝑝

2

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

00 QNS𝐾 (Ω) are called quasi-nearly subhar­ monic functions. Note that (3.3) implies sup 𝑢 ≤ 𝐾1 ∫ 𝑢 𝑑𝐴,

𝐷𝜀 (0)

𝐷𝛿 (0)

where 0 < 𝜀 < 𝛿 < 1 and 𝐾1 depend only on 𝐾, 𝜀, and 𝛿. The class QNS𝐾 is invariant under translations and dilations so the proof of Theo­ rem 3.1 gives the following result: Theorem 3.2. Let 0 < 𝑝 < ⬦. 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 that are closely related to Theorem 1.11. Unless otherwise stated, we consider real-valued functions defined on a proper subdomain Ω

76 | 3 Subharmonic behavior and mixed norm spaces of ℂ. We denote by 𝐻𝐶1𝐾 (Ω) the class of all locally Lipschitz functions 𝑓 satisfying |∇𝑓(𝑧)| ≤ 𝐾𝑟−1 sup |𝑓|,

𝐷𝑟 (𝑧) ⊂ Ω,

𝐷𝑟 (𝑧)

(3.4)

where 𝐾 ≥ 0 is a constant independent of 𝐷𝑟 (𝑧) ⊂ Ω, and let 𝐻𝐶1 (Ω) = ⋃ 𝐻𝐶1𝐾(Ω). 𝐾≥0

It should be noted that a locally Lipschitz function is differentiable almost every­ where and therefore ∇𝑓 is defined a.e. We define |∇𝑓| everywhere by |∇𝑓(𝑎)| = lim sup 𝑧→𝑎

|𝑓(𝑧) − 𝑓(𝑎)| . |𝑧 − 𝑎|

The function |∇𝑓| is Borel measurable; see Note 3.2. The proof of Theorem 1.11 (p. 13) can be easily modified to obtain Theorem 3.3. If 𝑓 ∈ 𝐻𝐶1𝐾 (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),

(3.5)

which is a restriction on the growth of 𝑓 and is therefore stronger than (3.4). For ex­ ample, the function 𝑓(𝑥 + 𝑖𝑦) = 𝑒𝑥 is in 𝐻𝐶1 (Ω), where Ω is the right half-plane, but 𝑓 does not satisfy (3.5). It is a simple but important fact that condition (3.5) 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 |∇𝑢(𝑧)| ≤

2𝑢(𝑧) . 1 − |𝑧|

A consequence of Theorem 3.3: Corollary 3.1. Let 0 < 𝑝 < ⬦. A function 𝑓 locally Lipschitz on Ω belongs to 𝐻𝐶1𝐾 (Ω) if and only if there is a constant 𝐶 depending only on 𝐾 and 𝑝 such that |∇𝑓(𝑧)|𝑝 ≤ 𝐾𝑟−2−𝑝 ∫ |𝑓|𝑝 𝑑𝐴,

𝐷𝑟 (𝑧) ⊂ Ω.

𝐷𝑟 (𝑧)

3.2 (The classes 𝑂𝐶1𝐾 (Ω) and RO). The class 𝑂𝐶1𝐾 (Ω) is the subclass of 𝐻𝐶1 (Ω) con­ sisting of those 𝑓 for which |∇𝑓(𝑧)| ≤ 𝐾𝑟−1 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) }.

3.2 Regularly oscillating functions |

77

We put RO = ⋃𝐾≥0 𝑂𝐶1𝐾 (Ω). Members of RO are called in [386] regularly oscillating functions. Theorem 3.4. If 𝑓 ∈ 𝑂𝐶1𝐾 (Ω), then both |𝑓| and |∇𝑓| belong to QNS𝐶 (Ω), where 𝐶 de­ pends only on 𝐾. Let

1/𝑝

{ } { 1 } 𝑂𝑝 𝑓(𝑧, 𝑟) = { ∫ |𝑓(𝑤) − 𝑓(𝑧)|𝑝 𝑑𝐴(𝑤)} { |𝐷𝑟 (𝑧)| } 𝐷𝑟 (𝑧) { } 𝑝 the 𝐿 -oscillation of 𝑓 over 𝐷𝑟 (𝑧).

,

Corollary 3.2. Let 0 < 𝑝 < ⬦. A function 𝑓 belongs to 𝑂𝐶1 (Ω) if and only if |∇𝑓(𝑧)| ≤ 𝐾𝑟−1 𝑂𝑝 𝑓(𝑧, 𝑟), 𝐷𝑟 (𝑧) ⊂ Ω, for some constant 𝐾. This is deduced from Corollary 3.1 by considering the functions 𝑓 − const. Proof of Theorem 3.4. The inclusion 𝑂𝐶1𝐾 (Ω) ⊂ QNS𝐶 (Ω) follows from Theoerem 3.3 and the inclusion 𝑂𝐶1𝐾 ⊂ 𝐻𝐶12𝐾 . It remains to prove that 𝑓 ∈ 𝑂𝐶1 (Ω) implies |∇𝑓| ∈ QNS(Ω). Let 𝑓 ∈ 𝑂𝐶1𝐾 (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 |∇𝑓(0)|𝑞 ≤ 𝐶 ∫ |∇𝑓|𝑞 𝑑𝐴. 𝔻

By Corollary 3.2, 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

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

0

whence

1

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

0

𝔻

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

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

78 | 3 Subharmonic behavior and mixed norm spaces ̄ A 𝜕-condition It is interesting that condition (3.4) can be weakened. Then condition (3.4) implies that |𝑓| is QNS (Remark 3.1). Before stating the theorem note that |∇𝑓| can also be expressed ̄ namely: via 𝜕𝑓 and 𝜕𝑓, ̄ 2 ). |∇𝑓|2 = 2(|𝜕𝑓|2 | + |𝜕𝑓| Theorem ([386]). If 𝑓 ∈ 𝐶1 (Ω) satisfies the condition ̄ |𝜕𝑓(𝑧)| ≤ 𝐾𝑟−1 sup |𝑓|,

whenever 𝐷𝑟 (𝑧) ⊂ Ω,

𝐷𝑟 (𝑧)

where 𝐾 is a constant, then |𝑓| is QNS. Proof. We use the formula 2𝜋

𝑓(0) = ∫ − 𝑓(𝑒𝑖𝜃 ) 𝑑𝜃 − 0

̄ 1 𝜕𝑓(𝑧) ∫ 𝑑𝐴(𝑧), 𝜋 𝑧 𝔻

valid for 𝑓 ∈ 𝐶1 (𝔻) (a special case of the Cauchy–Goursat formula). Hence |𝑓(0)| ≤ ∫ |𝑓| 𝑑𝑙 + 𝕋

̄ 1 |𝜕𝑓(𝑤)| ∫ 𝑑𝐴(𝑤) 𝜋 |𝑤| 𝔻

1 ̄ ∫ 1 𝑑𝐴(𝑤) ≤∫ − |𝑓| 𝑑𝑙 + sup |𝜕𝑓| 𝜋 𝔻 |𝑤| 𝔻

𝕋

̄ = ∫ |𝑓| 𝑑𝑙 + 2 sup |𝜕𝑓|. 𝕋

𝔻

We leave the details to the interested reader. 3.3 (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 |∇𝑓(𝑧)| ≤

𝐾 sup (𝑓(𝑤) − 𝑓(𝑧)) 𝑟 𝑤∈𝐷𝑟 (𝑧)

(where 𝐾 = 1). A function 𝑓 : Ω 󳨃→ ℝ satisfying this inequality for some 𝐾 > 0 will be 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 3.2. If a function 𝑢 : Ω 󳨃→ ℝ is harmonic, then |𝑢| ∈ NC12 (Ω).

3.2 Regularly oscillating functions

| 79

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)| ≤ 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| we see from Theorem 3.4 that |𝑢|𝑠−1 |∇𝑢| ∈ QNS, which can be expressed as |𝑢(0)|𝑝 |∇𝑢(0)|𝑞 ≤ 𝐶𝑝,𝑞 ∫ |𝑢|𝑝 |∇𝑢|𝑞 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 real-valued 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 3.3 (Schwarz modulus lemma). Let, as before, 𝐷𝜀 (𝑧) = {𝑤 : |𝑤 − 𝑧| < 𝜀}, 0 < 𝜀 ≤ 1 − |𝑧|, and 𝑓 ∈ 𝐻∞ . Then |𝑓󸀠 (𝑧)| ≤

2 sup (|𝑓(𝑤)| − |𝑓(𝑧)|) 𝜀 𝑤∈𝐷𝜀 (𝑧)

(𝑧 ∈ 𝔻).

Proof. Let 𝑀𝑧 = sup{|𝑓(𝑤)| : 𝑤 ∈ 𝐷𝜀 (𝑧)}. If 𝑧 = 0 and 𝑀0 = 1, then Schwarz’s lemma gives |𝑓󸀠 (0)| ≤ 1 − |𝑓(0)|2 ≤ 2(1 − |𝑓(0)|), which is the required inequality in a special case. In the general case we apply this special 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 3.2 and the following statement. Proposition 3.1. Let 𝑢 be a nonnegative, nearly convex function on Ω, and let 𝜙 be an increasing 𝐶1 -function such that 𝜙(𝑡)/𝑡𝛽 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 is also denoted by 𝜙 ∈ Δ[1, 𝛽]; see Section 3.7, p. 100. As always, it is enough to prove that |∇𝑣(0)| ≤ 𝐶 sup (𝑣(𝑧) − 𝑣(0)) 𝑧∈𝔻

under the hypothesis 𝔻 ⊂ Ω.

80 | 3 Subharmonic behavior and mixed norm spaces Assuming that 𝑢(0) ≠ 0, we have |∇𝑣(0)| = 𝜙󸀠 (𝑢(0))|∇𝑢(0)| ≤ 𝐶𝛽 = 𝐶𝛽 (

𝜙(𝑢(0)) (𝑢(𝑧) − 𝑢(0)) 𝑢(0)

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 so that 𝑢(𝑧) > 0. Then |∇𝑣(0)| = 𝜙󸀠 (0)|∇𝑢(0)| ≤ 𝐶𝜙󸀠 (0)𝑢(𝑧) = 𝐶𝑢(𝑧) lim+ 𝑡→0

𝜙(𝑡) 𝑡

𝜙(𝑢(𝑧)) = 𝐶𝜙(𝑢(𝑧)), ≤ 𝐶𝑢(𝑧) 𝑢(𝑧) which completes the proof. Example 3.1. Let 𝜙(𝑡) = 𝑡𝛼 log𝛾 (1 + 𝑡). Then 𝑡𝜙󸀠 (𝑡) 𝛾𝑡 =𝛼+ . 𝜙(𝑡) (1 + 𝑥) log(1 + 𝑥) 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 3.1 and Lemma 3.2, the function 𝑣(𝑧) = |𝑢(𝑧)|𝛼 log𝛾 (1 + |𝑢(𝑧)|) is nearly convex if 𝑢 is harmonic, 𝛼 ≥ 1 and 𝛾 ≥ 0. Subharmonic behavior and conformal mappings The classes QNS, RO, and NC are invariant under conformal mappings. More precisely: Theorem 3.6 ([319]). If 𝜑 : Ω 󳨃→ 𝜑(Ω) is a conformal mapping and 𝑢 is in 𝑆, where 𝑆 is one of the classes QNS, RO, and NC on 𝜑(Ω), then 𝑢 ∘ 𝜑 ∈ 𝑆. Moreover, if 𝑓 ∈ QNS𝐾 , then 𝑓 ∘ 𝜑 ∈ QNS𝐶 , where 𝐶 depends only on 𝐾, and similarly for 𝑂𝐶1𝐾 and NC𝐾 . Proof. Consider first the case of 𝑆 = QNS𝐾 . Assuming that 𝔻 ⊂ Ω, we have to prove that ∫ 𝑢(𝜑(𝑧)) 𝑑𝐴(𝑧) ≥ 𝑐𝐾 𝑢(𝜑(0)). 𝔻

3.2 Regularly oscillating functions

| 81

Suppose, for simplicity, that 𝜑(0) = 0. Let 𝜓 denote the inverse of 𝜑. By the Köbe onequarter theorem (see Theorem 10.3), we have ∫ 𝑢(𝜑(𝑧)) 𝑑𝐴(𝑧) = ∫ 𝑢(𝑤)|𝜓󸀠 (𝑤)|2 𝑑𝐴(𝑤) ≥ 𝔻

𝜑(𝔻)

∫ 𝑢(𝑤)|𝜓󸀠 (𝑤)|2 𝑑𝐴(𝑤), 𝐷𝜀/2 (0)

where 𝜀 = |𝜑󸀠 (0)|/4. On the other hand, by using the distortion theorem (Theorem 10.2), we get |𝜓󸀠 (𝑤)| ≥ |𝜓󸀠 (0)|(𝜀 − |𝑤)/8𝜀 ≥ |𝜓󸀠 (0)|/16, for |𝑤| < 𝜀/2. It follows that ∫ 𝑢(𝜑(𝑧)) 𝑑𝐴(𝑧) ≥ ∫ 𝑢(𝑤)|𝜓󸀠 (0)|2 𝑑𝐴(𝑤)/16 ≥ 𝑐𝑢(0)𝜀2 |𝜓󸀠 (0)|2 . 𝔻

𝐷𝜀/2

This finishes the proof because 𝜀2 |𝜓󸀠 (0)|2 = |𝜑󸀠 (0)𝜓󸀠 (0)|2 /16 = 1/16, and 𝑢(0) = 𝑢(𝜑(0)). Let 𝑢 ∈ 𝑂𝐶1𝐾 . Assuming as above that 𝔻 ⊂ Ω, we have, by the one-quarter theo­ rem, sup |𝑢(𝜑(𝑧)) − 𝑢(𝜑(0))| = sup |𝑢(𝑤) − 𝑢(𝜑(0))| 𝑧∈𝔻

𝑤∈𝜑(𝔻)

≥

sup

|𝑢(𝑤) − 𝑢(𝜑(0))| ≥ 𝑐𝜀|∇𝑢(𝜑(0))|

𝑤∈𝐷𝜀 (𝜑(0))

= (𝑐/4)|𝜑󸀠 (0)| |∇𝑢(𝜑(0))| = (𝑐/4)|∇(𝑢 ∘ 𝜑)(0)|, where 𝜀 was as above. This proves the theorem in the case where 𝑆=RO. The case of NC is treated in the same way.

Moduli of vector-valued functions The facts that the moduli of analytic or harmonic functions are nearly convex can be extended to functions with values in a Banach space. Theorem 3.7. If 𝑓 : Ω 󳨃→ 𝑋 is either an analytic function with values in a complex Banach space 𝑋 or a harmonic function with values in a real 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, ‖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) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ≤ |𝜎𝑎 (𝑧)|. 󵄨󵄨 1 − 𝜙(𝑎)𝜙(𝑧) 󵄨󵄨󵄨 󵄨 󵄨 Assuming that ‖𝑓(𝑧)‖ > ‖𝑓(𝑎)‖, we get |Λ(𝑓(𝑧))| ≤

|Λ(𝑓(𝑎))| + |𝜎𝑎 (𝑧)| . 1 + |Λ(𝑓(𝑎))| |𝜎𝑎 (𝑧)|

82 | 3 Subharmonic behavior and mixed norm spaces 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(𝑧)‖ ≤ This implies

‖𝑓(𝑎)‖ + |𝜎𝑎 (𝑧)| . 1 + ‖𝑓(𝑎)‖ |𝜎𝑎 (𝑧)|

󵄨󵄨 1 − 𝑎𝑧̄ 󵄨󵄨 ‖𝑓(𝑧)‖ − ‖𝑓(𝑎)‖ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ 1. 󵄨󵄨 󵄨 󵄨 󵄨 𝑎 − 𝑧 󵄨󵄨 1 − ‖𝑓(𝑎)‖ ‖𝑓(𝑧)‖ 󵄨󵄨

Letting 𝑧 tend to 𝑎, we obtain |∇‖𝑓‖(𝑧)| ≤

1 − ‖𝑓(𝑎)‖2 . 1 − |𝑎|2

Now, we proceed as in the case of Lemma 3.3 to finish the proof for analytic functions. The case of harmonic functions is similar and we omit the proof. Remark 3.1 (Vector-valued functions). The preceding notions and results can be ex­ tended to vector-valued functions. For example, if 𝑓 is a 𝐶1 -function with values in a Banach space 𝑋, then we call it regularly oscillating in Ω ⊂ ℂ if there is a constant 𝐾 such that 𝐾 sup ‖𝑓(𝑤) − 𝑓(𝑧)‖, 𝐷𝑟 (𝑧) ⊂ Ω. ‖𝑓󸀠 (𝑧)‖ ≤ 𝑟 𝑤∈𝐵𝑟 (𝑧) Here 𝑓󸀠 (𝑧) = 𝑑𝑓(𝑧) denotes the derivative at 𝑧 treated as a real-linear operator from ℂ to 𝑋. It turns out that if 𝑓 is RO, then the functions ‖𝑓(𝑧)‖ and ‖𝑓󸀠 (𝑧)‖, 𝑧 ∈ Ω, are QNS.

Polyharmonic functions A 𝐶∞ -function 𝑓 is polyharmonic of order 𝑘, where 𝑘 is a positive integer, if Δ𝑘 𝑓 ≡ 0. For the theory of polyharmonic functions we refer to Aronszajn–Creese–Lipkin [29]. Theorem 3.8. 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 is QNS. For a complex-valued function 𝑓 = 𝑢 + 𝑖𝑣 we define |∇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 [29]). This im­ plies that 2𝜋

𝑘−1

− 𝑓(𝑟𝑒𝑖𝜃 ) 𝑑𝜃 = ∑ 𝑢𝑚 (0)𝑟2𝑘 , ∫ 0

𝑚=0

3.3 Mixed norm spaces: definition and basic properties

| 83

and hence, in view of the inequality 󵄨󵄨 𝑘−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 |𝑎0 | ≤ 𝐶𝑘 ∫ 󵄨󵄨󵄨 ∑ 𝑎𝑚 𝑟2𝑚 󵄨󵄨󵄨 𝑟 𝑑𝑟, 󵄨󵄨𝑚=0 󵄨󵄨 󵄨 0 󵄨 1

{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 the fact that the partial derivatives of first order are polyharmonic.

3.3 Mixed norm spaces: definition and basic properties 𝑞

3.4 (The class 𝐿 −1 ). For a measurable function 𝐹 on (0, 1), and 0 < 𝑞 ≤ ∞, we write 𝑞 𝑞 𝐹 ∈ 𝐿 −1 = 𝐿 −1 (0, 1) if 1/𝑞

1

‖𝐹‖−1,𝑞

𝑟 𝑑𝑟 ) := (∫ |𝐹(𝑟)| 1 − 𝑟2 𝑞

< ∞.

(3.6)

0

In the case 𝑞 = ∞ this means that 𝐹 ∈ 𝐿∞ (0, 1), and if 𝑞 = ⬦, 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 the class defined by (3.6). The “true” definition reads lim𝑟→1− |𝐹(𝑟)|1[𝑟,1] = 0, where 1𝑆 denotes the characteristic function of 𝑆. 𝑝,𝑞 𝑝,𝑞 3.5 (The spaces ℎ𝑝,𝑞 𝛼 and 𝐻𝛼 ). Let 𝐿 𝛼 (0 < 𝑝, 𝑞 ≤ ∞, 𝛼 ∈ ℝ) denote the space of all Borel measurable functions 𝑢 on 𝔻 such that the function 𝑞

𝐹(𝑟) = (1 − 𝑟2 )𝛼 𝑀𝑝 (𝑟, 𝑢) belongs to 𝐿 −1 (0, 1).

(3.7)

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 − 𝑟2 )𝑀𝑝 (𝑟, 𝑓). 𝛼 0 0, 𝑞 < ∞, and such that ℎ𝑝,𝑞 𝛼 𝑝,𝑞 𝛼 ≤ 0, then 𝐻𝛼 = {0}, because of the increasing property of 𝑀𝑝 (𝑟, 𝑓). Also, 𝐻𝛼𝑝,∞ = {0} if 𝛼 < 0. The analogous facts for ℎ𝑝,𝑞 𝛼 hold but only for 𝑝 ≥ 1. In contrast to the other spaces, the space ℎ𝑝,𝑞 𝛼 (𝑝 < 1, 𝛼 < 0) does not contain 𝑝,𝑞 any nontrivial polynomial and moreover ℎ(𝔻) ∩ ℎ𝑝,𝑞 𝛼 = {0}, therefore in this case ℎ𝛼 is not an admissible space, although the continuous inclusion ℎ𝑝,𝑞 𝛼 ⊂ ℎ(𝔻) holds. See Note 3.5.

A generalized version of Theorem 1.12 Theorem 1.12 is a special case of the following. Theorem 3.10 (Hardy–Littlewood). Let 𝑢 = Re 𝑓, where 𝑓 ∈ 𝐻(𝔻), and Im 𝑓(0) = 0. If 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 𝑢 ∈ ℎ𝑝,𝑞 𝛼 (0 < 𝑝 ≤ ∞, 0 < 𝑞 ≤ ∞, 𝛼 > 0), then 𝑓 ∈ 𝐻𝛼 and ‖𝑓‖𝐿 𝛼 ≤ 𝐶‖𝑢‖𝐿 𝛼 , for some constant 𝐶 independent of 𝑓. We will deduce this theorem from the following assertion valid for all 𝛼 ∈ ℝ. 𝑝,𝑞

󸀠 Proposition 3.2. Let 𝑢 = Re 𝑓, where 𝑓 ∈ 𝐻(𝔻). If 𝑢 ∈ ℎ𝑝,𝑞 𝛼 (𝛼 ∈ ℝ), then 𝑓 ∈ 𝐻𝛼+1 and 󸀠 ‖𝑓 ‖𝐻𝑝,𝑞 ≤ 𝐶‖𝑢‖ℎ𝑝,𝑞 , for some constant 𝐶 independent of 𝑓. 𝛼 𝛼+1

Before the proof note the following consequence. Corollary 3.3. Let 0 < 𝑝 < 1. If either 𝑞 < ∞ and 𝛼 ≤ −1, or 𝑞 = ∞ and 𝛼 < −1, then ℎ𝑝,𝑞 𝛼 = {0}. Proof of Proposition 3.2. Let 𝑝, 𝑞 < ⬦, and 𝜀 = 1/2. We start from the inequality |𝑓󸀠 (0)|𝑝 = |∇𝑢(0)|𝑝 ≤ 𝐶 sup |𝑢(𝑧)| ≤ 𝐶 ∫ |𝑢|𝑝 𝑑𝜏, |𝑧| 𝑝. Since ∫𝐻 (𝑎) 𝑑𝜏 = ∫𝐻 (0) 𝑑𝜏, we can apply Jensen’s inequality to obtain 𝜀 𝜀 from (3.10) 𝑀𝑝𝑞 (|𝑎|, 𝑓󸀠 )(1 − |𝑎|2 )𝑞 ≤ 𝐶 ∫ 𝑀𝑝𝑞 (|𝑧|, 𝑢) 𝑑𝜏(𝑧).

(3.11)

𝐻𝜀 (𝑎)

If 𝑝 > 𝑞, then we replace 𝑝 with 𝑞 in (3.9) and then apply Minkowsky’s 𝐿𝑝/𝑞 -in­ equality to conclude that (3.11) holds in this case as well. Now we multiply (3.11) by (1 − |𝑎|2 )𝑞𝛼−1 , and then integrate over 𝑎 ∈ 𝔻 and use Fubini’s theorem to get ∫ 𝑀𝑝𝑞 (|𝑎|, 𝑓󸀠 )(1 − |𝑎|2 )𝑞(𝛼+1)−1 𝑑𝐴 ≤ 𝐶 ∫ 𝑀𝑝𝑞 (|𝑎|, 𝑢)(1 − |𝑎|2 )𝑞𝛼−1 𝑑𝐴. 𝔻

𝔻

This proves the inequality ‖𝑓󸀠 ‖ℎ𝑝,𝑞 ≤ 𝐶‖𝑢‖ℎ𝑝,𝑞 in the case when 𝑝 < ⬦ and 𝑞 < ⬦. If 𝛼 𝛼+1

𝑞 = ∞, then we use inequalities (3.10) with 𝑞 = 1, and 𝑀𝑝 (|𝑧|, 𝑢) ≤ 𝐶(1 − |𝑧|2 )−𝛼 to get 𝑀𝑝 (|𝑎|, 𝑓󸀠 )(1 − |𝑧|2 ) ≤ 𝐶 ∫ (1 − |𝑧|2 )−𝛼 𝑑𝜏(𝑧) 𝐻𝜀 (𝑎)

≤ 𝐶(1 − |𝑎|2 )−𝛼 ∫ 𝑑𝜏(𝑧), 𝐻𝜀 (𝑎)

which gives the result because ∫𝐻 (𝑎) 𝑑𝜏 = ∫𝐻 (0) 𝑑𝜏. The remaining cases are discussed 𝜀 𝜀 similarly. Remark 3.4 (Local-to-global estimates). In the proof of Proposition 3.2 we used the local estimate (3.11) along with Fubini’s theorem to obtain the “global” inequality (3.3). This operation will be applied several times in the following, and we shall not repeat it every time. In the general case we have the inequality of the form ℎ1 (𝑎) ≤ 𝐶1 ∫ ℎ2 (𝑧)(1 − |𝑧|2 )𝛾 𝑑𝜏(𝑧),

where ℎ1 , ℎ2 ≥ 0.

(3.12)

𝐻𝜀 (𝑎)

Taking into account that 1 − |𝑎|2 ≍ 1 − |𝑧|2 , for 𝑧 ∈ 𝐻𝜀 (𝑎), this inequality can be written in various forms. In any case, (3.12) implies ∫ ℎ1 (𝑎)(1 − |𝑎|2 )𝛽 𝑑𝐴(𝑧) ≤ 𝐶2 ∫ ℎ2 (𝑧)(1 − |𝑧|2 )𝛽+𝛾 𝑑𝐴(𝑧). 𝔻

𝔻

3.3 Mixed norm spaces: definition and basic properties | 87 𝑞

𝑞

In the proof of Proposition 3.2 we had ℎ1 (𝑎) = 𝑀𝑝 (|𝑎|, 𝑓󸀠 )(1 − |𝑎|2 )𝑞 , ℎ2 (𝑧) = 𝑀𝑝 (|𝑧|, 𝑓), 𝛾 = 0, and 𝛽 = 𝑞𝛼 − 1. Sometimes, we shall use the Euclidean disks 𝐸𝜀 (𝑎) = {𝑧 : |𝑧 − 𝑎| < 𝜀(1 − |𝑧|)}, where 0 < 𝜀 < 1. Then the inequality ℎ1 (𝑎) ≤ 𝐶1 ∫ ℎ2 (𝑧)(1 − |𝑧|2 )𝛾 𝑑𝐴(𝑧) 𝐸𝜀 (𝑎)

implies ∫ ℎ1 (𝑎)(1 − |𝑎|2 )𝛽 𝑑𝐴(𝑧) ≤ 𝐶2 ∫ ℎ2 (𝑧)(1 − |𝑧|2 )𝛽+𝛾+2 𝑑𝐴(𝑧). 𝔻

𝔻

When applying Fubini’s theorem in this case, we use the inclusion {𝑎 : |𝑧 − 𝑎| < 𝜀(1 − |𝑎|)} ⊂ 𝐸𝜀/(1−𝜀) (𝑧). Another possibility can occur, namely ∫ ℎ1 (𝑧) 𝑑𝜏(𝑧) ≤ 𝐶 ∫ ℎ2 (𝑧) 𝑑𝜏(𝑧), 𝐻𝛿 (𝑎)

𝐻𝜀 (𝑎)

where 0 < 𝛿 < 𝜀 < 1 are constants. From this, by integration and using Fubini’s theorem on both sides, we obtain ∫ ℎ1 (𝑧) 𝑑𝜏(𝑧) ≤ 𝐶2 ∫ ℎ2 (𝑧) 𝑑𝜏(𝑧). 𝔻

𝔻

The analogous fact for 𝐸𝛿 (𝑎), 𝐸𝜀 (𝑎), and 𝑑𝐴 hold. Remark 3.5. The passage from (3.10) to (3.11), by using Jensen’s and Minkowski’s in­ equalities, is also typical and will be applied several times in the book, with (or with­ out) appealing to this remark. In order to deduce Theorem 3.10 from Proposition 3.2, we need an elementary lemma. Lemma 3.5. Let {𝐴 𝑛}∞ 0 be a sequence of complex numbers and let 𝛾 > 0 and 𝛿 > 0. Then 1/𝛾

∞

( ∑ 2−𝑛𝛿 |𝐴 𝑛|𝛾 )

∞

≤ 𝐶|𝐴 0 |𝛾 + 𝐶 ( ∑ 2−𝑛𝛿 |𝐴 𝑛 − 𝐴 𝑛−1 |𝛾 )

𝑛=0

1/𝛾

,

𝑛=1

where 𝐶 depends only on 𝛿 and 𝛾. Proof. Let 𝛾 ≤ 1. Assuming that 𝐴 𝑛 = 0 for 𝑛 large enough, we have ∞

∞

𝑆 := ∑ 2−𝑛𝛿 |𝐴 𝑛 |𝛾 ≤ |𝐴 0 |𝛾 + ∑ 2−𝑛𝛿 (|𝐴 𝑛 − 𝐴 𝑛−1 |𝛾 + |𝐴 𝑛−1 |𝛾 ) 𝑛=0

𝑛=1 ∞

∞

𝑛=1 ∞

𝑛=0

= |𝐴 0 |𝛾 + ∑ 2−𝑛𝛿 |𝐴 𝑛 − 𝐴 𝑛−1 |𝛾 + ∑ 2−(𝑛+1)𝛿 |𝐴 𝑛|𝛾 = |𝐴 0 |𝛾 + ∑ 2−𝑛𝛿 |𝐴 𝑛 − 𝐴 𝑛−1 |𝛾 + 2−𝛿 𝑆. 𝑛=1

88 | 3 Subharmonic behavior and mixed norm spaces Since 𝑆 < ∞, we see that ∞

(1 − 2−𝛿 )𝑆 ≤ |𝐴 0 |𝛾 + ∑ 2−𝑛𝛿 |𝐴 𝑛 − 𝐴 𝑛−1 |𝛾 , 𝑛=1

which implies the desired inequality for 𝛾 ≤ 1. If 𝛾 > 1, then use Minkowski’s inequal­ ity to complete the proof. Remark 3.6. If 𝛾 = ∞, an analog of Lemma 3.5 holds, namely sup 2−𝑛𝛿 |𝐴 𝑛| ≤ 𝐶|𝐴 0 | + 𝐶 sup 2−𝑛𝛿 |𝐴 𝑛 − 𝐴 𝑛−1 |. 𝑛≥0

𝑛≥1

Also, if lim𝑛→∞ 2−𝑛𝛿 |𝐴 𝑛 − 𝐴 𝑛−1 | = 0, then lim𝑛→∞ 2−𝑛𝛿 |𝐴 𝑛 | = 0. To prove this note first that the hypothesis implies that the sequence 2−𝑛𝛿 𝐴 𝑛 is bounded. Then we have sup2−𝑛𝛿 |𝐴 𝑛 | ≤ sup 2−𝑛𝛿 |𝐴 𝑛 − 𝐴 𝑛−1 | + sup 2−𝑛𝛿 |𝐴 𝑛−1 | 𝑛≥𝑚

𝑛≥𝑚

−𝑛𝛿

= sup 2 𝑛≥𝑚

𝑛≥𝑚

−𝛾

|𝐴 𝑛 − 𝐴 𝑛−1 | + 2

sup 2−𝑛𝛾 |𝐴 𝑛 |.

𝑛≥𝑚−1

Hence 𝐿 := lim sup𝑛→∞ 2−𝑛𝛿 |𝐴 𝑛 | ≤ 0 + 2−𝛾 𝐿. Since 𝐿 ∈ ℝ+ we see that 𝐿 = 0. We also need another elementary fact. We define the maximal function 𝑢× by 𝑢× (𝑟𝜁) =

sup

|𝑢(𝜌𝜁)|,

𝜁 ∈ 𝕋.

𝑟≤𝜌≤(1+𝑟)/2

Proposition 3.3. Let 0 < 𝑝, 𝑞 ≤ ∞, 𝛼 ∈ ℝ, and 𝐾 be a positive constant. An upper × 𝑝,𝑞 semicontinuous function 𝑢 ∈ QNS𝐾 belongs to 𝐿𝑝,𝑞 𝛼 if and only if and only if 𝑢 ∈ 𝐿 𝛼 . Moreover we have ‖𝑢× ‖𝐿𝑝,𝑞 ≤ 𝐶‖𝑢‖𝐿𝑝,𝑞 , where 𝐶 is independent of 𝑢. 𝛼 𝛼 The function 𝑢× is measurable but proving this is irrelevant for us because in applica­ tions we mainly encounter continuous QNS functions; see, however, Remark 3.8 (p. 97) and Note 3.6. ⇒ 𝑢× ∈ 𝐿𝑝,𝑞 Proof. We have to prove the validity of the implication 𝑢 ∈ 𝐿𝑝,𝑞 𝛼 𝛼 . Let 𝑝, 𝑞 < ⬦, and let 𝐷𝜀 (𝑧) = {𝑤 ∈ 𝔻 : |𝑧 − 𝑤| < 𝜀(1 − |𝑧|)}, where 𝜀 < 1. We have 𝑢× (𝑧)𝑝 ≤ 𝐶(1 − |𝑧|)−2 ∫ 𝑢(𝑤)𝑝 𝑑𝐴(𝑤),

where 𝜀 = 3/4.

(3.13)

𝐷𝜀 (𝑧)

Now we proceed in the same way as in the proof of Proposition 3.2 to obtain 𝑀𝑝𝑞 (|𝑧|, 𝑢× ) ≤ 𝐶(1 − |𝑧|)−2 ∫ 𝑀𝑝𝑞 (|𝑤|, 𝑢) 𝑑𝐴(𝑤). 𝐷𝜀 (𝑧)

We have

1 5 (1 − |𝑧|) ≤ 1 − |𝑤| ≤ (1 − |𝑧|) for 𝑧 ∈ 𝐸𝜀 (𝑤), 4 4

(3.14)

3.3 Mixed norm spaces: definition and basic properties |

89

and 𝑀𝑝𝑞 (|𝑧|, 𝑢× )(1 − |𝑧|)𝑞𝛼−1 ≤ 𝐶 ∫ 𝑀𝑝𝑞 (|𝑤|, 𝑢)(1 − |𝑧|)𝑞𝛼−3 𝑑𝐴(𝑤). 𝐷𝜀 (𝑧)

Integrating this inequality over 𝔻, then changing the order of integration, and using (3.14), we get the desired result in the case 𝑝, 𝑞 < ⬦. In other cases the proof is similar and simpler. What we need to prove Theorem 3.10 is the inequality 𝑀𝑝 (𝑟𝑛 , 𝑢× ) ≤ 𝐶 sup 𝑀𝑝 (𝑟, 𝑢),

(3.15)

𝑢 ∈ QNS,

0≤𝑟≤𝑟𝑛+2

where 𝑟𝑛 = 1 − 2−𝑛 , 𝑛 ≥ 0. This is an immediate consequence of (3.13). Proof of Theorem 3.10. Let 𝑟𝑛 = 1 − 2−𝑛 , and 𝑝 < 1. We have, by the increasing property of 𝑀𝑝 , Proposition 3.2, and (3.15), 1

∞

∫ 𝑀𝑝𝑞 (𝑟, 𝑓)(1 − 𝑟2 )𝑞𝛼−1 𝑟 𝑑𝑟 ≤ 𝐶 ∑ 𝑀𝑝𝑞 (𝑟𝑛 , 𝑓)2−𝑛𝑞𝛼 0

𝑛=0 ∞

≤ 𝐶|𝑢(0)|𝑞 + 𝐶 ∑ 2−𝑛𝑞𝛼 (𝑀𝑝𝑝 (𝑟𝑛+1 , 𝑓) − 𝑀𝑝𝑝 (𝑟𝑛 , 𝑓))

𝑞/𝑝

𝑛=0 𝑞

∞

≤ 𝐶|𝑢(0)| + 𝐶 ∑ 2

2

󸀠

∞

0

𝑟𝑛 ≤𝑟≤𝑟𝑛+1 𝑞/𝑝

2𝜋 −𝑛𝑞𝛼 −𝑛𝑞

= 𝐶|𝑢(0)| + 𝐶 ∑ 2

2

󸀠 × 𝑝

𝑖𝜃

(− ∫ (|𝑓 | ) (𝑟𝑛 𝑒 ) 𝑑𝜃)

𝑛=0

0

∞

𝑞/𝑝

≤ 𝐶|𝑢(0)|𝑞 + 𝐶 ∑ 2−𝑛𝑞𝛼 2−𝑛𝑞 (𝑀𝑝𝑝 (𝑟𝑛+2 , 𝑓󸀠 )) 𝑛=0 1 𝑞

≤ 𝐶|𝑢(0)| + 𝐶 ∫ 𝑀𝑝𝑞 (𝑟, 𝑓󸀠 )(1 − 𝑟)𝑞𝛼−1 𝑑𝑟 0 1

≤ 𝐶|𝑢(0)|𝑝 + 𝐶 ∫ 𝑀𝑝𝑞 (𝑟, 𝑢)(1 − 𝑟)𝑞𝛼−1 𝑑𝑟 0 1

≤ 𝐶 ∫ 𝑀𝑝𝑞 (𝑟, 𝑢)(1 − 𝑟)𝑞𝛼−1 𝑑𝑟. 0

𝑖𝜃 𝑝

(− ∫ sup |𝑓 (𝑟𝑒 )| 𝑑𝜃)

𝑛=0

𝑞

𝑞/𝑝

2𝜋 −𝑛𝑞𝛼 −𝑛𝑞

90 | 3 Subharmonic behavior and mixed norm spaces In the last step we used Lemma 3.4. This completes the proof in the case 𝑝 < 1, 𝑞 < ⬦. If 𝑝 ≥ 1, then we use the inequality 1

∫𝑀𝑝𝑞 (𝑟, 𝑓)(1 − 𝑟2 )𝑞𝛼−1 𝑟 𝑑𝑟 0 ∞

𝑞

≤ 𝐶|𝑓(0)|𝑞 + 𝐶 ∑ 2−𝑛𝑞𝛼 (𝑀𝑝 (𝑟𝑛+1 , 𝑓) − 𝑀𝑝 (𝑟𝑛 , 𝑓)) . 𝑛=0

and proceed as above to complete the proof of Theorem 3.10 in the case when 𝑝 ≤ ∞ and 𝑞 < ⬦. For the case 𝑞 ∈ {⬦, ∞}, see Remark 3.6. As a byproduct of the above proofs we have the well-know equivalence ‖𝑓‖𝐻𝛼𝑝,𝑞 ≍ |𝑓(0)| + ‖𝑓󸀠 ‖𝐻𝑝,𝑞 , 𝛼+1

𝛼 > 0, 𝑓 ∈ 𝐻(𝔻).

(3.16)

due to Hardy and Littlewood. Remark 3.7. The Hardy–Littlewood complex maximal theorem (Theorem B.11) can be used to give a slightly shorter proof of Theorem 3.10.

Digression: an asymptotic form of the Hardy–Stein identity The singularity in the Hardy–Stein identity induced by log(1/|𝑧|) may cause technical difficulties, so it is of some importance to replace log(1/|𝑧|) with 1 − |𝑧|. We use (3.16) to prove that this is possible. Lemma 3.6. A function 𝑓 ∈ 𝐻(𝔻) belongs to 𝐻𝑝 (0 < 𝑝 < ⬦) if and only if ̂𝑝 (𝑓) := ∫ |𝑓(𝑧)|𝑝−2 |𝑓󸀠 (𝑧)|2 (1 − |𝑧|2 ) 𝑑𝐴(𝑧) < ∞, 𝐻 𝔻

and we have of 𝑓.

𝑝 ‖𝑓‖𝑝

̂𝑝 (𝑓), where the equivalence constants are independent ≍ |𝑓(0)|𝑝 + 𝐻

Proof. If 𝑝 ≥ 2, then the proof is simple because the function |𝑓|𝑝−2 |𝑓󸀠 |2 is log-subhar­ ̂𝑝 (𝑓). monic, and we can state somewhat more: 𝐻𝑝 (𝑓) ≍ 𝐻 Let 0 < 𝑝 < 2. Multiplying the second identity in Theorem 2.18 by 𝑟2 and then integrating the resulting identity from 𝑟 = 0 to 𝑟 = 1 we get ‖𝑓‖𝑝𝑝 =

𝑝2 ̂ 1 ∫ |𝑓|𝑝 𝑑𝐴 + 𝐻 (𝑓) = (1/𝜋)𝐼1 + (𝑝2 /4𝜋)𝐼2 . 𝜋 4𝜋 𝑝 𝔻

It remains to prove that 𝐼1 ≤ 𝐶(𝐼2 + |𝑓(0)|𝑝 ).

(3.17)

3.3 Mixed norm spaces: definition and basic properties | 91

Assume first that 𝑓 is analytic in a neighborhood of 𝔻. We apply the reverse Hölder’s inequality to get 𝐼2 ≥ ∫ |𝑓(𝑧)|𝑝−2 |𝑓󸀠 (𝑧)|2 (1 − |𝑧|2 )2 𝑑𝐴(𝑧) 𝔻 (𝑝−2)/𝑝 𝑝

2/𝑝 󸀠

≥ (∫ |𝑓(𝑧)| 𝑑𝐴(𝑧))

𝑝

2 𝑝

(∫ |𝑓 (𝑧)| (1 − |𝑧| ) 𝑑𝐴(𝑧))

𝔻

𝔻

≥ 𝑐(|𝑓(0)|𝑝 + 𝐵)(𝑝−2)/𝑝 𝐵2/𝑝 , where

𝐵 = ∫ |𝑓󸀠 (𝑧)|𝑝 (1 − |𝑧|2 )𝑝 𝑑𝐴(𝑧). 𝔻

Here we have used the equivalence ∫ |𝑓|𝑝 𝑑𝐴 ≍ |𝑓(0)|𝑝 + 𝐵,

(3.18)

𝔻

a special case of (3.16). Thus we have 𝐼2 ≥ 𝑐(|𝑓(0)|𝑝 + 𝐵) (

2/𝑝 𝐵 ) . 𝐵 + |𝑓(0)|𝑝

Now we use this inequality to deduce (3.17). If |𝑓(0)|𝑝 ≥ 𝐵, then 𝐼2 + |𝑓(0)|𝑝 ≥ (1/2)(|𝑓(0)|𝑝 +𝐵), whence, by (3.18), we have 𝐼2 ≥ 𝑐1 ∫𝔻 |𝑓|𝑝 𝑑𝐴 ≥ 𝑐1 𝐼1 . Let 𝐵 ≥ |𝑓(0)|𝑝 ≠ 0. Then 𝐼2 ≥ 𝑐(|𝑓(0)|𝑝 + 𝐵) (

|𝑓(0)|𝑝 ) |𝑓(0)|𝑝 + |𝑓(0)|𝑝

2/𝑝

= 𝑐2−2/𝑝 (|𝑓(0)|𝑝 + 𝐵) ≥ 𝑐2 ∫ |𝑓|𝑝 𝑑𝐴 ≥ 𝑐2 𝐼1 , 𝔻

which completes the proof of (3.17). It follows that ‖𝑓‖𝑝𝑝 ≤ 𝐶|𝑓(0)|𝑝 + 𝐶 ∫ |𝑓󸀠 (𝑧)|2 |𝑓(𝑧)|𝑝−2 (1 − |𝑧|) 𝑑𝐴(𝑧) 𝔻

under the above conditions. Applying this to the functions 𝑓𝜌 and letting 𝜌 → 1− we get the same inequality for arbitrary 𝑓 ∈ 𝐻𝑝 .

Harmonic conjugates and self-conjugate spaces Theorem 3.10 can be stated in an equivalent way: Theorem 3.11 (H–L conjugate functions theorem). Under the conditions of Theorem 𝑝,𝑞 3.10, we have that 𝑢 ∈ ℎ𝑝,𝑞 𝛼 if and only if 𝑢̃ ∈ ℎ𝛼 .

92 | 3 Subharmonic behavior and mixed norm spaces Observe that if 1 < 𝑝 < ⬦, this is a consequence of the Riesz conjugate functions theorem. Following Shields–Williams [444], we call a space 𝑋 for which “𝑢 ∈ 𝑋 ⇒ 𝑢̃ ∈ 𝑋” holds a self-conjugate space. Another equivalent form of Theorem 3.10 reads. Theorem 3.12 (H–L projection theorem). Under the conditions of Theorem 3.10, the 𝑝,𝑞 Riesz projection maps ℎ𝑝,𝑞 𝛼 onto 𝐻𝛼 .

3.4 Embedding theorems In contrast to Theorem 2.21, the embedding theorems which will be proved in this section do not depend so much on the hypothesis that the underlying functions are analytic or harmonic; see Propositions 3.4 and 3.7 below. If 𝑓 ∈ 𝐻(𝔻), then 𝑀𝑝 (𝑟, 𝑓) increases with 𝑟 and this can be used to prove that 𝐻𝛼𝑝,𝑞 ⊂ 𝐻𝛼𝑝,𝑞1 ,

𝑞 < 𝑞1 ≤ ∞,

𝑝 > 0,

𝛼 > 0.

The analogous inclusion holds for ℎ𝑝,𝑞 𝛼 , for 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 3.13 (Increasing inclusion theorem). If 𝑞 < 𝑞1 ≤ ∞, and 𝛼 ∈ ℝ, then ℎ𝑝,𝑞 𝛼 ⊂ 𝑝,𝑞 𝑝,⬦ 1 . In particular, ℎ ⊂ ℎ when 𝑞 < ⬦. If 𝛼 > 0, then the inclusions are proper. ℎ𝑝,𝑞 𝛼 𝛼 𝛼 1 Proof. To prove that the inclusions ℎ𝑝,𝑞 ⊂ ℎ𝑝,𝑞 are proper when 𝑞 < 𝑞1 , and 𝛼 > 0, 𝛼 𝛼 consider the functions

𝑓(𝑧) = (1 − 𝑧)−𝛽−1 (log

4 −𝛾 ) 1−𝑧

(see [301, p. 93]). Let 𝑞 < 𝑞1 , and let 𝛽 = 𝛼 + 1/𝑝 − 1, and 𝛾 = 1/𝑞. An elementary but tedious computation shows that 𝑓 ∈ 𝐻𝛼𝑝,𝑞1 \ 𝐻𝛼𝑝,𝑞 . We omit 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 3.4. Let 0 < 𝑞 < 𝑞1 ≤ ∞, and 𝛼 ∈ ℝ. If an upper semicontinuous function 𝑢 × 𝑝,𝑞1 × 𝑝,𝑞 𝑝,𝑞 belongs to QNS𝐾 ∩ 𝐿𝑝,𝑞 𝛼 , then 𝑢 ∈ 𝐿 𝛼 , and we have ‖𝑢 ‖𝐿 𝛼 1 ≤ 𝐶‖𝑢‖𝐿 𝛼 , where 𝐶 is independent of 𝑢. Proof. Let 𝑢 ∈ QNS𝐾 , and 𝑟𝑛 = 1 − 2−𝑛 . The desired result follows from the equivalence ∞

‖𝑢‖

𝑝,𝑞 𝐿𝛼

1/𝑞 −𝑛𝛼𝑞

≍ (∑ 2 𝑛=0

𝑀𝑝𝑞 (𝑟𝑛 , 𝑢× ))

=: 𝑄,

(3.19)

3.4 Embedding theorems | 93

where 𝑄 for 𝑞 = ⬦, ∞ is interpreted in a similar way as in the case of 𝐿𝑝,𝑞 𝛼 . Let 𝑝 < ⬦, 𝑞 < ⬦. Then 𝑞 ‖𝑢‖𝐿𝑝,𝑞 𝛼

𝑟𝑛+1

∞

𝑛(1−𝑞𝛼)

≍ ∑2 𝑛=0

∫ 𝑀𝑝𝑞 (𝑟, 𝑢)𝑟 𝑑𝑟 =: 𝑆. 𝑟𝑛

×

Since obviously 𝑀𝑝 (𝑟, 𝑢) ≤ 𝑀𝑝 (𝑟𝑛 , 𝑢 )(𝑟𝑛+1 − 𝑟𝑛 ), we have 𝑄 ≤ 𝐶 𝑆. 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)2𝑛 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 3.5) 𝑀𝑝𝑞 (𝑟𝑛 , 𝑢× ) ≤ 𝐶2𝑛

∫

𝑀𝑝𝑞 (|𝑤|, 𝑢) 𝑑𝐴(𝑤),

𝑟𝑛−2 ≤|𝑤|≤𝑟𝑛+2

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

(𝑟 → 1− , 𝑛 → ∞),

or what is the same sup 𝑀𝑝 (𝑟, 𝑢) = 𝑜(2𝑛𝛼 ) ⇔ 𝑀𝑝 (𝑟𝑛 , 𝑢× ) = 𝑜(2𝑛𝛼 ).

𝑟𝑛 ≤𝑟≤𝑟𝑛+1

94 | 3 Subharmonic behavior and mixed norm spaces Since sup𝑟𝑛 ≤𝑟≤𝑟𝑛+1 𝑀𝑝 (𝑟, 𝑢) ≤ 𝑀𝑝 (𝑟𝑛 , 𝑢× ), we see that “⇐” holds. The converse holds because of the inequality 𝑟𝑛+2 ×

𝑛

𝑀𝑝 (𝑟𝑛 , 𝑢 ) ≤ 𝐶2 ∫ 𝑀𝑝 (𝑟, 𝑢)𝑟 𝑑𝑟. 𝑟𝑛−2

The case 𝑞 = ∞ is treated in the same way. Finally, note that if 𝑢 ∈ 𝐿𝑝,𝑞 𝛼 ∩ QNS, 𝑞 < ⬦, then 𝑢 ∈ 𝐿𝑝,⬦ , by (3.19). The proof is now complete. 𝛼 Theorem 3.14 (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 3.4. Let 𝑝 < 1. If either 𝛼 ≤ 1 − 1/𝑝 and 𝑞 < ∞ or 𝛼 < 1 − 1/𝑝 and 𝑞 = ∞, then ℎ𝑝,𝑞 𝛼 = {0}. 1,𝑞

1,𝑞

Proof. This follows from the inclusion ℎ𝑝,𝑞 ⊂ ℎ𝛼+1/𝑝−1 , and the fact that ℎ𝛽 = {0} if 𝛼 either 𝑞 < ∞ and 𝛽 ≤ 0 or 𝑞 = ∞ and 𝛽 < 0. Proof of Theorem 3.14. Let 𝑒𝑛 (𝑧) = 𝑧𝑛 , 𝑛 ≥ 0. That the inclusion is strict can be obtained from the relations ‖𝑒𝑛 ‖𝐻𝛼𝑝,𝑞 ≍ 𝑛−𝛼 (𝛼 > 0) and ‖𝑒𝑛 ‖𝐻𝑠,𝑞 ≍ 𝑛−𝛽 by using the closed graph 𝛽

𝑠,𝑞

theorem. The embedding ℎ𝑝,𝑞 𝛼 ⊂ ℎ𝛽 is a consequence of the following proposition. Proposition 3.5. Let 0 < 𝑝 < 𝑠 ≤ ∞, 0 < 𝑞 ≤ ∞, and 𝛼 ∈ ℝ. If an upper semicontinuous 𝑠,𝑞 function 𝑢 belongs to QNS𝐾 ∩ 𝐿𝑝,𝑞 𝛼 , then it belongs to 𝐿 𝛽 , 𝛽 = 𝛼 + 1/𝑝 − 1/𝑠, and we have ‖𝑢‖𝐿𝑝,𝑞 ≤ 𝐶‖𝑢‖𝐿𝑠,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

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

we obtain

1

𝑀𝑠 (𝑟, 𝑢) ≤ 𝐶(1 − 𝑟) 𝑠 and hence

1

𝑀𝑠 (𝑟, 𝑢) ≤ 𝐶(1 − 𝑟) 𝑠 i.e.

1

𝑀𝑠 (𝑟, 𝑢)(1 − 𝑟) 𝑝

− 1𝑠

− 𝑝1

− 𝑝1

𝐵𝑝 (𝑟, 𝑢),

(𝑀𝑝 (2𝑟 − 1, 𝑢× ) + 𝑀𝑝 (𝑟, 𝑢× )) ,

≤ 𝐶𝑀𝑝 (2𝑟 − 1, 𝑢× ) + 𝐶𝑀𝑝 (𝑟, 𝑢× ).

Now the inclusion is obtained by use of Proposition 3.3.

(3.20)

3.5 Fractional integration

| 95

Most properties of QNS functions are independent of the hypothesis that the function is upper semicontinuous. For instance, inequality (3.20) holds for any QNS function. Using the “increasing” property of the integral means of subharmonic functions we obtain from (3.20) the following lemma. Lemma 3.7. 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

1 𝑠−1 ∫ (log(1/𝑡)) 𝑢(𝑡𝑧) 𝑑𝑡, (J𝑠 𝑢)(𝑧) = 𝛤(𝑠)

𝑧 ∈ 𝔻,

0

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

|𝑛| 𝑖𝑛𝜃 ̂ 𝑒 . J𝑠 𝑢(𝑟𝑒𝑖𝜃 ) = ∑ (|𝑛| + 1)−𝑠 𝑢(𝑛)𝑟 𝑛=−∞

The formula J𝑠 J𝜂 𝑓 = J𝑠+𝜂 𝑓 holds whenever 𝑓 is a nonnegative Borel functions. The proof can be reduced to radial functions; then it is enough to verify the formula for 𝑓(𝑟) = 𝑟𝑘 , 𝑘 ≥ 0. Proposition 3.6 (Fractional integration proposition). Let 𝑢 be an upper semicontinu­ 𝑝,𝑞 ous function of class QNS𝐾 , and 𝛼 > 0. The 𝑢 ∈ 𝐿 𝛼+𝑠 implies J𝑠 𝑢 ∈ 𝐿𝑝,𝑞 𝛼 . Moreover, we have ‖J𝑠 𝑢‖𝐿𝑝,𝑞 ≤ 𝐶‖𝑢‖𝐿𝑝,𝑞 , 𝛼 𝛼+𝑠 where 𝐶 is independent of 𝑢. In proving this we use the maximal function 𝑢⋎ (𝑟𝜁) = sup |𝑢(𝑡𝜁)|,

𝜁 ∈ 𝕋.

0≤𝑡≤𝑟

Proposition 3.7. Let 𝛼 > 0. If an upper semicontinuous function 𝑢 belongs to QNS𝐾 ∩ ⋎ 𝑝,𝑞 ⋎ 𝑝,𝑞 𝑝,𝑞 𝐿𝑝,𝑞 𝛼 , then 𝑢 ∈ 𝐿 𝛼 , and ‖𝑢 ‖𝐿 𝛼 ≤ 𝐶‖𝑢‖𝐿 𝛼 , where 𝐶 is independent of 𝑢.

96 | 3 Subharmonic behavior and mixed norm spaces Proof. Let 𝑞 < ⬦. We have, by Lemma 3.5, 1

∞

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

0

∞

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

≤ 𝐶 sup |𝑢(𝑧)|𝑞 + 𝐶 ∑ 2−𝑛𝑞𝛼 𝑀𝑝𝑞 (𝑟𝑛 , 𝑢× ) . |𝑧| 0. Remark 3.8. The requirement of the upper semicontinuity of 𝑢 is needed only to guar­ antee 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 va­ lidity of Propositions 3.4, 3.5, and 3.6 for all 𝑢 ∈QNS. Then, applying this generalized Proposition 3.6 and imitating the proof of Proposition 3.2, we obtain

98 | 3 Subharmonic behavior and mixed norm spaces Proposition 3.8. Let 𝑢 ∈ 𝑂𝐶1𝐾 , and 𝛼 > 0. Then ≍ |𝑢(0)| + ‖ |∇𝑢| ‖𝐿𝑝,𝑞 , ‖𝑢‖𝐿𝑝,𝑞 𝛼 𝛼+1

where the equivalence constants depend only on 𝑝, 𝑞, 𝛼, and 𝐾. An interesting corollary: Corollary 3.5. Let 𝑢 and 𝑣 be regularly oscillating functions and 𝛼 > 0. If |∇𝑢| ≤ |∇𝑣| 𝑝,𝑞 and 𝑣 ∈ 𝐿𝑝,𝑞 𝛼 , then 𝑢 ∈ 𝐿 𝛼 . This corollary can be viewed as a generalization of Theorem 3.11 because if 𝑓 = 𝑢 + 𝑖𝑣 is analytic, then |∇𝑢| = |∇𝑣| = |𝑓󸀠 |. Example 3.2. Let 𝑓(𝑧) = |𝑢(𝑧)| log(1 + |𝑢(𝑧|), where 𝑢 is real valued and harmonic in 𝔻. Then 𝑓 is RO, by Example 3.1. Since |∇𝑓| = |∇𝑢| log(1 + |𝑢|) + |∇𝑢|

|𝑢| , 1 + |𝑢|

we have, by Proposition 3.8, that ∫ |𝑢|𝑝 [(log(1 + |𝑢|)]𝑝 𝑑𝐴 ≍ ∫ |∇𝑢(𝑧)|𝑝 [(log(1 + |𝑢(𝑧)|)]𝑝 (1 − |𝑧|)𝑝 𝑑𝐴(𝑧). 𝔻

𝔻

Exercise 3.2. Let 𝛼 > 0, and 𝐹 : 𝔻 󳨃→ 𝑋 is an analytic function, where 𝑋 is a Banach space. Denote by ‖𝐹‖ the function 𝑧 󳨃→ ‖𝐹(𝑧)‖. Then the following three conditions are 𝑝,𝑞 𝑝,𝑞 󸀠 equivalent: (a) ‖𝐹‖ ∈ 𝐿𝑝,𝑞 𝛼 ; (b) ‖𝐹 ‖ ∈ 𝐿 𝛼+1 ; (c) |∇‖𝐹‖ | ∈ 𝐿 𝛼+1 .

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, ∞

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

(3.21)

𝑛=−∞

The analog of Theorem 3.15 can be proved by using the preceding results: Theorem 3.16 (Tangential differentiation theorem). 𝑝,𝑞 𝑁 (i) If 𝛼 ∈ ℝ and 𝑢 ∈ ℎ𝑝,𝑞 𝛼 , then 𝐷 𝑢 ∈ ℎ𝛼+𝑁 . 𝑝,𝑞 𝑁 𝑝,𝑞 (ii) If 𝛼 > 0, then 𝑢 ∈ ℎ𝑝,𝑞 𝛼 if and only if 𝐷 𝑢 ∈ ℎ𝛼+𝑁 . Moreover, we have ‖𝑢 − 𝑢(0)‖𝐿 𝛼 ≍ 𝑁 ‖𝐷 𝑢‖𝐿𝑝,𝑞 . 𝛼+𝑁

3.6 Weighted mixed norm spaces | 99

In fact, using the H–L projection theorem, we can reduce the proof to the case of 𝐻𝛼𝑝,𝑞 . Then, it is easy to see that the equivalence (3.16) remains true if we replace 𝑓󸀠 by 𝐷1 𝑓; applying this new equivalence 𝑁 times we obtain the desired result. The theorem will be proved again by using the decomposition method; see Corollary 5.2. Another important theorem was originated by Hardy and Littlewood. Here Δ𝑁𝑡 = 𝑁 (Δ 𝑡 ) (𝑡 ∈ ℝ) denotes the symmetric difference operator of order 𝑁; and Δ 𝑡 𝑔(𝜁) = 𝑔(𝜁𝑒𝑖𝑡 ) − 𝑔(𝜁) for 𝑔 ∈ 𝐿𝑝 (𝕋), 𝜁 ∈ 𝕋. 𝑝,𝑞

Theorem 3.17. Let 0 < 𝛼 < 𝑁, where 𝑁 is a positive integer, and 𝑋 = 𝐿 𝑁−𝛼 . If 𝑓 ∈ 𝐻(𝔻) and 𝐷𝑁 𝑓 ∈ 𝑋, then 𝑓 ∈ 𝐻𝑝 and 1 𝑁

‖𝐷 𝑓‖𝑋 ≍ (∫ ( 0

‖Δ𝑁𝑡 𝑓‖𝑝 𝑡𝛼

𝑞

𝑑𝑡 ) ) 𝑡

1/𝑞

=: 𝐾(𝑓).

Conversely, if 𝑓 ∈ 𝐻𝑝 and 𝐾(𝑓) < ∞, then 𝐷𝑁 𝑓 ∈ 𝑋. If 𝑝 ≥ 1, then the analogous assertions for 𝑓 ∈ ℎ(𝔻) hold. See Hardy–Littlewood [185, Theorem 3], [186, Theorem 23], and [191, Theorem 48], for the case 𝑁 = 1. The case 𝑝 < 1 for analytic functions was treated by Gwilliam [179]. The case when 𝑁 = 2, and 𝑝 = 𝑞 = ∞ is of special interest and was treated by Zyg­ mund [534]. A proof in the general case can be found in [350]. We postpone the discussion on this theorem to Chapters 8 and 9, where more gen­ eral results will be proved.

3.6 Weighted mixed norm spaces Classes of real functions 3.6 (Almost monotone functions). According to Bernstein [54], 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], or 𝐽 = [1, ∞), or 𝐽 = (0, ∞). If 𝜓 is such a functions, then in most cases we assume that there are real numbers 𝛼 and 𝛽 (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 with (‡); this “Δ 2 -condition” occurs naturally in the theory of Orlicz spaces [278]. The following statement enables us to replace 𝜓 with a function that has better properties. Proposition 3.9. If 𝜓 possesses the above properties, then there exists a function 𝜓2 ≍ 𝜓 on 𝐽 such that 𝜓2 (𝑥)/𝑥𝛼 is increasing and 𝜓2 (𝑥)/𝑥𝛽 is decreasing on 𝐽.

(3.22)

100 | 3 Subharmonic behavior and mixed norm spaces Proof. Consider the case 𝐽 = [1, ∞). We define 𝜓1 (𝑥) = 𝑥𝛽 sup 𝑡≥𝑥

𝜓(𝑡) 𝜓(𝑡𝑥) = inf 𝛽 , 𝑡≥1 𝑡𝛽 𝑡

𝜓0 (𝑥) = 𝑥𝛼 inf 𝑡≥𝑥

𝜓1 (𝑡) 𝜓 (𝑥𝑡) = inf 1 𝛼 . 𝑡≥1 𝑡𝛼 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. 3.7 (The classes Δ[𝛼, 𝛽]). We write “𝜓 ∈ Δ[𝛼, 𝛽] on 𝐽” if 𝜓 satisfies (3.22), and 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.

(3.23)

If an absolutely continuous function satisfies (3.23), then it belongs to Δ[𝛼, 𝛽]. 3.8 (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 in [442]. We call 𝜓 subnormal if 𝜓 ∈ Δ[0, 𝛽] for some 𝛽 > 0. The notions of almost normal and almost subnormal functions are defined in an obvious way. 3.9 (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 called the Hardy class and is denoted by (H), and is called the Hardy field¹. A remark­ able result of Hardy (see [74, 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 𝐿 𝜓 := lim𝑥→∞ 𝑥𝜓󸀠 (𝑥)/𝜓(𝑥) exists. If 𝐿 𝜓 > 0 is finite, then 𝜓 is normal near infinity and can be changed on some [0, 𝑡0 ] to be normal on [0, ∞). 𝑝,𝑞

3.10. For a positive continuous function 𝜙 on (0, 1] define the space 𝐿 𝜙 by replac­ ing (1 − 𝑟2 )𝛼 in (3.7) with 𝜙(1 − 𝑟). The corresponding spaces of analytic or harmonic functions are defined by 𝑝,𝑞

𝑝,𝑞

𝐻𝜙 = 𝐿 𝜙 ∩ 𝐻(𝔻)

𝑝,𝑞

𝑝,𝑞

and ℎ𝜙 = 𝐿 𝜙 ∩ ℎ(𝔻).

(3.24)

𝛼

In this context the function 𝜙 is called a weight. If 𝜙(𝑥) = 𝑥 for some 𝛼 > 0, then 𝜙 is called a standard weight and the corresponding spaces are called standard mixed 𝑝,∞ norm spaces, so the Hardy space 𝐻𝑝 = 𝐻0 is not a standard mixed norm space. If 𝜙 is normal (see Section 3.7), then most of the results of this chapter remain valid. In particular:

1 The “Hardy field” as defined in [74] is much larger.

3.6 Weighted mixed norm spaces | 101

Theorem 3.18. If 𝑓 ∈ 𝐻(𝔻) and 𝜙 is normal, then the following conditions are equiva­ 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 lent: (a) 𝑓 ∈ 𝐻𝜙 ; (b) 𝑓󸀠 ∈ 𝐻𝜓 , where 𝜓(𝑥) = 𝑥𝜙(𝑥); (c) Re 𝑓 ∈ ℎ𝜙 . 𝑝,𝑞

Corollary 3.6. If 𝜙 is normal, then ℎ𝜙 is self-conjugate. 𝑝,𝑞

Corollary 3.7. If 𝜙 is normal, then the Riesz projection is bounded on ℎ𝜙 . The implication (c) ⇒ (b) is independent of the hypothesis that 𝜙 is normal. Namely, we can assume that 𝜙 ∈ Δ[𝛼, 𝜂] (see Section 3.7). Then from (3.11) we get 𝑀𝑝𝑞 (|𝑎|, ℎ)(1 − |𝑎|2 )𝑞 ≤ 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 3.4. To prove that (b) implies (a), with the hypothesis that 𝜙 is normal, we need to make somewhat more changes to the proof of Theorem 3.10. Actually, we need to show that Lemma 3.5 remains true if 2−𝑛𝛾 is replaced with 𝜙(2−𝑛 ), and then replace 2−𝑛𝛼 with 𝜙(2−𝑛 ) in the chain of inequalities on p. 89. That is all. 𝑝,𝑞 That Propositions 3.3 and 3.7 hold for 𝐿 𝜙 can be proved by application of these assertions to the QNS function 𝑢(𝑧)𝜙(1 − |𝑧|)/(1 − |𝑧|)𝛾 , where 0 < 𝛾 < 𝛼. A generalization of Proposition 3.6 reads: 𝑝,𝑞

Proposition 3.10. If an upper semicontinuous QNS function 𝑢 belongs to 𝐿 𝜓 , 𝜓(𝑡) = 𝑝,𝑞 𝜙(𝑡)𝑡𝑠 , where 𝜙 is normal and 𝑠 > 0, then J𝑠 𝑢 belongs to 𝐿 𝜙 . Proof. Let 𝜔(𝑡) = 𝜙(𝑡)/𝑡𝛾 , 0 < 𝛾 < 𝛼, where 𝜙(𝑡)/𝑡𝛼 increases with 𝑡. Then 𝜔(𝑡) is in­ creasing 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 3.6 (applied to 𝐿𝑝,𝑞 𝛾 ) and the fact that the func­ tion 𝑢(𝑧)𝜔(1 − |𝑧|) is QNS. From this we can deduce a generalized version of Theorem 3.15. Theorem 3.19. Let 𝜙 be a normal function, 𝑠 > 0, and 𝜓(𝑡) = 𝑡𝑠 𝜙(𝑡). Then a function 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 𝑢 ∈ ℎ(𝔻) is in ℎ𝜓 if and only if J𝑠 𝑢 ∈ ℎ𝜙 . Equivalently, 𝑢 ∈ ℎ𝜙 if and only if J𝑠 𝑢 ∈ ℎ𝜓 . Exercise. If 𝑢 ∈ QNS(𝔻) and 𝜙 is subnormal on [0, 1] then 𝑢(𝑧)𝜙(1 − |𝑧|) ∈ QNS(𝔻).

102 | 3 Subharmonic behavior and mixed norm spaces 3.6.1 Lacunary series in mixed norm spaces As in the case of 𝐻𝑝 -spaces, membership in 𝐻𝛼𝑝,𝑞 of a function with Hadamard gaps is independent of 𝑝. 𝑘𝑛 Theorem 3.20. Let 𝜙 be a normal weight. If 𝑓(𝑧) = ∑∞ 𝑛=0 𝑐𝑛 𝑧 , where {𝑘𝑛 } is lacunary. 𝑝,𝑞 𝑞 Then 𝑓 ∈ 𝐻𝜙 if and only if {𝑐𝑛 𝜙(1/𝑘𝑛 )} ∈ ℓ .

Proof. In one direction, we have 1

1

𝑞

∞ 𝑑𝑟 𝑑𝑟 𝑑𝑟 ≤ ∫ ( ∑ |𝑐𝑛 |𝑟𝑘𝑛 𝜙(1 − 𝑟)) , ∫ (𝑀𝑝 (𝑟, 𝑓)𝜙(1 − 𝑟)) 1−𝑟 1 −𝑟 𝑛=0 𝑞

0

0

which via Lemma 3.11 proves the “if” part. In the other direction, we use Paley’s the­ orem and the mentioned lemma to finish the proof of the theorem in the case 𝑞 < ⬦. The other cases are discussed in the same way. As it is seen, this result is rather elementary and can be easily deduced from some results of [327], although special cases can be found in many other papers. We mention 𝑝,𝑝 here that a deeper result when 𝑝 = 𝑞 holds. The space ℎ𝜙 is called a diagonal weighted space. Theorem 3.21. Let 0 < 𝑝 ≤ ∞ and 𝜁 ∈ 𝕋. Then, for 𝑓 as above, 1/𝑝

1

𝑑𝑟 (∫ (|𝑓(𝑟𝜁)|𝜙(1 − 𝑟)) 𝑑𝑟) 1−𝑟 𝑝

≍ ‖{𝑐𝑛 𝜙(1/𝑘𝑛 )}‖ℓ𝑝 ,

0

where the equivalence constants are independent not only of 𝑓 but also of 𝜁. This is a consequence of a generalized version of a theorem of Gurariy and Mat­ saev [177]; see Theorem 3.26. As an immediate consequence of Theorem 3.21 is an improved version of a result of Anderson and Girela [25, Theorems 6, 7]. In the case where 𝑝 = 1 and 𝜑(𝑥) ≡ 1, the theorem is due to Zygmund [533] and appears to be the first result of this kind; in fact, it coincides with the case 𝑝 = 1 of the Gurariy– Matsaev theorem. Zygmund’s theorem was generalized in a more serious direction by Gnuschke and Pommerenke [168].

3.6.2 Bergman spaces with rapidly decreasing weights A positive function 𝜑 ∈ 𝐿1 (0, 1) 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 which 1

∫ 𝑀𝑝𝑞 (𝑟, 𝑓)𝜑(𝑟)𝑟 𝑑𝑟 < ∞. 0

3.6 Weighted mixed norm spaces | 103

If 𝑝 = 𝑞, this is the same as ∫ |𝑓(𝑧)|𝑝 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 𝑝 way. The “standard” Bergman space 𝐴 𝛼 (𝛼 > −1) is defined by choosing 𝜑(𝑟) = (1−𝑟2 )𝛼 . The function 1

1 D𝜑 (𝑟) = ∫ 𝜑(𝑡) 𝑑𝑡 𝜑(𝑟) 𝑟

is called the distortion function of 𝜑. Assume that 1

𝜑󸀠 (𝑟) 𝜑󸀠 (𝑟) D𝜑 (𝑟) = sup ∫ 𝜑(𝑥) 𝑑𝑥 ≤ 𝐿, sup 2 0 0 are real 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 𝑞

𝑝

≍ |𝑓(0)| + ∫ (𝑀𝑝 (𝑟, 𝑓󸀠 )(1 − 𝑟)𝛼+1 ) exp (− 0

𝜅 ) 𝑑𝑟. (1 − 𝑟)𝛼

This is unusual because in “usual” cases 𝑀𝑝 (𝑟, 𝑓) behaves like 𝑀𝑝 (𝑟, 𝑓󸀠 )(1 − 𝑟), while here 𝑀𝑝 (𝑟, 𝑓) is transformed into 𝑀𝑝 (𝑟, 𝑓󸀠 )(1 − 𝑟)𝛼+1 . It is not easy to guess what is the analog to Theorem 3.22 when 𝑞 = ∞. An earlier result of the author helps to reformulate Theorem 3.22 in, maybe, clearer way. Theorem 3.23 (Pavlović [371]). Let 0 < 𝑝 ≤ ∞ and let 𝜔 > 0 be a 𝐶2 -function on [0, 1), strictly increasing near 1, such that lim sup 𝑟→1−

𝜔󸀠󸀠 (𝑟)𝜔(𝑟) < ∞. 𝜔󸀠 (𝑟)2

(3.28)

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 ∈ 𝐶1 (0, 1), let 1

1 = 𝑞 ∫ 𝜑(𝑥) 𝑑𝑥, 𝜔(𝑟)𝑞

0 < 𝑞 < ⬦.

𝑟

Then the condition (3.26) is equivalent to (3.28). Define the measure 𝑑𝑚𝜔 on (0, 1) by 𝑑𝑚𝜔 (𝑟) =

𝜔󸀠 (𝑟) 𝑑𝑟. 𝜔(𝑟)

Now Theorems 3.22 and 3.23 can be stated in a unique way. Theorem 3.24 (with the hypotheses of Theorem 3.23). 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 (𝑑𝑚𝜔 ) ≍ |𝑓(0)| + ‖𝐹2 ‖𝐿𝑞 (𝑑𝑚𝜔 ) .

3.6 Weighted mixed norm spaces | 105

The proof of this fact is rather long and is based on similar ideas as the proof of Theo­ rem 3.10, but a more complicated technique is involved. These are reasons for which we omit it. It was proved in [371, 385] that under condition (3.28) the space A𝜑𝑝,𝑞 is self-con­ jugate for 𝑞 ≥ 1. If (3.28) 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 , where 𝜑 is given by (3.27) (𝜅 > 0), is self-conjugate.

3.6.3 Mixed norm spaces with subnormal weights Let 𝜙 be a subnormal weight on (0, 1). We consider two types of spaces: 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 – The spaces ℎ𝜙 = ℎ(𝔻) ∩ 𝐿 𝜙 and 𝐻𝜙 ∩ 𝐿 𝜙 have been defined in Section 3.10; see (3.24). These spaces occur in the study of Lipschitz spaces in a natu­ ral way (see Chapter 9). If 𝜙 is normal, then they are nontrivial and moreover infinite-dimensional because 1/𝑞

1

𝑑𝑡 ) (∫ 𝜙(𝑡) 𝑡 𝑞

(3.29)

0) if 1

∫ 𝑥

𝜙(𝑡) 𝜙(𝑡) 𝑑𝑡 ≤ 𝐶 𝑚 , 𝑡𝑚+1 𝑥

0 < 𝑥 ≤ 1.

(3.31)

It turns out that condition (3.30) forces 𝜙 to have better “increasing” property, while “𝑏𝑚 -condition” forces 𝜙(𝑡) to growth strictly slower than 𝑡𝑚 . Proposition 3.11. (i) The function 𝜙 satisfies (3.30) if and only if there is a real constant 𝛼 > 0 such that 𝜙(𝑡)/𝑡𝛼 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)/𝑡𝛽 is almost decreasing. Consequently, if 𝜙 satisfies (3.30) and (3.31), then there is 𝜙1 ∈ Δ(0, 𝑚) such that 𝜙1 ≍ 𝜙 (see Section 3.7, p. 100). Such assertions are often encountered in the theory of regularly varying functions (cf. [22, 437]). Proof. Proof of (i). The “if” part is too simple. In the other direction, (3.30) implies 𝜙(𝜆𝑥) log(1/𝜆) ≤ 𝐶𝜙(𝑥),

0 < 𝜆 < 1, 0 < 𝑥 ≤ 1.

Choose 𝜆 so small 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 so 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 )

where 𝛼 = −1/ log 𝜆 > 0. This concludes the proof of (i).

2 We use the term “Dini” in a different sense; see p. 242.

𝜙(𝑥) = 𝑒 𝜂𝛼 𝜙(𝑥),

108 | 3 Subharmonic behavior and mixed norm spaces Proof of (ii). Again, the “if” part is simple. In the other direction, we have 1

𝐹(𝑥) := ∫ 𝑥

1

𝜙(𝑡) 1 𝑑𝑡 ≥ 𝜙(𝑥) ∫ 𝑡−𝑚−1 𝑑𝑡 ≥ 𝜙(𝑥)𝑥−𝑚 , 𝑡𝑚+1 𝑚

0 < 𝑥 ≤ 1.

𝑥

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 (𝑥𝑏 𝐹(𝑥))󸀠 shows that we can take 𝑏 = 1/𝐶, where 𝐶 satisfies (3.31). Finally, we mentioned 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 sup ∫ |𝑓󸀠 (𝑧)|2 𝐾 (1 − |𝜎𝑎 (𝑧)|2 ) 𝑑𝐴(𝑧) < ∞. 𝑎∈𝔻

𝔻

For information, results, and references we refer to [505]. We note that, under the above conditions on 𝐾, the space 𝑄𝐾 contains all polynomials; read at least the first page of [505]. Given such a 𝐾, the auxiliary function 𝜑𝐾 is defined by 𝜑𝐾 (𝑠) = sup

0 0. Although these conditions look nice, their verification can cause techni­ cal difficulties. However, it is not hard to prove that (3.32)(1) is equivalent to the exis­ tence of 𝜀 > 0 such that 𝐾(𝑥)/𝑥𝛾−1+𝜀 is almost increasing in 𝑥 ∈ (0, 1], while (3.32)(2) is equivalent to the existence of 𝛽 < 𝛾 such that 𝐾(𝑥)/𝑥𝛽 is almost decreasing in 𝑥. In both cases the “if” part is easy. In proving the “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 3.11(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 of this consideration we have: If 𝐾 satisfies (1), with 𝛾 ≥ 1, and 𝜑𝐾 (2) < ∞, then there is a normal function 𝐾1 (𝑡), 0 < 𝑡 ≤ 1, such that 𝐾1 ≍ 𝐾; if 𝛾 < 1, then then we can only guarantee that 𝐾1 is subnormal. The reader is referred to Blasco’s paper [66] for various types of weight functions and a list of papers where they were introduced.

3.7 𝐿𝑞 -integrability of lacunary power series |

109

𝑝,𝑞

Isomorphic classification of 𝐻𝜙 That the ordinary Bergman space 𝐴𝑝 = 𝐴𝑝𝜑 (𝜑 ≡ 1) is isomorphic to ℓ𝑝 (1 < 𝑝 < ⬦) was proved by Lindenstrauss and Pełczińsky (Theorem A.3) by using the bounded­ ness of the Bergman projection and the so-called Pełczińsky’s decomposition method; for the latter see [391] and [298, p. 54]. This method is nonconstructive. Shields and Williams [442] used the same method to prove that 𝐻𝜙∞,∞ , resp. 𝐻𝜙1,1 (𝜙 normal) is iso­ morphic to ℓ∞ , resp. ℓ1 ; in [443], they proved that ℎ1,1 𝜙 , where 𝜙 is subnormal, is isomor­ 𝑝,𝑞

phic to ℓ1 . In [327], an explicit isomorphism was constructed between 𝐻𝜙 (1 < 𝑝 < ⬦, 𝜙 normal) and the space ℓ(𝑝, 𝑞), the direct ℓ𝑞 -sum of the sequence of 𝑛-dimensional ℓ𝑝 -spaces. This proof works for subnormal weights also; recently the author realized that the same method was used by Lizorkin [305] seven years before [327] so this re­ sult should be attributed to him. Wojtaszczyk [501] used spline systems to construct an isomorphism from 𝐴𝑝 onto ℓ𝑝 for 0 < 𝑝 ≤ 1. In [502], he constructed and isomor­ 1,𝑞 phism from 𝐻𝜙 onto ℓ(1, 𝑞), where 𝜙 is normal; a duality argument shows that then ∞,𝑞 𝐻𝜙 is isomorphic to ℓ(∞, 𝑞) for 𝑞 ≥ 1. However, Lusky went furthest in isomorphic 𝑝,𝑞

classification of 𝐻𝜙 . One of his results [310, Corollary 2.7] states the following: Theorem 3.25 (Lusky). Let 𝜙 be subnormal, 1 ≤ 𝑞 ≤ ∞ and 𝑝 ∈ {1, ∞}. The following 𝑝,𝑞 𝑝,𝑞 three conditions are equivalent: (a) 𝐻𝜙 is isomorphic to ℓ(𝑝, 𝑞); (b) ℎ𝜙 is self-conju­ gate; (c) 𝜙 is normal. The equivalence (b) ⇔ (c) had been proved in [360, Part II, Theorem 3.3] for all 𝑞 > 0. 𝑝,𝑞 𝑝,𝑞 In contrast to 𝐻𝜙 , the space ℎ𝜙 (𝜙 subnormal) is isomorphic to ℓ(𝑝, 𝑞) for 𝑝 ∈ {1, ∞} (see [310, Corollary 2.6]), which generalizes the above-mentioned result of Shields and Williams. It follows from [312, Corollary 1.3] that if 𝜙 is subnormal but is not normal, then 𝐻𝜙∞,∞ is isomorphic to 𝐻∞ . For further investigations in this direction, see [197, 311, 313], where 𝐿1 -spaces and 𝐿∞ -spaces with exponential weights are considered.

3.7 𝐿𝑞 -integrability of lacunary power series Let

∞

𝐺(𝑟) = ∑ 𝑎𝑛 𝑟𝑛 ,

0 < 𝑟 < 1,

𝑛=0

where 𝑎𝑛 ≥ 0 and the series converges for 0 < 𝑟 < 1. If 𝜙 is a strictly increasing, subnormal function on (0, 1], and {𝐵𝑛}∞ 𝑛=0 a sequence of positive real numbers such that 𝐵1 ≤ 𝜙(1), and 𝐵 𝐵 1 < inf 𝑛 ≤ sup 𝑛 < ∞, 𝑛≥1 𝐵𝑛+1 𝐵 𝑛≥1 𝑛+1 then there exists a unique sequence {𝜆 𝑛}∞ 1 such that 𝜙(1/𝜆 𝑛 ) = 1/𝐵𝑛 for 𝑛 ≥ 1. Then let 𝐼0 = 𝐼0 (𝜙, 𝐵) = [0, 𝜆 1 ),

and 𝐼𝑛 = 𝐼𝑛 (𝜙, 𝐵) = [𝜆 𝑛 , 𝜆 𝑛+1 )

for 𝑛 ≥ 1.

110 | 3 Subharmonic behavior and mixed norm spaces This notation will be used throughout this section, unless otherwise is specified. The following lemma appears to be useful in the so-called blocking technique [176]. Lemma 3.8. Let 0 < 𝑞 ≤ ∞, and let 𝜙 be normal. With the above notation, we have 1/𝑞

1

𝑑𝑟 ) (∫[𝐺(𝑟)𝜙(1 − 𝑟)] 1−𝑟 𝑞

1/𝑞

𝑞

∞

≍ ( ∑ (𝜙(1/𝜆 𝑛 ) ∑ 𝑎𝑘 ) ) 𝑛=0

0

,

𝑗∈𝐼𝑛

where the equivalence constants are independent of 𝐺. If 𝑞 = ∞, this is to be interpreted as sup 𝐺(𝑟)𝜙(1 − 𝑟) ≍ sup ∑ 𝜙(1/𝜆 𝑛 )𝑎𝑗 . 0 1 and 𝐾 > 1 are constants. This implies that the last quantity is ≤ 𝐶𝐵𝑗 ≤ 𝐶/𝜙(1 − 𝑟). We are done.

112 | 3 Subharmonic behavior and mixed norm spaces Lemma 3.12 is a generalization of Lemma 3.9. There is, however, a substantial im­ provement: Lemma 3.11 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 Mat­ saev [177], 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 3.26. Let 𝜙 be normal function on [0, 1], and L(𝑟) = ∑∞ 𝑛=0 𝑐𝑛 𝑟 (𝑐𝑛 ∈ ℂ) a series converging for 𝑟 ∈ (0, 1), where {𝜆 𝑛} (𝜆 0 ≥ 1) is a lacunary sequence of real numbers. Then ‖L(𝑟)𝜙(1 − 𝑟)‖𝐿𝑝 ≍ ‖{𝜙(1/𝜆 𝑛)𝑐𝑛 }‖ℓ𝑝 , −1

where the equivalence constants are independent of {𝑐𝑛 }. An even more general fact will be proved later on; see Theorem A.8.

3.7.1 Lacunary series in 𝐶[0, 1] We continue to denote by {𝑐𝑛 } a sequence of complex numbers, and consider the series ∞

L(𝑟) = ∑ 𝑐𝑛 𝑟𝜆 𝑛 , 𝑛=0

where 𝜆 𝑛 is a lacunary sequence, i.e. a sequence satisfying inf

𝑛≥0

𝜆 𝑛+1 = 𝜌 > 1. 𝜆𝑛

(3.35)

The following theorem was established by Hardy and Littlewood [183]. Theorem 3.27. If there exists the limit 𝑆 := lim𝑟→1− L(𝑟), and is finite, then the series ∑∞ 𝑛=0 𝑐𝑛 converges and its sum is equal to 𝑆. Proof. First we prove that the hypotheses imply sup |𝑐𝑛 | ≤ 𝐶 𝑀, 𝑛≥0

where 𝑀 = sup0 −1) if and only if 𝑝,𝑞 𝑢𝑗 ∈ 𝐿 𝛼+𝑗 for all 𝑗. For further results in this direction, see [55, 118, 204]. 𝑝,𝑞 3.5 (The space ℎ𝑝,𝑞 𝛼 , 𝑝 < 1). It follows from Corollaries 3.3 and 3.4 that ℎ𝛼 = {0} in the following cases: (1) 𝑞 ≤ ⬦, 𝛼 ≤ 1 − 1/𝑝, 1/2 ≤ 𝑝 < 1. (2) 𝑞 ≤ ⬦, 𝛼 ≤ −1, 0 < 𝑝 < 1/2. (3) 𝑞 = ∞, 𝛼 < 1 − 1/𝑝, 1/2 < 𝑝 < 1. (4) 𝑞 = ∞, 𝛼 < −1, 0 < 𝑝 < 1/2. (5) 𝑞 = ∞, 𝛼 < −1, 𝑝 = 1/2. The estimates of the integral means of the Poisson kernel (Proposition 1.2) show that in cases (1)–(4) the bounds of 𝛼 are optimal; this means that in the following cases the space is nontrivial and actually 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. Using Aleksandrov’s results (see Problem 2.3), one proves that ℎ1/2,∞ = {0} and −1 moreover, if a function 𝑢 ∈ ℎ(𝔻) satisfies

𝑀1/2 (𝑟, 𝑢) = 𝑜(1 − 𝑟) (log

2 2 ) , 1−𝑟

𝑟 → 1− ,

then 𝑢 ≡ 0. 3.6 (Maximal functions). The proofs of Propositions 3.3 and 3.7 can be modified to show that they remain valid if we replace 𝑢× and 𝑢⋎ by the functions 𝑢#(𝑧) = sup |𝑢(𝑤)|,

where 𝜀 = 𝛿(1 − |𝑧|)

(0 < 𝛿 < 1),

𝑤∈𝐷𝜀 (𝑧)

and 𝑢& (𝑧) = sup |𝑢(𝑤)|, 𝑤∈𝐵𝑧

𝐵𝑟𝜁 = {𝑤 : |𝑤| < (1 + 𝑟)/2} ∩ 𝑈𝜁,𝑐 ,

where, recall, 𝑈𝜁,𝑐 is the open convex hull of the set 𝑐𝔻 ∪ {𝜁}, 𝜁 ∈ 𝕋. These functions are measurable for any 𝑢 (measurable or not) and, moreover, they are lower semicon­ tinuous, i.e. 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. 3.7 (Spaces with rapidly decreasing weights). The first result on spaces with rapidly decreasing weights is Theorem 3.23 in [371], where it was also proved that the space ℎ𝑝,∞ [𝜔] := {𝑢 ∈ ℎ(𝔻) : 𝑀𝑝 (𝑟, 𝑢) = O(𝜔(𝑟))}

3.7 Further notes and results |

117

is self-conjugate if (3.28) holds. The notion of the distortion function was introduced by Siskakis [447], who proved Theorem 3.22 for 1 ≤ 𝑝 = 𝑞 < ⬦ but under three hy­ potheses, the most complicated of which is that for all sufficiently small 𝛿 > 0 lim sup 𝑟→1−

𝜑(𝑟) < ∞. 𝜑(𝑟 + 𝛿D𝜑 (𝑟))

The other two read 1

𝐴 ∫ 𝜑(𝑥) 𝑑𝑥, 1−𝑟

and 𝜑󸀠 (𝑟) ≤

𝑟

𝐵 𝜑(𝑟), 1−𝑟

(3.39)

where 𝐴 and 𝐵 are positive constant independent of 𝑟 ∈ (0, 1). It is easy to show that (3.39) implies (3.25) with 𝐿 = 𝐵/𝐴. 3.8 (Characterization of Bergman spaces). The following statement was proved in [285], under the hypothesis 𝑓(0) = 0, and in [388], for an arbitrary 𝑓. Let 0 < 𝑝 < ⬦, 0 < 𝑞 < 𝑝 + 2, 𝛽 > −1, and 𝑓 ∈ 𝐻(𝔻). Then ∫ |𝑓|𝑝 (1 − |𝑧|2 )𝛽 𝑑𝐴 ≍ |𝑓(0)|𝑝 + ∫ |𝑓|𝑝−𝑞 |𝑓󸀠 |𝑞 (1 − |𝑧|2 )𝛽+𝑞 𝑑𝐴, 𝔻

𝔻

where the equivalence constants are independent of 𝑓. In fact, in [388] an analog for several variables was proved, which includes the or­ dinary and the invariant gradients. The paper [379] contains an analogous result for (complex-valued) harmonic functions on the real ball. 3.9 (Several variables). The definitions of quasi-nearly subharmonic and regularly oscillating functions in a domain Ω ⊂ ℝ𝑛 is similar to that in the case 𝑛 = 2: the disk 𝐷𝑟 (𝑎) should be replaced by the ball 𝐵𝑟 (𝑎) = {𝑥 ∈ ℝ𝑛 : |𝑥 − 𝑎| < 𝑟}; the unit disk by the unit ball 𝔹𝑛 ; 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 3.2, continue to be true in this general situation.

4 Taylor coefficients with applications In this chapter, we present Hardy–Littlewood’s theorems on the Taylor coefficients of an 𝐻𝑝 -function and applications of these theorems to the partial sums and the ̂ 𝑝 ≤ Cesàro means of the Taylor series. In particular, we use the inequality |𝑓(𝑛)| 𝑝 𝑐(𝑛 + 1)1−𝑝 ‖𝑓‖𝑝 (0 < 𝑝 < 1) and the Hardy–Littlewood complex maximal theorem to prove the Hardy–Littlewood–Sunouchi theorem. Some improvements of the coeffi­ cient theorems are also presented.

4.1 Using interpolation of operators on 𝐻𝑝 Beside the Housdorff–Young theorem and the inequality ̂ |𝑓(𝑛)| ≤ 𝐶𝑝 (𝑛 + 1)1/𝑝−1 ‖𝑓‖𝑝 ,

𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 ≤ 1

(4.1)

(see Corollary 1.24), there is another very important result on the coefficients of an 𝐻𝑝 -function due to Hardy and Littlewood [184]. Theorem 4.1 (Hardy–Littlewood Σ-inequality). If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 ≤ 2, then ∞

̂ 𝑝 ≤ 𝐶 ‖𝑓‖𝑝 . 𝐾𝑝 (𝑓) := ∑ (𝑛 + 1)𝑝−2 |𝑓(𝑛)| 𝑝 𝑝

(4.2)

𝑛=0 𝑝

If 𝐾𝑝 (𝑓) < ∞, for some 𝑝 ≥ 2, then 𝑓 ∈ 𝐻𝑝 and ‖𝑓‖𝑝 ≤ 𝐶𝑝 𝐾𝑝 (𝑓). 𝑝

The inequality ‖𝑓‖𝑝 ≤ 𝐶𝑝 𝐾𝑝 (𝑓) is, of course, obtained from (4.2) by a duality argu­ ment. We have already proved (4.2) 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 𝐻𝑝 -spaces holds. Instead of Zyg­ mund’s theorem we shall use a Marcinkiewicz-type theorem to deduce (4.2) from (4.1) for all 𝑝 < 2. It should be stressed, however, that (4.2) is of elementary character, up to the Hardy–Littlewood decomposition lemma; this follows from Note 4.1 and the proof of Theorem 4.6(𝑞 = 2) below. Theorem 4.2 (Kislyakov and Xu [266]). Let 𝜇 be a sigma-finite measure over a set Ω, let 0 < 𝑝 < 𝑞 < ⬦, and let 𝑇 be a quasilinear operator from 𝐻𝑝 to the set of all nonnega­ tive 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 ∈ 𝐻𝑝 , 𝑞

𝑓∈𝐻 .

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

𝑓 ∈ 𝐻𝑠 .

(4.3) (4.4)

4.1 Using interpolation of operators on 𝐻𝑝

| 119

As before, we use “‖⋅‖𝑝,⋆ ” to denote the quasinorm in 𝐿𝑝,⋆ . “Quasilinear” means 𝑇(𝑓+𝑔) ≤ 𝐾(𝑇𝑓 + 𝑇𝑔); see p. 400. Observe that the case 𝑞 = ∞ is now excluded. In that case the things lie deeper, as one can see in [237] (cf. [47, Ch. 5]). Proof. The idea is the same as in the case of the classical Marcinkiewicz’s theorem (see Theorem B.2 and its proof). The obstacle is in that we cannot use the “old” decompo­ sition 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

‖ℎ𝜆 ‖𝑞𝑞

≤ 𝐴 ∫ |𝑓|𝑞 𝑑𝑙 + 𝐴𝜆2𝑞 ∫ |𝑓|−𝑞 𝑑𝑙, |𝑓|≤𝜆

|𝑓|>𝜆

where 𝐴 = const. (Lemma 4.1 below). 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 4.1 (Bourgain). If 𝑓 ∈ 𝐻𝑝 (0 < 𝑝 < ⬦) and 𝜆 > 0, then there are functions ℎ ∈ 𝐻∞ and 𝑔 ∈ 𝐻𝑝 such that |ℎ∗ | ≤ 𝐶𝜆 min ( ‖𝑔‖𝑝𝑝 ≤ 𝐶

|𝑓∗ | 𝜆 , ) 𝜆 |𝑓∗ |

∫ 𝜁∈𝕋, |𝑓∗ (𝜁)|>𝜆

where 𝐶 depends only on 𝑝.

and

|𝑓∗ (𝜁)|𝑝 |𝑑𝜁|,

120 | 4 Taylor coefficients with applications Proof. Let 𝜆 > 0 and define the functions 𝐴 on 𝕋, and 𝜙 on 𝔻 by 𝐴 = max (1, (

|𝑓∗ | 𝑝/2 ) ) 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. Since |𝜙∗ | ≤ 1/𝐴 and |𝜓| ≤ 𝐶|𝜙|4/𝑝 (by Schwarz’s lemma), we have |ℎ∗ | ≤ 𝐶|𝑓∗ | min (1, (

|𝑓∗ | −2 ) ), 𝜆

and this gives the desired estimate for ℎ. On the other hand, 󵄨󵄨 1 − 𝐴 + 𝑖𝐻(𝐴) 󵄨󵄨 󵄨 󵄨󵄨 |𝑔∗ | ≤ 𝐶|1 − 𝜙∗ |2/𝑝 |𝑓∗ | = 𝐶 󵄨󵄨󵄨 󵄨 󵄨󵄨 𝐴 + 𝑖𝐻(𝐴) 󵄨󵄨󵄨 𝐴 − 1 |𝐻(𝐴 − 1)| 2/𝑝 + ) |𝑓∗ | ≤ 𝐶( 𝐴 𝐴 ≤ 𝐶(1 − 1/𝐴)2/𝑝 |𝑓∗ | + 𝐶𝜆 |𝐻(𝐴 − 1)|2/𝑝 , 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 ‖𝑔‖𝑝𝑝 ≤ 𝐶 ∫(1 − 1/𝐴)2 |𝑓∗ |𝑝 𝑑𝑙 + 𝐶𝜆𝑝 ∫ |𝐻(𝐴 − 1)|2 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 4.1¹. 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 (4.4) is satisfied (𝑞 = 2). It remains to prove that ̂ 𝑎𝜆 := 𝜇 ({𝑛 : (𝑛 + 1)|𝑓(𝑛)| > 𝜆}) ≤ (𝐶2 /𝜆)𝑝 ‖𝑓‖𝑝𝑝 ,

1 See the proof of Theorem B.6.

𝜆 > 0.

4.1 Using interpolation of operators on 𝐻𝑝

| 121

We may assume that 𝜆 > 1 because 𝜇 is finite. Let ‖𝑓‖𝑝 = 1. We have, by (4.1), 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, (4.3) holds with 𝐶2 = 𝐶3 𝐾𝑝 . Now Theorem 4.2 tells us that ∞

𝑠 ̂ ∑ ((𝑛 + 1)|𝑓(𝑛)|) (𝑛 + 1)−2 ≤ 𝐶‖𝑓‖𝑠𝑠 , 𝑛=0

as desired. If 0 < 𝑝 < 1, then both inequalities (4.1) and (4.2) are contained in the following [324]: Theorem 4.3. If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 < 1, then ∞

̂ 𝑝 ≤ 𝐶 ‖𝑓‖𝑝 . ∑ (𝑛 + 1)𝑝−2 sup |𝑓(𝑘)| 𝑝 𝑝 0≤𝑘≤𝑛

𝑛=0

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

and then use (4.1) twice to show that (4.3) and (4.4) hold for 0 < 𝑝 < 𝑞 < 1. The result follows.

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

(𝑟 → 1)

for 𝑞 > 𝑝,

(4.5)

𝑝

see (1.35). Then, using the fact that the polynomials are dense in 𝐻 , we can prove that (4.5) remains valid if we replace “O” by “o”. This is further improved by the following. Theorem 4.4 (Hardy–Littlewood 𝑀𝑞𝑝 -theorem). If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 < ⬦ and ∞ ≥ 𝑞 > 𝑝, then 1

∫ 𝑀𝑞𝑝 (𝑟, 𝑓)(1 − 𝑟)−𝑝/𝑞 𝑑𝑟 ≤ 𝐶𝑝,𝑞 ‖𝑓‖𝑝𝑝 . 0

From this, (1.35) and the “increasing” property of the integral means we obtain Theorem 4.5. If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 < ⬦, ∞ ≥ 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.

(4.6)

122 | 4 Taylor coefficients with applications In the case 𝑠 = 𝑞, this can be written in the form: Corollary 4.1. If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 < ⬦ and ⬦ > 𝑞 > 𝑝, then ∫ |𝑓(𝑧)|𝑞 (1 − |𝑧|2 )𝑞/𝑝−2 𝑑𝐴(𝑧) ≤ 𝐶𝑝,𝑞 ‖𝑓‖𝑞𝑝 . 𝔻

Proof of Theorem 4.4. We can suppose that 𝑞 is finite because 𝑀∞ (𝑟, 𝑓) ≤ (1 − 𝑟)−1/𝑞 𝑀𝑞 (√𝑟, 𝑓). Define the (quasilinear) operator 𝑇 : 𝐻𝜌 󳨃→ 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 𝐻𝑠 into 𝐿𝑠,⋆ (0, 1) for 𝑠 = 𝜌 and 𝑠 = 𝑞. By Theorem 4.2, this will imply that 𝑇 maps 𝐻𝑝 into 𝐿𝑝 (0, 1) for 𝜌 < 𝑝 < 𝑞, which will conclude the proof. In the case 𝑠 = 𝜌 we use inequality (1.35); we get 𝑇𝑓(𝑟) ≤ 𝐴(1 − 𝑟)−1/𝜌 ‖𝑓‖𝜌 ,

(4.7)

where 𝐴 is a positive constant. If ‖𝑓‖𝜌 = 1, then it follows from (4.7) that |{𝑟 ∈ (0, 1) : 𝑇𝑓(𝑟) > 𝜆}| ≤ |{𝑟 ∈ (0, 1) : 𝐴(1 − 𝑟)−1/𝜌 > 𝜆}| = min(1, (𝐴/𝜆)𝜌 ), 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 4.1. If 𝑓 is harmonic in 𝔻, then (4.6) holds for 𝑞 > 𝑝 > 1. This was proved by Flett [154] using Marcinkiewicz’s theorem.

Some improvements of Theorem 4.1 In order 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 [258], consists of complex sequences {𝑎𝑘 }∞ 0 such that { 𝑛𝛼 𝑝 {2 ( ∑ |𝑎𝑗 | ) 𝑗∈𝐼𝑘 {

1/𝑝

∞

} ∈ ℓ𝑞 . } }𝑘=0

4.1 Using interpolation of operators on 𝐻𝑝

|

123

The quasinorm in 𝑋 = ℓ𝛼𝑝,𝑞 is given by 󵄩󵄩 1/𝑝 ∞ 󵄩 󵄩 󵄩󵄩{ } 󵄩󵄩󵄩 󵄩󵄩 𝑛𝛼 𝑝 ‖{𝑎𝑗 }‖𝑋 = 󵄩󵄩󵄩{2 ( ∑ |𝑎𝑗 | ) } 󵄩󵄩󵄩󵄩 . 󵄩󵄩 󵄩 𝑗∈𝐼𝑘 󵄩󵄩{ }𝑘=0 󵄩󵄩󵄩ℓ𝑞 𝑝,𝑝

It follows that ℓ0 is identical with ℓ𝑝 . These spaces often appear in the theory of coefficient multipliers, which will be considered later on. As a special case of Kellogg’s theorem we have Lemma 4.2. Let 1/𝑠󸀠 + 1/𝑠 = 1 for 1 ≤ 𝑠 ≤ ∞, and 𝑠󸀠 = ∞ for 0 < 𝑠 < 1. If 0 < 𝑞 ≤ ⬦, 𝑝󸀠 ,𝑞󸀠

then 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. In the case 1 < 𝑝 < 2, there is an improvement of Theorem 4.1; see Theorem B.5. The following theorem contains, however, an improvement of (4.2) in other direction. Theorem 4.6. Let 1 < 𝑝 < 2 and 0 < 𝑞 < 𝑝󸀠 . If 𝑓 ∈ 𝐻𝑝 , then ∞

𝑝/𝑞 𝑛(𝑝−1)

∑2

−𝑛

(2

𝑛=0

̂ 𝑞) ∑ |𝑓(𝑘)|

≤ 𝐶𝑝,𝑞 ‖𝑓‖𝑝𝑝 ,

(4.8)

𝑘∈𝐼𝑛

or, what is the same, 𝑞,𝑝

𝐻𝑝 ⊂ ℓ1/𝑝󸀠 −1/𝑞 . Before proving this theorem note that if 𝑞 = 𝑝, then (4.8) reduces to (4.2). On the other hand, combining the Hausdorff–Young inequality and Theorem 2.21 we get 1

‖𝑓‖𝑝𝑝

∞

𝑝󸀠

󸀠

󸀠

2/𝑝󸀠

̂ 𝑝 𝑟𝑛𝑝 ) ≥ 𝑐𝑝 ∫(1 − 𝑟) ( ∑ 𝑛 |𝑓(𝑛)|

𝑑𝑟

𝑛=0

0

and hence, by Lemma 3.10, 𝑝󸀠 ,2

𝐻𝑝 ⊂ ℓ0

(1 ≤ 𝑝 < 2),²

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

subset of 𝑋, can be deduced from the fact that the function 𝑓(𝑧) = ∑ 𝑐𝑛 𝑧2 belongs to

2 This inclusion was proved by Kellogg [258].

124 | 4 Taylor coefficients with applications 𝐻𝑝 if and only if 𝑓 ∈ 𝑌. On the other hand, it is easy to see that if 𝑓(𝑧) = ∑ 𝑏𝑛 𝑧𝑛 , 𝑏𝑛 ↓ 0, 𝑝 then the condition {𝑏𝑛 } ∈ 𝑌 is equivalent with ∑ 2𝑛(𝑝−1) 𝑏2𝑛 < ∞, which is equivalent 𝑝 with 𝑓 ∈ 𝐻 , by Theorem 4.9 below. ̂ 𝑞 )1/𝑞 increases with 𝑞, we may assume that Proof of Theorem 4.6. Since (2−𝑛 ∑𝐼𝑛 |𝑓(𝑘)| 󸀠 󸀠 2 < 𝑞 < 𝑝 . Then we have 𝑝 < 𝑞 < 2 so we can apply Theorem 4.4 and the Hausdorff– Young inequality to obtain 1

‖𝑓‖𝑝𝑝

1 −𝑝/𝑞󸀠

≥ 𝑐 ∫(1 − 𝑟)

𝑝 𝑀𝑞󸀠 (𝑟, 𝑓) 𝑑𝑟

−𝑝/𝑞󸀠

≥ 𝑐𝑝 ∫(1 − 𝑟)

0 ∞

∞

󸀠

𝑛=0

𝑛=0

𝑝/𝑞

̂ 𝑞 𝑟𝑛𝑞 ) ( ∑ |𝑓(𝑛)| 𝑛=0

0 𝑝/𝑞

̂ 𝑞) ≥ 𝑐 ∑ 2𝑛(𝑝/𝑞 −1) ∑ ( ∑ |𝑓(𝑘)|

∞

.

𝑘∈𝐼𝑛

The result follows. In the same way one proves the following. Theorem 4.7 (Mateljević–Pavlović [324]). If 𝑓 ∈ 𝐻1 , then 1/𝑞

∞

̂ 𝑞) ∑ (2−𝑛 ∑ |𝑓(𝑘)| 𝑛=0

≤ 𝐶‖𝑓‖1 ,

0 < 𝑞 < ⬦,

(4.9)

𝑘∈𝐼𝑛

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

J𝛼 𝑓(𝑧) =

1 1 𝛼−1 ∫ (log ) 𝑓(𝜌𝑧) 𝑑𝜌, 𝛤(𝛼) 𝜌

𝑧 ∈ 𝔻.

0

If 𝑓 is analytic in 𝔻, then ∞

𝑛 ̂ . J𝛼 𝑓(𝑧) = ∑ (𝑛 + 1)−𝛼 𝑓(𝑛)𝑧 𝑛=0

The following theorem was proved by Hardy and Littlewood [191] and Sobolev [455]. Theorem 4.8. Let 0 < 𝑝 < 𝑞 < ⬦ and 𝛼 =

1 𝑝

− 1𝑞 . Then J𝛼 maps 𝐻𝑝 into 𝐻𝑞 .

It may be interesting that the operator J𝛼 is not compact; see Theorem 11.21 and Exer­ cise 11.6.

4.1 Using interpolation of operators on 𝐻𝑝

|

125

Proof [15]. 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 ), i.e. |𝑓(𝜌𝑒𝑖𝜃 )| ≤ min{(1 − 𝜌)−1/𝑝 , 𝑀rad 𝑓(𝑒𝑖𝜃 )}. 1/2

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

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

where 𝑀 = 𝑀rad 𝑓(𝑒𝑖𝜃 ). Hence 1−𝑀−𝑝

1

𝑖𝜃

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

0

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

It follows that 𝑞

𝑞

𝑝

|J𝛼 𝑓(𝑒𝑖𝜃 )|𝑞 ≤ 𝐶1 + 𝐶2 (𝑀rad 𝑓(𝑒𝑖𝜃 )) . Now the desired result is obtained by integration and using the complex maximal the­ orem. Taking 𝑞 = 2 we get Corollary 4.2. If 0 < 𝑝 < 2 and 𝑓 ∈ 𝐻𝑝 , then ∞

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

(4.10)

𝑛=0

Consequently, by duality, ∞

̂ 2, ‖𝑓‖2𝑞 ≤ 𝐶𝑞 ∑ (𝑛 + 1)1−2/𝑞 |𝑓(𝑛)|

(4.11)

2 ≤ 𝑞 < ⬦.

𝑛=0

It is interesting to compare (4.10) with (4.2). Let 𝑋𝑝 (resp. 𝑌𝑝 ) be the space of the sequences {𝑎𝑛} such that 𝐾𝑝󸀠 ({𝑎𝑛 }) < ∞ (resp. 𝐾𝑝 ({𝑎𝑛 }) < ∞, see (4.2)) normed in the obvious way. First we show that they are incomparable (treated independently of 𝐻𝑝 -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 (4.10) is a consequence of (4.1) and (4.2). Indeed, if ‖𝑓‖𝑝 = 1, then ∞

∞

𝑛=0

𝑛=0 ∞

∑ (𝑛 + 1)1−2/𝑝 |𝑎𝑛|2 ≤ 𝐶 ∑ (𝑛 + 1)1−2/𝑝 |𝑎𝑛 |𝑝 ((𝑛 + 1)1/𝑝−1 ) = 𝐶 ∑ (𝑛 + 1)𝑝−2 |𝑎𝑛 |𝑝 . 𝑛=0

2−𝑝

126 | 4 Taylor coefficients with applications It is not clear whether such a simple trick is possible when 1 < 𝑝 < 2. In this case we can deduce (4.10) from (4.8). To see how it goes, we take 𝑞 = 2 and rewrite (4.8) as 𝑝/2

∞

̂ 2) ∑ (2𝑛(1−2/𝑝) ∑ |𝑓(𝑘)| 𝑛=0

≤ 𝐶𝑝 ‖𝑓‖𝑝𝑝 .

𝑘∈𝐼𝑛

𝑝/2 (𝑥𝑛 ≥ 0) to obtain (4.10). Now we use the inequality ∑𝑛 𝑥𝑝/2 𝑛 ≥ (∑𝑛 𝑥𝑛 ) In a similar way, starting from (4.8) and adding the Hausdorff–Young inequality, we obtain ∞

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

1 < 𝑝 < 2, 𝑝 ≤ 𝑞 ≤ 𝑝󸀠 ,

𝑛=0

which unifies (4.2) and (4.10); see [184, p. 161]. Remark 4.2. If 0 < 𝑝 ≤ 1, then Theorem 4.8 holds for 𝑞 = ∞. To see this we write J1/𝑝 𝑓 = J1 J1/𝑝−1 𝑓. It follows from Theorem 4.8 that J1/𝑝−1 𝑓 ∈ 𝐻1 , if 𝑓 ∈ 𝐻𝑝 . 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 ̂ decreases to zero, not hold for 𝑞 = ∞. This can be deduced from the fact that if 𝑓(𝑛) 𝑝 𝑝−2 ̂ 𝑝 then 𝑓 ∈ 𝐻 if and only if ∑(𝑛 + 1) |𝑓(𝑛)| < ∞ (see Theorem 4.9). In particular, the 1/𝑝−1 𝑛 𝑛 function 𝑓(𝑧) = ∑∞ 𝑧 / log 𝑛 belongs to 𝐻𝑝 , while J1/𝑝 𝑓(𝑧) = ∑∞ 𝑛=2 𝑛 𝑛=2 𝑧 /(𝑛 log 𝑛) ∞ is not in 𝐻 . It should be noted that in the case 𝑝 = 1, 𝑞 = ∞, Privalov’s theorem 1.32 can be reformulated to get a result stronger then that J1 maps 𝐻1 into 𝐻∞ : actually J1 maps 𝐻1 into the subclass of 𝐴(𝔻) consisting of functions absolutely continuous on 𝕋.

4.1.2 The case of monotone coefficients ̂ If the sequence {𝑓(𝑛)} is decreasing, then both implications in Theorem 4.1 become equivalences. Theorem 4.9 (H–L monotone coefficients theorem [188]). Let 1 < 𝑝 < ⬦, 𝑓 ∈ 𝐻(𝔻), ̂ and let the sequence {𝑓(𝑛)} (→ 0) be real and decreasing. Then, 𝑓 ∈ 𝐻𝑝 if and only if ∞

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

and we have ‖𝑓‖𝑝 ≍ 𝐾𝑝 (𝑓).

4.1 Using interpolation of operators on 𝐻𝑝

| 127

In view of Theorem 4.1, we have to prove two inequalities: ‖𝑓‖𝑝 ≤ 𝐶𝐾𝑝 (𝑓), 𝐾𝑝 (𝑓) ≤ 𝐶‖𝑓‖𝑝 ,

(4.12)

1 < 𝑝 < 2, 2 < 𝑝 < ∞.

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

To prove (4.12) we use the Riesz projection theorem, Lemmas 2.2 and 3.9, and an in­ equality of Littlewood and Paley (which will be proved in Chapter 5, Theorem 5.14). We get 1

‖𝑓‖𝑝𝑝

𝑝

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

≤ 𝐶|𝑓(0)|𝑝 + 𝐶 ∑ 2−𝑛 ‖Δ 𝑛 (𝑓󸀠 )‖𝑝𝑝 . 𝑛=1

Hence, by Exercise A.4, ∞

‖𝑓‖𝑝𝑝 ≤ 𝐶 ∑ ‖Δ 𝑛 𝑓‖𝑝𝑝 ,

1 < 𝑝 < 2.

(4.13)

𝑛=0 𝑝

𝑝

Hence, by duality, ∑∞ 𝑛=0 ‖Δ 𝑛 𝑓‖𝑝 ≤ 𝐶‖𝑓‖𝑝 , 2 < 𝑝 < ∞. In order to finish the proof we need another lemma. 𝑘 Lemma 4.3. If 𝑃𝑛 (𝑧) = ∑2𝑛−1 𝑘=𝑛 𝑧 , then, for 𝑝 > 1, 1− 𝑝1

‖𝑃𝑛 ‖𝑝 ≍ 𝑛

,

𝑛 ≥ 1.

Proof. By Corollary 1.10, we have, for 𝑟 = 1 − 1/𝑛, 1

1

‖𝑃𝑛 ‖𝑝 ≥ 𝑀∞ (𝑟, 𝑃𝑛 )(1 − 𝑟2 ) 𝑝 ≥ 𝑟2𝑛−1 ‖𝑃𝑛 ‖∞ (1 − 𝑟2 ) 𝑝 1− 𝑝1

≥ (1 − 1/𝑛)2𝑛−1 𝑛(1/𝑛)1/𝑝 ≥ 𝑐𝑛

.

On the other hand, let 𝑓𝑟 (𝑧) = (1 − 𝑟𝑧)−1 . Then, by Exercise A.4 and Lemma 2.2, and the projection theorem, 󵄩󵄩2𝑛−1 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ 𝑟𝑘 𝑒𝑘 󵄩󵄩󵄩 (𝑒𝑘 (𝑧) = 𝑧𝑘 ) 󵄩󵄩 𝑟2𝑛−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.

1

128 | 4 Taylor coefficients with applications ̂ We continue the proof of the theorem. Let 1 < 𝑝 < 2, and let the sequence {𝑓(𝑛)} be decreasing. Then, by (4.13), Lemma 4.3, and Exercise A.4, ∞

̂ 𝑛 )𝑝 ‖𝑃 𝑛 ‖𝑝 ‖𝑓‖𝑝𝑝 ≤ 𝐶|𝑓(0)|𝑝 + 𝐶 ∑ 𝑓(2 2 𝑝 𝑛=0 ∞

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

̂ 𝑝. ≤ 𝐶 ∑ (𝑛 + 1)𝑝−2 𝑓(𝑛) 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 𝐻1 , it is perhaps surprising that Theorem 4.9 continues to be true for 𝑝 = 1. ̂ ∈ ℝ and 𝑓(𝑛) ̂ ↓ 0, then Theorem 4.10. If 𝑓(𝑛) ̂ 𝑓(𝑛) , 𝑛=0 𝑛 + 1 ∞

‖𝑓‖1 ≤ 𝐶 ∑ where 𝐶 is an absolute constant. Proof. To verify this, we write 𝑓(𝑧) as ∞

𝑛

𝑛=0

𝑘=0

̂ − 𝑓(𝑛 ̂ + 1)) ∑ 𝑧𝑘 , 𝑓(𝑧) = ∑ (𝑓(𝑛) ̂ → 0. It follows that which holds because 𝑓(𝑛) 󵄨󵄨 𝑛 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 ̂ − 𝑓(𝑛 ̂ + 1)) ∫ − 󵄨󵄨󵄨 ∑ 𝑟𝑘 𝜁𝑘 󵄨󵄨󵄨 |𝑑𝜁| 𝑀1 (𝑟, 𝑓) ≤ ∑ (𝑓(𝑛) 󵄨 󵄨󵄨 󵄨 𝑛=0 󵄨 𝕋 󵄨𝑘=0 ∞

∞ ̂ 𝑓(𝑘) 1 =𝐶∑ , 𝑘+1 𝑘+1 𝑘=0 𝑘=0 𝑛

̂ − 𝑓(𝑛 ̂ + 1)) ∑ ≤ 𝐶 ∑ (𝑓(𝑛) 𝑛=0

where 𝐶 is an absolute constant. The following statement might be also of some interest. Corollary 4.3. If {𝑎𝑛 } is a monotone convergent sequence of real numbers, then the func­ tion 𝑓(𝑧) = ∑ 𝑎𝑛 𝑧𝑛 belongs to 𝐻𝛼1,𝑞 for all 𝛼 > 0 and 0 < 𝑞 ≤ ∞. Proof. Let 𝑙 = lim𝑛 𝑎𝑛 . Let 𝑏𝑛 = 𝑙 − 𝑎𝑛 if the sequence is increasing, and 𝑏𝑛 = 𝑎𝑛 − ℓ if it is decreasing. In any case 𝑏𝑛 ↓ 0. The function ∑𝑛 𝑙𝑧𝑛 belongs to 𝐻𝛼1,𝑞 , so it is sufficient to prove that 𝑔(𝑧) = ∑ 𝑏𝑛 𝑧𝑛 belongs to 𝐻𝛼1,𝑞 . Since 𝑏𝑛 𝑟𝑛 ↓ 0 for 0 < 𝑟 < 1, we have, by Theorem 4.10, ∞ 𝑏 𝑟𝑛 𝑀1 (𝑟, 𝑔) ≤ 𝐶 ∑ 𝑛 , 𝑛=0 𝑛 + 1 where 𝐶 is independent of 𝑟. The rest is obtained by integration and using Lemma 3.8.

4.2 Strong convergence in 𝐻1

|

129

4.2 Strong convergence in 𝐻1 For a function 𝑓 analytic in 𝔻 let 𝑃𝑛 𝑓 =

1 𝑛 1 𝑠 𝑓, ∑ 𝐿 𝑛 𝑗=0 𝑗 + 1 𝑗

𝑛

1 𝑗 + 1 𝑗=0

where 𝐿 𝑛 = ∑

(𝑛 ∈ ℕ)

and 𝑠𝑗 𝑓 are the partial sums of the Taylor series of 𝑓. It is well known that ‖𝑠𝑛 𝑓‖ ≤ 𝐶 𝐿 𝑛 ‖𝑓‖ and that 𝐿 𝑛 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).

(4.14)

where 𝐶 is an absolute constant. It turns out, however, that the stronger inequality holds, namely 1 𝑛 1 ‖𝑠 𝑓‖ ≤ 𝐶‖𝑓‖ ∑ log 𝑛 𝑗=0 𝑗 + 1 𝑗

(𝑓 ∈ 𝐻1 , 𝑛 ≥ 2).

(4.15)

Moreover, we have the following characterization of the space 𝐻1 . Theorem 4.11 ([367, 452]). A function 𝑓 ∈ 𝐻(𝔻) belongs to 𝐻1 if and only if one of the following two conditions is satisfied: sup 𝑛

1 𝑛 1 ‖𝑠 𝑓‖ < ∞ ; ∑ 𝐿 𝑛 𝑗=0 𝑗 + 1 𝑗

(4.16)

sup ‖𝑃𝑛 𝑓‖ < ∞.

(4.17)

𝑛

Remark 4.3. It follows from the proof that the quantities occurring in (4.16) and (4.17) are equivalent to the original norm in 𝐻1 ; in particular (4.15) holds. Since the polynomials are dense in 𝐻1 , we have the following consequence: Theorem 4.12. If 𝑓 ∈ 𝐻1 , then lim 𝑛

1 𝑛 1 ‖𝑓 − 𝑠𝑗 𝑓‖ = 0 ∑ 𝐿 𝑛 𝑗=0 𝑗 + 1

lim 𝑛

1 𝑛 1 ‖𝑠 𝑓‖ = ‖𝑓‖. ∑ 𝐿 𝑛 𝑗=0 𝑗 + 1 𝑗

and, consequently,

Corollary 4.4 ([367]). If 𝑓 ∈ 𝐻1 , then lim inf 𝑛→∞ ‖𝑓 − 𝑠𝑛 𝑓‖ = 0. There are functions 𝜙 ∈ 𝐿1 such that lim𝑛 ‖𝜙 − 𝑠𝑛 𝜙‖ = ∞; such an example is given −1/2 by 𝜙(𝑒𝑖𝜃 ) = ∑∞ cos 𝑗𝜃. Since the sequence (log 𝑗)−1/2 is convex, the function 𝑗=2 (log 𝑗)

130 | 4 Taylor coefficients with applications belongs to 𝐿1 (see 1.1). Furthermore, one 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 4.4 we obtain Corollary 4.5. If 𝜙 ∈ 𝐻1 (𝕋), then lim inf 𝑛→∞ |𝜙(𝑒𝑖𝜃 ) − 𝑠𝑛𝜙(𝑒𝑖𝜃 )| = 0 a.e. On the other hand, there exists a function 𝜙 ∈ 𝐻1 (𝕋) whose Fourier series diverges almost everywhere (see [537, Ch. VIII (3.6)]). This result is due to Hardy and Rogosin­ ski [195]; see Theorem 4.15 below. Proof of Theorem 4.11. It is obvious that (4.16) implies (4.17). To prove that 𝑓 ∈ 𝐻1 implies (4.16) let 𝑓 ∈ 𝐻1 and for fixed 𝑛 ≥ 2 and 𝑤 ∈ 𝔻 define the function 𝑔 ∈ 𝐻1 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 (The­ orem 2.15) we get ∞

∑ 𝑗=0

∞ 1 1 ̂ |𝑠𝑗 𝑓(𝑤)|𝑟𝑗 = ∑ |𝑔(𝑗)| ≤ 𝜋‖𝑔‖. 𝑗+1 𝑗+1 𝑗=0

Since 𝑟𝑗 = (1 − 1/𝑛)𝑗 ≥ 𝑐 for 0 ≤ 𝑗 ≤ 𝑛, where 𝑐 > 0 is an absolute constant, we have 2𝜋

𝑛

1 |𝑠 𝑓(𝑤)| ≤ (𝜋/𝑐)‖𝑔‖ = (1/2𝑐) ∫ |1 − 𝑟𝑒𝑖𝑡 |−1 |𝑓(𝑟𝑤𝑒𝑖𝑡 )|𝑑𝑡. ∑ 𝑗+1 𝑗 𝑗=0 0

Integrating this inequality over the circle |𝑤| = 1 we find 2𝜋

𝑛

1 ‖𝑠𝑗 𝑓‖ ≤ (1/2𝑐)‖𝑓‖ ∫ |1 − 𝑟𝑒𝑖𝑡 |−1 𝑑𝑡, ∑ 𝑗=0 𝑗 + 1 0

where we have used Fubini’s theorem. Finally, using the estimate 2𝜋

∫ |1 − 𝑟𝑒𝑖𝑡 |−1 𝑑𝑡 ≤ 𝐶 log 0

1 = 𝐶 log 𝑛, 1−𝑟

we see that (4.15) holds and therefore that (4.16) is implied by 𝑓 ∈ 𝐻1 . 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 (4.17) implies 𝑓 ∈ 𝐻1 , and this completes the proof.

4.2 Strong convergence in 𝐻1

| 131

4.2.1 Generalization to (𝐶, 𝛼)-convergence The Cesàro means of order 𝛼 > −1 of an analytic function 𝑓 are defined by 𝜎𝑛𝛼 𝑓(𝑧) =

𝛤(𝑛 + 1) 𝑛 𝛤(𝛼 + 𝑛 + 1 − 𝑘) ̂ 𝑓(𝑘)𝑧𝑘 , ∑ 𝛤(𝛼 + 𝑛 + 1) 𝑘=0 𝛤(𝑛 + 1 − 𝑘)

where 𝛤 is the Euler gamma function. This can be written as 𝜎𝑛𝛼 𝑓(𝑧) = where

1 𝑛 𝛼 𝑘 ̂ ∑ 𝐴 𝑓(𝑘)𝑧 , 𝐴𝛼𝑛 𝑘=0 𝑛−𝑘

𝑛+𝛼 ) ≍ (𝑛 + 1)𝛼 , 𝐴𝛼𝑛 = ( 𝑛

(4.18)

𝑛 ≥ 0.

In particular 𝑛

𝜎𝑛1 𝑓(𝑧) = ∑ (1 − 𝑘=0

𝑘 𝑘 ̂ ) 𝑓(𝑘)𝑧 . 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)𝑧𝑛 ,

𝑧 ∈ 𝔻, 𝜁 ∈ 𝕋.

(4.19)

𝑛=0

This means that the sequence 𝐴𝛼𝑛𝜎𝑛𝛼 𝑓(𝜁) (𝑛 ≥ 0) coincides with the sequence of the Taylor coefficients of the function 𝑧 󳨃→ (1 − 𝑧)−𝛼−1 𝑓(𝜁𝑧). From this, (4.2) and (4.18), we obtain ∞

∑ (𝑛 + 1)𝑝−2 (𝑛 + 1)𝛼𝑝 |𝜎𝑛𝛼 𝑓(𝜁)|𝑝 𝑟𝑛𝑝 𝑛=0 2𝜋

≤ 𝐶𝑝,𝛼 ∫ |1 − 𝑟𝑒𝑖𝜃 |−(𝛼+1)𝑝 |𝑓(𝜁𝑟𝑒𝑖𝜃 )|𝑝 𝑑𝜃,

0 < 𝑟 < 1.

0

After integration over 𝜁 ∈ 𝕋 we obtain ∞

∑ (𝑛 + 1)𝑝−2+𝛼𝑝 ‖𝜎𝑛𝛼 𝑓‖𝑝𝑝 𝑟𝑛𝑝 ≤ 𝐶𝑝,𝛼 𝑀𝑝𝑝 (𝑟, 𝑓) 𝛾(𝑟), 𝑛=0

where (1 − 𝑟)−(𝛼+1)𝑝+1 , { { { 2 𝛾(𝑟) = {log 1−𝑟 , { { {1,

𝛼 > 1/𝑝 − 1, 𝛼 = 1/𝑝 − 1, 𝛼 < 1/𝑝 − 1.

The most interesting case is 𝛼 = 1/𝑝 − 1. Then we have:

132 | 4 Taylor coefficients with applications Theorem 4.13. Let 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 ≤ 2, and 𝛼 = 1/𝑝 − 1. Then 1 𝑛 1 ‖𝜎𝛼 𝑓‖𝑝 ≤ 𝐶𝑝 ‖𝑓‖𝑝𝑝 , ∑ 𝐿 𝑛 𝑘=0 𝑘 + 1 𝑛 𝑝 lim

𝑛→∞

1 𝑛 1 ‖𝜎𝛼 𝑓 − 𝑓‖𝑝𝑝 = 0, ∑ 𝐿 𝑛 𝑘=0 𝑘 + 1 𝑛 lim inf ‖𝜎𝑛𝛼 𝑓 − 𝑓‖𝑝 = 0, 𝑛→∞

lim inf |𝜎𝑛𝛼 𝑓(𝜁) − 𝑓(𝜁)| = 0, 𝑛→∞

for a.e. 𝜁 ∈ 𝕋.

(4.20)

Relation (4.20) deserves to be commented. Namely, it is a theorem of Zygmund [532] that if 0 < 𝑝 < 1, then (4.20) remains true if we replace “lim inf” with “lim sup,” i.e. the 𝑛 ̂ series ∑∞ 𝑛=0 𝑓(𝑛)𝜁 is (𝐶, 𝛼) summable for a.e. 𝜁 ∈ 𝕋. If 𝑝 = 1, then by the Hardy–Ro­ gosinski theorem (Theorem 4.15 below) there exists 𝑓 ∈ 𝐻𝑝 such that 𝑠𝑛 𝑓(𝜁) = 𝜎𝑛0 𝑓(𝜁) diverges a.e. on 𝕋. On the other hand, by the Carleson–Hunt theorem, if 𝑓 ∈ 𝐻𝑝 and 𝑝 > 1, then 𝑠𝑛𝑓(𝜁) → 𝑓(𝜁) a.e. on 𝕋. Since (𝐶, 𝛼) summability implies (𝐶, 𝛽) summa­ bility for 𝛼 < 𝛽 (see [537, Ch. III (1.1)], and 1/𝑝 − 1 < 0 for 1 < 𝑝 ≤ 2, the following question arises naturally. Problem 4.1. Whether there exists a function 𝑓 ∈ 𝐻𝑝 (𝕋) (1 < 𝑝 ≤ 2) such that the sequence 𝜎𝑛1/𝑝−1 𝑓(𝜁) (𝑛 → ∞) diverges almost everywhere? Remark 4.4. In the case 2 ≤ 𝑝 ≤ ∞, the above proof shows only that Theorem 4.13 remains true with 𝛼 = 1/2.

4.3 A (𝐶, 𝛼)-maximal theorem In contrast to the case 1 < 𝑝 < ⬦, the sequence {𝑒𝑛} is not a Schauder basis in 𝐻𝑝 for 𝑝 ∈ (0, 1]. Hardy and Littlewood [192] proved that this sequence is a (𝐶, 𝛼) basis in 𝐻𝑝 for 𝛼 > 1/𝑝 − 1 (𝑝 ≤ 1). Define the maximal operator 𝜎∗𝛼 by (𝜎∗𝛼 𝑓)(𝜁) = sup |𝜎𝑛𝛼 𝑓(𝜁)| 𝑛

(𝜁 ∈ 𝕋).

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 |𝑓(𝜁𝑧)| ≤

|1 − 𝑧|2 (𝜎1 𝑓)(𝜁), (1 − |𝑧|)2 ∗

while the latter follows from ∞

𝑓(𝜁𝑧) = (1 − 𝑧)2 ∑ (𝑛 + 1)(𝜎𝑛1 𝑓)(𝜁) 𝑧𝑛 . 𝑛=0

4.3 A (𝐶, 𝛼)-maximal theorem | 133

Theorem 4.14 (Hardy–Littlewood–Sunouchi). If 𝑓 ∈ 𝐻𝑝 , 0 < 𝑝 < ∞, and 𝛼 > max{0, 1/𝑝 − 1}, then ‖𝜎∗𝛼 𝑓‖𝑝 ≤ 𝐶𝑝 ‖𝑓‖𝑝 , lim ‖𝑓 −

𝑛→∞

𝜎𝑛𝛼 𝑓‖𝑝

lim 𝜎𝛼 𝑓(𝑒𝑖𝜃 ) 𝑛→∞ 𝑛

(4.21) (4.22)

= 0, 𝑖𝜃

a.e.

= 𝑓(𝑒 )

(4.23)

Proof. The beginning of our proof differs from that of Oswald [350, pp. 410–411]. We start from identity (4.19). By the inequality ̂ 𝑠 ≤ 𝐶(𝑛 + 1)1−𝑠 ‖𝑔‖𝑠𝑠 , |𝑔(𝑛)|

𝑔 ∈ 𝐻𝑠 , 𝑠 ≤ 1,

we have 𝜋

𝑟𝑛𝑠 |𝐴𝛼𝑛 𝜎𝑛𝛼 𝑓(𝜁)|𝑠 ≤ 𝐶(𝑛 + 1)1−𝑠 ∫ |1 − 𝑟𝑒𝑖𝑡 |−(𝛼+1)𝑠 |𝑓(𝑟𝑒𝑖𝑡 𝜁)|𝑠 𝑑𝑡 −𝜋 𝜋

≤ 𝐶(𝑛 + 1)1−𝑠 ∫ (|1 − 𝑟 + |𝜃|)−(𝛼+1)𝑠 |𝑓(𝑟𝑒𝑖𝑡 𝜁)|𝑠 𝑑𝑡,

1/2 ≤ 𝑟 < 1, 𝑠 < 1.

−𝜋

From now on our proof is almost identical to Oswald’s proof. We take 𝑟 = 1−1/(𝑛+2) and use (4.18) to obtain |𝜎𝑛𝛼 𝑓(𝜁)|𝑠 ≤ 𝐶(𝑛 + 1)

|𝑓(𝑟𝜁𝑒𝑖𝑡 )|𝑠 𝑑𝑡

∫ |𝑡| 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 maps 𝐻1 (𝕋) into L0 (𝕋) continuously, we have, by Lemma B.1, |{𝜁 ∈ 𝕋 : 𝑠max 𝑔(𝜁) > 𝜆‖𝑔‖1 }| ≤ 𝑐(𝜆),

where 𝑐(𝜆) → 0 as 𝜆 → ∞,

4.3 A (𝐶, 𝛼)-maximal theorem | 135

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 maps 𝐿1 (𝕋) to L0 , continuously. 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 [265, 264]. Lemma 4.1 is essentially due to Bourgain [76], and its presentation is taken from Kislyakov–Xu [266]. The reader should read the Hardy–Littlewood’s memoir [184], where further inter­ esting and deep results can be found. On p. 162 they wrote: “Our proof of Theorem 5 (= Theorem 4.1, 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 [537, Ch. XII (3.19)]), but Hardy and Lit­ tlewood wrote their paper 13 years before the appearance of Marcinkiewicz’s theorem. However, it is possible to prove Theorem 4.1 without using interpolation theorems. See Notes 4.1 and 4.2 below. Hardy and Littlewood [192] proved (4.22) for all 𝛼 > 1/𝑝 − 1 (0 < 𝑝 < 1) but (4.23) only for 𝛼 > [ 𝑝1 ]. As remarked after Theorem 4.13, Zygmund [532]³ proved a much stronger result: the validity of (4.23) for 𝛼 = 1/𝑝 − 1 (0 < 𝑝 < 1). The maximal in­ equality (4.21) was established by Sunouchi [480] (see also Flett [152, 153]). Of course, in the case 𝑝 > 1, Theorem 4.14 states that the Fourier series of 𝑓(𝑒𝑖𝜃 ) is summable (𝐶, 𝜀) for all 𝜀 > 0. It is interesting that Zygmund in [532] at the beginning of page 327 wrote: “The problem whether in this result we may replace summability (𝐶, 𝜀) by or­ dinary convergence remains open, but the answer is probably negative.” As already mentioned, Carleson proved that the answer is positive. Concerning other results on the Cesàro means we refer to Stein’s paper [459]. 4.1 (Elementary proof of Theorem 4.4). Theorem 4.4, from which we deduced The­ orem 4.1(a) very easily, 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 ∈ 𝐻2 , 𝑞 > 2.

0

3 This paper should be read by everyone who wants to see how subtle and deep the Fourier analysis can be.

136 | 4 Taylor coefficients with applications Write 𝑓 as 𝑓(𝑧) = ∑∞ 𝑘=0 𝑃𝑘 (𝑧), where 2𝑘 −1

𝑗 ̂ 𝑃𝑘 (𝑧) = ∑ 𝑓(𝑗)𝑧

(𝑘 ≥ 1),

𝑃0 (𝑧) = 𝑓(0).

𝑗=2𝑘−1

Then, by the 𝐿2 -integrability lemma, Lemmas 2.2, and 2.3, 1

∞

∫ 𝑀𝑞2 (𝑟, 𝑓)(1 − 𝑟)−2/𝑞 𝑑𝑟 ≤ 𝐶1 |𝑓(0)|2 + 𝐶2 ∑ 2𝑘(2/𝑞−1) ‖𝑃𝑘 ‖2𝑞 𝑘=1

0

∞

2

≤ 𝐶1 |𝑓(0)|2 + 𝐶2 ∑ 2−𝑘(1−2/𝑞) (2𝑘(1/2−1/𝑞) ‖𝑃𝑘 ‖2 ) 𝑘=0 ∞

= 𝐶1 |𝑓(0)|2 + 𝐶2 ∑ ‖𝑃𝑘 ‖22 ≤ 𝐶‖𝑓‖22 . 𝑘=0

4.2. Theorem 4.3 is easily deduced from Theorem 4.4. Take 𝑞 = 1 in (4.6), then use the 𝑛 ̂ and the 𝐿𝑝 -integrability Lemma 3.8. inequality 𝑀1 (𝑟, 𝑓) ≥ 𝑐 sup𝑛 |𝑓(𝑛)|𝑟 4.3. Kolmogorov’s theorem is usually stated in the following way (see Note 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 [537, pp. 305–306]). However, Theorem B.23 shows that (a) implies (b). 4.4 (Konyagin’s theorem). Konyagin [272] proved the following improvement of Kol­ mogorov’s theorem: Theorem. 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) = ∞ for all 𝑚→∞

𝜃 ∈ 𝕋.

4.5. Inequality (4.14) 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 one takes 𝑓 to be the Poisson kernel, then let 𝑟 tend to 1 and use the norm estimate for the Dirichlet kernel. 4.6. Using Fej’er’s theorem one shows, by summation by parts, that if 𝑓 ∈ ℎ1 , then sup𝑛 ‖𝑃𝑛 𝑓‖ < ∞, where 𝑃𝑛 is extended to harmonic functions in the obvious way. Con­ versely, if 𝑓 is harmonic in 𝔻 and sup𝑛 ‖𝑃𝑛 𝑓‖ < ∞, then 𝑓 ∈ ℎ1 . 4.7 (Littlewood’s conjecture). In connection with the so-called Littlewood’s conjec­ ture, 2𝜋 󵄨 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 𝑖𝜆 𝑘 𝜃 󵄨󵄨󵄨 ∫ 󵄨󵄨 ∑ 𝑒 󵄨󵄨 𝑑𝜃 ≥ 𝑐 log 𝑛, (†) 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨𝑘=1 the following extension of Theorem 4.1 (𝑝 = 1) was proved in [334].

4.3 A (𝐶, 𝛼)-maximal theorem | 137 1 Theorem. If {𝜆 𝑛}∞ 1 is a strictly increasing sequence of positive integers, and 𝑓 ∈ 𝐻 is ̂ such that supp 𝑓 ⊂ {𝜆 𝑛 : 𝑛 ≥ 1}, then ∞

̂ )| ≤ 𝐶‖𝑓‖ , ∑ 𝑛−1 |𝑓(𝜆 𝑛 1

(4.25)

𝑛=1

where 𝐶 is independent of {𝜆 𝑛} and 𝑓. Taking 𝑓(𝑧) = ∑𝑛𝑘=1 𝑧𝜆 𝑘 we get (†). In [500], Wittman improved (4.25) by proving that ∞

∑ 𝑛=0

̂ 2 ∑𝑛𝑗=0 |𝑓(𝑗)| ̂ (∑𝑛𝑗=0 |𝑓(𝑗)|)

2

̂ |𝑓(𝑛)| ≤ 66‖𝑓‖1 ,

𝑓 ∈ 𝐻1 .

5 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 𝑞 × 𝑀𝑝 (𝑟, J𝑠 𝑓), 0 < 𝑟 < 1, belongs to 𝐿 −1 , where J𝑠 is a radial derivative of order 𝑠. The 𝑝,𝑞 scale of Besov spaces contains mixed norm spaces 𝐻−𝛼 (𝛼 < 0), “Hardy–Bloch” spaces (𝛼 = 0), and Lipschitz spaces of functions on the unit circle (𝛼 > 0). We prove that B𝑝,𝑞 𝛼 can be decomposed into an ℓ𝑞 -sum of finite-dimensional spaces if 1 < 𝑝 < ∞, and “quasi”-ℓ𝑞 -sum in other cases, and use these decompositions to describe the dual of B𝑝,𝑞 𝛼 . Besides, the existence of radial limits of Hardy–Bloch functions is considered. In Section 5.5 we consider 3 × 2 important embedding theorems due to Hardy–Little­ wood, Littlewood–Paley, and Flett. Section 5.6 is devoted to characterizations of B𝑝,𝑞 𝛼 (𝛼 > 0) via best approximation by polynomials. In the last section we give optimal es­ timates for best approximations of a singular inner function and use them to discuss the membership of an inner function in Besov spaces.

5.1 Decomposition of Besov spaces: case 1 < 𝑝 < ⬦ Various results of Hardy, Littlewood, and Paley show that quantities of the form 1 𝑞 ∫0 (1 − 𝑟)𝛽 𝑀𝑝 (𝑟, 𝑓(𝜈) ) 𝑑𝑟 occur in a natural way. This serves as a motivation for intro­ ducing the mixed norm spaces ℎ𝑝,𝑞 𝛼 , which was done in Section 3.3. An application of the maximum modulus principle shows that the original norm in 𝑋 = ℎ𝑝,𝑞 𝛼 (inherited from 𝐿𝑝,𝑞 ) is equivalent to 𝛼 1/𝑞

1

‖𝑓‖󸀠𝑋

:=

(∫ 𝑀𝑝𝑞 (𝑟, 𝑓)(1

𝑞𝛼−1

− 𝑟)

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 has been denoted by 𝐻𝛼 . 𝑝,𝑞 For 0 < 𝑝, 𝑞 ≤ ∞ and 𝛼 ∈ ℝ, we define the harmonic Besov spaces space B𝛼 to 𝑝,𝑞 be the subclass of ℎ(𝔻) which consists of those 𝑓 for which J𝜎 𝑓 ∈ ℎ𝑠−𝛼 , where 𝑠 is any real number greater than 𝛼, and where ∞

|𝑛| 𝑖𝑘𝜃 ̂ 𝑒 , J𝜎 𝑓(𝑟𝑒𝑖𝜃 ) = ∑ (|𝑘| + 1)𝜎 𝑓(𝑘)𝑟

𝜎 ∈ ℂ.

𝑘=−∞ 𝑝,𝑞

𝑝,𝑞

In particular, we have ℎ𝛽 = B−𝛽 (𝛽 > 0). This definition is independent of the choice of 𝑠 > 𝛼, which can be deduced from Theorem 3.15; however, we shall give an alterna­ tive proof based on decompositions of the spaces. Since the space B𝛼𝑝,𝑞 is “self-conjugate” (Theorem 3.10), we can reduce the proof of this and other facts to the case of analytic Besov spaces; see Theorems 5.1 and 5.4.

5.1 Decomposition of Besov spaces: case 1 < 𝑝 < ⬦

| 139

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 a nonnegative integer), and the second – the operator R𝑠 defined for all 𝑠 ∈ ℝ by ∞ 𝑛 ̂ . (5.1) R𝑠 𝑓(𝑧) = ∑ 𝑛𝑠 𝑓(𝑛)𝑧 𝑛=1

Observe that if 𝑁 is an integer, then 𝐷𝑁 𝑓 = 𝑖𝑁 R𝑁 𝑓; see (3.21). It will be clear from a decomposition theorem that these definitions are equivalent; see Corollary 5.2. Let, as before, 𝐼𝑛 = {𝑘 : 2𝑛 ≤ 𝑘 < 2𝑛+1 } for 𝑛 ≥ 1 and 𝐼0 = {0, 1}, and 𝑘 ̂ . Δ 𝑛 𝑓(𝑧) = ∑ 𝑓(𝑘)𝑧 𝑘∈𝐼𝑛

We define

Δ𝑝,𝑞 𝛼

to be the space of functions 𝑓 ∈ 𝐻(𝔻) for which ∞

𝑞

1/𝑞

( ∑ (2𝑛𝛼 ‖Δ 𝑛 𝑓‖𝑝 ) )

(5.2)

< ∞,

𝑛=0

with the obvious quasinorm, and the usual interpretation when 𝑞 = ∞ or 𝑞 = ⬦. The following fact appears to be useful in various situations (see, for instance, [327, 167]). 𝑝,𝑞 Theorem 5.1. Let 1 < 𝑝 < ⬦, and 𝛼 ∈ ℝ. Then B𝑝,𝑞 𝛼 ≃ Δ𝛼 .

Proof. This theorem can be deduced from Lemma 3.10 in the following way. We have ∞

∞

𝑛

𝑀𝑝 (𝑟, J𝑠 𝑓) ≤ ∑ 𝑀𝑝 (𝑟, Δ 𝑛 J𝑠 𝑓) ≤ ‖Δ 0 J𝑠 𝑓‖𝑝 + ∑ ‖Δ 𝑛 J𝑠 𝑓‖𝑝 𝑟2 𝑛=0

𝑛=1 ∞

𝑛

≤ 𝐶‖Δ 0 𝑓‖𝑝 + 𝐶 ∑ 2(𝑛+1)𝑠 ‖Δ 𝑛 𝑓‖𝑝 𝑟2 , 𝑛=1

where we have applied Lemma 2.2, Lemma 3.10, and the fact that ‖Δ 𝑛J𝑠 𝑓‖𝑝 ≍ 2𝑛𝑠 ‖Δ 𝑛 𝑓‖𝑝 , which follows from Lemma 5.4. In a similar way we use the Riesz theorem and the above facts to obtain 𝑛+1 𝑀𝑝 (𝑟, J𝑠 𝑓) ≥ 𝑐𝑝 sup 2𝑛𝑠 ‖Δ 𝑛 𝑓‖𝑝 𝑟2 . 𝑛≥0

It remains to apply Lemma 3.10 or Lemma 3.11.

Duality in the case 1 < 𝑝 < ⬦ As an application of Theorem 5.1 we have the following description of the dual of B𝑝,𝑞 𝛼 . Theorem 5.2. Let 1 < 𝑝 < ⬦, 𝑞 ≤ ⬦, and 𝛼 ∈ ℝ. Then the dual of B𝑝,𝑞 𝛼 is isomorphic to 𝑝󸀠 ,𝑞󸀠

the space B−𝛼 under the pairing ∞

̂ 𝑔(𝑛), ̂ ⟨𝑓, 𝑔⟩ = ∑ 𝑓(𝑛)

󸀠

󸀠

𝑝 ,𝑞 𝑓 ∈ B𝑝,𝑞 𝛼 , 𝑔 ∈ B−𝛼 .

𝑛=0

where the series converges in the ordinary sense. Here 1/𝑠 + 1/𝑠󸀠 = 1 for all 𝑠 ∈ (0, ∞] and in particular ⬦󸀠 = 1.

(5.3)

140 | 5 Besov spaces Proof. Let 𝑋𝑛,𝑝 ⊂ 𝐻𝑝 consist of the polynomials with support contained in 𝐼𝑛 . Using the Riesz projection theorem it can easily be proved that (𝑋𝑛,𝑝 )󸀠 is isomorphic to 𝑋𝑛,𝑝󸀠 , under the pairing (1.41), as well as that ‖𝑓‖(𝑋𝑛,𝑝 )󸀠 ≍ ‖𝑓‖𝑋𝑛,𝑝󸀠 , where the equivalence constants are independent of 𝑛 and 𝑓. On the other hand, the 𝑞 space B𝑝,𝑞 𝛼 is a “weighted” ℓ -sum of the spaces 𝑋𝑛,𝑝 . (In the case 𝑞 = ⬦, we assume 𝑞 that ℓ = c0 , the space of sequences tending to zero.) The dual of this sum is equal to 󸀠 the corresponding ℓ𝑞 -sum of the spaces (𝑋𝑛,𝑝 )󸀠 . From these two facts and Theorem 5.1 we get the desired conclusion.

Hardy–Bloch spaces 𝑝,𝑞 The spaces B𝑝,𝑞 := B0 are called in [167] Hardy–Bloch spaces. In particular we have ∞,∞ B ≃ B, where B denotes the standard Bloch space (see Section 6.6), and 1

{ } B𝑝,𝑝 = {𝑓 ∈ 𝐻(𝔻) : ∫ 𝑀𝑝𝑝 (𝑟, J1 𝑓)(1 − 𝑟)𝑝−1 𝑑𝑟 < ∞} . 0 { } The quantity (|𝑓(0)|𝑝 + ∫𝔻 |𝑓󸀠 (𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧))1/𝑝 is an equivalent norm on B𝑝,𝑝 . Another important space is 1

B

𝑝,2

{ } = {𝑓 ∈ 𝐻(𝔻) : ∫ 𝑀𝑝2 (𝑟, 𝑓󸀠 )(1 − 𝑟) 𝑑𝑟 < ∞} . 0 { }

As a special case of Theorem 5.2 we have 󸀠

󸀠

Proposition 5.1. If 0 < 𝑞 ≤ ⬦, 1 < 𝑝 < ⬦, the dual of B𝑝,𝑞 is isomorphic to B𝑝 ,𝑞 under the pairing (5.3).

5.2 Maximal function Polynomials 𝑊𝑛 We will construct the “smooth” Cesàro basis, which has an advantage in that it is “uni­ versal”, i.e. independent of 𝑝 (Theorem 5.3). Let 𝜓 be a complex-valued 𝐶∞ -function with compact support in ℝ. Define the trigonometric polynomials 𝑊𝑛, 𝑛 ≥ 1, by 𝑘 𝑊𝑛(𝑒𝑖𝑡 ) = 𝑊𝑛𝜓 (𝑒𝑖𝑡 ) = ∑ 𝜓 ( ) 𝑒𝑖𝑘𝑡 . 𝑛 |𝑘| 0 there is a constant 𝐶 = 𝐶𝑁 such that |𝑊𝛹 (𝑒𝑖𝜃 )| ≤ 𝐶‖𝛹(𝑁) ‖∞ (𝑏 − 𝑎) min{|𝜃|−𝑁 , (𝑏 − 𝑎)𝑁 },

|𝜃| < 𝜋.

Here ‖ ⋅ ‖∞ stands for the sup-norm. Proof. We have ∞

(1 − 𝑒−𝑖𝜃 )𝑁 𝑊(𝑒−𝑖𝜃 ) = ∑ 𝛹(𝑘)(1 − 𝑒−𝑖𝜃 )𝑁 𝑒𝑖𝑘𝜃 𝑘=−∞ ∞ 𝑁 𝑁 = ∑ 𝛹(𝑘) ∑ ( )(−1)𝑚 𝑒𝑖(𝑚−𝑘)𝜃 𝑚=0 𝑚 𝑘=−∞ 𝑁 𝑁 ∞ = ∑ (−1)𝑚 ( ) ∑ 𝛹(𝑘)𝑒𝑖(𝑚−𝑘)𝜃 𝑚 𝑘=−∞ 𝑚=0 𝑁 𝑁 ∞ = ∑ (−1)𝑚 ( ) ∑ 𝛹(𝑘 + 𝑚)𝑒𝑖𝑘𝜃 𝑚 𝑘=−∞ 𝑚=0 𝑁 ∞ 𝑁 = ∑ ∑ (−1)𝑚 ( )𝛹(𝑘 + 𝑚)𝑒𝑖𝑘𝜃 𝑚 𝑘=−∞ 𝑚=0

=

𝑁 𝑁 𝑒𝑖𝑘𝜃 ∑ (−1)𝑚 ( )𝛹(𝑘 + 𝑚). 𝑚 𝑚=0 𝑎≤𝑘≤𝑏+𝑁

∑

The inner sum in the last expression, denoted by 𝑆𝑘 , is equal to the symmetric difference of order 𝑁 of the sequence 𝛹(𝑛), 𝑛 ∈ ℤ. Therefore, by Lagrange’s theorem, |𝑆𝑘 | ≤ ‖𝛹(𝑁) ‖∞ . It follows that |(1 − 𝑒−𝑖𝜃 )𝑁 𝑊(𝑒𝑖𝜃 )| ≤ (𝑁 + 𝑏 − 𝑎)‖𝛹(𝑁) ‖∞ , whence |𝑊(𝑒𝑖𝜃 )| ≤ 𝐶𝑁 (𝑏 − 𝑎)‖𝛹(𝑁) ‖∞ |𝜃|−𝑁 .

142 | 5 Besov spaces On the other hand, from the definition of 𝑊 it follows that |𝑊(𝑒𝑖𝜃 )| ≤ (𝑏 − 𝑎)‖𝛹‖∞ . And Taylor’s formula gives 𝑥

1 ∫(𝑥 − 𝑡)𝑁−1 𝛹(𝑁) (𝑡) 𝑑𝑡, 𝛹(𝑥) = (𝑁 − 1)! 𝑎

whence ‖𝛹‖∞ ≤

1 (𝑏 − 𝑎)𝑁 ‖𝛹(𝑁) ‖∞ . 𝑁!

This completes the proof. As an application of Lemma 5.2, we have Lemma 5.3. Let 𝑝 > 0, 𝑏 − 𝑎 ≥ 1, and 𝑁𝑝 > 1. Then, with the hypotheses of Lemma 5.2, there is a constant 𝐶 = 𝐶𝑝,𝑁 such that 2𝜋

∫ |𝑊𝛹 (𝑒𝑖𝜃 )|𝑝 𝑑𝜃 ≤ 𝐶‖𝛹(𝑁) ‖𝑝∞ (𝑏 − 𝑎)𝑁𝑝+𝑝−1 . 0

Another useful lemma is a consequence of Lemma 5.1. Lemma 5.4. Let 𝜎 = 𝑠 + 𝑖𝑡 be a complex number and 𝑝 > 0. Then 󵄩󵄩 󵄩󵄩 4𝑛 󵄩󵄩 󵄩󵄩 4𝑛 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ∑ 𝑘𝜎 𝑎𝑘 𝑒𝑘 󵄩󵄩󵄩 ≍ 𝑛𝑠 󵄩󵄩󵄩 ∑ 𝑎𝑘 𝑒𝑘 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑝 󵄩󵄩𝑘=𝑛 󵄩󵄩𝑝 󵄩󵄩𝑘=𝑛

(where 𝑒𝑘 (𝑧) = 𝑧𝑘 ),

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].

The operator 𝑊max For a function 𝑓 ∈ 𝐻(𝔻), we define the maximal function 𝑊max 𝑓 by 𝜓 𝑓)(𝜁) = sup |𝑊𝑛𝜓 ∗ 𝑓(𝜁)| (𝑊max 𝑓)(𝜁) = (𝑊max 𝑛

(𝜁 ∈ 𝕋).

Lemma 5.5. If 0 < 𝑞 ≤ 1, then (𝑊max 𝑓)𝑞 ≤ 𝐶𝑞 M(|𝑓|𝑞 ),

𝑓 ∈ 𝐻𝑞 .

Proof. Let supp(𝜓) ⊂ [−2, 2]. Then 󵄨󵄨 𝜋 󵄨󵄨 2𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 𝑖𝑡 −𝑘 −𝑖𝑘𝑡 ∫ 𝑓(𝑟𝜁𝑒 ) ∑ 𝑟 𝜓(𝑘/𝑛)𝑒 𝑑𝑡󵄨󵄨󵄨 |𝑊𝑛 ∗ 𝑓(𝜁)| ≤ 󵄨󵄨− 󵄨󵄨 󵄨󵄨 𝑘=−2𝑛 󵄨󵄨 󵄨󵄨−𝜋 𝜋 󵄨 2𝑛 󵄨󵄨 −2𝑛 󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 𝑟 ∫ 󵄨󵄨󵄨󵄨𝑓(𝑟𝜁𝑒𝑖𝑡 )󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ∑ 𝑟2𝑛−𝑘 𝜓(𝑘/𝑛)𝑒(2𝑛−𝑘)𝑖𝑡 󵄨󵄨󵄨 𝑑𝑡. ≤ 󵄨 󵄨󵄨 2𝜋 󵄨󵄨−2𝑛 󵄨 −𝜋

5.3 Decomposition of Besov spaces: case 0 < 𝑝 ≤ ∞ | 143

For fixed 𝜁 , 𝑛 put 2𝑛

𝑔(𝑧) = 𝑓(𝑧𝜁) ∑ 𝜓(𝑘/𝑛) 𝑧2𝑛−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 − 𝑟2 )1−1/𝑞 𝑀𝑞 (1, 𝑔) (see Corollary 1.10). Putting 𝑟 = 1 − 1/(𝑛 + 1), we get 𝜋

|𝑊𝑛 ∗ 𝑓(𝜁)|𝑞 ≤ 𝐶 𝑛1−𝑞 ∫ |𝑔(𝑒𝑖𝜃 )|𝑞 𝑑𝜃 −𝜋 𝜋

= 𝐶 𝑛1−𝑞 ∫ |𝑓(𝜁𝑒𝑖𝜃 )|𝑞 |𝑊𝑛(𝑒𝑖𝜃 )|𝑞 𝑑𝜃. −𝜋

Hence, by Lemma 5.1, |𝑊𝑛 ∗ 𝑓(𝑒𝑖𝑡 )|𝑞 ≤ 𝐶𝑛1−𝑞 ∫ |𝑓(𝑒𝑖(𝜃+𝑡) )|𝑞 𝑛𝑞 𝑑𝜃 |𝜃| 0, then each 𝑓 ∈ B𝑝,𝑞 𝛼 has radial limits almost ev­ erywhere. 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 3.17 and the H–L con­ jugate functions theorem. From Lemma 5.4 and Theorem 5.4 we obtain Corollary 5.2. Let 𝜎 = 𝑠 + 𝑖𝜂 ∈ ℂ and 𝑠 > 𝛼. A function 𝑓 ∈ 𝐻(𝔻) belongs to B𝑝,𝑞 𝛼 if and 𝑝,𝑞 only if R𝜎 𝑓 ∈ 𝐻𝑠−𝛼 . If 𝑁 is a nonnegative integer, then the equivalence remains true if we replace R𝑁 with 𝑓(𝑁) . 󰜚,𝑞

𝑝,𝑞

Exercise 5.1. If 𝜆 ∈ ℝ+ and 𝑞 ∈ {⬦, ∞}, then B ∩ B𝜆/󰜚 ⊂ B𝜆/𝑝 for 0 < 󰜚 < 𝑝 < ⬦. The following assertion is an immediate consequence of Theorem 5.4, although it can be easily deduced from the definition of B𝑝,𝑞 𝛼 . 𝑝,⬦ Corollary 5.3. If 𝑞 < ⬦ and 𝛼 ∈ ℝ, then B𝑝,𝑞 𝛼 ⊂ B𝛼 . 𝑠,𝑞

Theorem 5.5 (Mixed embedding theorem). If 𝑠 > 𝑝, then B𝑝,𝑞 𝛼 ⫋ B𝛽 , where 𝛽 = 1/𝑠 − 1/𝑝. This is a reformulation of Theorem 3.14 but can also be deduced from Theorem 5.4 and Lemma 2.3. As an application of Theorem 5.4 and Lemma 5.4 we have a fact which is, again, a reformulation of a result obtained by means of quasi-nearly subharmonic functions (Theorem 3.15): 𝑝,𝑞

Theorem 5.6. Let 𝜅 ∈ ℝ. Then for all 𝛼 ∈ ℝ, we have J𝜅 𝑓 ∈ B𝑝,𝑞 𝛼 if and only if 𝑓 ∈ B𝛼+𝜅 , and the corresponding norms are equivalent. The most simple case occurs if 𝛼 < 0 and 𝜅 = 1. Then we get (3.16) again. The case 𝛼 > 0 of Theorem 3.13 is easily deduced from Theorem 5.4 (and the “selfconjugacy” of ℎ𝑝,𝑞 𝛼 ). In terms of Besov spaces it reads 𝑝,𝑞1 Theorem 5.7 (Increasing inclusion theorem). We have B𝑝,𝑞 for all 𝛼 ∈ ℝ. 𝛼 ⫋ B𝛼

That the inclusion is proper is verified by using lacunary series: 𝑘𝑛 Theorem 5.8. Let 𝑓(𝑧) = ∑∞ 𝑛=0 𝑐𝑛 𝑧 be a function from 𝐻(𝔻) such that {𝑘𝑛 } is lacunary. 𝑝,𝑞 𝛼 Then 𝑓 ∈ B𝛼 if and only if {𝑘𝑛 𝑐𝑛 } ∈ ℓ𝑞 .

Proof. This follows from Theorem 3.20.

5.3.1 Radial limits of Hardy–Bloch functions By the Littlewood–Paley theorem, we have B𝑝,𝑝 ⊂ 𝐻𝑝 for 𝑝 ≤ 2, 𝐻𝑝 ⊂ B𝑝,𝑝 for 𝑝 > 2, and, by Theorem 2.21, we have 𝐻𝑝 ⊂ B𝑝,2 for 𝑝 < 2, and B𝑝,2 ⊂ 𝐻𝑝 for 𝑝 ≥ 2. It follows

146 | 5 Besov spaces 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 the implication 𝑓 ∈ B𝑝,𝑞 ⇒ 𝑓 has radial limits a.e.

(5.11)

holds. Theorem 5.9. Implication (5.11) does not hold if and only if one of the following two con­ ditions are 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 (5.11) holds if and only if 0 < 𝑞 ≤ 𝑝 ≤ 2. For the case 𝑝 = 𝑞 > 2, see [262], where the existence of a function 𝑓 for which radial limits exists almost nowhere was given based on a different idea. Proof. Consider first the case 1∘ . Let (5.11) hold. Define the operators 𝑇𝑛 : B𝑝,𝑞 󳨃→ L0 (𝕋) by 𝑇𝑛 𝑓(𝑒𝑖𝜃 ) = 𝑓(𝑟𝑛 𝑒𝑖𝜃 ), where 𝑟𝑛 ↑ 1. Because of (5.11), we have 𝑇max 𝑓 = sup |𝑇𝑛 𝑓(𝑒𝑖𝜃 )| < ∞ 𝑎.𝑒. 𝑛

𝑝,𝑞

for every 𝑓 ∈ B . Hence by Banach’s principle (Theorem B.19) the sublinear operator 𝑇max maps B𝑝,𝑞 into L0 (𝕋) continuously. The hypotheses of the Nikishin–Stein theo­ rem (B.17) are satisfied with 𝑋 = B𝑝,𝑞 and 𝑇 = 𝑇max , because every 𝑠-Banach space (0 < 𝑠 ≤ 1) is of type 𝑠. It follows from this and Corollary B.2 that 𝑇max maps B𝑝,𝑞 into 𝐿󰜚 , for some 󰜚 > 0, which means that B𝑝,𝑞 ⊂ 𝐻󰜚 . Let 𝑞 > 2. Then the preceding inclusion does not hold, which can be seen by 𝑛 considering lacunary series; for instance, the function 𝑓(𝑧) = ∑(𝑛 + 1)−1/2 𝑧2 belongs to B𝑝,𝑞 (see Theorem 5.8), while 𝑓 ∈ ̸ 𝐻󰜚 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 B.21. The case 2∘ . This does not seem so easy. Assume that we have proved the following lemmas: Lemma 5.6. The space B𝑝,𝑞 , where 0 < 𝑝 < 𝑞 ≤ 2, is of type 𝑝. Lemma 5.7. If a linear operator maps B𝑝,𝑞 into 𝐿𝑝,⋆ and B𝑞,𝑞 into 𝐿𝑞,⋆ , where 𝑝 < 𝑞, then 𝑇 maps B󰜚,𝑞 into 𝐻󰜚 , where 1 1 1 = + . 󰜚 2𝑝 2𝑞 By the same reasoning as above and using Lemma 5.6 we conclude that the identity operator, 𝐼𝑑, maps B𝑝,𝑞 into 𝐿𝑝,⋆ . Since 𝐼𝑑 maps B𝑞,𝑞 into 𝐿𝑞 ⊂ 𝐿𝑝,⋆ , by Littlewood– Paley’s theorem (Theorem 5.14 below), we have, according to Lemma 5.7, that 𝐼𝑑 maps B󰜚,𝑞 into 𝐻󰜚 (continuously). To see that this is impossible, consider the functions 𝑓𝜉 (𝑧) = (1 − 𝜉𝑧)−1/󰜚 ,

0 < 𝜉 < 1.

5.3 Decomposition of Besov spaces: case 0 < 𝑝 ≤ ∞ | 147

It is easy to check that ‖𝑓𝜉 ‖󰜚 ≍ (log

1/󰜚 4 ) . 1−𝜉

On the other hand, since 𝑀󰜚 (𝑟, 𝑓𝜉󸀠 ) ≤ 𝐶(1 − 𝑟𝜉)−1 , we have 1/𝑞

1 𝑞−1

‖𝑓𝜉 ‖B󰜚,𝑞 ≤ 𝐶 (∫(1 − 𝑟)

−𝑞

(1 − 𝑟𝜉)

𝑑𝑟)

≤ 𝐶 (log

0

1/𝑞 4 ) . 1−𝜉

From this and the inclusion B󰜚,𝑞 ⊂ 𝐻𝜌 we find that (log

1/󰜚 1/𝑞 4 4 ) ≤ 𝐶 (log ) , 1−𝜉 1−𝜉

which does not hold because 1/󰜚 > 1/𝑞. This completes the proof. Proof of Lemma 5.6. If 𝑝 ≤ 1, then the space is of type 𝑝 because it is a 𝑝-Banach space. Let 1 < 𝑝 < 𝑞 ≤ 2. Consider a more general case. Let 𝑋 be a Banach space of type 𝑝, and 𝑌 the space of continuous functions 𝑓 : (0, 1) 󳨃→ 𝑋 such that 1/𝑞

1

‖𝑓‖𝑌 :=

𝑞 (∫ ‖𝑓(𝑟)‖𝑋

𝑑𝜇(𝑟))

< ∞,

0

where 𝑑𝜇 is a finite measure on (0, 1). The space B𝑝,𝑞 is isometric to a subspace of 𝑌 with 𝑋 = 𝐿𝑝 (𝕋) and 𝑑𝜇(𝑟) = (1 − 𝑟)𝑞−1 𝑑𝑟. Therefore, it suffices to prove that 1/𝑝 1/𝑝 󵄩󵄩 𝑁 󵄩󵄩𝑝 𝑁 󵄩󵄩 󵄩󵄩 𝑝 𝐴(𝑝) := (∫ 󵄩󵄩󵄩 ∑ 𝑟𝑘 (𝑡)𝑓𝑘 󵄩󵄩󵄩 𝑑𝑡) ≤ 𝐶 ( ∑ ‖𝑓𝑘 ‖𝑌 ) , 󵄩󵄩 󵄩󵄩 𝑘=0 󵄩𝑌 0 󵄩𝑘=0 1

where 𝐶 is independent of 𝑁. We have 1/𝑞 󵄩󵄩𝑞 󵄩󵄩 𝑁 󵄩󵄩 󵄩󵄩 𝐴(𝑝) ≤ 𝐴(𝑞) = (∫ ∫ 󵄩󵄩󵄩 ∑ 𝑟𝑘 (𝑡)𝑓𝑘 (𝑟)󵄩󵄩󵄩 𝑑𝑡 𝑑𝜇(𝑟)) . 󵄩󵄩 󵄩󵄩 󵄩𝑋 0 0 󵄩𝑘=0 1 1

Now we use Kahane’s inequality 1/𝑞 1/𝑝 1󵄩 󵄩󵄩𝑞 󵄩󵄩𝑝 󵄩󵄩 𝑁 󵄩󵄩 𝑁 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 (∫ 󵄩󵄩 ∑ 𝑟𝑘 (𝑡)𝑓𝑘 (𝑟)󵄩󵄩 𝑑𝑡) ≤ 𝐶 (∫ 󵄩󵄩 ∑ 𝑟𝑘 (𝑡)𝑓𝑘 (𝑟)󵄩󵄩 𝑑𝑡) , 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩𝑋 󵄩𝑋 0 󵄩𝑘=0 0 󵄩𝑘=0 1

see [240] and [299, p. 74]. We obtain 1

𝐴(𝑝) ≤

𝑝 𝐶 (∫ (∑ ‖𝑓𝑘 (𝑟)‖𝑋 𝑘 0

𝑞/𝑝

𝑑𝜇(𝑟))

1/𝑞

)

= 𝐶 𝐵(𝑝, 𝑞).

148 | 5 Besov spaces On the other hand, by 𝐿𝑞/𝑝 -Minkowski’s inequality, we have 𝑝/𝑞

1 𝑝

𝐵(𝑝, 𝑞) ≤

𝑞 ∑ (∫ ‖𝑓𝑘 (𝑟)‖𝑋 𝑘 0

𝑑𝜇(𝑟))

𝑝

= ∑ ‖𝑓𝑘 ‖𝑌 , 𝑘

which implies the result. Proof of Lemma 5.7. In order to prove this lemma, we need some facts from the theory of interpolation spaces. Given two quasi-Banach spaces 𝑋0 and 𝑌0 that are continu­ ously embedded into a linear topological space, one uses the 𝐾-method (of the real interpolation) to construct the spaces [𝑋0 , 𝑌0 ]󰜚,𝜂 (0 < 󰜚 < ∞, 0 < 𝜂 < 1). Let 𝑋1 , 𝑌1 be another couple of such spaces. Then the following assertion holds: If 𝑇 is a linear operator that maps continuously 𝑋0 into 𝑌0 and 𝑋1 into 𝑌1 , then 𝑇 maps (continuously) [𝑋0 , 𝑌0 ]󰜚,𝜂 into [𝑋1 , 𝑌1 ]󰜚,𝜂 . It is known that if 1 1−𝜂 𝜂 = + , 󰜚 𝑝 𝑞 then [𝐻𝑝 , 𝐻𝑞 ]󰜚,𝜂 = 𝐻󰜚

and [𝐿𝑝,⋆ , 𝐿𝑞,⋆ ]󰜚,𝜂 = 𝐿󰜚 .

(5.12)

For the first relation, see [509, 510]¹. The second relation is well known and can be found in many books, e.g. [47, 49]. Another fact is needed: [ℓ𝑞 (𝑋), ℓ𝑞 (𝑌)]󰜚,𝜂 = ℓ𝑞 ([𝑋, 𝑌]󰜚,𝜂 ).

(5.13)

From (5.12) and (5.13) it follows that [ℓ𝑞 (𝐻𝑝 ), ℓ𝑞 (𝐻𝑞 )]󰜚,𝜂 = ℓ𝑞 (𝐻󰜚 ). Now let 𝐹 = {𝑓𝑗 } ∈ ℓ𝑞 (𝐻𝑝 ) and 𝐹 ∈ ℓ𝑞 (𝐻𝑞 ). 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 ). From this we infer that 𝑇 maps ℓ𝑞 (𝐻𝑝 ) into 𝐿𝑝,⋆ and, similarly ℓ𝑞 (𝐻𝑞 ) into 𝐿𝑞,⋆ . Now we take 𝐹 = {𝑄𝑗 ∗ 𝑓}, where 𝑄𝑗 = 𝑉𝑗−1 + 𝑉𝑗 + 𝑉𝑗+1 , and 𝑓 ∈ 𝐵𝑝,𝑞 ∩ 𝐵𝑞,𝑞 . Since 𝑄𝑛 ∗ 𝑉𝑛 = 𝑉𝑛 we see that 𝑇𝐹 = 𝑓, and hence 𝑓 ∈ 𝐵󰜚,𝑞 , which completes the proof of the lemma and of the theorem.

1 Kislyakov–Xu’s paper [265] contains much more general results.

5.4 Duality in the case 0 < 𝑝 ≤ ∞

| 149

Remark 5.1. The reader will found results, explanations, and references concerning interpolation for Hardy and other spaces in Kislyakov’s paper [264]. Remark 5.2. The above brutal proof of Theorem 5.9 is based on a few facts that doubt­ less lie deeper than the theorem. Therefore it would be desirable to find a “civilized” proof. Problem 5.1. It may be of some importance to characterize B𝑝,𝑞 via the boundary func­ tions (when they exists). In the case 𝑝 = 𝑞 = 1 this was done by de Souza and Samp­ son [111]. 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, i.e. a real-valued function 𝑏 defined on 𝕋 such that either 1 1 𝑏(𝑡) ≡ 1/2𝜋 or 𝑏 = − |𝐼|𝑅 + |𝐼|𝐿 , 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 [111] 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 𝜁+𝑧 𝑓(𝑧) = ∫ − 𝑔(𝜁)| 𝑑𝜁| 𝜁−𝑧 𝕋

belongs to B1,1 . The proof of this result is based on earlier investigations of de Souza [108, 109, 110].

5.4 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 (5.3) with the series converging in the ordinary sense since otherwise {𝑒𝑛} is a Schauder basis in the space; for the case 𝑝 = 1, see Note 5.1. 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 𝑋𝐴 of an admissible space 𝑋 is, by definition, the set of all 𝑔 ∈ ℎ(𝔻) such that the limit ∞ 𝑛 ̂ 𝑔(𝑛)𝑟 ̂ ⟨𝑓, 𝑔⟩ := lim− ∑ 𝑓(𝑛) (5.14) 𝑟→1 𝑛=−∞

exists (and is finite) for all 𝑓 ∈ 𝑋. Theorem 5.10. Let 𝑋 be a minimal space. Then (i) If 𝑔 ∈ 𝑋𝐴 , then the functional 𝑓 󳨃→ ⟨𝑓, 𝑔⟩, 𝑓 ∈ 𝑋, belongs to 𝑋󸀠 . (ii) Conversely, if 𝛷 ∈ 𝑋󸀠 , then there is a unique function 𝑔 ∈ 𝑋𝐴 such that 𝛷(𝑓) = ⟨𝑓, 𝑔⟩. We have ‖𝛷‖ = sup{|⟨𝑓, 𝑔⟩| : 𝑓 ∈ 𝑋, ‖𝑓‖𝑋 ≤ 1}.

150 | 5 Besov spaces Proof. Assertion (i) is proved by using the Banach–Steinhaus principle. To prô = 𝛷(𝑒𝑛 ). Since lim𝑟→1− ‖𝑓 − 𝑓𝑟 ‖ = 0 we have ve (ii) define 𝑔 by 𝑔(𝑛) ∞

𝑛 ̂ 𝛷(𝑓) = lim− 𝛷(𝑓𝑟 ) = lim− 𝛷 ( ∑ 𝑓(𝑛)𝑒 𝑛𝑟 ) 𝑟→1

𝑟→1

𝑛=−∞

∞

𝑛 ̂ = lim− ∑ 𝑓(𝑛)𝛷(𝑒 𝑛 )𝑟 = ⟨𝑓, 𝑔⟩. 𝑟→1 𝑛=−∞

𝑛 ̂ The series ∑∞ 𝑛=−∞ 𝑓(𝑛)𝑒𝑛 𝑟 converges in 𝑋 for 0 < 𝑟 < 1 because of (1.17). The last assertion is obvious.

There are several ways to express ⟨𝑓, 𝑔⟩. For example, by Parseval’s formula, we have ∞ 2𝑛 ̂ 𝑔(𝑛)𝑟 ̄ |𝑑𝜁|. ̂ ⟨𝑓, 𝑔⟩ = lim− ∑ 𝑓(𝑛) = lim− ∫ − 𝑓(𝑟𝜁)𝑔(𝑧𝜁) (5.15) 𝑟→1 𝑟→1 𝑛=−∞

𝕋

In order to represent ⟨𝑓, 𝑔⟩ in another ways 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 𝜌 1 ∫ ∇𝑢(𝑧)∇𝑣(𝑧)̄ log 𝑑𝐴(𝑧). ∫ − 𝑢(𝜌𝜁)𝑣(𝜌𝜁)̄ |𝑑𝜁| = 𝑢(0)𝑣(0) + 𝜋 |𝑧|

𝕋

|𝑧| 2. and 𝜆 𝑘 = 𝑘1/𝑝−1 (log 𝑘)−1/2 . Then, by Lemma 5.13, the function 𝑓(𝑧) = 𝑘 𝑝,2 ∑∞ 𝑛=2 𝜆 𝑘 𝑧 is not in B . On the other hand, we have ∞

∞

∑ 𝑛𝑝−2 𝜆𝑝𝑛 = ∑ 𝑛−1 (log 𝑛)−𝑝/2 < ∞, 𝑛=2

𝑛=2 𝑝

because 𝑝/2 > 1. Thus 𝑓 ∈ 𝐻 , by the Hardy–Littlewood inequality. This proves that the inclusion (1) is proper. The proof that the inclusion (2) is proper is similar and we omit the details. There is an interesting fact concerning B𝑝,𝑝 . While B𝑝,2 increases when 𝑝 decreases, the “function” B𝑝,𝑝 is not monotone. Namely: Proposition 5.3. If 𝑝 < 𝑞, then B𝑝,𝑝 ⊄ B𝑞,𝑞 and B𝑞,𝑞 ⊄ B𝑝,𝑝 . 𝑛

−1/𝑝 2 Proof. By Theorem 5.8, the function 𝑓(𝑧) = ∑∞ 𝑧 belongs to B𝑞,𝑞 , but not to 𝑛=1 𝑛 ∞ 𝑝,𝑝 1/𝑞−1 𝑛 𝑧 belongs to B𝑝,𝑝 but not to B𝑞,𝑞 . B . By Lemma 5.13, the function 𝑔(𝑧) = ∑𝑛=1 𝑛

𝑞,𝑝

𝑞,𝑝

Proposition 5.4. If 2 < 𝑝 < ⬦ and 𝑞 < 𝑝, then B1/𝑞−1/𝑝 ⊄ B𝑝,2 and B𝑝,2 ⊄ B1/𝑞−1/𝑝 . The same holds if 𝑝 < 2 and 𝑞 > 𝑝.

160 | 5 Besov spaces 𝑛

2 Proof. Let 2 < 𝑝 < ⬦ and 𝑞 < 𝑝. Let 𝑓(𝑧) = ∑∞ 𝑛=1 (1/𝑛)𝑧 . By Theorem 5.8, 𝑓 be­ 𝑞,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 5.13, the function 𝑛=1 2 ∞

∑ 𝑛1/𝑝−1 (log 𝑛)−1/2 𝑧𝑛 𝑛=2 𝑞,𝑝

belongs to B1/𝑞−1/𝑝 but not to B𝑝,2 . In the case 𝑝 < 2 we proceed in a similar way but slightly different functions are to be used.

5.6 Best approximation by polynomials 𝑝 As noted in Remark to Corollary 5.1, the space B𝑝,𝑞 𝛼 , 𝛼 > 0, consists of 𝐻 -functions sat­ isfying a Lipschitz condition. Here we prove a characterization via best approximation by polynomials.

Lemma 5.14. 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 (5.9). From the definition of 𝑊2𝑛 it follows that it is a polynomial of degree 2𝑛+1 , which implies the right-hand ̂2𝑛 (𝑘) = 1 for 0 ≤ 𝑘 ≤ 2𝑛 , we see that if 𝑃 is a polynomial of degree inequality. Since 𝑊 𝑛 ≤ 2 , then 𝑊2𝑛 ∗ 𝑃 = 𝑃, so ‖𝑅𝑛 ∗ 𝑓‖𝑝 = ‖𝑅𝑛 ∗ (𝑓 − 𝑃)‖𝑝 ≤ 𝐶𝑝 ‖𝑓 − 𝑃‖𝑝 . This completes the proof. Theorem 5.15. Let 0 < 𝑝 ≤ ∞, 0 ≤ 𝑞 ≤ ∞, and 𝛼 > 0. Then 𝑓 ∈ B𝑝,𝑞 𝛼 if and only if one of the following conditions are satisfied: 𝑓∈𝐻

𝑝

1/𝑞

∞

and

( ∑ (𝑛 + 1)𝛼𝑞−1 𝐸𝑛 (𝑓)𝑞𝑝 )

< ∞,

(5.35)

𝑛=0

𝑓∈𝐻

𝑝

∞

and

𝑛𝑞𝛼

(∑ 2

1/𝑞 𝑞 𝐸2𝑛 (𝑓)𝑝 )

< ∞,

(5.36)

< ∞.

(5.37)

𝑛=0

𝑓∈𝐻

𝑝

∞

and

1/𝑞 𝑛𝑞𝛼

(∑ 2

‖𝑅𝑛 𝑓‖𝑞𝑝 )

𝑛=0

Proof. Let 0 < 𝑞 < ∞. The sequence 𝐸𝑛(𝑓)𝑝 (𝑛 ≥ 1) is decreasing and this shows that (5.35) is equivalent to (5.36). By Lemma 5.14, in order to prove that the condition

5.6 Best approximation by polynomials

|

161

𝑝,𝑞 𝑓 ∈ B𝑝,𝑞 𝛼 is equivalent to (5.36), it is enough to prove that 𝑓 ∈ B𝛼 if and only if (5.37) holds. Since ∞

𝑉𝑛 ∗ 𝑓 = 𝑉𝑛 ∗ ∑ 𝑉𝑗 ∗ 𝑓,

𝑛 ≥ 0,

𝑗=𝑛−1

we have

󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 ‖𝑉𝑛 ∗ 𝑓‖𝑝 ≤ 𝐶 󵄩󵄩󵄩󵄩 ∑ 𝑉𝑗 ∗ 𝑓󵄩󵄩󵄩󵄩 , 󵄩󵄩𝑗=𝑛−1 󵄩󵄩 󵄩 󵄩𝑝 𝑞

𝑞

from which, by Theorem 5.4, the inequality ‖𝑓‖B𝑝,𝑞 ≤ 𝐶 ∑ 2𝑛𝑞𝛼 ‖𝑅𝑛 𝑓‖𝑝 follows. 𝛼 In other direction, we start from the inequality ∞

‖𝑅𝑛 𝑓‖𝑠𝑝 ≤ ∑ ‖𝑉𝑘 ∗ 𝑓‖𝑠𝑝 ,

where 𝑠 = min{𝑝, 1}.

𝑘=𝑛+1

Now the following lemma shows that 𝑓 ∈ B𝑝,𝑞 𝛼 . Lemma 5.15. Let {𝑐𝑛 }∞ 0 be a scalar sequence such that 𝑐𝑛 = 0 for 𝑛 large enough, and 𝛽, 𝛾 positive real numbers. Then 󵄨󵄨 ∞ 󵄨󵄨𝛾 ∞ ∞ 󵄨󵄨 󵄨󵄨 ∑ 2𝑛𝛽𝛾 󵄨󵄨󵄨 ∑ 𝑐𝑘 󵄨󵄨󵄨 ≤ 𝐶 ∑ 2𝑛𝛽𝛾 |𝑐𝑛 |𝛾 , 󵄨󵄨𝑘=𝑛 󵄨󵄨 𝑛=0 𝑛=0 󵄨 󵄨 where 𝐶 is a constant independent of {𝑐𝑛 }. Proof. First we rewrite the inequality in the equivalent form 󵄨󵄨 ∞ 󵄨󵄨𝛾 ∞ ∞ 󵄨󵄨 󵄨󵄨 ∑ 2𝑛𝛽𝛾 󵄨󵄨󵄨 ∑ 2−𝑘𝛽 𝑐𝑘 󵄨󵄨󵄨 ≤ 𝐶 ∑ |𝑐𝑛 |𝛾 , 󵄨󵄨𝑘=𝑛 󵄨󵄨 𝑛=0 𝑛=0 󵄨 󵄨 Let 𝛾 ≤ 1. Then ∞

𝑛𝛽𝛾

∑2 𝑛=0

󵄨󵄨 ∞ 󵄨󵄨𝛾 ∞ ∞ 󵄨󵄨 󵄨 󵄨󵄨 ∑ 2−𝑘𝛽 𝑐𝑘 󵄨󵄨󵄨 ≤ ∑ 2𝑛𝛽𝛾 ∑ 2−𝑘𝛽𝛾 |𝑐𝑘 |𝛾 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑘=𝑛 󵄨󵄨 𝑛=0 𝑘=𝑛 ∞

𝑘

= ∑ 2−𝑘𝛽𝛾 |𝑐𝑘 |𝛾 ∑ 2𝑛𝛽𝛾 𝑘=0 ∞

𝑛=0

≤ 𝐶 ∑ |𝑐𝑘 |𝛾 , 𝑘=0

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

(†)

162 | 5 Besov spaces and consider the action of 𝑇 on the spaces 𝐿𝛾 (𝜇, ℕ), where 𝜇({𝑛}) = 2𝑛𝛽 . By the preced­ ing 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 above lemma can be proved in an elementary way, i.e. without appeal­ ing to the Riesz–Thorin theorem. Remark 5.4. If 𝑞 = ∞, then condition (5.35) is interpreted as 𝐸𝑛 (𝑓)𝑝 = O(𝑛−𝛼 ); if 𝑞 = ⬦, then “O” should be replaced by “𝑜.” Remark 5.5. The proof of Theorem 5.15 and the Riesz projection theorem show that if 𝑝 1 < 𝑝 < ∞, then 𝑓 ∈ B𝑝,𝑞 𝛼 if and only if 𝑓 ∈ 𝐻 and 1/𝑝

∞

( ∑ 2𝑛𝑞𝛼 ‖𝑓 − 𝑠𝑛 𝑓‖𝑞𝑝 )

< ∞.

𝑛=0

5.7 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)/𝑡𝛼 , resp. 𝜙(𝑡)/𝑡𝛽 , is increasing, resp. decreasing, in 𝑡 ∈ (0, 1]. We define the space 𝑝,𝑞 B𝜙 = {𝑓 ∈ 𝐻(𝔻) : . . . } by the requirement that for some 𝑠 > 𝛽, 𝑝,𝑞

J𝑠 𝑓 ∈ 𝐻𝜙 , 𝑠

where 𝜙𝑠 (𝑡) =

𝑡𝑠 ; 𝜙(𝑡)

𝑝,𝑞

see Section 3.6. (The harmonic Besov space B𝜙 is defined in the same way².) More precisely 𝑓 ∈

𝑝,𝑞 B𝜙

if

1

∫ [𝜙𝑠 (1 − 𝑟)𝑀𝑝 (𝑟, J𝑠 𝑓)] 0

𝑞

𝑑𝑟 < ∞, 1−𝑟 𝑝,∞

with the usual interpretation for 𝑞 = ⬦, ∞. For instance, 𝑓 ∈ B𝜙 𝑀𝑝 (𝑟, J𝑠 𝑓) ≤ 𝐶

𝜙(1 − 𝑟) (1 − 𝑟)𝑠

if and only if

for some 𝑠 > 𝛽.

It is important here that the function 𝜙𝑠 is normal and, more precisely, belongs to the class Δ[𝑠 − 𝛽, 𝑠 − 𝛼]. Working exactly as in the case when 𝜙(𝑡) = 𝑡𝛼 , we arrive at the following generalization of Theorem 5.4.

2 An alternative way is to use the tangential derivative 𝐷𝑠 .

5.7 Normal Besov spaces | 163 𝑝,𝑞

Theorem 5.16. A function 𝑓 ∈ 𝐻(𝔻) belongs to B𝜙 if and only if {[1/𝜙(2−𝑛 )]‖𝑉𝑛 ∗ 𝑓‖𝑝 }𝑛∈ℕ ∈ ℓ𝑞 . In proving this fact, Lemma 3.10 is to be used. As a corollary we have that the definition of these spaces are independent of 𝑠 > 𝛽. By using Theorem 5.16 one could generalize some of the preceding assertions. We note two theorems. 𝑝󸀠 ,𝑞󸀠

𝑝,𝑞

Theorem 5.17. If 𝑞 ≤ ⬦ and 𝜙 is normal, then (B𝜙 )𝐴 = B1/𝜙 , where 𝜙𝑝 = 𝜙 for 𝑝 ≥ 1, 𝑝

and 𝜙𝑝 (𝑡) = 𝜙(𝑡)𝑡1−1/𝑝 for 𝑝 < 1. Theorem 5.18. Let 𝜙 be normal. Then Theorem 5.15 remains true if we replace (𝑛+1)𝑞𝛼−1 𝑝,∞ and 2𝑛𝑞𝛼 with 𝜙(1/(𝑛 + 1))−𝑞 /(𝑛 + 1) and (𝜙(2−𝑛 ))−𝑞 , respectively. In particular, 𝑓 ∈ B𝜙 if and only if 𝐸𝑛 (𝑓) = O(𝜙(1/𝑛)), 𝑛 → ∞.

Isomorphisms between Besov spaces The following theorem can be viewed as an extension of Theorem 5.6. ̂ Theorem 5.19. There exists a function 𝜆 independent of 𝑝 and 𝑞 such that 1/𝜆(𝑛) ≍ 𝑝,𝑞 𝑝,𝑞 𝜙(1/(𝑛 + 1)), and that 𝑓 ∈ B𝜙 if and only if 𝜆 ∗ 𝑓 ∈ B , with equivalent norms. Proof. Since ‖J𝛾 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) = ∫ 𝑟𝑡 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 3.12. Since 1

1

|𝛹(𝑚) (𝑡)| = ∫ 𝑟𝑡 | log 𝑟|𝑚 0

𝜙(1 − 𝑟) 𝑑𝑟 ≍ ∫ 𝑟𝑡 (1 − 𝑟)𝑚−1 𝜙(1 − 𝑟) 𝑑𝑟, 1−𝑟 0

for 𝑚 a positive integer, we have |𝛹(𝑚) (𝑡)| ≤ 𝐶

𝛹(𝑡) , (𝑡 + 1)𝑚

𝑡 ≥ 0.

Using this and the identity (𝑚)

𝛹(1/𝛹)

𝑚−1 𝑚 = − ∑ ( )𝛹(𝑚−𝑗) (1/𝛹)(𝑗) 𝑗=0 𝑗

we obtain, by induction, |(1/𝛹)(𝑚) (𝑡)| ≤ 𝐶

1/𝛹(𝑡) , (𝑡 + 1)𝑚

𝑡 ≥ 0.

Now we can copy the proof of Lemma 5.12 to obtain the desired conclusion.

164 | 5 Besov spaces Remark 5.6. Taking 𝜙(𝑡) = 𝑠𝑡𝑠 , 𝑠 > 0, we can obtain the original version of the Hardy–Littlewood fractional integral theorem [191]. They defined the operators 𝐷[𝑠] and 𝐷[−𝑠] = 𝐷[𝑠] as 𝐷[𝑠] 𝑓(𝑧) =

∞ 𝛤(𝑛 + 𝑠 + 1) ̂ 1 ∑ 𝑓(𝑛)𝑧𝑛 𝛤(𝑠 + 1) 𝑛=1 𝛤(𝑛 + 1)

(5.38)

∞

𝛤(𝑛 + 1) 𝑛 ̂ 𝑓(𝑛)𝑧 . 𝛤(𝑛 + 𝑠 + 1) 𝑛=0

𝐷[𝑠] 𝑓(𝑧) = 𝛤(𝑠 + 1) ∑ Since

1

𝑠 ∫ 𝑟𝑛 (1 − 𝑟)𝑠−1 𝑑𝑟 = 𝑠𝐵(𝑛 + 1, 𝑠) = 0

(5.39)

𝛤(𝑛 + 1)𝛤(𝑠 + 1) 𝛤(𝑛 + 1 + 𝑠)

we have, as a consequence of Theorem 5.19, another result of Hardy and Littlewood: 𝐷[𝑠] 𝑓 ∈ B𝑝,𝑞 𝛼

if and only if 𝑓 ∈ B𝑝,𝑞 𝛼+𝑠

(𝑠 ∈ ℝ).

The most important property of 𝐷[𝑠] is 𝐷[𝑠] 𝑓(𝑧) = (1 − 𝑧)−𝑠−1 ,

where 𝑓(𝑧) = (1 − 𝑧)−1 .

Note also that 𝐷[𝑠] 𝐷[𝑠] 𝑓 = 𝐷[𝑠] 𝐷[𝑠] 𝑓 = 𝑓.

5.8 Inner functions in Besov and Hardy–Sobolev spaces For 0 < 𝑝 ≤ ∞, we define the Hardy–Sobolev space 𝑆𝑝𝛼 (𝛼 > 0) by 𝑆𝑝𝛼 = {𝑓 ∈ 𝐻(𝔻) : J𝛼 𝑓 ∈ 𝐻𝑝 },

(5.40)

with the norm ‖𝑓‖𝑆𝑝 = ‖J1/𝑝 𝑓‖𝑝 . The decomposition theorem (Theorem 5.4) together 1/𝑝

with the density of P in 𝑆𝑝𝛼 (𝑝 ≤ ⬦) can be used to show that 𝑆𝑝𝛼 ⊂ B𝑝,⬦ 𝛼 (𝑝 ≤ ⬦). In this } for which the following holds: If an inner section we detect the spaces 𝑋 ∈ {𝑆𝑝𝛼 , B𝑝,𝑞 𝛼 function 𝐼 belongs to 𝑋, then 𝐼 is a Blaschke product. We present two methods. The first one is based on best approximation of singular inner functions, and the second, due to Ahern and Jevtić [7], – on a simple but powerful inequality combined with the complex maximal theorem; see Lemma 5.17 below.

5.8.1 Approximation of a singular inner function Recall that

1+𝑧 ) , where 𝜆 > 0. 1−𝑧 Here we use Theorem 2.31 to prove the following: 𝐴 𝜆 (𝑧) = exp (−𝜆

5.8 Inner functions in Besov and Hardy–Sobolev spaces | 165

Theorem 5.20. If 𝑆 is a singular inner function, and 𝑝 ∈ ℝ+ , then 𝐸𝑛(𝑆)𝑝 ≥ 𝑐𝑝,𝑆 𝑛−1/2𝑝 ,

𝑛 ≥ 1.

The exponent −1/2𝑝 is the best possible. Lemma 5.16. If 𝑝 > 1, then 𝐸𝑛 (𝐴 𝜆 )𝑝 ≍ 𝑛−1/2𝑝 . Proof. First we write 𝐴 𝜆 as 𝐴 𝜆 (𝑧) = 𝑒−𝜆 exp (

−2𝜆𝑧 ). 1−𝑧

From the theory of orthogonal polynomials (Szöge [483]) we know that ∞

𝑛 𝐴 𝜆 (𝑧) = 𝑒−𝜆 ∑ 𝐿(−1) 𝑛 (2𝜆)𝑧 , 𝑛=0

𝐿(−1) 𝑛

are the Laguerre polynomials (a special case). By Fejér’s theorem [483, where Theorem 8.22.1], we have, for a fixed 𝑥 ∈ ℝ+ , −3/4 𝐿(−1) cos (2√𝑛𝑥 + 𝜋/4) + O(𝑛−1 ), 𝑛 (𝑥) = 𝐶𝑥 𝑛

𝑛 ≥ 1.

It follows that 2

∞

2 (𝐸𝑛(𝐴 𝜆 )2 ) = ∑ |𝐿(−1) 𝑘 (2𝜆)| 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 ∈ 𝐴 𝑚 := [𝐾(𝑚 − 1/2)2 , 𝐾𝑚2 ],

where 𝐾 = 𝜋2 /8𝜆 and 𝑚 an integer.

A segment [𝑎, 𝑏] ⊂ ℝ contains at least 𝑏 − 𝑎 integers. Hence 𝐴 𝑚 contains at least 𝐾(2𝑚 − 1/2)/2 integers so can choose 𝑚0 so that 𝐴 𝑚 ≠ 0 for 𝑚 ≥ 𝑚0 . For a sufficiently large 𝑛, choose 𝑚 so that 𝑛 + 1 ∈ 𝐴 𝑚 . Since 𝐴 𝑚 are mutually disjoint, we have 2

(𝐸𝑛 (𝐴 𝜆 )) ≥ 𝑐𝜆 ∑

∑ 𝑘−3/2 (1/2) − 𝑐𝑛−1

𝑗=𝑚+1 𝑘∈𝐴 𝑗 ∞

≥ 𝑐𝜆 ∑ (𝐾𝑗2 )−3/2 (1/2)𝐾(2𝑗 − 1/2) − 𝑐𝑛−1 𝑗=𝑚+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 (𝐴 𝜆 )2 ≍ 𝑛−1/4 , where the equivalence constants depend only on 𝜆.

(5.41)

166 | 5 Besov spaces In order to treat the case 𝑝 ≠ 2, we need the following estimates for the 𝜈th deriva­ tive of 𝐴 𝜆 : (1 − 𝑟)1/2𝑝−𝜈 , 𝜈 > 1/2𝑝, { { { { 2 2𝜈 (𝜈) 𝑀𝑝 (𝑟, 𝐴 𝜆 ) ≍ {(log ) , 𝜈 = 1/2𝑝, { 1−𝑟 { { 𝜈 < 1/2𝑝. {1,

(5.42)

𝑝,∞

From this we infer that 𝐴 𝜆 ∈ B1/2𝑝 when 𝑝 > 0, and hence, by Theorem 5.15, ‖𝑅𝑛 𝐴 𝜆 ‖𝑝 ≤ 𝐶2−𝑛/2𝑝 ,

if 𝑝 > 0.

Let 1 < 𝑝 < ⬦. Then, by (5.42) and (5.41), 󸀠

𝑐2−𝑛/2 ≤ ‖𝑅𝑛 𝐴 𝜆 ‖22 ≤ ‖𝑅𝑛 𝐴 𝜆 ‖𝑝 ‖𝑅𝑛 𝐴 𝜆 ‖𝑝󸀠 ≤ 𝐶‖𝑅𝑛 𝐴 𝜆 ‖𝑝 2−𝑛/2𝑝 , and hence ‖𝑅𝑛 𝐴 𝜆 ‖𝑝 ≥ 𝑐2−𝑛/2𝑝 . This proves the lemma. Remark 5.7. Estimates (5.42) can be obtained in a similar way as those used in the proof of Theorem 2.26 so we omit the details; see [325] or treat this as an exercise. Proof of Theorem 5.20. The function 𝑆 is subordinate to 𝐴 𝜆 , where, as we may assume, 𝜆 > 0. Hence, by Theorem 2.31 and Lemma 5.16, 𝐸𝑛 (𝑆)𝑝 ≥ 𝑐𝑛−1/2𝑝 for 𝑝 > 1. Let 𝑝 ≤ 1. Then − |𝑅𝑛 𝑆|−1 |𝑅𝑛𝑆|𝑝+1 𝑑𝑙 ≥ 𝑐− ∫ |𝑅𝑛 𝑆|𝑝+1 𝑑𝑙 ≥ 𝑐2−𝑛/2 . ‖𝑅𝑛 𝑆‖𝑝𝑝 = ∫ 𝕋

𝕋

That the exponent −1/2𝑝 is optimal follows from Theorem 5.15 and the fact that, ac­ 𝑝,∞ cording to (5.42), 𝐴 𝜆 ∈ B1/2𝑝 . This concludes the proof. Corollary 5.5. Let 𝑆 be a singular inner function and let 0 < 𝑝 < ⬦. If either 0 < 𝑞 ≤ ⬦, and 𝛼 ≥ 1/2𝑝, or 𝑞 = ∞ and 𝛼 > 1/𝑝, then 𝑆 does not belong to B𝑝,𝑞 𝛼 . The result is the best possible in the sense that the atomic (singular) function 𝐴 𝜆 belongs to B𝑝,𝑞 𝛼 for 𝑝,∞ 𝑞 ≤ ⬦, 𝛼 < 1/𝑝, and to B𝛼 for 𝛼 ≤ 1/2𝑝. The last assertion follows from relations (5.42). 𝑝,𝑞

Proof. Firstly, we can assume that 𝛼 = 1/2𝑝 because B𝑝,𝑞 𝛼 ⊂ B𝛽 for 𝛼 > 𝛽. Secondly, 𝑝,⬦ we can assume that 𝑞 = ⬦ because B𝑝,𝑞 𝛼 ⊂ B𝛼 . Assuming these, we use the bound­ 𝑛 −𝑛 edness of 𝑆 to get |J 𝑆(𝑧)| ≤ 𝐶(1 − |𝑧|) and so

𝑀𝑝𝑝 (𝑟, J𝑛 𝑆) ≤ 𝐶(1 − 𝑟)−𝑛(𝑝−󰜚) 𝑀󰜚󰜚 (𝑟, J𝑛 𝑆),

󰜚 < 𝑝. 𝜌,⬦

Hence, after a simple calculation, we conclude that the implication 𝑆 ∈ B1/2𝜌 ⇒ 𝑝,⬦

𝑆 ∈ B1/2𝑝 holds (see Exercise 5.1), and therefore we can assume that 𝑝 > 1. Now the result follows from Theorems 5.15 and 5.20.

5.8 Inner functions in Besov and Hardy–Sobolev spaces |

167

Remark 5.8. Although the above corollary is an immediate consequence of Theorems 5.15 we gave a slightly longer proof because it can be used to prove that Theorem 5.20 and Corollary 5.5 remain true if we assume that 𝑆 is an inner function with a singular factor. Namely the inequality 𝐸𝑛 (𝑓𝐼)𝑝 ≥ 𝑐𝑝 𝐸𝑛 (𝑓) holds, where 𝑓 ∈ 𝐻𝑝 , 1 < 𝑝 < ⬦, and 𝐼 is an inner function; see [332, Proposition 1.1]. Thus the following theorem holds. Theorem 5.21. Let 0 < 𝑝 < ⬦. 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 actually an improvement of the preceding theorem. Theorem 5.22 (Ahern–Jevtić [7]). If 𝐼 is an inner function such that 𝑀𝑝𝑝 (𝑟, J1/2𝑝 𝐼) = 𝑜 (log

1 ) 1−𝑟

for some 𝑝, 0 < 𝑝 < ⬦,

then 𝐼 is a Blaschke product. Lemma 5.17 (Ahern–Jevtić [7]). If 0 < 𝛼 < 𝛽 < ⬦, and 𝑓 ∈ 𝐻∞ with ‖𝑓‖∞ ≤ 1, then |J𝛼 𝑓(𝑧)| ≤ 𝐶𝛼,𝛽 𝑀𝛼/𝛽 (𝑧, J𝛽 𝑓), where 𝑀(𝑧, 𝑓) = sup0 1. We need another fact: Lemma 5.18. 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.

168 | 5 Besov spaces Proof. For 𝑟 ∈ (0, 1) choose a positive integer 𝑛 so 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 − 𝑟𝑛+1 ‖𝑅𝑛 𝐼‖𝑝 ), where we have used the inequality ‖𝑅𝑛 𝐼𝑟 ‖𝑝 ≤ 𝑟𝑛+1 ‖𝑅𝑛 𝐼‖𝑝 , which is proved in the same way as Lemma 2.2. Since 𝑟𝑛+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 5.20 and Remark 5.8, that ‖𝐼 − 𝐼𝑟 ‖𝑝 ≥ 𝑐𝑛−1/2𝑝 , from which the result follows. Proof of Theorem 5.22. In view of (5.43), we may assume that 𝑝 > 1. Then, by the pre­ vious lemma, 2𝜋 1/2

𝑐(1 − 𝑟)

≤ ‖𝐼 −

𝐼𝑟 ‖𝑞𝑞

𝑞

1 󸀠

𝑖𝜃

(5.44)

≤∫ − (∫ |𝐼 (𝜌𝑒 )| 𝑑𝜌) 𝑑𝜃. 𝑟

0

Let 0 < 𝛼 < 1. Then, by Hölder’s inequality, 𝑞

1 󸀠

𝑞

1

𝑖𝜃

󸀠

𝑖𝜃

𝛼

−𝛼

(∫ |𝐼 (𝜌𝑒 )| 𝑑𝜌) = (∫ |𝐼 (𝜌𝑒 )|(1 − 𝜌) (1 − 𝜌) 𝑟

𝑑𝜌)

𝑟 1 𝑞−1

≤ 𝐶𝑞 ∫ |𝐼󸀠 (𝜌𝑒𝑖𝜃 )|𝑞 (1 − 𝜌)𝛼𝑞 𝑑𝜌 ((1 − 𝑟)1−𝛼𝑞/(𝑞−1) ) 𝑟 1 𝑞−1−𝛼𝑞

∫ |𝐼󸀠 (𝜌𝑒𝑖𝜃 )|𝑞 (1 − 𝜌)𝛼𝑞 𝑑𝜌.

= 𝐶𝑞 (1 − 𝑟)

𝑟

From this and (5.44) 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, J1/2𝑝 𝐼), we see that 1 −1

𝑐(1 − 𝑟)

𝑞−5/2−𝛼𝑞

≤ (1 − 𝑟)

∫ 𝑀𝑞𝑞 (𝜌, J1/2𝑝 𝐼)(1 − 𝜌)𝑞/2𝑝−𝑞+𝛼𝑞 𝑑𝜌. 𝑟

Now we take 𝑞 = 2𝑝 and 𝛼 = 1 − 1/2𝑝 and so this inequality becomes 1 2𝑝

𝑐(1 − 𝑟)−1 ≤ (1 − 𝑟)−3/2 ∫ 𝑀2𝑝 (𝜌, J1/2𝑝 𝐼) 𝑑𝜌. 𝑟

5.8 Inner functions in Besov and Hardy–Sobolev spaces | 169

Integrating then this inequality from 0 to 𝑡 and using Fubini’s theorem, we obtain 𝜌

𝑡

1 2𝑝 ≤ ∫ 𝑀2𝑝 (𝜌, J1/2𝑝 𝐼) 𝑑𝜌 ∫(1 − 𝑟)−3/2 𝑑𝑟 𝑐 log 1−𝑡 0

0

1

𝑡 2𝑝

+ ∫ 𝑀2𝑝 (𝜌, J1/2𝑝 𝐼) 𝑑𝜌 ∫(1 − 𝑟)−3/2 𝑑𝑟 𝑡

0

𝑡 2𝑝

= 𝐶 ∫ 𝑀2𝑝 (𝜌, J1/2𝑝 𝐼)(1 − 𝜌)−1/2 𝑑𝜌 0 1 2𝑝

+ 𝐶(1 − 𝑡)−1/2 ∫ 𝑀2𝑝 (𝜌, J1/2𝑝 𝐼) 𝑑𝜌 𝑡

= 𝐶𝐼1 (𝑡) + 𝐶(1 − 𝑡)−1/2 𝐼2 (𝑡). We have 1

2𝜋

∫ |J1/2𝑝 𝐼(𝜌𝑒𝑖𝜃 )|2𝑝 𝑑𝜃 𝐼2 (𝑡) = ∫ 𝑑𝑡− 𝑡

0 1

2𝜋

∫ |J1/2𝑝 𝐼(𝜌𝑒𝑖𝜃 )|𝑝 (1 − 𝜌)−1/2 𝑑𝜌 ≤ 𝐶 ∫ 𝑑𝑡− 𝑡

0

1

= 𝐶 ∫ 𝑀𝑝𝑝 (𝜌, J1/2𝑝 𝐼)(1 − 𝜌)−1/2 𝑑𝑡. 𝑡

Now we estimate 𝐼1 (𝑡) by using Lemma 5.17. For a fixed 𝜃 we have 𝑡

𝜆 1/2𝑝

𝐼1 (𝑡, 𝜃) := ∫ |J

𝑖𝜃 2𝑝

−1/2

𝐼(𝜌𝑒 )| (1 − 𝜌)

𝑡

𝑑𝜌 = ∫ . . . 𝑑𝜌 + ∫ . . . 𝑑𝜌

0

0

𝜆

𝜆

1

≤ 𝐶 ∫(1 − 𝜌)−3/2 𝑑𝜌 + [𝑀(𝑡𝑒𝑖𝜃 , J1/2𝑝 𝐼)] 0

2𝑝

∫(1 − 𝜌)−1/2 𝑑𝜌 𝜆

−1/2

≤ 𝐶(1 − 𝜆)

𝑖𝜃

1/2𝑝

+ 2 [𝑀(𝑡𝑒 , J

𝐼)]

2𝑝

(1 − 𝜆)1/2 .

If 𝑀 := 𝑀(𝑡𝑒𝑖𝜃 , J1/2𝑝 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) ≤ 𝐶𝑀𝑝 , and hence, by integration and using the complex maximal theorem, 𝐼1 (𝑡) ≤ 𝐶 + 𝐶𝑀𝑝𝑝 (𝑡, J1/2𝑝 𝐼).

170 | 5 Besov spaces All in all, log

1 ≤ 𝐶 + 𝐶𝑀𝑝𝑝 (𝑡, J1/2𝑝 𝐼) 1−𝑡 1

+ 𝐶(1 − 𝑡)−1/2 ∫ 𝑀𝑝𝑝 (𝜌, J1/2𝑝 𝐼)(1 − 𝜌)−1/2 𝑑𝜌. 𝑡

The result follows. Theorem 5.22 is sharp in the sense that the atomic function satisfies 𝑀𝑝𝑝 (𝑟, J1/2𝑝 𝐴 𝜆 ) = 𝑂 (log

1 ). 1−𝑟

By means of (5.43), this is reduced to the case where 1/2𝑝 is an integer; then (5.42) is used. As a consequence we have the following: Corollary 5.6. If an inner function 𝐼 belongs to 𝑆𝑝𝛼 , where 0 < 𝑝 < ⬦ and 𝛼 ≥ 1/2𝑝, then 𝐼 is a Blaschke product. This result is also sharp because 𝐴 𝜆 ∈ 𝑆𝑝𝛼 for 𝛼 < 1/2𝑝, which can be proved by using (5.42) and (5.43).

𝑝

5.8.2 Hardy–Sobolev space 𝑆1/𝑝 Problem 5.2. The Hardy–Littlewood–Sobolev theorem can be expressed via Hardy– 𝑝 𝑞 Sobolev spaces in a symmetric form: 𝑆1/𝑝 ⊂ 𝑆1/𝑞 , 0 < 𝑝 < 𝑞 < ⬦. If 𝑝 = 1 or 2, then 𝑝

𝑆1/𝑝 is Möbius invariant in the sense that ‖𝑓 ∘ 𝜎 − 𝑓(𝑎)‖𝑋 ≤ 𝐶‖𝑓‖𝑋 for all 𝜎 ∈ Möb(𝔻), where 𝐶 is independent of 𝑓 and 𝜎. Therefore it is natural to ask what is about the other values of 𝑝. Some interpolation theorem for the pair (𝑆11 , 𝑆21/2 ) would solve this problem for 1 < 𝑝 < 2, 𝑝 but the author does not know any such theorem. The spaces 𝑆1/𝑝 have some nice prop­ erties which suggest that they might be invariant. To show some of them we use the (Möbius) duality pairing (𝑓, 𝑔)m = 𝑓(0)𝑔(0) +

1 ∫ 𝑓󸀠 (𝑧)𝑔󸀠 (𝑧) 𝑑𝐴(𝑧), 𝜋 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 pair­ ing, then 𝑌 is also invariant. Denote 𝑌 by 𝑋m and call it the Möbius dual of 𝑋. Then

5.8 Inner functions in Besov and Hardy–Sobolev spaces |

171

the three duality theorems for 𝐻𝑝 can be reformulated as

𝑝 (𝑆1/𝑝 )m

for 𝑝 < 1, {B, { { = {BMOA, for 𝑝 = 1, { { 𝑝󸀠 for 1 < 𝑝 < ⬦. {𝑆1/𝑝󸀠 ,

𝑝

Thus, if 𝑆1/𝑝 is invariant for 1 < 𝑝 < 2, then so is for 2 < 𝑝 < ⬦. Is this all coincidence?

5.8.3 f-property and K-property Using the fact that ‖𝑔𝐼‖𝑝 = ‖𝑔‖𝑝 (1 ≤ 𝑝 ≤ ∞) (𝐼 is inner) and the well-known for­ mula for computing 𝐸𝑛(𝑓) (Duren [130, Ch. VII]) one proves that 𝐸𝑛 (𝑓𝐼)𝑌 ≥ 𝐸𝑛 (𝑓)𝑌 , where 𝑌 = 𝑋𝐴 and 𝑋 = 𝐻𝑝 (1 < 𝑝 < ∞); see the proof of Proposition 1.1 in [332] for 󸀠 the case 𝑋 = 𝐻𝑝 , 1 < 𝑝 < ⬦ – if we replace 𝐻𝑝 with (𝐻𝑝 )𝐴 in this proof we obtain 𝐸𝑛(𝑓𝐼)𝑝 ≥ 𝐸𝑛 (𝑓)𝑝 for 1 < 𝑝 < ∞; does this remains true for 𝑝 = 1? It is probably true that 𝐸𝑛 (𝑓𝐼)K ≥ 𝐸𝑛(𝑓)K , where K = 𝐴(𝔻)𝐴 is the space of Cauchy transforms (see Note 1.1), but the author could not proved this; Vinogradov [495] proved that K has the f -property (Havin [201]). An admissible space 𝑋 has this property if the following holds: If 𝑓 ∈ 𝐻𝑠 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 possibly positive answers to these ques­ tions is of some interest because it follows from Lemma 5.10 and the proof of Theorem 5.15 that if 𝑝 ∈ {1, ∞}, then the latter remains true if we replace ‖ ⋅ ‖∞ , resp. ‖ ⋅ ‖1 , with ‖ ⋅ ‖(𝐻1 )𝐴 , resp. ‖ ⋅ ‖𝐴(𝔻)𝐴 . Then we could extend the following theorem to the case 𝑝 ∈ {1, ∞}. Theorem 5.23. Let 𝛼 > 0, 1 < 𝑝 < ⬦, and 0 < 𝑞 ≤ ∞. Then 𝑋 = B𝑝,𝑞 𝛼 has the f -property. Proof. This follows from Theorem 5.15 and aforementioned Proposition 1.1 of [332]. The first result of this kind (excluding Hardy spaces) was proved by Carleson [87] (Dirichlet spaces). In [178], Gurariy 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.: 𝐻𝑝 ∩ B [165]. Kahane [239] was the first who used the best approximation in proving that some spaces have the f -property. In [201], Havin considered the case 𝑞 = 𝑝 = ∞ of The­ orem 5.23 as well as some other spaces, and proved that all they have the so-called Kproperty, which means that the Töplitz operators − 𝑇𝜑 (𝑓)(𝑧) = ∫ 𝕋

𝜑(𝜁)𝑓(𝜁) |𝑑𝜁| 1 − 𝜁𝑧̄

map the space under consideration into itself, for all 𝜑 ∈ 𝐻∞ . Havin noted that Ko­ renblum [274] was the first who observed that the K-property implies the f -property.

172 | 5 Besov spaces That the spaces B𝑝,𝑞 𝛼 (𝑝 ≥ 1, 𝑞 ≥ 1, 𝛼 > 0) have the K-property can also be deduced from the three facts: (a) 𝑇𝜑 is the adjoint to the operator of pointwise multiplication 󸀠

󸀠

by 𝜑; (b) 𝐵𝑝𝛼 ,𝑞 is equal to the dual of Bergman space 𝐻𝛼𝑝,𝑞 (𝑞 ≤ ⬦); (c) 𝑇𝜑 maps P into P . One of powerful methods, developed by Dyn’kin [142, 143], in detecting the Kproperty, and in various other questions, is that of pseudoanalytic continuation. However, there are many important classes, e.g. 𝐻1 , 𝑆11 , 𝐴(𝔻), that have f -property without having the K-property. In such cases the proofs can be much more delicate. We refer the reader to the Shirokov’s book [445] for further results on various spaces of this kind, as well as for references before 1988. Among the papers after 1988 we mention [496], 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 𝐻𝑝 \ B𝑝,𝑝 . In the case 𝑝 = 1, the latter was proved by Rudin [427] 40 years earlier. As far the author knows, the most recent expository paper concerning the f - and K-properties was written by Girela et al. [166].

Further notes and results Theorem 5.12, sometimes in a different form, was proved, and reproved, by several authors: Zakharyta and Yudovich [523] (1 < 𝑝 = 𝑞 < ∞), Taibleson [484] (1 < 𝑝, 𝑞 < ∞) (nonperiodic spaces), Flett [156] (0 < 𝑝 ≤ ∞, 0 ≤ 𝑞 ≤ 1), Anderson et al.[24] (the case of the little Bloch space b := B∞,⬦ ), and Ahern and Jevtić [6] (𝑝 = 1 and 1 < 𝑞 < ∞). Our approach is borrowed from [360], where a more general class of spaces were considered (see [229] for partial simplifications of [360]). Theorem 5.15 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 3.17 and some results of Soviet mathematicians on relationships between moduli of smoothness and best approximation by polynomials (see, e.g. [42, 489]). The case 0 < 𝑝 < 1 (and 𝑝 ≥ 1) is treated in [350]. For information and references in the nonperiodic case we refer to Nikol󸀠 ski˘ı’s monograph [345]. Corollary 5.5 is a result of Ahern [4], who proved it by using his Theorem 2.25; the proof in the text is based on the ideas from [332]. 1,𝑞

5.1 (Non-Schauder bases in 𝐻𝛽 ). An interesting difference between Hardy and Berg­ 1,𝑞

man spaces should be noted. The spaces ℎ𝑝 (1 < 𝑝 < ⬦) and ℎ𝛽 are self-conjugate. In the case of ℎ𝑝 this was used to show that the sequence {𝑒𝑛} is a Schauder basis in 𝑝,𝑞 𝐻𝑝 . In contrast, this sequence is not a Schauder basis in 𝐻𝛽 , 𝑝 ≤ 1. To check this let

5.8 Inner functions in Besov and Hardy–Sobolev spaces | 173 𝑝,𝑞

𝑋 = 𝐻𝛽 , and assume that, for instance, 𝑞 < ⬦. Define two spaces: 1

𝑞 𝑋𝛽

{ } 𝑞 = {𝑓 ∈ 𝐻(𝔻) : ∫ ‖𝑓𝜌 ‖𝑋 (1 − 𝜌)𝑞𝛽−1 𝑑𝜌 < ∞} 0 { } ∞

𝑞

and

𝑞

Δ𝑋𝛽 = {𝑓 ∈ 𝐻(𝔻) : ∑ 2−𝑛𝛽𝑞 ‖Δ 𝑛 𝑓‖𝑋 < ∞} . 𝑛=0 𝑞

𝑝,𝑞

𝑞

𝑝,𝑞

A relatively simple calculation shows that 𝑋𝛽 = 𝐻2𝛽 and Δ𝑋𝛽 = Δ 2𝛽 ; see (5.2). As­ suming that {𝑒𝑛} is a Schauder basis in 𝑋 we can use Lemmas 2.2 and 3.11 to show that 𝑝,𝑞 𝑝,𝑞 (†) 𝐻2𝛽 ⊂ Δ 2𝛽 . We will prove that this is impossible and moreover that the following holds: 𝑝,𝑞

Proposition ([327]). If 𝑝 ≤ 1, then 𝐻2𝛽 ⊈ Δ1,∞ 𝛾 , where 𝛾 = 2𝛽 + 1/𝑝 − 1. That this improves the negation of (†) follows from the inclusions 𝑝,𝑞

𝑝,∞

Δ 2𝛽 ⊂ Δ 2𝛽 ⊂ Δ1,∞ 𝛾

(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 [43, p. 599]: 𝑛

𝑛

𝑛

𝑛

∞

𝑓𝑛 (𝑧) = 𝑧2 (1 − 𝑧)−2/𝑝 (1 − 𝑧2 )2/𝑝 = 𝑧2 (1 − 𝑧2 )2/𝑝 ∑ 𝑐𝑘 𝑧𝑘 , 𝑘=0

where 𝑐𝑘 = 𝛤(𝑘 + 2/𝑝)/𝑘!𝛤(2/𝑝) ≍ (𝑘 + 1)2/𝑝−1 . We have Δ 𝑛 𝑓𝑛 (𝑧) = 𝑧

2𝑛

2𝑛 −1

∑ 𝑐𝑘 𝑧𝑘

𝑘=0

and hence, by Hardy’s inequality, 󵄨󵄨2𝑛 −1 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 |𝑐 | ‖Δ 𝑛 𝑓𝑛 ‖1 = 󵄨󵄨󵄨 ∑ 𝑐𝑛−𝑘 𝑒𝑘 󵄨󵄨󵄨 ≥ ∑ 𝑛−𝑘 ≥ 𝑐(𝑛 + 1)2𝑛(2/𝑝−1) , 󵄨󵄨 𝑘=0 󵄨󵄨 󵄨 󵄨1 𝑘=0 𝑘 + 1 i.e. ‖𝑓𝑛 ‖𝑍 ≥ 𝑐(𝑛 + 1)2−𝑛𝛾 2𝑛(2/𝑝−1) = 𝑐(𝑛 + 1)2𝑛(−2𝛽+1/𝑝) . On the other hand, 1 𝑞 ‖𝑓𝑛 ‖𝑌

2𝛽𝑞−1 𝑞2𝑛

≍ ∫(1 − 𝑟) 0

𝑟

󵄨󵄨2 𝑞/𝑝 󵄨󵄨2𝑛 −1 󵄨󵄨 𝑘 𝑖𝑘𝜃 󵄨󵄨󵄨 ( ∫ 󵄨󵄨 ∑ 𝑟 𝑒 󵄨󵄨 ) 𝑑𝑟 𝑑𝜃 ≤ 𝐶2𝑛𝑞/𝑝 2−2𝑛𝛽𝑞 , 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨 𝑘=0 2𝜋 󵄨

(+)

174 | 5 Besov spaces where the inequality 2𝜋 󵄨 𝑛 󵄨󵄨2 2𝑛 −1 󵄨󵄨2 −1 󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨 ∑ 𝑟𝑘 𝑒𝑖𝑘𝜃 󵄨󵄨󵄨 = ∑ 𝑟2𝑘 ≤ 2𝑛 󵄨󵄨 𝑘=0 󵄨󵄨 𝑘=0 󵄨 0 󵄨

has been used. 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 [48, Lemma 1.14]: There ex­ ists 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.) 5.2. If 1 < 𝑝 < ⬦, then, due to Theorem 5.1, inequalities (5.22) and (5.23) can be ex­ pressed in the form ∞

∞

𝑛=0

𝑛=0

2/𝑝

𝑐𝑝 ∑ ‖Δ 𝑛 𝑓‖𝑝𝑝 ≤ ‖𝑓‖𝑝𝑝 ≤ 𝐶𝑝 ( ∑ ‖Δ 𝑛 𝑓‖2𝑝 ) ∞

𝑐𝑝 ( ∑ 𝑛=0

2/𝑝

‖Δ 𝑛 𝑓‖2𝑝 )

(2 ≤ 𝑝 < ⬦)

∞

≤ ‖𝑓‖𝑝𝑝 ≤ 𝐶𝑝 ∑ ‖Δ 𝑛 𝑓‖𝑝𝑝

(2 ≤ 𝑝 < ⬦).

𝑛=0

These inequalities were deduced from the Littlewood–Paley 𝑔-theory by Sledd [448], who, however, did not recognized their connection with Theorems 2.21 and 5.14.

6 The dual of 𝐻1 and some related spaces We define the space BMO (of functions of bounded mean oscillation) via a Garsia-type 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, (𝐻1 )𝐴 ≃ BMOA, without using the Carleson measures and Carleson’s theo­ rem. 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, in Section 6.8, recent results on compact composition operators on the Bloch space and BMOA.

6.1 Norms on BMOA A function 𝑔 ∈ 𝐿1 (𝕋) is called, after John and Nirenberg [235], a function of bounded mean oscillation if sup 𝐼⊂𝕋

1 󵄨󵄨 󵄨 ∫ 󵄨𝑔 − 𝑔𝐼 󵄨󵄨󵄨 𝑑𝑙 = ‖𝑔‖∗ < ∞, |𝐼| 󵄨 𝐼

where

𝑔𝐼 =

1 ∫ 𝑔 𝑑𝑙, |𝐼| 𝐼

and the supremum is taken over all subarcs of 𝕋. The class BMO = {𝑔 : ‖𝑔‖∗ < ∞} is normed by ‖𝑔‖ = ‖𝑔‖𝐿1 + ‖𝑔‖∗ . It is not easy to see that BMO coincides with the class BMO2 which consists of the functions 𝑔 ∈ 𝐿2 (𝕋) such that 1 ‖𝑔‖2∗∗ := sup ∫ |𝑔(𝑒𝑖𝑡 ) − 𝑔𝐼 |2 𝑑𝑡 < ∞. 𝐼⊂𝕋 𝐼 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‖∗ while this is implied by the strong John–Nirenberg inequality: there exists 𝑐 > 0 such that 𝜀 < 𝑐/‖𝑔‖∗ implies 𝑖𝑡 1 ∫ 𝑒𝜀|𝑔(𝑒 )−𝑔𝐼 | 𝑑𝑡 < ∞. sup 𝐼⊂𝕋 |𝐼| 𝐼

We are mainly concerned with the intersection of BMO with 𝐻1 (𝕋), which is denoted by BMOA. Since BMOA ⊂ 𝐻1 (𝕋), we can treat BMOA as a subset of 𝐻(𝔻). It is known that (see [159, Ch. VI, Th. 1.2]) − |𝑓∗ (𝜁) − 𝑓(𝑧)| 𝑃(𝑧, 𝜁) |𝑑𝜁| =: ‖𝑓‖∗1 . ‖𝑓∗ ‖∗ ≍ sup ∫ 𝑧∈𝔻

𝕋

176 | 6 The dual of 𝐻1 and some related spaces We do not reproduce the proof of this fact here. Instead, we define BMOA by the re­ quirement ‖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 1/2 2

‖𝑓‖BMO2 := |𝑓(0)| + sup (− ∫ |𝑓(𝜉) − 𝑓(𝑎)| 𝑃(𝑎, 𝜉) |𝑑𝜉|) 𝑎∈𝔻

𝕋

= |𝑓(0)| + ‖𝑓‖∗2 , (see [159, Ch. VI]). Here we start from “our” definition and prove Garsia’s theorem, and then use it to prove Fefferman’s duality theorem. First observe that ‖𝑓‖∗1 can be expressed as − |𝑓(𝜎𝑎 (𝜁)) − 𝑓(𝜁)| |𝑑𝜁|. ‖𝑓‖∗1 = sup ∫ 𝑎∈𝔻

𝕋

As follows from the following lemma, the seminorm ‖𝑓‖∗2 can be expressed in several ways. Lemma 6.1. We have 𝐵1 (𝑎, 𝑓) = 𝐵2 (𝑎, 𝑓) = 𝐵3 (𝑎, 𝑓) = 𝐵4 (𝑎, 𝑓) = 𝐵5 (𝑎, 𝑓) (𝑎 ∈ 𝔻, 𝑓 ∈ 𝐻2 ), where − |𝑓(𝜉) − 𝑓(𝑎)|2 𝑃(𝑎, 𝜉) |𝑑𝜉|, 𝐵1 (𝑎, 𝑓) = ∫ 𝕋

𝐵2 (𝑎, 𝑓) = ∫ − (|𝑓(𝜉)|2 − |𝑓(𝑎)|2 )𝑃(𝑎, 𝜉) |𝑑𝜉|, 𝑇

𝐵3 (𝑎, 𝑓) = ∫ − |𝑓(𝜎𝑎 (𝜁)) − 𝑓(𝑎)|2 |𝑑𝜁|, 𝕋

𝐵4 (𝑎, 𝑓) =

2 1 ∫ |𝑓󸀠 (𝜎𝑎 (𝑧))|2 |𝜎𝑎󸀠 (𝑧)|2 log 𝑑𝐴(𝑧), 𝜋 |𝑧| 𝔻

2 1 𝑑𝐴(𝑧). 𝐵5 (𝑎, 𝑓) = ∫ |𝑓󸀠 (𝑧)|2 log 𝜋 |𝜎𝑎 (𝑧)| 𝔻

Recall that 𝜎𝑎 (𝑧) =

𝑎−𝑧 . 1 − 𝑎𝑧̄

Proof. The identity 𝐵1 = 𝐵2 holds because − (|𝑓(𝜁)|2 + |𝑓(𝑎)|2 − 2 Re(𝑓(𝜁)𝑓(𝑎)) 𝑃(𝑎, 𝜁) |𝑑𝜁| 𝐵1 (𝑎, 𝑓) = ∫ 𝕋

and ∫ − (Re(𝑓(𝜁)𝑓(𝑎))𝑃(𝑎, 𝜁) |𝑑𝜁| = |𝑓(𝑎)|2 . 𝕋

6.1 Norms on BMOA |

177

The identity 𝐵1 = 𝐵3 is proved by the change 𝜎𝑎 (𝜁) = 𝜉. That 𝐵1 = 𝐵4 follows from the Green’s applied to 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 𝐵6 (𝑓) := sup(𝐵6 (𝑎, 𝑓))1/2 , 𝑎∈𝔻

where 𝐵6 (𝑎, 𝑓) = ∫ |𝑓󸀠 (𝜎𝑎 (𝑧))|2 |𝜎𝑎󸀠 (𝑧)|2 (1 − |𝑧|2 ) 𝑑𝐴(𝑧) 𝔻

= ∫ |𝑓󸀠 (𝑧)|2 (1 − |𝜎𝑎 (𝑧)|2 ) 𝑑𝐴(𝑧)

(6.1)

𝔻

= ∫ |𝑓󸀠 (𝑧)|2 𝔻 1/2

That 𝐵6 (𝑓) ≍ sup𝑎∈𝔻 (𝐵4 (𝑎, 𝑓))

(1 − |𝑎|2 )(1 − |𝑧|2 ) 𝑑𝐴(𝑧). ̄ 2 |1 − 𝑎𝑧|

= ‖𝑓‖∗2 , is a consequence of the following lemma.

Lemma 6.2. 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 one shows that 1 𝑑𝐴(𝑧) ≍ ∫ 𝑢(𝑧)(1 − |𝑧|2 ) 𝑑𝐴(𝑧), ∫ 𝑢(𝑧) log |𝑧| 𝔻

𝔻

where the equivalence constants are independent of 𝑢. Taking 𝑢(𝑧) = |𝑓󸀠 (𝜎𝑎 (𝑧))|2 |𝜎𝑎󸀠 (𝑧)|2 , we get the result. Lemma 6.3. The space BMOA𝑝 := {𝑓 ∈ 𝐻𝑝 : ‖𝑓‖∗𝑝 < ∞} (𝑝 = 1, 2), normed by, 1/𝑝 𝑝

∫ |𝑓(𝜁) − 𝑓(𝑎)| 𝑃(𝑎, 𝜁) |𝑑𝜁|) ‖𝑓‖BMO𝑝 = |𝑓(0)| + sup (− 𝑎∈𝔻

𝕋

is “homogeneous and has the Fatou property.” “Homogeneous” means: If 𝑓 ∈ 𝑋, then 𝑓𝜁 ∈ 𝑋 (|𝜁| ≤ 1) and sup𝑤∈𝔻 ‖𝑓𝑤 ‖ = ‖𝑓‖; “has the Fatou property” – if sup𝑤∈𝔻 ‖𝑓𝑤 ‖ ≤ 1, then 𝑓 ∈ 𝑋 and ‖𝑓‖ ≤ 1. The discussion on such spaces is postponed to Chapter 11; see Section 11.1.

178 | 6 The dual of 𝐻1 and some related spaces Proof. Consider the case 𝑝 = 1. We have sup ‖𝑓𝑤 ‖BMO1 = |𝑓(0)| + sup sup ∫ − |𝑓(𝑤𝜁) − 𝑓(𝑤𝑎)|𝑃(𝑎, 𝜁)| |𝑑𝜁| 𝑎∈𝔻 |𝑤| 1, is increasing, we have (1 − 1/𝜆 𝑛)𝜆 𝑛 ≥ (1 − 1/𝜆 1 )𝜆 1 ≥ 1/4, where 󰜚 is chosen so that 𝜆 1 ≥ 2, and (1 −

𝜆𝑗 1 𝜆𝑗 ) ≤ exp (− ), 𝜇𝑛 𝜇𝑛

we see that ∞

𝜆𝑗

𝑗=𝑛+1

𝜇𝑛

|ℎ1 (𝑧)| ≥ 󰜚𝑛 (1/4 − 1/󰜚𝑛 − 1/(󰜚 − 1)) − ∑ 󰜚𝑗 exp (− To estimate the last sum we proceed as follows: ∞

𝜆𝑗

𝑗=𝑛+1

𝜇𝑛

∑ 󰜚𝑗 exp (−

∞

𝜆 𝑗+𝑛

𝑗=1

𝜇𝑛

) = 󰜚𝑛 ∑ 󰜚𝑗 exp (−

)

∞

≤ 󰜚𝑛 ∑ 󰜚𝑗 exp (−2−𝑗−1/2 󰜚𝑗−1 √󰜚). 𝑗=1

Since

∞

lim ∑ 󰜚𝑗 exp (−2−𝑗−1/2 󰜚𝑗−1 √󰜚) = 0,

󰜚→∞

𝑗=1

we can choose 󰜚 > 101 so that ∞

∑ 󰜚𝑗 exp (−2−𝑗−1/2 󰜚𝑗−1 √󰜚) < 1/100. 𝑗=1

Since 1/󰜚𝑛 < 1/100, we have 1/4 − 1/󰜚𝑛 − 1/(󰜚 − 1) > 1/4 − 1/100 − 1/100. Combining all these estimates we get (6.34).

).

206 | 6 The dual of 𝐻1 and some related spaces In the case of (6.35) we have 𝑛−1

∞

𝑗=1

𝑗=𝑛+1

|ℎ2 (𝑧)| ≥ 󰜚𝑛 |𝑧|𝜇𝑛 − 1 − ∑ 󰜚𝑗 − ∑ 󰜚𝑗 |𝑧|𝜇𝑗 ≥ 󰜚𝑛 (1 −

𝜇𝑛

∞ 󰜚𝑛 1 1 𝜇𝑗 − ∑ 󰜚𝑗 (1 − ) −1− ) 𝜇𝑛 󰜚 − 1 𝑗=𝑛+1 𝜆 𝑛+1 ∞

𝜇𝑗

𝑗=𝑛+1

𝜆 𝑛+1

≥ 󰜚𝑛 (1/4 − 1/󰜚𝑛 − 1/(󰜚 − 1)) − ∑ 󰜚𝑗 exp (−

)

∞

𝜇𝑗+𝑛

𝑗=1

𝜆 𝑛+1

= 󰜚𝑛 (1/4 − 1/󰜚𝑛 − 1/(󰜚 − 1)) − 󰜚𝑛 ∑ 󰜚𝑗 exp (−

)

∞

≥ 󰜚𝑛 (1/4 − 1/󰜚𝑛 − 1/(󰜚 − 1)) − 󰜚𝑛 ∑ 󰜚𝑗 exp (−23/2−2𝑗 󰜚𝑗−1 √󰜚) . 𝑗=1

Now the proof of (6.35) is completed as in the case of (6.34). Finally, we have to prove that |ℎ(𝑧)| ≤ 𝐶𝜙 (

1 ), 1 − |𝑧|

|𝑧| < 1,

where ℎ = ℎ1 or ℎ2 , but this follows from the modified Lemma 6.10. Thus the theorem is proved.

Further notes and results Our approach to BMOA in Section 6.2 (except Subsection 6.2.1) is essentially the same as in Knese’s paper [267], although we have made some technical simplifications. Also, Knese does not treat the case 𝑝 < 1 of Theorems 6.2 and 6.3. Beside Knese’s paper, the reader could read [378] for various, rather paradoxical characterizations of ℎ1 . A rather complicated proof of Theorem 6.3 (𝑝 ≥ 1) and Theorem 6.2 (𝑝 > 0) can be found in Girela’s survey paper [163, Theorems 4.1, 5.1, and 5.4]. This paper and also Baernstein’s paper [40] contain a lot of information and references. The first characterization of compact operators on B, the equivalence (a) ⇔ (c) of Theorem 6.17, was found by Madigan and Matheson [316]. The equivalence (a) ⇔ (d) is due to Maria Tjani [490], and (a) ⇔ (b) was proved in Wulan–Zheng–Zhu [504]. The relations (a) ⇔ (b) ⇒ (c) of Theorem 6.18 were proved by Wulan et al. [504]. Their proof is based on an earlier result of Wulan [503], who proved that (a) is equiva­ lent to (b) and (c); in [504], it is only observed that (b) implies (c). Theorem 6.19 from which the validity of “(c) ⇒ (a)” follows immediately was proved by Laitila et al. [290]. For further results, references, and the history of the subject, see [504, 290, 289]. Lemma 6.11 was proved in [288, 1]; for a substantial improvement, see [175].

6.9 Proof of the bi-Bloch lemma | 207

6.1 (Fefferman’s theorem and the Carleson measures). Theorem 6.4 is usually proved by using the famous Carleson’s theorem on the Carleson measures. A positive mea­ sure 𝜇 on 𝔻 is called a Carleson measure if there is a constant 𝐶 such that 𝜇(𝑊(𝐼)) ≤ 𝐶|𝐼| for every arc 𝐼 ⊂ 𝕋, where 𝑧 ∈ 𝐼}. 𝑊(𝐼) = {𝑧 ∈ 𝔻 : 1 − |𝐼| < |𝑧| < 1, |𝑧| The set 𝑊(𝐼) is called a Carleson window. Theorem (Carleson [86, 88]). Let 𝜇 be a positive Borel measure on 𝔻. Then the follow­ ing assertions are equivalent: – 𝜇 is a Carleson measure. – There is a constant 𝐶1 such that 𝜇{|𝑓| > 𝜆} ≤ 𝐶1 |{𝑀∗ 𝑓 > 𝜆}| for every Borel func­ tion 𝑓 on 𝔻. – There is a constant 𝐶2 such that (∫𝔻 |𝑓|𝑝 𝑑𝜇)1/𝑝 ≤ 𝐶2 ‖𝑓‖𝑝 for all 𝑝 > 0 and 𝑓 ∈ 𝐻𝑝 . – 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 . A relatively easy, geometrically obvious proof is in Garnett [159]. Having proved Carleson’s theorem, one can easily prove Fefferman’s theorem; see [526, Section 5.4] for the case of several variables. 6.2 (𝑄𝑝 -spaces). For 0 < 𝑝 < ⬦, the space 𝑄𝑝 ⊂ 𝐻(𝔻) is defined by the requirement sup ∫ |𝑓󸀠 (𝑧)|2 (1 − |𝜎𝑎 (𝑧)|2 )𝑝 𝑑𝐴(𝑧) < ∞. 𝑎∈𝔻

𝔻

The reader is referred to the Xiao’s books [506, 507] for the theory, history, and refer­ ences. Here we note that 𝑄𝑝 = B for 𝑝 > 1 and, obviously 𝑄1 = BMOA. 6.3 (Higher order derivatives). It follows from our definition of BMOA, Lemma 6.2, and Carleson’s theorem that a function 𝑓 ∈ 𝐻(𝔻) belongs to BMOA if and only if 𝑑𝜇(𝑧) := (1 − |𝑧|2 )|𝑓󸀠 (𝑧)|2 𝑑𝐴(𝑧) is a Carleson measure. Here 𝑓󸀠 can be replaced by J1 . In [226], Jevtić considered the measures (on the unit ball of ℂ𝑛 ) 𝑑𝜇𝑠 (𝑧) = (1 − |𝑧|2 )2𝑠−1 |J𝑠 (𝑧)|2 𝑑𝐴(𝑧),

𝑠 > 0,

and proved that 𝑑𝜇𝑠 is a Carleson measure if and only if so is 𝑑𝜇. As a consequence, we have that 𝑓 ∈ BMOA if and only if sup ∫ |J𝑠 𝑓(𝑧)|2 (1 − |𝑧|2 )2𝑠−1 𝑎∈𝔻

𝔻

1 − |𝑎|2 𝑑𝐴(𝑧) < ∞ ̄ 2 |1 − 𝑎𝑧|

for some (for all) 𝑠 > 0. The analogous result but with 𝑓(𝑠) (𝑠 ∈ ℕ+ ) instead of J𝑠 was later proved by Aulaskari–Nowak–Zhao [33] in the context of 𝑄𝑝 -spaces.

208 | 6 The dual of 𝐻1 and some related spaces 6.4 (Inner functions in the little Bloch space). There are inner functions in b that are not finite Blaschke product (Stephenson [469]). An infinite Blaschke product belong­ ing to b was constructed by Bishop [58, 59]. Each of these constructions are highly nontrivial. See, however, Garnett [159, Ch. X, Further results 11], where a simpler proof is outlined. Aleksandrov, Anderson, and Nicolau [17, Theorem 3], substantially im­ proved Stephenson’s theorem by proving that there is an inner function 𝐼 ∈ b such that |𝐼󸀠 (𝑧)|(1 − |𝑧|2 ) = 0. lim− |𝑧|→1 1 − |𝐼(𝑧)|2 As shown in [17], such a function cannot be extended analytically to any point of 𝕋; on the other hand, there is an inner function in b which extends analytically to a.e. point of 𝕋. 𝑘𝑛 6.5 (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 Pom­ merenke [400] and then generalized by Yamashita [513]. This result was extended to some special cases of weighted Bloch spaces by several authors (see, e.g. [277]). Finally, Yang and Xu [518] proved a result which covers all these special cases. In the terms of weighted Bloch spaces, defined by (6.18), 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 3.28 we obtain much more [380]: If 𝜔(𝑡)/𝑡 is normal, then

sup 0 𝑝, then 1

𝐼𝑝,𝑞 (𝑓) := ∫ (𝑀𝑝 (𝑟, 𝑓) /log1/𝑠 0

𝑞 2 𝑑𝑟 ) 1 was established in [167]. The proof was, however, unnecessarily complicated. A simple proof, which also works in the case of harmonic functions of several variables, goes as follows. Assume that 𝑓(0) = 0, and consider the case when 𝑝 ≥ 2 and 𝑞 > 2. By Theo­ rem 2.21 and Hölder’s inequality with indices 𝑞/𝑠, 𝑞/(𝑞 − 𝑠), 𝑞/𝑠

𝑟 𝑞/𝑠 (𝑀𝑝𝑠 (𝑟, 𝑓))

≤

𝐶 (∫ 𝑀𝑝𝑠 (𝑡, 𝑓󸀠 )(𝑟

𝑠−1

− 𝑡)

𝑑𝑡)

0 𝑟

≤ 𝐶 ∫ 𝑀𝑝𝑞 (𝑡, 𝑓)(1 − 𝑡)(𝑠−1)𝑞/𝑠 𝜙(𝑡)𝑞/𝑠 𝑑𝑡 0 (𝑞−𝑠)/𝑠

𝑟 −𝑞/(𝑞−𝑠)

× (∫ 𝜙(𝑡)

𝑑𝑡)

,

0

where

2 . 1−𝑡 After computing the inner integral we use the resulting inequality and Fubini’s the­ orem to 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 𝜙(𝑡) = (1 − 𝑡)(𝑞−𝑠)/𝑞 log

1

‖𝑓‖𝑝𝑝 ≤ 𝐶𝑝 ∫ 𝑀𝑝𝑝 (𝜌, J1/𝑝 𝑓) 𝑑𝜌,

0 < 𝑝 < 2,

0

and, at the very end, Theorem 3.19. 6.8. It was proved by Holland and Walsh [217] that a function 𝑓 ∈ 𝐻(𝔻) belongs to B if and only if 𝛽(𝑓) := sup 𝑤,𝑧∈𝔻

|𝑓(𝑤) − 𝑓(𝑧)|√(1 − |𝑧|2 )(1 − |𝑤|2 ) < ∞. |𝑤 − 𝑧|

It maybe surprising that the identity 𝛽(𝐹) = B(𝐹) := sup𝑧∈𝔻 (1 − |𝑧|2 )‖𝐹󸀠 (𝑧)‖ holds for any 𝐶1 -function 𝐹 with values in a Banach space [377]. The hypothesis 𝐹 ∈ 𝐶1 can be weakened so one can use the relation |∇|𝑓| | = |𝑓󸀠 | (𝑓 ∈ 𝐻(𝔻)) to show that B(𝑓) = 𝛽(|𝑓|), for 𝑓 ∈ B. Another example is sup (1 − 𝑟2 )𝑀𝑝 (𝑟, 𝑓󸀠 ) = sup

0 0 and 𝑞 > 0, then we choose 𝛾 > 0 so that 𝑝/𝛾 > 1 and 𝑞/𝛾 > 1 and apply Theorem 7.1 to the functions 𝑢𝛾𝑛 and the indices 𝑝/𝛾 and 𝑞/𝛾 instead of 𝑝 and 𝑞. If the functions 𝑢𝑛 are continuous on the closed disk, then the right-hand side of (7.5) can be replaced with 𝑝/𝑞

∞

∫ ( ∑ 𝑢𝑛 𝑞 )

𝑑ℓ.

𝑛=0

𝕋

This will be used in the sequel. 𝑞 Theorem 7.3. Let {𝑓𝑗 }∞ 0 be a sequence in 𝐻 . If

and 𝑚 > max{0, 1/𝑝 − 1, 1/𝑞 − 1},

𝑝 > 0, 𝑞 > 0, then

𝑝/𝑞

∞

𝑞 ∫ (∑ (𝜎∗𝑚 𝑓𝑗 ) ) 𝑗=0 𝕋

∞

𝑝/𝑞 𝑞

𝑑ℓ ≤ 𝐶 ∫ (∑ |𝑓𝑗 | ) 𝕋

𝑑ℓ.

𝑗=0

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}. We have by (4.24), 𝑝/𝑞

∞

Σ := ∫ ( ∑ (𝜎∗𝑚 𝑓𝑛 )𝑞 ) 𝑛=0

𝕋

𝑑ℓ

𝑛=0 𝕋 𝑝0 /𝑞0

∞

= 𝐶 ∫ ( ∑ [M(𝑀rad |𝑓𝑛 |𝑠 )]𝑞0 ) 𝕋

𝑝/𝑞

∞

𝑑ℓ ≤ 𝐶 ∫ ( ∑ [M(𝑀rad |𝑓𝑛 |𝑠 )]𝑞/𝑠 )

𝑑ℓ,

𝑛=0

where 𝑝0 = 𝑝/𝑠 > 1 and 𝑞0 = 𝑞/𝑠 > 1. Using Theorem B.24 with the indices 𝑝0 and 𝑞0 we obtain ∞

Σ ≤ 𝐶 ∫ ( ∑ (𝑀rad |𝑓𝑛 |𝑠 )𝑞0 (𝜁)) 𝕋

𝑝0 /𝑞0

𝑑ℓ.

𝑛=0

Now we can apply Theorem 7.1 or Theorem 7.2 to conclude the proof.

7.2 Littlewood–Paley 𝑔-theorem |

213

Theorem 7.4 (Calderón). Let 𝐷𝜁 , 𝜁 ∈ 𝕋, be one of the sets: S𝜁,𝐵 , 𝑈𝜁,𝑐 , or 𝑇𝜁 . If 𝑓 is analytic in the unit disk, 𝑓(0) = 0, 𝑝 > 0, 𝑞 > 0, then 𝑝/𝑞

‖𝑓‖𝑝𝑝

≍ ∫ (∫ |𝑓|

𝑞−2

󸀠 2

|𝑓 | 𝑑𝐴)

|𝑑𝜁|.

𝐷𝜁

𝕋

Calderón stated this theorem for upper half-plane. However, in order to apply his proof (postponed to Section 7.5, Theorem 7.10) to the unit disk, only one fact must be explain (Lemma 7.7). The function A𝑓(𝜁) = ∫𝐷 |𝑓󸀠 (𝑧)|2 𝑑𝐴(𝑧), where 𝐷𝜁 is S𝜁,𝐵 or 𝑈𝜁,𝑐 , introduced 𝜁

in [309], 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 [537, Ch. XIV]. As a special case of Calderón’s theorem we have Theorem 7.5. Let 𝑝 > 0, and 𝑓 ∈ 𝐻(𝔻) with 𝑓(0) = 0. Then 𝑓 ∈ 𝐻𝑝 if and only if A𝑓 ∈ 𝐿𝑝 (𝕋), and we have ‖𝑓‖𝑝 ≍ ‖A𝑓‖𝑝 .

7.2 Littlewood–Paley 𝑔-theorem For 𝑓 ∈ 𝐻(𝔻), the function 1/2

1 󸀠

2

G[𝑓](𝜁) = (∫(1 − 𝑟)|𝑓 (𝑟𝜁)| 𝑑𝑟)

,

𝜁 ∈ 𝕋 := 𝜕𝔻,

0

is called the (Littlewood–Paley) 𝑔-function associated to 𝑓. The corresponding “max­ imal function” is defined as 1/2

1

G∗ [𝑓](𝜁) = (∫(1 − 𝑟) sup |𝑓󸀠 (𝜌𝜁)|2 𝑑𝑟)

.

0 0), then lim (1 − 𝑟) sup |𝑓󸀠 (𝜌𝜁)| = 0 for almost all 𝜁 ∈ 𝕋.

𝑟→1−

𝜌 1. 1 This step is not needed in the case of the half-plane.

226 | 7 Littlewood–Paley theory Lemma 7.9. If the inequality (all 𝐺 ∈ G, 𝐶 independent of 𝐺),

‖𝑆𝐺‖𝑠 ≤ 𝐶‖𝐺‖𝑠

(7.15)

holds for some 𝑠 > 0, then it holds for all 𝑞 ∈ (0, 𝑠). Proof. Here and elsewhere we assume that 𝐺 = |𝑓𝜌 |𝛿 , where 𝑓𝜌 (𝑧) = 𝑓(𝜌𝑧) (0 < 𝜌 < 1) is analytic a neighborhood of the closed disk. The final conclusion is obtained by let­ ting 𝜌 → 1− and using Fatou’s lemma together the fact that the constant 𝐶 is indepen­ dent of 𝜌. Let 0 < 𝑞 < 𝑠, 𝑝 = 𝑞𝑠 > 1. Then by (7.14) applied to 𝐺1/𝑝 we have 𝑆𝐺 ≤ 𝑝 (𝑀∗ (𝐺1/𝑝 ))

𝑝−1

𝑆[𝐺1/𝑝 ] = 𝑝(𝑀∗ 𝐺)(𝑝−1)/𝑝 𝑆[𝐺1/𝑝 ].

Hence by Hölder’s inequality with the indices 𝑝 = 𝑠/𝑞 and 𝑝/(𝑝 − 1) = 𝑠/(𝑠 − 𝑞), 󵄩 󵄩 ‖𝑆𝐺‖𝑞𝑞 ≤ 𝑝𝑞 󵄩󵄩󵄩󵄩(𝑀∗ 𝐺)𝑞(𝑝−1)/𝑝 (𝑆[𝐺1/𝑝 ])𝑞 󵄩󵄩󵄩󵄩1 󵄩 󵄩 ≤ 𝑝𝑞 󵄩󵄩󵄩󵄩(𝑀∗ 𝐺)𝑞(𝑝−1)/𝑝 󵄩󵄩󵄩󵄩𝑝/(𝑝−1) ‖𝑆[𝐺1/𝑝 ]𝑞 ‖𝑝 = 𝑝𝑞 ‖𝑀∗ 𝐺‖(𝑝−1)𝑞/𝑝 ‖𝑆[𝐺1/𝑝 ]‖𝑞𝑠 . 𝑞 From this and (7.15), via the maximal theorem, it follows that ‖𝐺1/𝑝 ‖𝑞𝑠 = 𝐶‖𝐺‖𝑞𝑞 . ‖𝑆𝐺‖𝑞𝑞 ≤ 𝐶‖𝐺‖(𝑝−1)𝑞/𝑝 𝑞 This finishes the proof. In view of Lemma 7.9, the following fact completes the proof of the inequality ‖𝑆𝐺‖𝑞 ≤ 𝐶‖𝐺‖𝑞 for all 𝑞 > 0. Lemma 7.10. The inequality ‖𝑆𝐺‖𝑞 ≤ 𝐶‖𝐺‖𝑞 ,

𝑞 > 4,

(7.16)

holds. Proof. Let 𝑞 > 2. Let ℎ(𝜁) and 𝜁 ∈ 𝕋 be any bounded positive measurable function. Then ∫(𝑆𝐺)2 ℎ 𝑑ℓ = ∫ ℎ(𝜁) |𝑑𝜁| ∫ 𝐾(𝑧, 𝜁)|∇𝐺(𝑧)|2 𝑑𝐴(𝑧) 𝕋

𝔻

𝕋 2

= ∫ |∇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) = (1 − |𝑧|2 )/|1 − 𝑧|2 , we see that ∫(𝑆𝐺)2 𝑑ℓ ≤ 𝐶 ∫ |∇𝐺(𝑧)|2 (1 − |𝑧|)𝑢(𝑧) 𝑑𝐴(𝑧) 𝕋

𝔻

≤ 𝐶 ∫ Δ(𝐺2 )𝑢(𝑧)(1 − |𝑧|) 𝑑𝐴, 𝔻

where 𝑢 = P[ℎ].

7.5 Proof of Calderón’s theorem | 227

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|

(7.17)

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 𝔻

𝕋 2

≤ 𝐶1 ∫ 𝐺 ℎ 𝑑ℓ + 𝐶4 ∫(𝑀∗ 𝐺)(𝑆𝐺)(𝑆𝑢) 𝑑ℓ. 𝕋

𝕋

Now let 𝑝 = 𝑞/(𝑞 − 1) (< 2) and apply the three-term Hölder inequality with the 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 , where we have used Lemma 7.8 (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 ,

228 | 7 Littlewood–Paley theory which implies 𝐶 + √𝐶2 + 4𝐶 ‖𝐺‖2𝑞 . 2

‖𝑆𝐺‖2𝑞 ≤

This completes the proof of (7.16) and therefore of the inequality ‖𝑆𝐺‖𝑞 ≤ 𝐶‖𝐺‖𝑞 for all 𝑞 > 0.

Proof of the inequality ‖𝐺‖𝑞 ≤ 𝐶‖𝑆𝐺‖𝑞 (𝑞 > 0) In proving this we use the reverse inequality, which we have proved, and a simple lemma (see [82, (2)].) Lemma 7.11. If 𝛼𝜎 + 𝛽(1 − 𝜎) = 1, where 0 < 𝜎 < 1 and 𝛼, 𝛽 > 0, then (1−𝜎) 𝜎 1 1 𝑆(𝐺) ≤ [ 𝑆(𝐺𝛼 )] [ 𝑆(𝐺𝛽 )] . 𝛼 𝛽

The proof is simple and we omit it. Assume first that 𝐺 = |𝑓|𝛿 , where 𝑓 is analytic on the closed disk. We have ‖𝐺‖𝑞𝑞 = ‖𝐺𝑞/2 ‖22 ≍ ‖𝑆[𝐺𝑞/2 ]‖22 ≤ 𝐶‖𝑆[𝐺𝛼𝑞/2 ]2𝜎 𝑆[𝐺𝛽𝑞/2 ]2(1−𝜎) ‖1 = 𝐶‖𝑆[𝐺𝛼𝑞/2 ]2𝜎 (𝑆𝐺)2(1−𝜎) ‖1 , where we have used Lemma 7.11 with 𝛼=

2𝑞 , 𝑞+2

𝛽=

2 , 𝑞

𝜎=

𝑞+2 , 2(𝑞 + 1)

1−𝜎=

𝑞 . 2(𝑞 + 1)

Hence, by Hölder’s inequality with the indices (𝑞 + 1)/𝑞 and 𝑞 + 1, ‖𝐺‖𝑞𝑞 ≤ 𝐶‖𝑆[𝐺𝛼𝑞/2 ]2𝜎 ‖(𝑞+1)/𝑞 ‖(𝑆𝐺)2(1−𝜎) ‖𝑞+1 . Since 2𝜎(𝑞 + 1)/𝑞 = (𝑞 + 2)/𝑞, we have ‖𝑆[𝐺𝛼𝑞/2 ]2𝜎 ‖(𝑞+1)/𝑞 = ‖𝑆[𝐺𝛼𝑞/2 ]‖2𝜎 (𝑞+2)/𝑞 , and, since 2(1 − 𝜎) = 𝑞/(𝑞 + 1), we also have ‖(𝑆𝐺)2(1−𝜎) ‖𝑞+1 = ‖𝑆𝐺‖2(1−𝜎) 𝑞 so 2(1−𝜎) . ‖𝐺‖𝑞𝑞 ≤ 𝐶‖𝑆[𝐺𝛼𝑞/2 ]‖2𝜎 (𝑞+2)/𝑞 ‖𝑆𝐺‖𝑞

On the other hand, we have proved that , ‖𝑆[𝐺𝛼𝑞/2 ]‖(𝑞+2)/𝑞 ≤ 𝐶‖𝐺𝛼𝑞/2 ‖(𝑞+2)/𝑞 = 𝐶‖𝐺‖𝛼𝑞/2 𝑞

7.6 Littlewood–Paley inequalities |

229

whence 2𝜎

) ‖𝑆𝐺‖2(1−𝜎) , ‖𝐺‖𝑞𝑞 ≤ 𝐶 (‖𝐺‖𝛼𝑞/2 𝑞 𝑞 and hence ‖𝐺‖𝑞(1−𝛼𝜎) ≤ 𝐶‖𝑆𝐺‖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‖𝑆𝐺‖𝑞 , which completes the proof of Theorem 7.10 in the case where 𝑓 is analytic on the closed disk. If 𝑓 ∈ 𝐻(𝔻) is arbitrary, then we apply the result to the functions 𝐺𝜌 = |𝑓𝜌 |𝛿 , 𝜌 < 1. By the preceding reasoning, we have 𝑞/2 2

2

‖𝐺𝜌 ‖𝑞 ≤ 𝐶 ∫ (∫ 𝜌 |∇𝐺(𝜌𝑧)| 𝑑𝐴(𝑧)) 𝕋

|𝑑𝜁|,

𝑈𝜁

whence, by the substitution 𝜌𝑧 = 𝑤, 𝑞/2

‖𝐺𝜌 ‖𝑞 ≤ 𝐶 ∫ ( ∫ 𝜌2 |∇𝐺(𝑤)|2 𝑑𝐴(𝑤)) 𝕋

|𝑑𝜁|,

𝜌𝐷𝜁

where 𝜌𝐷𝜁 := {𝜌𝑤 : 𝑤 ∈ 𝐷𝜁 } ⊂ 𝐷𝜁 . It follows that ‖𝐺𝜌 ‖𝑞 ≤ 𝐶‖𝑆𝐺‖𝑞 , where 𝐶 is inde­ pendent of 𝜌. Letting 𝜌 tend to 1 and using Fatou’s lemma we conclude the proof of Theorem 7.10.

7.6 Littlewood–Paley-type inequalities The Littlewood–Paley inequalities (Theorem 5.14) are closely related to the Little­ wood–Paley theory and in fact Littlewood and Paley deduced them from the 𝑔-theorem ([304, Theorems 6 and 7]). However, the term “the Littlewood–Paley inequality” usu­ ally means just inequality (5.24) perhaps because Zygmund does not mention the case 𝑝 < 2 (see [537, Ch. XIV (3.24)], where (5.24) is proved by means of the Riesz–Thorin theorem). As we saw, (5.24) lies not so deep as the 𝑔-theorem. Besides, we need not 󸀠 󸀠 use the dual of B𝑝 ,𝑝 when 1 < 𝑝 < 2; see [374, Theorem 3.5.1]. In this section we prove some Littlewood–Paley-type inequalities for subharmonic functions with quasi-nearly subharmonic (QNS) Laplacian, and then obtain (5.24) in a stronger form; see (7.20). For a function 𝑢 defined in 𝔻 we write 𝐼(𝑢) = sup0 0,

𝐸𝜀 (𝑧)

which leads to the following: Theorem 7.12. Let 𝑢 ≥ 0 be a subharmonic function of class 𝐶2 (𝔻) such that its Lapla­ cian is a quasi-nearly subharmonic function. If 𝑞 ≥ 1 and 𝐼(𝑢𝑞 ) < ∞, then ∫(1 − |𝑧|)2𝑞−1 (Δ𝑢(𝑧))𝑞 𝑑𝐴(𝑧) ≤ 𝐶𝑞 (𝐼(𝑢𝑞 ) − 𝑢(0)𝑞 ) .

(7.19)

𝔻

Inequality (5.24) is a special case of (7.19): if 𝑝 ≥ 2 and 𝐼(|ℎ|𝑝 ) < ∞, where ℎ is a realvalued function harmonic in 𝔻, then we take 𝑢 = ℎ2 and 𝑞 = 𝑝/2 to get ∫ |∇ℎ(𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧) ≤ 𝐶(‖𝑢‖𝑝𝑝 − |𝑢(0)|𝑝 ).

(7.20)

𝔻

In the case 𝑞 < 1 we have the following theorem. Theorem 7.13. Let 0 < 𝑞 < 1 and let 𝑢 ≥ 0 be a 𝐶2 -function such that 𝑢𝑞 is subharmonic and Δ𝑢 quasi-nearly subharmonic. If ∫(1 − |𝑧|)2𝑞−1 (Δ𝑢(𝑧))𝑞 𝑑𝐴(𝑧) < ∞, 𝔻

then 𝐼(𝑢𝑞 ) < ∞ and we have 𝐼(𝑢𝑞 ) − 𝑢(0)𝑞 ≤ 𝐶𝑞 ∫(1 − |𝑧|)2𝑞−1 (Δ𝑢(𝑧))𝑞 𝑑𝐴(𝑧).

(7.21)

𝔻

If 𝑓 is analytic, then we can take 𝑢 = |𝑓|2 in Theorems 7.12 and 7.13 to get the following variant of the Littlewood–Paley theorem and also of Flett’s theorem (𝑝 ≤ 1).

7.6 Littlewood–Paley inequalities |

231

Theorem 7.14. Let 𝑓 ∈ 𝐻(𝔻), and 𝐾𝑝 (𝑓) = ∫(1 − |𝑧|)𝑝−1 |𝑓󸀠 (𝑧)|𝑝 𝑑𝐴(𝑧). 𝔻 𝑝

If 𝐾𝑝 (𝑓) < ∞ and 0 < 𝑝 ≤ 2, then 𝑓 ∈ 𝐻𝑝 and ‖𝑓‖𝑝 − |𝑓(0)|𝑝 ≤ 𝐶𝑝 𝐾𝑝 (𝑓). If 𝑝 ≥ 2, and 𝑝 𝑓 ∈ 𝐻𝑝 , then 𝐾𝑝 (𝑓) < ∞ and 𝐾𝑝 (𝑓) ≤ 𝐶𝑝 (‖𝑓‖𝑝 − |𝑓(0)|𝑝 ). See Exercise 7.1 for more general results. It can happened that Δ𝑢 in Theorems 7.12 and 7.13 need not be subharmonic: the function 𝑢(𝑧) = 4|𝑥|5/2 is subharmonic and its Laplacian Δ𝑢(𝑥) = 15|𝑥|1/2 is QNS, but is not subharmonic. Remark 7.2. Stoll’s paper [475] contains interesting (not only) extensions of Theo­ rems 7.12 and 7.13 to a domain in ℝ𝑛 the Green function of which satisfies some mild conditions. We point out that Stoll obtained his results without knowing Theorems 7.12 and 7.13. On the other hand, when writing this section the author was not aware of his paper. In fact, both were motivated by the paper [372]. One of Stoll’ s results states that if 𝑢2 (𝑢 ≥ 0) is subharmonic and |∇𝑢| is QNS, then ∫ |∇𝑢(𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧) ≤ 𝐶𝐼(𝑟, |𝑢|𝑝 ),

𝑝 ≥ 2.

𝔻

It follows from our Theorem 7.12 that this inequality remains true if 𝐼(𝑟, 𝑢𝑝 ) is replaced with 𝐼(𝑟, 𝑢𝑝 ) − 𝑢(0)𝑝 . In particular, this improved inequality holds if 𝑢 is subharmonic and regularly oscillating.

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 ∫ log 𝑑𝜇(𝑧) 2𝜋 |𝑧|

(0 < 𝑟 < 1)

(7.22)

𝑟𝔻

holds (see Theorem 2.6). Lemma 7.12. We have 𝐼(𝑢) − 𝑢(0) =

1 1 ∫ log 𝑑𝜇(𝑧). 2𝜋 |𝑧| 𝔻

Proof. This follows from (7.22). Lemma 7.13. Let 𝑞 ≥ 1 and let 𝜇𝑞 be the Riesz measure of 𝑢𝑞 . Then {𝜇(𝐸)}𝑞 ≤ 𝐶𝑞 𝜇𝑞 (5𝐸) for every disk 𝐸 such that 6𝐸 ⊂ 𝔻. The constant 𝐶𝑞 depends only of 𝑞.

(7.23)

232 | 7 Littlewood–Paley theory If 𝐸 is a disk of radius 𝑅, then 𝑟𝐸 denotes the concentric disk 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, e.g. 𝐸 = 𝜀𝔻, 𝜀 = 1/6. Using the simple inequalities (𝐼(𝑟, 𝑢) − 𝑢(0))𝑞 ≤ (𝐼(𝑟, 𝑢))𝑞 − 𝑢(0)𝑞 and (𝐼(𝑟, 𝑢))𝑞 ≤ 𝐼(𝑟, 𝑢𝑞 ), which hold because 𝑞 > 1, we see from (7.22) (applied to 𝑢 and 𝑢𝑞 ) that 𝑞

𝑟 1 𝑟 1 ∫ log 𝑑𝜇(𝑧)) ≤ ∫ log 𝑑𝜇 (𝑧). ( 2𝜋 |𝑧| 2𝜋 |𝑧| 𝑞 𝑟𝔻

𝑟𝔻

Letting 𝑟 = 4𝜀, we get {𝜇(2𝜀𝔻)}𝑞 ≤ 𝐶 ∫ |𝑧|−1 𝑑𝜇𝑞 (𝑧),

(7.24)

4𝜀𝔻

where we have applied the estimate log(4𝜀/|𝑧|) ≥ log 2 for |𝑧| < 2𝜀, and log(4𝜀/|𝑧|) ≤ 1/|𝑧|. Therefore, in order to prove (7.23) we have to remove |𝑧|−1 . To do this, we translate the “center” of (7.24) to get {𝜇(2𝜀𝐷𝑎 )}𝑞 ≤ 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 {𝜇(𝜀𝔻)}𝑞 ≤ 𝐶 ∫ |𝑧 − 𝑎|−1 𝑑𝜇𝑞 (𝑧). 5𝜀𝔻

Now we integrate this inequality over the disk 𝜀𝔻, with respect to 𝑑𝐴(𝑎), and apply Fubini’s theorem, which finishes the proof because sup 𝑧∈𝔻 ∫5𝜀𝔻 |𝑧−𝑎|−1 𝑑𝐴(𝑎) < ∞. Proof of Theorem 7.11. From (7.23) it follows that 𝑞

∫(1 − |𝑧|)−1 {𝜇(𝐸𝜀 (𝑧))} 𝑑𝐴(𝑧) ≤ 𝐶 ∫(1 − |𝑧|)−1 𝜇𝑞 (𝐸5𝜀 (𝑧)) 𝑑𝐴(𝑧). 𝔻

𝔻

Further, from 𝜇𝑞 (𝐸5𝜀 (𝑧)) = ∫ 𝑑𝜇𝑞 (𝑤) 𝐸5𝜀 (𝑧)

and Fubini’s theorem it follows that the right-hand side of (7.25) equals ∫ 𝑑𝜇𝑞 (𝑤) ∫ (1 − |𝑧|)−1 𝑑𝐴(𝑧), 𝔻

Ω5𝜀 (𝑤)

(7.25)

7.6 Littlewood–Paley inequalities | 233

where (7.26)

Ω𝜀 (𝑤) = {𝑧 : |𝑧 − 𝑤| < 𝜀(1 − |𝑧|)}.

Since 𝑧 ∈ Ω5𝜀 (𝑤) implies |𝑧| − |𝑤| < 5𝜀(1 − |𝑧|) and so 1 − |𝑤| < (1 + 5𝜀)(1 − |𝑧|), we see that ∫ (1 − |𝑧|)−1 𝑑𝐴(𝑧) ≤ (1 + 5𝜀) |Ω5𝜀 (𝑤)| (1 − |𝑤|)−1 . Ω5𝜀 (𝑤)

And since (1 − 5𝜀)(1 − |𝑧|) < 1 − |𝑤| for 𝑧 ∈ Ω(𝑤), we have |Ω5𝜀 (𝑤)| ≤ 𝐶󸀠 (1 − |𝑤|)2 , where 𝐶󸀠 = 𝜋(5𝜀/(1 − 5𝜀))2 . Combining all these results we get 𝑞

∫(1 − |𝑧|)−1 {𝜇(𝐸𝜀 (𝑧))} 𝑑𝐴 ≤ 𝐶𝑞 ∫(1 − |𝑤|) 𝑑𝜇𝑞 (𝑤). 𝔻

𝔻

This completes the proof of (7.18) because of Lemma 7.12 and the inequality 1 − |𝑤| ≤ log(1/|𝑤|). Proof of Theorem 7.13. Fix 𝜀 < 1/6. Applying Lemma 7.13 to the pair 𝑢𝑞 , (𝑢𝑞 )1/𝑞 we ob­ tain, because 1/𝑞 > 1, 𝑞 𝜇𝑞 (𝐸𝜀 (𝑧)) ≤ 𝐶𝑞 (𝜇(𝐸5𝜀 (𝑧))) , where 𝜇𝑞 and 𝜇 are the Riesz measure 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|)2𝑞 sup (Δ𝑢(𝑤))𝑞 𝑤∈𝐸5𝜀 (𝑧)

𝐸5𝜀 (𝑧)

≤ 𝐶2 ∫ (1 − |𝑧|)2𝑞−2 (Δ𝑢(𝑧))𝑞 𝑑𝐴(𝑧). 6𝜀𝔻

It follows that ∫(1 − |𝑧|)−1 𝜇𝑞 (𝐸𝜀 (𝑧)) 𝑑𝐴(𝑧) ≤ 𝐶 ∫(1 − |𝑧|)2𝑞−3 𝑑𝐴(𝑧) ∫ (Δ𝑢)𝑞 𝑑𝐴, 𝔻

𝔻

𝐸6𝜀 (𝑧)

where 𝐶 depends only on 𝑞. Hence, as in the proof of Theorem 7.11, ∫(1 − |𝑧|) 𝑑𝜇𝑞 (𝑧) ≤ 𝐶 ∫(1 − |𝑧|)2𝑞−1 (Δ𝑢)𝑞 𝑑𝐴. 𝔻

𝔻

𝑞

This implies that 𝐼(𝑢 ) < ∞ because of Lemma 7.12 applied to 𝑢𝑞 . In order to deduce (7.19) from (7.27) we rewrite (7.21) as 1/𝑞

1 𝑟 ( ∫ log 𝑑𝜇 (𝑧)) 2𝜋 |𝑧| 𝑞 𝑟𝔻

≤

1 𝑟 ∫ log 𝑑𝜇(𝑧). 2𝜋 |𝑧| 𝑟𝔻

(7.27)

234 | 7 Littlewood–Paley theory Hence ∫ log 𝜀𝔻

𝜀 𝑑𝜇 (𝑧) ≤ 𝐶 sup(Δ𝑢)𝑞 ≤ 𝐶 ∫ (Δ𝑢)𝑞 𝑑𝐴 ≤ 𝐶𝑀, |𝑧| 𝑞 𝜀𝔻 2𝜀𝔻

where 𝑀 = ∫(1 − |𝑧|)2𝑞−1 (Δ𝑢(𝑧))𝑞 𝑑𝐴(𝑧), 𝔻

and hence 𝐼(𝑢𝑞 ) − 𝑢(0)𝑞 = ∫ log 𝔻

= ∫ log 𝜀𝔻

1 𝑑𝜇 (𝑧) |𝑧| 𝑞 𝜀 1 𝑑𝜇 (𝑧) + log ∫ 𝑑𝜇𝑞 (𝑧) + |𝑧| 𝑞 𝜀 𝜀𝔻

∫ log 𝜀≤|𝑧| 𝑝 in (7.28), see [281]. Luecking [308] proved a more delicate result: if 2 ≤ 𝑞 ≤ 𝑝 + 2 and 𝑓 ∈ 𝐻𝑝 , then ∫ |𝑓(𝑧)|𝑝−𝑞 |𝑓󸀠 (𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧) ≤ 𝐶‖𝑓‖𝑝𝑝 . 𝔻

(Compare Note 3.8.) The proof is based on the fact that if 𝐵(𝑧) is a Blaschke product, then the measure 𝑑𝜈 = |𝐵|𝑝−𝑞 |𝐵󸀠 |𝑞 (1 − |𝑧|)𝑞−1 𝑑𝐴 (2 ≤ 𝑞 < 𝑝 + 2) is Carleson. Exercise 7.2. Under the hypothesis of Theorem 7.12, we have ∫ 𝑢𝑞 𝑑𝐴 − 𝑢(0)𝑞 ) . ∫(Δ𝑢(𝑧))𝑞 (1 − |𝑧|)2𝑞 𝑑𝐴 ≤ 𝐶𝑞 (− 𝔻

𝔻

The reverse inequality holds under the hypothesis of Theorem 7.13.

2 This was proved by Kim and Kwon [261] under the additional hypothesis 𝑓(0) = 0.

7.7 Hyperbolic Hardy classes | 235

7.7 Hyperbolic Hardy classes For 𝑝 > 0 we define H𝑝 to be the class of the self-mappings 𝜑 of 𝔻 for which 𝑝

− (𝜅𝜑 (𝑟𝜁)) |𝑑𝜁| < ∞, H𝑝 (𝜑) := sup ∫ 0 0}, where h(𝔻) denotes the class of all analytic self-mappings of 𝔻. Then, let 1/2

𝑆[𝐺](𝜁) = (∫ Δ(𝐺2 ) 𝑑𝐴)

.

𝐷𝜁

If 𝐺 = 𝑢, where 𝑢 is harmonic and real valued, this agrees (up to a multiplicative constant) with the notation used in Section 7.5. However, if 𝐺 ∈ G, then 1/2

𝑆[𝐺](𝜁) ≍ (∫ |∇𝐺|2 𝑑𝐴 𝜑 )

,

where 𝑑𝐴 𝜑 =

𝐷𝜁

1 1 log 𝑑𝐴, 2 |𝜑| 1 − |𝜑|2

(7.34)

where the equivalence constants are independent of 𝐺. Theorem 7.15. For 𝑝 > 0 and 𝜑 ∈ h(𝔻), with 𝜑(0) = 0, we have ‖𝐺‖𝑝 ≍ ‖𝑆𝐺‖𝑝 , 𝐺 ∈ G. Proof: A sketch. The analog of (7.14) holds in the form 𝑆[𝐺𝑝 ] ≤ 𝐶𝑝(𝑀∗ 𝐺)𝑝−1 (𝑆𝐺), where 𝐶 is an absolute constant. Using this one proves the analog of Lemma 7.9. In proving the analog of Lemma 7.16, we start from (7.17) and get ∫(𝑆𝐺)2 ℎ 𝑑ℓ ≤ 𝐶1 ∫ 𝐺2 ℎ 𝑑ℓ + 𝐶4 ∫(𝑀∗ 𝐺)(𝑆1 𝐺)(𝑆𝑢) 𝑑ℓ, 𝕋

𝕋

𝕋

where

1/2

(𝑆1 𝐺)(𝜁) := (∫ |∇𝐺|2 𝑑𝐴) 𝐷𝜁

≤ 𝐶 𝑆𝐺(𝜁).

7.7 Hyperbolic Hardy classes |

237

The last inequality follows from (7.34) and the inequality 1 1 ) ≥ 1. log ( |𝜑|2 1 − |𝜑|2 Then we proceed exactly as in the case of Lemma 7.16 to obtain ‖𝐺‖𝑝 ≤ 𝐶‖𝑆𝐺‖𝑝 for all 𝑝 > 0; the inequality ‖𝑀∗ 𝐺‖𝑝 ≤ 𝐶‖𝐺‖𝑝 holds because the functions 𝐺 ∈ G are logsubharmonic. In proving the reverse inequality, we first use (7.34) to prove Lemma 7.11 with 𝑆𝐺 ≤ 𝐶 . . . , and then continue in the same way as before to complete the proof of the theo­ rem. Theorem 7.15 can be stated as follows. Theorem 7.16. If 𝑝 > 0 and 𝑞 > 0, then 𝑝/𝑞

H𝑝 (𝜑) ≍ ∫ (∫

𝜅𝜑𝑞−1 |𝜑[h] |2

𝑑𝐴)

|𝑑𝜁|,

𝐷𝜁

𝕋

where 𝜑[h] =

𝜑󸀠 , 1 − |𝜑|2

the hyperbolic derivative of 𝜑. Next we introduce the hyperbolic 𝑔-functions g𝑞 by 1/𝑞

1 𝑞−1

g𝑞 [𝜑](𝜁) = (∫ 𝜅𝜑 (𝑟𝜁)

[h]

2

|𝜑 (𝑟𝜁)| 𝑑𝑟)

,

𝜑 ∈ h(𝔻), 𝜁 ∈ 𝕋.

0

The following theorem is deduced from Theorem 7.15 in the same way as Theorem 7.6 from Calderon’s theorem. 𝑝

Theorem 7.17. If 𝑝 > 0 and 𝑞 ≥ 1, and 𝜑(0) = 0, then h𝑝 (𝜑) ≍ ‖g𝑞 [𝜑]‖𝑝 . Remark 7.3. If 𝜑 is a univalent function, then the integral Ah (𝜑)(𝜁) := ∫ |𝜑[h] |2 𝑑𝐴 𝐷𝜁

is the hyperbolic area of 𝜑(𝐷𝜁 ). Exercise 7.3. Using Theorems 7.12 and 7.13 one can prove the hyperbolic version of the Littlewood–Paley theorem. Let 𝑝 ≥ 1. If 𝜑 ∈ H𝑝 , then ∫(1 − |𝑧|)2𝑝−1 |𝜑[ℎ] (𝑧)|2𝑝 𝑑𝐴(𝑧) < ∞. 𝔻

In the case 𝑝 ≤ 1, the converse holds.

238 | 7 Littlewood–Paley theory Theorem 7.18. Let 𝜑 ∈ h(𝔻) and 0 < 𝑝 < ⬦. Then the following conditions are equiva­ lent: (a) C𝜑 : B 󳨃→ 𝐻𝑝 is bounded; (b) C𝜑 : B 󳨃→ 𝐻𝑝 is compact; (c) 𝜑 belongs to the hyperbolic Hardy class h𝑝/2 . Proof. Assume, as we may, that 𝜑(0) = 0. The implication (c) ⇒ (a) is a consequence of, e.g. Theorems 7.5 and 7.16 (𝑞 = 1). To prove the converse we use the bi-Bloch lemma to find two Bloch functions 𝑓1 and 𝑓2 such that |𝑓1󸀠 (𝑧)| + |𝑓2󸀠 (𝑧)| ≥ (1 − |𝑧|2 )−1 . From this we obtain 𝑝/2 󸀠 2

(∫ |(𝑓1 ∘ 𝜑) | 𝑑𝐴)

𝑝/2

+ (∫ |(𝑓2 ∘ 𝜑) | 𝑑𝐴)

𝐷𝜁

𝑝/2

|𝜑󸀠 |2 ≥ 𝑐𝑝 (∫ ) (1 − |𝜑|2 )2

󸀠 2

𝐷𝜁

,

𝐷𝜁

where 𝐷𝜁 is a Stoltz-type domain. Integrating this inequality in 𝜁 ∈ 𝕋, and then using the boundedness of C𝜑 together with Theorems 7.5 and 7.16 we conclude that (a) im­ plies (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 6.16, it is sufficient to prove that ‖𝐶𝜑 (𝑓𝑛 )‖𝑝 → 0. This also follows from the two theorems. Namely, by (c) the function 𝑝/2

𝐴(𝜁) := (∫ |𝜑[h] |2 𝑑𝐴) 𝐷𝜁

is finite for all 𝜁 ∈ 𝐸 ⊂ 𝕋 with |𝐸| = |𝕋|. Let 𝜁 ∈ 𝐸. Since |(𝑓𝑛 ∘ 𝜑)󸀠 | ≤ |𝜑[h] |, we have, by the dominant convergence theorem, lim𝑛 ℎ𝑛 (𝜁) = 0, for all 𝜁 ∈ 𝐸, where 𝑝/2

ℎ𝑛 (𝜁) = (∫ |(𝑓𝑛 ∘ 𝜑)󸀠 |2 𝑑𝐴)

.

𝐷𝜁

On the other hand, {ℎ𝑛} has the integrable dominant 𝐴(𝜁), 𝜁 ∈ 𝐸. So the dominated convergence theorem concludes the proof.

Further notes and results The material of this chapter, except Sections 7.6 and 7.7, is based on the author’s pa­ per [383]. Littlewood and Paley proved the implications (a) ⇔ (b) (𝑝 > 1) of Theorem 7.6 (see [304, Theorem 7] and, for the case 𝑝 > 0, [537, Ch. XIV (3.5)]) and (b)⇒(a) (𝑝 > 1) (see [304, Theorem 7] and [537, Ch. XIV (3.19)]). The equivalence (a) ⇔ (c) ⇔ (d) is,

7.7 Hyperbolic Hardy classes

| 239

maybe, new. As noted in [350, 5], the case 𝑝 > 0 can be treated by the methods of Fefferman and Stein’s paper [149]³. The validity of Theorem 7.3 was noted by Oswald [350]. In the case 𝑝 > 1, Theo­ rem 7.5 is due to Marcinkiewicz and Zygmund [321]. The equivalence 𝑓 ∈ 𝐻𝑝 ⇔ 𝑄2 (𝑓) < ∞ ⇔ 𝑄∗2 (𝑓) < ∞ of Theorem 7.9 was proved by Oswald [350, pp. 417–421]; our proof is somewhat simpler in technical details. The validity of Corollary 7.2 was conjectured by MacGregor and Sterner [315], who proved that this assertion does not hold in 𝐻∞ . The hyperbolic Hardy classes were introduced by Yamashita [512, 514]. The im­ portance of these classes and the “𝑔-theorems” in the study of composition operators from the Bloch space to 𝐻𝑝 were recognized by Kwon, who proved Theorem 7.17 for 𝑞 = 1 [282, 284]. Theorem 7.18 was proved by Kwon in [282, 283], and subsequently by Pérez-González and Xiao [394]. 7.1. It may be interesting to deduce some other classical, relatively simple inequalities from the equivalence (a) ⇔ (d) of Theorem 7.6. Proof of Theorem 5.14. Let 𝑝 ≥ 2. We use the inequality (𝑥 + 𝑦)𝑝/2 ≥ 𝑥𝑝/2 + 𝑦𝑝/2 (𝑥 ≥ 0, 𝑦 ≥ 0) to conclude that if 𝑓 ∈ 𝐻𝑝 , then ‖𝑓‖𝑝𝑝 ≥

∞

𝑐𝑝 2𝜋

∫ ∑ 2−𝑛𝑝 |𝑓󸀠 (𝑟𝑛 𝜁)|𝑝 |𝑑𝜁| 𝕋

𝑛=0 1

∞

= 𝑐𝑝 ∑ 2−𝑛𝑝 𝑀𝑝𝑝 (𝑟𝑛 , 𝑓󸀠 ) ≥ 𝑐𝑝 ∫(1 − 𝑟)𝑝−1 𝑀𝑝𝑝 (𝑟, 𝑓󸀠 ) 𝑑𝑟, 𝑛=0

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, we use the Minkowski inequality in the normed space 𝐿𝑝/2 to get ‖𝑓‖2𝑝

−2𝑛

≤ 𝐶𝑝 (∫ ( ∑ 2 𝕋

2

|𝑓 (𝑟𝑛 𝜁)| |𝑑𝜁|)

) 2/𝑝

−2𝑛

≤ 𝐶𝑝 ∑ (∫ (2 ∞

󸀠

𝑛=0

∞ 𝑛=0

2/𝑝

𝑝/2

∞

󸀠

2 𝑝/2

|𝑓 (𝑟𝑛 𝜁)| )

|𝑑𝜁|)

𝕋 1

= 𝐶𝑝 ∑ 2−2𝑛 𝑀𝑝2 (𝑟𝑛 , 𝑓󸀠 ) ≤ 𝐶𝑝 ∫(1 − 𝑟)𝑀𝑝2 (𝑟, 𝑓󸀠 ) 𝑑𝑟. 𝑛=0

0

3 Flett [151] proved this inequality in the case where 𝑓 has no zeros in 𝔻.

240 | 7 Littlewood–Paley theory In the case 𝑝 < 2 the proof is similar: we use the reverse Minkowski inequality, which is valid because the summands are nonnegative. 7.2. Concerning the case 𝑝 > 1 of Theorem 7.8, it should be noted that Littlewood and Paley proved a theorem that is deeper than the equivalence 𝑓 ∈ 𝐻𝑝 ⇔ 𝑄1 (𝑓) < ∞ (𝑝 > 1). Theorem (LP). Let 𝜆 = {𝜆 𝑛}∞ 0 (𝜆 0 > 0) be a lacunary sequence of integers, and let 𝜆 0 −1

𝑗 ̂ , Δ 0,𝜆 𝑓 = ∑ 𝑓(𝑗)𝑧

𝜆 𝑛 −1

𝑗 ̂ Δ 𝑛,𝜆 𝑓 = ∑ 𝑓(𝑗)𝑧 .

𝑗=0

𝑗=𝜆 𝑛−1

Then 𝑓 ∈ 𝐻𝑝 (𝑝 > 1) if and only if ∞

󵄨2 󵄨 ∫ ( ∑ 󵄨󵄨󵄨Δ 𝑛,𝜆 𝑓(𝜁)󵄨󵄨󵄨 ) 𝕋

𝑝/2

|𝑑𝜁| < ∞

𝑛=0

For a proof see [537, Ch. XIV (4.24)]. As an application we have Theorem (LP1). With the above notation we have ∞

1/𝑝

𝐶−1 ( ∑ ‖Δ 𝑛,𝜆 𝑓‖𝑝𝑝 )

∞

1/2

≤ ‖𝑓‖𝑝 ≤ 𝐶 ( ∑ ‖Δ 𝑛,𝜆 𝑓‖2𝑝 )

𝑛=0

,

𝑝 ≥ 2.

𝑛=0

The reverse inequalities hold for 1 < 𝑝 ≤ 2. 7.3. The following result from [114] shows that Flett’s inequality is rather elementary. Theorem (A). Let 𝑢 be a regularly oscillating (real-valued) function on 𝔻, 𝑢 ∈ 𝑂𝐶1𝐾 . Then the inequality ∫ − sup |𝑢(𝑡𝜁)|𝑝 |𝑑𝜁| ≤ |𝑢(0)|𝑝 + ∫ |∇𝑢(𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧), 𝕋

0 0, or, what is the same, Lipschitz space |𝑔(𝑧) − 𝑔(𝑤)| ≤ 𝐶 |𝑧 − 𝑤|𝛼

(𝑧, 𝑤 ∈ 𝐾),

where 𝐶 is a constant independent of 𝑧, 𝑤. The space Λ 𝜔 (𝐾) More generally, let 𝜔 be a positive function on (0, 𝑡0 ], with 𝜔(0+) = 0, where 𝑡0 is large enough. Then the space Λ 𝜔 (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 Λ 𝜔 (𝐾) is a Banach space. If 𝑔 ∈ Λ 𝜔 (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).

242 | 8 Lipschitz spaces of first order We will assume that 𝜔 is a majorant, i.e. an increasing function on [0, 𝑡0 ] such that 𝜔(0+ ) = 𝜔(0) = 0 and that 𝜔(𝑡)/𝑡 decreases (𝑡 ∈ (0, 𝑡0 )). By Proposition 8.3 below, the hypothesis that 𝜔(𝑡)/𝑡 is decreasing can be replaced by an apparently weaker one: 𝜔(𝑡)/𝑡 is almost decreasing. We write 𝜔(𝑔, 𝛿) = 𝜔(𝑔, 𝛿; 𝕋) if 𝑔 ∈ 𝐶(𝕋);

and Ω(𝑔, 𝛿) = 𝜔(𝑔, 𝛿; 𝔻),

if 𝑔 ∈ 𝐶(𝔻).

The reason for which we assume that 𝜔(𝑡)/𝑡 is (almost) decreasing lies in the fol­ lowing. 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) ≤ 𝐶𝑠 𝜔(𝑔, 𝛿),

and

Ω(𝑔, 𝑠𝛿) ≤ 𝐶𝑠 Ω(𝑔, 𝛿) ,

𝑠 ≥ 1, 𝛿 > 0,

where 𝐶 is independent of 𝑠, 𝛿. (Moreover, 𝐶 can also be chosen to be independent of 𝑔.) 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 in the interval [0, 2]. We say that 𝜔 is a Dini majorant if 2

∫ 0

𝜔(𝑥) 𝑑𝑥 < ∞. 𝑥

Following Dyakonov [140] we call a majorant 𝜔 fast if 𝑥

∫ 0

𝜔(𝑡) 𝑑𝑡 ≤ 𝐶𝜔(𝑥), 𝑡

0 < 𝑥 < 2,

(8.2)

𝜔(𝑡) 𝜔(𝑥) , 𝑑𝑡 ≤ 𝐶 2 𝑡 𝑥

0 < 𝑥 < 2,

(8.3)

where 𝐶 is a constant. If 2

∫ 𝑥

then 𝜔 is called 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.

8.1 Definitions and basic properties | 243

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)

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 3.11. 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 8.1. Note that the equivalence (8.3) ⇔ (8.5) is independent of the hypothesis that 𝜔(𝑡)/𝑡 is (almost) decreasing in 𝑡. Proposition 8.3. 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) = inf 𝑡≥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, 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

𝑥

𝑥/𝑒

𝑥/𝑒

𝛷(𝑡) 1 𝑑𝑡 ≥ 𝛷(𝑥/𝑒) ∫ 𝑑𝑡 = 𝛷(𝑥/𝑒). 𝛷(𝑥) ≥ 𝛷1 (𝑥) ≥ ∫ 𝑡 𝑡 It follows that 𝜙 := the inverse of 𝛷1 is concave and satisfies 𝜑3 (𝑥) ≤ 𝜙(𝑥) ≤ 𝑒 𝜑3 (𝑥). Exercise 8.1. 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 .

244 | 8 Lipschitz spaces of first order 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). Lemma 8.1. Let 𝜔 be a Dini majorant, and let 𝑢 : 𝔻 󳨃→ ℂ be a 𝐶1 -function such that |𝑢󸀠 (𝑧)| ≤

𝜔(1 − |𝑧|) , 1 − |𝑧|

Then 𝑢 ∈ Λ(𝜔[1] , 𝔻), where

𝑥

𝜔[1] (𝑥) = ∫ 0

𝑧 ∈ 𝔻.

(8.6)

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 ∈ Λ 𝜔 (𝔻). Proof of Lemma. Let (8.6) be satisfied. Let |𝑎| ≤ |𝑏| ≤ 1. By Lagrange’s theorem, |𝑢(𝑎) − 𝑢(𝑏)| ≤

𝜔(1 − |𝑐|) |𝑎 − 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| 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 above we get |𝑢(𝑎) − 𝑢(𝑏󸀠 )| ≤

𝜔(1 − |𝑏󸀠 |) 𝜔(𝛿) |𝑎 − 𝑏󸀠 | = |𝑎 − 𝑏󸀠 | ≤ 𝜔(𝛿) ≤ 𝜔[1] (𝛿). 󸀠 1 − |𝑏 | 𝛿

In the case of |𝑢(𝑏󸀠 ) − 𝑢(𝑏)|, we have |𝑏| 󸀠

|𝑢(𝑏 ) − 𝑢(𝑏)| ≤ ∫ |𝑏󸀠 |

1

𝜔(1 − 𝑡) 𝜔(1 − 𝑡) 𝑑𝑡 ≤ ∫ 𝑑𝑡 = 𝜔[1] (𝛿). 1−𝑡 1−𝑡 1−𝛿

Finally, if 𝛿 > 1 − |𝑎|, 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 above. As an application of Lemma 8.1, we have the following result of Hardy and Little­ wood [191]: Theorem 8.1 (Hardy–Littlewood). Let 𝜔 be a fast majorant. A function 𝑢 ∈ ℎ(𝔻) belongs to Λ 𝜔 (𝔻) if and only if 𝜔(1 − 𝑟) . 𝑀∞ (𝑟, |∇𝑢|) ≤ 𝐶 1−𝑟 Moreover, if 𝐶(𝑢) denotes the smallest 𝐶 ≥ 0 satisfying the above condition, then 𝐶(𝑢) ≍ ‖𝑢‖Λ 𝜔 (𝔻) − ‖𝑢‖𝐿∞ (𝔻) .

8.1 Definitions and basic properties | 245

Proof. The “only if” part is a direct consequence of Lemma 8.1 and the hypothesis that 𝜔 is fast. The converse follows from the inequality |∇𝑢(𝑧)| ≤ 𝐶 sup 𝑤∈𝔻

𝜔(1 − |𝑧|) |𝑢(𝑧 + (1 − |𝑧|)𝑤) − 𝑢(𝑧)| ≤𝐶 , 1 − |𝑧| 1 − |𝑧|

where 𝐶 is an absolute constant. (𝐶 = 2 if 𝑢 is real valued.)

Lipschitz spaces on the circle and the disk The space ℎΛ 𝜔 (𝔻) is the (closed) subspace of Λ 𝜔 (𝔻) consisting of harmonic functions: ℎΛ 𝜔 (𝔻) = ℎ(𝔻) ∩ Λ 𝜔 (𝔻). Theorem 8.2 (Hardy–Littlewood). If 𝜔 is a fast majorant, then the space ℎΛ 𝜔 (𝔻) is selfconjugate. Proof. Assuming, as we may, that 𝑢 is real valued, we have |∇𝑢| = |∇𝑢|̃ so the result follows from Theorem 8.1. The space Λ 𝜔 (𝕋) can be identified, via the Poisson integral, with the space ℎΛ 𝜔 (𝕋) of harmonic functions: ℎΛ 𝜔 (𝕋) = {P[𝑔] : 𝑔 ∈ Λ 𝜔 (𝕋)}. Equivalently, a function 𝑢 ∈ ℎ(𝔻) belongs to ℎΛ 𝜔 (𝕋) if and only if 𝑢 has a continuous extension to the closed unit disk and the boundary function 𝑢∗ belongs to Λ 𝜔 (𝕋). By definition, ‖𝑢‖ℎΛ 𝜔 (𝕋) = ‖𝑢∗ ‖Λ 𝜔 (𝕋) . If 𝜔(𝑡) = 𝑡𝛼 , then we write ℎΛ 𝜔 ( ⋅ ) = ℎΛ 𝛼 ( ⋅ ). It is clear that ℎΛ 𝜔 (𝔻) ⊂ ℎΛ 𝜔 (𝕋). However, Theorem 8.3. If 𝜔 is a slow majorant then ℎΛ 𝜔 (𝔻) ≃ ℎΛ 𝜔 (𝕋). Proof. Let condition (8.3) be satisfied, and extend 𝜔 to the interval [2, 𝜋 + 1) by 𝜔(𝑡) = 𝜔(2)𝑡/2. Let |𝑢(𝜁) − 𝑢(𝜂)| ≤ 𝜔(|𝜁 − 𝜂|), for 𝜁, 𝜂 ∈ 𝕋, and |𝑧| < |𝑤| ≤ 1. Assume first that |𝑤| = 1. Then the function 𝜑(𝑧) = |𝑢(𝑤) − 𝑢(𝑧)| is subharmonic in 𝔻 and continuous on 𝔻, whence (1 − |𝑧|2 )𝜑(𝜁) |𝑑𝜁|. 𝜑(𝑧) ≤ ∫ − |𝜁 − 𝑧|2 𝕋

Since 𝜑(𝜁) ≤ 𝜔(|𝑤 − 𝑧| + |𝜁 − 𝑧|) ≤ 𝜔(|𝑤 − 𝑧|) + |𝑢(𝜁) − 𝑢(𝑧)|, we have 𝜑(𝑧) ≤ 𝜔(|𝑤 − 𝑧|) + ∫ − 𝕋

(1 − |𝑧|2 )𝜔(|𝜁 − 𝑧|) |𝑑𝜁|. |𝜁 − 𝑧|2

246 | 8 Lipschitz spaces of first order But simple calculation shows that 𝜋

(1 − |𝑧|2 ) 𝜔(|𝜁 − 𝑧|) (1 − 𝑟) 𝜔(1 − 𝑟 + 𝑡) |𝑑𝜁| ≤ 𝐶 ∫ 𝑑𝑡 ∫ |𝜁 − 𝑧|2 (1 − 𝑟 + 𝑡)2

(𝑟 = |𝑧|)

0

𝕋

1−𝑟

𝜋

= 𝐶(∫ + ∫) 0

1−𝑟

(1 − 𝑟) 𝜔(1 − 𝑟 + 𝑡) 𝑑𝑡 (1 − 𝑟 + 𝑡)2 𝜋

≤ 𝐶1 𝜔(1 − 𝑟) + 𝐶2 (1 − 𝑟) ∫ 1−𝑟

𝜔(𝑡) 𝑑𝑡 𝑡2

≤ 𝐶3 𝜔(1 − |𝑧|) ≤ 𝐶3 𝜔(|𝑤 − 𝑧|). Thus |𝑢(𝑤) − 𝑢(𝑧)| ≤ 𝐶𝜔(|𝑤 − 𝑧|) provided 𝑤 ∈ 𝕋. If 0 < |𝑤| < 1, we consider the function ℎ(𝜉) = 𝑢(𝜉𝑤/|𝑤|) − 𝑢(𝜉𝑧/|𝑤|), |𝜉| ≤ 1. This function is harmonic in 𝔻, continuous on 𝔻 (because |𝑧| < |𝑤|), and ℎ(0) = 0. Hence, by the harmonic Schwarz lemma (see Note 2.1) and the preceding case, |𝑢(𝑤) − 𝑢(𝑧)| = |ℎ(|𝑤|)| ≤ (4/𝜋)|𝑤| ‖ℎ‖∞ ≤ 𝐶|𝑤| 𝜔(|𝑤/|𝑤| − 𝑧/|𝑤| |) ≤ 𝐶 𝜔(|𝑤| |𝑤/|𝑤| − 𝑧/|𝑤| |) = 𝐶 𝜔(|𝑤 − 𝑧|), which completes the proof. From this theorem and Theorem 8.2, we obtain another theorem of Privalov [405]. Theorem 8.4 (Privalov). If 𝜔 is regular, then Λ 𝜔 (𝕋) is self-conjugate.

8.1.1 Lipschitz spaces of analytic functions Let 𝐻Λ 𝜔 (𝔻) resp. 𝐻Λ 𝜔 (𝕋) denote the (closed) subspace of ℎΛ 𝜔 (𝔻) resp. ℎΛ 𝜔 (𝕋) con­ sisting of analytic functions. If 𝜔(𝑡) = 𝑡𝛼 , then we write 𝐻Λ 𝜔 (⋅) = 𝐻Λ 𝛼 ( ⋅ ). These spaces are contained in 𝐴(𝔻), the space of all functions that are analytic in 𝔻 and continuous on 𝔻. Theorem 8.5. If 𝑓 ∈ 𝐴(𝔻), then 𝜔(𝑓, 𝛿) ≤ Ω(𝑓, 𝛿) ≤ 𝐶𝜔(𝑓, 𝛿),

𝛿 > 0,

where 𝐶 is an absolute constant. Consequently, if 𝜔 is an arbitrary majorant, then 𝐻Λ 𝜔 (𝔻) = 𝐻Λ 𝜔 (𝕋). Proof. For a fixed 𝜂 ∈ 𝕋 and 0 < 𝑞 < 1, the function 𝜑(𝑧) = |𝑓(𝜁)−𝑓(𝑧)|𝑞 is subharmonic in 𝔻 and continuous on 𝔻, and hence 𝜑(𝑧) ≤

(1 − |𝑧|2 )𝜑(𝜁) 1 ∫ |𝑑𝜁|. 2𝜋 |𝜁 − 𝑧|2 𝕋

8.1 Definitions and basic properties |

247

Arguing as in the proof of Theorem 8.3 we find that 𝜋 𝑞

|𝑓(𝜂) − 𝑓(𝑧)|𝑞 ≤ 𝐶1 (𝜔(𝑓, 1 − 𝑟)) + 𝐶2 (1 − 𝑟) ∫ 1−𝑟 𝑞

𝑞

(𝜔(𝑓, 𝑡)) 𝑑𝑡. 𝑡2

𝑞

Since (𝜔(𝑓, 𝑡)) /𝑡 is almost decreasing (Proposition 8.1) and 𝑞 < 1, we see that 𝜋

𝑞

𝑞

(𝜔(𝑓, 𝑡)) (𝜔(𝑓, 1 − 𝑟)) . ∫ 𝑑𝑡 ≤ 𝐶 𝑡2 1−𝑟

1−𝑟

This shows that |𝑓(𝜂) − 𝑓(𝑧)| ≤ 𝐶𝜔(𝑓, |𝜂 − 𝑧|), 𝑧 ∈ 𝔻. Finally, an application of the classical Schwarz lemma as in the proof of Theorem 8.3 yields |𝑓(𝑤) − 𝑓(𝑧)| ≤ 𝐶𝜔(𝑓, |𝑤 − 𝑧|),

𝑤, 𝑧 ∈ 𝔻,

which completes the proof. Combining Theorems 8.1 and 8.5 we get: Theorem 8.6. Let 𝜔 be a fast majorant and let 𝑓 ∈ 𝐻(𝔻). Then 𝑓 belongs to 𝐻Λ 𝜔 (𝕋) if and only if 𝑓 belongs to 𝐻Λ 𝜔 (𝔻) if and only if |𝑓󸀠 (𝑧)| ≤ 𝐶

𝜔(1 − |𝑧|) 1 − |𝑧|

(𝑧 ∈ 𝔻).

Exercise 8.2. The space 𝐻Λ 𝜔 (𝔻) is an algebra with respect to multiplication: (𝑓𝑔)(𝑧) = 𝑓(𝑧)𝑔(𝑧). Exercise 8.3. There are cases where a Lipschitz condition on the circle extends to the disk with saving the corresponding Lipschitz constant. For example, if 𝑓 ∈ 𝐴(𝔻), 0 < 𝛼 ≤ 1 and |𝑓(𝜁) − 𝑓(𝜂)| ≤ |𝜁 − 𝜂|𝛼 , 𝜁, 𝜂 ∈ 𝕋, then |𝑓(𝑧) − 𝑓(𝑤)| ≤ |𝑧 − 𝑤|𝛼 , 𝑧, 𝑤 ∈ 𝔻 (see Note 8.1). The latter is equivalent to |𝑓(𝑧) − 𝑓(𝑤)| ≤

|𝑧 − 𝑤| ̄ 1−𝛼 |1 − 𝑤𝑧|

(𝑧, 𝑤 ∈ 𝔻),

and this implies |𝑓󸀠 (𝑧)| ≤ (1 − |𝑧|2 )𝛼−1 , 𝑧 ∈ 𝔻.

8.1.2 Mean Lipschitz spaces Let 0 < 𝑝 < ⬦. The 𝐿𝑝 (𝕋)-modulus of continuity is defined as 𝜔(𝑡, 𝑔)𝑝 = sup{‖𝑔𝜁 − 𝑔𝜂 ‖𝐿𝑝 (𝕋) : 𝜂, 𝜁 ∈ 𝕋, |𝜁 − 𝜂| < 𝑡},

(8.7)

where 𝑔𝜁 (𝑒𝑖𝜃 ) = 𝑔(𝜁𝑒𝑖𝜃 ). If we require that 𝜂, 𝜁 ∈ 𝔻, then the resulting quantity is de­ noted by Ω(𝑡, 𝑔)𝑝 . For a majorant 𝜔, we define the spaces Λ𝑝𝜔 (𝕋), resp. Λ𝑝𝜔 (𝔻), by the

248 | 8 Lipschitz spaces of first order requirement 𝜔(𝑡, 𝑔)𝑝 = O(𝜔(𝑡)), resp. Ω(𝑡, 𝑔)𝑝 = O(𝜔(𝑡)). The corresponding spaces of analytic or harmonic functions will be denoted by 𝐻Λ𝑝𝜔 (⋅), resp. ℎΛ𝑝𝜔 (⋅). At the beginning of this chapter, we defined the space Λ 𝜔 (𝕋) as a subset of 𝐶(𝕋). On the other hand, if we take 𝑝 = ∞ in (8.7), then the corresponding Lipschitz space, ∞ ∞ Λ∞ 𝜔 (𝕋), is a subset of 𝐿 (𝕋). It turns out that if 𝑔 ∈ Λ 𝜔 (𝕋), then there exists a function 𝑔2 ∈ Λ 𝜔 (𝕋) such that 𝑔2 = 𝑔 a.e. To see this, we consider the function 𝑢 = P[𝑔]. Then, by the maximum principle, max |ℎ(𝑟𝑒𝑖(𝜃+𝑡) ) − ℎ(𝑟𝑒𝑖𝜃 )| ≤ 𝐶𝜔(𝑡), 𝜃∈ℝ

0 < 𝑟 < 1,

where 𝐶 is independent of 𝑟. Since also max𝜃 |𝑢(𝑟𝑒𝑖𝜃 )| ≤ ‖𝑔‖∞ , we see that the family {𝑢𝑟 : 0 < 𝑟 < 1} ⊂ 𝐶(𝕋), where 𝑢𝑟 (𝜁) = 𝑢(𝑟𝜁), is bounded in 𝐶(𝕋) and equicontinuous. Hence, by the Arzelà–Ascoli theorem, there exists a sequence 𝑟𝑛 → 1− such that 𝑢𝑟𝑛 tends to some 𝑔2 in 𝐶(𝕋). On the other hand, 𝑢(𝑟𝜁) → 𝑔(𝜁) a.e. on 𝕋, whence 𝑔 = 𝑔2 a.e. Thus we can identify Λ∞ 𝜔 (𝕋) with Λ 𝜔 (𝕋). (I) The analog of Theorem 8.3 is true for 𝑝 ≥ 1, which can be proved by considering the subharmonic function 1/𝑝 𝑝

𝜑(𝑧) = (− ∫ |𝑢(𝑤) − 𝑢(𝑧)| |𝑑𝑤|)

,

𝕋

and using assertion (A) of Note 2.1. (II) Theorem 8.5 extends to all 𝑝 > 0 by using the subharmonicity of the function 𝑞/𝑝 𝑝

𝜑(𝑧) = (− ∫ |𝑓(𝑧) − 𝑓(𝑤)| |𝑑𝑤|)

,

𝕋

(see Lemma 2.1) and assertion (B) of Note 2.1. So we have 𝜔(𝑡, 𝑓)𝑝 ≤ Ω(𝑡, 𝑓)𝑝 ≤ 𝐶𝑝 𝜔(𝑡, 𝑓)𝑝 ,

𝑓 ∈ 𝐻𝑝 , 0 < 𝑡 < 𝜋.

(8.8)

The fact that 𝜔(𝑡, 𝑓)𝑝 /𝑡1/𝑝 , where 𝑓 ∈ 𝐻𝑝 , 𝑝 < 1, is almost decreasing is to be used. In [479, Theorem 6], Storozhenko proved that if 𝑓 ∈ 𝐻𝑝 (0 < 𝑝 < ∞), then ‖𝑓∗ − 𝑓𝑟 ‖𝑝 ≍ 𝜔(𝑓∗ , 1 − 𝑟)𝑝 ,

0 < 𝑟 < 1.

(III) The generalized Theorem 8.6 asserts that 𝑓 is in 𝐻Λ𝑝𝜔 (𝕋), where 𝜔 is fast and 𝑝 > 0, if and only if 𝜔(1 − 𝑟) ). 𝑀𝑝 (𝑟, 𝑓󸀠 ) = 𝑂 ( 1−𝑟 In the case where 𝜔 is regular and 𝑝 ≥ 1, this was proved by Blasco and de Souza [68, Theorem 2.1(i)], and independently by the author [364] in the general case. In Chapter 9 we will prove a more general result (Theorem 9.9).

8.2 Lipschitz condition for the modulus | 249

8.2 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 conclude that the relation 𝑓 ∈ 𝐻(𝔻) 𝑓 ∈ Λ 1 (𝔻) ⇔ |𝑓| ∈ Λ 1 (𝔻), holds. This is the simplest case of the following theorem. Theorem 8.7. Let 𝜔 be a Dini majorant. If 𝑓 ∈ 𝐻(𝔻) and |𝑓| ∈ Λ 𝜔 (𝔻), then 𝑓 ∈ Λ(𝜔[1] , 𝔻). 𝑥

𝑑𝑡. Recall that 𝜔[1] (𝑥) = ∫0 𝜔(𝑡) 𝑡 The theorem states, in particular, that if |𝑓| ∈ Λ 𝜔 (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 disk. To show this, we use the known fact that there exists a bounded analytic function 𝑢 + 𝑖𝑣 such that 𝑢 is continuous on 𝔻, while 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 the desired function is 𝑓 = exp(𝑢 + 𝑖𝑣). Theorem 8.7 is a direct consequence of Lemma 8.1 and the inequality |∇|𝑓|(𝑧)| = |𝑓󸀠 (𝑧)| ≤ 2

sup

(|𝑓(𝑤)| − |𝑓(𝑧)|),

(8.9)

|𝑤−𝑧| 0. It can be proved that the existence of such an 𝛼 is implied by the hypothesis that 𝜔̄ is fast. The main properties of 𝜔̄ are given by the inequalities 𝛼

̄ ̄ 𝜔(𝑡) 𝜔(𝑡) ≤ 𝜔̄ 󸀠 (𝑡) ≤ , 𝑡 𝑡

0 < 𝑡 < 1,

(8.10)

252 | 8 Lipschitz spaces of first order and ̄ ̄ 𝜔̄ 󸀠 (𝑥)(𝑥 − 𝑦) ≤ 𝜔(𝑥) − 𝜔(𝑦),

0 < 𝑥, 𝑦 ≤ 1,

(8.11)

where 𝜔̄ 󸀠 is, say, the left derivative of 𝜔.̄ Theorem 8.11. For a function 𝜑 ∈ h(𝔻), the following conditions are equivalent: (A0) 𝜑 belongs to C(𝐻Λ 𝜔̄ , 𝐻Λ 𝜔 ). (A) 𝜔̄ 󸀠 (1 − |𝜑(𝑧)|)|𝜑󸀠 (𝑧)| ≤ 𝐶 𝜔(1−|𝑧| . 1−|𝑧| ̄ − |𝜑|) ∈ Λ 𝜔 (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 instance, ̄ = 𝑡 and 𝑓(𝑧) = 𝑧, then the equivalence (A) ⇔ (C) says that 𝜑 ∈ 𝐻Λ(𝜔, 𝔻) if we take 𝜔(𝑡) if and only if |𝜑| ∈ Λ 𝜔 (𝔻), which coincides with the equivalence (a) ⇔ (b) of Theo­ rem 8.8 in the case |𝜑| < 1. If 𝜑 ∈ 𝐴(𝔻) is arbitrary, then we apply this special case to the function 𝜑/𝑀, where 𝑀 > ‖𝜑‖∞ . Although Theorem 8.11 is a special case of a theorem which will be proved later on, we give an independent proof because it is simpler in that we need not use biBloch functions. Proof of Theorem 8.11. Note first that if (A0) holds, then we take 𝑓(𝑧) = 𝑧 to conclude that |𝜑| ∈ Λ 𝜔 (𝔻), which implies 𝜑 ∈ 𝐴(𝔻), by Theorem 8.7. First we prove that (A) implies (A0). Indeed, if (A) holds and 𝑓 ∈ 𝐻Λ 𝜔̄ (𝔻), then, by Theorem 8.6 and (8.10), ̄ − |𝜑(𝑧)|) 󸀠 𝜔(1 |𝜑 (𝑧)| 1 − |𝜑(𝑧)| 𝜔(1 − |𝑧|) . ≤ 𝐶𝜔̄ 󸀠 (1 − |𝜑(𝑧)|)|𝜑󸀠 (𝑧)| ≤ 𝐶 1 − |𝑧|

|(C𝜑 𝑓)󸀠 (𝑧)| = |𝑓󸀠 (𝜑(𝑧))| |𝜑󸀠 (𝑧)| ≤ 𝐶

Applying Theorem 8.6 again, we conclude that (A0) holds. In order to prove that (A0) implies (A), we consider two possible alternatives: the ̄ function 𝑠(𝑡) = 𝜔(𝑡)/𝑡 (0 < 𝑡 ≤ 1) is either bounded or unbounded. If 𝑠 is bounded, ̄ ≍ 𝑡, so 𝐻Λ 𝜔̄ (𝔻) = 𝐻Λ 1 (𝔻), and therefore we have to prove that (A) implies then 𝜔(𝑡) |𝜑󸀠 (𝑧)| ≤ 𝐶𝜔(1 − |𝑧|)/(1 − |𝑧|): it is enough to take 𝑓(𝑧) = 𝑧. ̄ If 𝑠 is not bounded, then 𝑠(𝑡) → ∞ as 𝑡 ↓ 0. Let 𝜓(𝑥) = 𝑠(1/𝑥) = 𝑥𝜔(1/𝑥), 𝑥 ≥ 1. This function is increasing and satisfies the condition 𝜓(2𝑥) ≤ 𝐶𝜓(𝑥), 𝑥 ≥ 1, because 𝜔̄ is fast. As a test function we take the function 𝑓𝜓,𝑎 from Lemma 6.10, ∞

𝑓𝜓,𝑎 (𝑧) = ∑ 2𝑛 𝑛=0

𝑎𝜆 𝑛 𝑧𝜆 𝑛+1 , 𝜆𝑛 + 1

𝑧 ∈ 𝔻, |𝑎| = 1,

where 𝜆 𝑛 is chosen as in Lemma 6.10. Since, by the same lemma, ∞

󸀠 |𝑓𝜓,𝑎 (𝑧)| ≤ ∑ 2𝑛 |𝑧|𝜆 𝑛 ≤ 𝐶𝜓 ( 𝑛=0

̄ − |𝑧|) 𝜔(1 1 )=𝐶 , 1 − |𝑧| 1 − |𝑧|

8.3 Composition operators

| 253

󸀠 ∈ Λ(𝜔,̄ 𝔻). By hypothesis, C𝜑 (𝑓𝜓,𝑎 ) belongs to Λ 𝜔 (𝔻), i.e. we see that 𝑓𝜓,𝑎

󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜔(1 − |𝑧|) |(C𝜑 𝑓𝜓,𝑎 )󸀠 (𝑧)| = 󵄨󵄨󵄨 ∑ 2𝑛 (𝑎𝜑(𝑧))𝜆 𝑛 󵄨󵄨󵄨 |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 ∞

∑ 2𝑛 |𝜑(𝑧)|𝜆 𝑛 |𝜑󸀠 (𝑧)| ≤ 𝐶 𝑛=0

𝜔(1 − |𝑧|) . 1 − |𝑧|

It remains to apply Lemma 6.10 and (8.10) to conclude that (A) holds. ̄ − 𝑧|). Since |∇𝑔(𝑧)| = 𝜔̄ 󸀠 (1 − |𝜑(𝑧)|)|𝜑󸀠 (𝑧)| we see by means of Let 𝑔(𝑧) = 𝜔(1 Lemma 8.1 that the implication (A) ⇒ (B) holds. In order to prove the implication (B) ⇒ (A) we use (8.11) in the form ̄ − |𝜑(𝑧)|) − 𝜔(1 ̄ − |𝜑(𝑤)|), 𝜔̄ 󸀠 (1 − |𝜑(𝑧)|)(|𝜑(𝑤)| − |𝜑(𝑧)|) ≤ 𝜔(1 and hence 𝜔̄ 󸀠 (1 − |𝜑(𝑧)|)(|𝜑(𝑤)| − |𝜑(𝑧)|) ≤ 𝐶𝜔(|𝑤 − 𝑧|). On the other hand, by Lemma 3.3, the inequality |𝜑󸀠 (𝑧)| ≤ 3

|𝜑(𝑤𝑧 )| − |𝜑(𝑧)| 1 − |𝑧|

holds, where |𝑤𝑧 − 𝑧| < 1 − |𝑧|, 𝑧 ∈ 𝔻, which implies 𝜔̄ 󸀠 (1 − |𝜑(𝑧)|)|𝜑󸀠 (𝑧)|(1 − |𝑧|) ≤ 3𝐶1 𝜔(|𝑤𝑧 − 𝑧|) ≤ 3𝐶1 𝜔(1 − |𝑧|). From this we conclude that (B) implies (A). It remains to prove that (C) implies (A). The proof of (B) ⇒ (A) shows that it is enough to prove that (C) implies ̄ − |𝜑(𝑧)|) − 𝜔(1 ̄ − |𝜑(𝑤)|) ≤ 𝐶𝜔(1 − |𝑧|) 𝜔(1 under the condition |𝑤 − 𝑧| < 1 − |𝑧|. To show this assume that |𝑤 − 𝑧| < 1 − |𝑧|, 𝑟 = |𝑧|, and 𝜁 = 𝑧/𝑟. Then ̄ − |𝜑(𝑧)|) − 𝜔(1 ̄ − |𝜑(𝑤)|) = 𝜔(1 ̄ − |𝜑(𝑧)|) − 𝜔(1 ̄ − |𝜑(𝜁)|) 𝜔(1 ̄ − |𝜑(𝜁)|) − 𝜔(1 ̄ − |𝜑(𝑤)|) + 𝜔(1 ≤ 𝐶𝜔(1 − |𝑧|) + 𝐶𝜔(|𝜁 − 𝑤|) ≤ 𝐶𝜔(1 − |𝑧|) + 𝐶𝜔(|𝜁 − 𝑧|) + 𝐶𝜔(|𝑧 − 𝜁|) ≤ 3𝐶𝜔(1 − |𝑧|). 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 following holds: hence 𝜔(1

254 | 8 Lipschitz spaces of first order Theorem 8.12. Let 𝜔 be a regular majorant. Then the condition (A0) of Theorem 8.11 is equivalent to each of the following: ̄ − |𝜑(𝑧)|) − P[𝜔(1 ̄ − |𝜑∗ |) ](𝑧) ≤ 𝐶𝜔(1 − |𝑧|) ; (D) 𝜔(1 ̄ − |𝜑∗ |) ∈ Λ 𝜔 (𝕋) and (E) 𝜔(1 ̄ − |𝜑(𝑟𝜁)|) − 𝜔(1 ̄ − |𝜑∗ (𝜁)|) ≤ 𝐶𝜔(1 − 𝑟) (𝜁 ∈ 𝕋, 0 < 𝑟 < 1). 𝜔(1 We omit the proof because it is similar to the proof of Theorem 8.9.

8.4 Composition operators into 𝐻Λ𝑝𝜔 In this section, we consider the class C(𝜔,̄ 𝜔, 𝑝) := C(𝐻Λ 𝜔̄ (𝔻), 𝐻Λ𝑝𝜔 (𝔻)). We also use the notation 𝑡 𝜔(𝑥) 𝑑𝑥. Ψ(𝑡) = ∫ 𝑥 0

Theorem 8.13. Let 0 < 𝑝 < ⬦. Then 𝜑 is in C(𝜔,̄ 𝜔, 𝑝) if and only if one of the following three conditions is satisfied: (A𝑝) There is a constant 𝐶 such that 2𝜋 𝑝

∫ (|𝜑󸀠 (𝑟𝑒𝑖𝜃 )| 𝜔̄ 󸀠 (1 − |𝜑(𝑟𝑒𝑖𝜃 )|)) 𝑑𝜃 ≤ 𝐶Ψ󸀠 (1 − 𝑟)𝑝 ,

0 < 𝑟 < 1.

0

̄ − |𝜑|) belongs to the space Λ𝑝𝜔 (𝔻). (B𝑝) The function 𝜔(1 ̄ − |𝜑|) belongs to the space Λ𝑝𝜔 (𝕋) and there is a constant 𝐶 such (C𝑝) The function 𝜔(1 that 2𝜋

󵄨 󵄨𝑝 ̄ − |𝜑(𝑟𝑒𝑖𝜃 )|) − 𝜔(1 ̄ − |𝜑(𝑒𝑖𝜃 )|)󵄨󵄨󵄨󵄨 𝑑𝜃 ≤ 𝐶𝜔(1 − 𝑟)𝑝 , ∫ 󵄨󵄨󵄨󵄨𝜔(1

0 < 𝑟 < 1.

0

Theorem 8.14. If 𝜔 is regular, and 𝑝 ≥ 1, then 𝜑 ∈ C(𝜔,̄ 𝜔, 𝑝) if and only if one of the following two conditions is satisfied: ̄ − |𝜑|) belongs to the space Λ𝑝𝜔 (𝕋), and there is a constant 𝐶 such (D𝑝) The function 𝜔(1 that 2𝜋 𝑝

̄ − |𝜑(𝑟𝑒𝑖𝜃 )|) − 𝜔(1 ̄ − |𝜑(𝑒𝑖𝜃 )|)}+ 𝑑𝜃 ≤ 𝐶𝜔(1 − 𝑟)𝑝 , ∫ {𝜔(1

0 < 𝑟 < 1,

0

where {𝑥}+ = max{𝑥, 0}, ̄ − |𝜑|) belongs to the space (E𝑝) The function 𝜔(1 that

𝑥 ∈ ℝ.

Λ𝑝𝜔 (𝕋),

and there is a constant 𝐶 such

2𝜋

𝑝

̄ − |𝜑(𝑟𝑒𝑖𝜃 )|) − P[𝜔(1 ̄ − |𝜑|)](𝑟𝑒𝑖𝜃 )) 𝑑𝜃 ≤ 𝐶𝜔(1 − 𝑟)𝑝 . ∫ (𝜔(1 0

𝑝

8.4 Composition operators into 𝐻Λ 𝜔

| 255

Proof of Theorem 8.13. The proof that (A𝑝) implies 𝜑 ∈ C(𝜔,̄ 𝜔, 𝑝) is simple and we omit it. The proof that 𝜑 ∈ C(𝜔,̄ 𝜔, 𝑝) implies (A𝑝) is more complicated than in the case of Theorem 8.11. We shall use Lemma 6.11 (bi-Bloch lemma). ̄ There are two cases: (1) 𝜔(1)𝑡 ≤ 𝜔(𝑡) ≤ 𝐶𝑡 for 0 < 𝑡 < 1; (2) 𝜔(𝑡)/𝑡 → ∞ as 𝑡 → 0. 𝑝 In case (1) we have Λ𝑝 (𝜔)̄ = Λ 1 . Then, if 𝐶𝜑 maps 𝐻Λ 𝜔̄ into 𝐻Λ𝑝𝜔 , 2𝜋

∫ |𝑓󸀠 (𝜑(𝑟𝑒𝑖𝜃 ))|𝑝 |𝜑󸀠 (𝑟𝑒𝑖𝜃 )|𝑝 𝑑𝜃 ≤ 𝐶Ψ󸀠 (1 − 𝑟)𝑝 ,

0 < 𝑟 < 1,

0

̄ − |𝑧|)/(1 − |𝑧|) ≍ 1. Taking 𝑓(𝑧) = 𝑧 we get for any 𝑓 for which |𝑓󸀠 (𝑟𝑒𝑖𝜃 )| ≤ 𝐶𝜔(1 2𝜋 ∫0 |𝜑󸀠 (𝑟𝑒𝑖𝜃 )|𝑝 𝑑𝜃 ≤ 𝐶Ψ󸀠 (1 − 𝑟)𝑝 , which gives the result. In the case (2) we use Lemma 6.11 and proceed in the same way as in the proof of Theorem 6.21 to conclude that (A𝑝) holds. That (A𝑝) implies (B𝑝) follows from the ̄ − |𝜑|). following lemma and Lemma 8.3 below applied to the function 𝑔 = 𝜔(1 ̄ − |𝜑|) is regularly oscillating. Lemma 8.2. The function 𝑔 = 𝜔(1 Proof. Let 0 < 𝜀 < 1 − |𝑧|, 𝑧 ∈ 𝔻. By Lemma 3.3, we may choose 𝑤𝑧 ∈ 𝐷𝜀 (𝑧) so that |𝜑󸀠 (𝑧)| ≤

3 (𝑥 − 𝑦), 𝜀

where 𝑥 = 1 − |𝜑(𝑧)|, 𝑦 = 1 − |𝜑(𝑤𝑧 )|.

(8.12)

From this and the inequality |∇𝑔(𝑧)| = 𝜔̄ 󸀠 (𝑥)|𝜑󸀠 (𝑧)| ≤

̄ 𝜔(𝑥) |𝜑󸀠 (𝑧)|, 𝑥

we see that it is enough to prove that ̄ 𝜔(𝑥) 𝐶 ̄ ̄ |𝜑󸀠 (𝑧)| ≤ (𝜔(𝑥) − 𝜔(𝑦)). 𝑥 𝜀 𝛼 ̄ increases with 𝑡. Then, by (8.12), Choose 𝛼 > 0 so that the function 𝜔(𝑡)/𝑡

̄ 1/𝛼 ̄ 1/𝛼 󸀠 𝜔(𝑥) 𝜔(𝑥) 3 ̄ 1/𝛼 − |𝜑 (𝑧)| ≤ (𝜔(𝑥) 𝑦) 𝑥 𝜀 𝑥 3 3 ̄ 1/𝛼 − 𝜔(𝑦) ̄ 1/𝛼−1 (𝜔(𝑥) ̄ 1/𝛼 ) ≤ 𝜔(𝑥) ̄ ̄ − 𝜔(𝑦)) . ≤ (𝜔(𝑥) 𝜀 𝛼 This implies the desired result. Lemma 8.3. Let 𝑔 : 𝔻 󳨃→ ℝ be a regularly oscillating function such that, for some 0 < 𝑝 < ⬦, 2𝜋

∫ |∇𝑔(𝑒𝑖𝜃 𝑧)|𝑝 𝑑𝜃 ≤ Ψ󸀠 (1 − |𝑧|)𝑝 , 0

Then 𝑔 ∈ Λ𝑝𝜔 (𝔻).

𝑧 ∈ 𝔻.

256 | 8 Lipschitz spaces of first order Proof. Let 𝐺 = |∇𝑔|. Let 𝑎, 𝑏 ∈ 𝔻, |𝑎| ≤ |𝑏| < 1, and 𝛿 = |𝑎−𝑏|. As in the proof of Lemma 8.1, 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

𝜀 = (1 − |𝑎|)/2.

𝑧∈𝐷𝜀 (𝑎)

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

∫ |𝑔(𝑎𝑒𝑖𝜃 ) − 𝑔(𝑏𝑒𝑖𝜃 )|𝑝 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) ≤ 𝜔(1−𝑟)/(1−𝑟) implies the same with 𝐺# instead of 𝐺 (see Note 3.6) so we have 2𝜋

∫ |𝑔(𝑎𝑒𝑖𝜃 ) − 𝑔(𝑏𝑒𝑖𝜃 )|𝑝 𝑑𝜃 ≤ 𝐶Ψ󸀠 (1 − |𝑎|)𝑝 𝛿𝑝 + 𝐶Ψ󸀠 (1 − |𝑏|)𝑝 𝛿𝑝 ≤ 𝐶𝜔(𝛿). 0

In the last step we used the inequalities 1 − |𝑎| ≥ 1 − |𝑏| ≥ 𝛿 and the condition that the function Ψ󸀠 (𝑡) = 𝜔(𝑡)/𝑡 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 above case that 2𝜋

∫ |𝑔(𝑎𝑒𝑖𝜃 ) − 𝑔(𝑏󸀠 𝑒𝑖𝜃 )|𝑝 𝑑𝜃 ≤ 𝐶𝜔(𝛿󸀠 )𝑝 ≤ 𝐶𝜔(𝛿)𝑝 = 𝐶𝜔(|𝑎 − 𝑏|)𝑝 .

(8.13)

0

(Here we have used the geometrically obvious inequality 𝛿󸀠 ≤ 𝛿.) On the other hand, |𝑏|

1

|𝑔(𝑏𝑒𝑖𝜃 ) − 𝑔(𝑏󸀠 𝑒𝑖𝜃 )| ≤ ∫ 𝐺((𝑏/|𝑏|)𝑠𝑒𝑖𝜃 ) 𝑑𝑠 ≤ ∫ |𝐺((𝑏/|𝑏|)𝑠𝑒𝑖𝜃 )| 𝑑𝑠. |𝑏󸀠 |

1−𝛿

(8.14)

𝑝

8.4 Composition operators into 𝐻Λ 𝜔

257

|

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

∞

∞

( ∫ ℎ(𝑠) 𝑑𝑠) ≤ ∑ 2−𝑘𝑝 sup ℎ(𝑠)𝑝 = ∑ 2−𝑘𝑝 𝐺× (𝑟𝑘 𝑒𝑖𝜃 )𝑝 . 𝑟𝑘 ≤𝑠≤𝑟𝑘+1

𝑘=𝑛−1

1−𝛿

𝑘=𝑛−1

Now, arguing as in the proof of Proposition 3.4 (see (3.19)) we conclude that the last quantity is less or equal to 𝑟𝑛−1

1 𝑝−1

∫ (1 − 𝑠)

𝑀𝑝𝑝 (𝑠, 𝐺) 𝑑𝑠

2−𝑛+1

= ∫ 𝑡𝑝−1 0

𝜔(𝑡)𝑝 𝑑𝑡 𝑡𝑝

≤ 𝐶𝜔(𝑟𝑛−1 )𝑝 ≤ 𝐶𝜔(𝛿)𝑝 , where we have used the hypothesis that 𝜔 is fast. Combining this with (8.13) we get 2𝜋

∫ |𝑔(𝑎𝑒𝑖𝜃 ) − 𝑔(𝑏𝑒𝑖𝜃 )|𝑝 𝑑𝜃 ≤ 𝐶𝜔(|𝑎 − 𝑏|)𝑝 .

(8.15)

0

If 𝑝 ≥ 1, then we apply Minkowski’s inequality to (8.14) 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|, 𝑏󸀠 = (1 − 𝛿)𝑏/|𝑏|, and proceed as above to get (8.15). This completes the proof of the lemma. We continue the proof of the equivalence of the conditions (A𝑝)–(C𝑝). We have proved that (A𝑝) implies (B𝑝). Since the implication (B𝑝) ⇒ (C𝑝) is trivial, we see that it re­ mains to prove the implication (C𝑝) ⇒ (A𝑝). To do this we need some more facts. Lemma 8.4. Let 𝑝 > 0 and 0 < 𝜀 ≤ 1 − |𝑧|, 𝑧 ∈ 𝔻. If 𝜑 ∈ 𝐻(𝔻), then the inequality 𝑝

|𝜑󸀠 (𝑧)|𝑝 ≤ 𝐶𝑝 𝜀−𝑝−2 ∫ {|𝜑(𝑤)| − |𝜑(𝑧)|}+ 𝑑𝐴(𝑤), 𝐷𝜀 (𝑧)

holds. Proof. It follows from Schwarz’s modulus lemma that |𝜑󸀠 (𝑧)| ≤

4 𝜀

sup {|𝜑(𝑤)| − |𝜑(𝑧)|}+ .

𝑤∈𝐷𝜀/2 (𝑧)

For a fixed 𝑧, the function 𝑤 󳨃→ {|𝜑(𝑤)| − |𝜑(𝑧)|}+ is subharmonic, which along with the Fefferman–Stein theorem gives the desired result.

258 | 8 Lipschitz spaces of first order The inequality ̄ 𝜔̄ 󸀠 (𝑥){𝑥 − 𝑦}+ ≥ {𝜔(𝑥) − 𝜔(𝑦)} +,

0 < 𝑥, 𝑦 < 1,

(8.16)

will also be used. ̄ − |𝜑|), and Lemma 8.5. Let 𝜑 ∈ h(𝔻), 𝑔 = 𝜔(1 ∫{𝑔(𝜁𝑧) − 𝑔(𝜁𝑤)}𝑝+ |𝑑𝜁| ≤ 𝜔(1 − |𝑧|)𝑝

(8.17)

𝕋

if |𝑤 − 𝑧| < (1 − |𝑧|)/2. Then ∫ |∇𝑔(𝜁𝑧)|𝑝 |𝑑𝜁| ≤ 𝐶Ψ󸀠 (1 − |𝑧|)𝑝 . 𝕋

Proof. Let 𝐷(𝑧) = {𝑤 : |𝑤 − 𝑧| < (1 − |𝑧|)/2}. We have, by Lemma 8.3, inequality (8.16) and condition (8.17), ∫ |∇𝑔(𝜁𝑧)|𝑝 |𝑑𝜁| = ∫ |𝜑󸀠 (𝜁𝑧)|𝑝 𝜔̄ 󸀠 (1 − |𝜑(𝜁𝑧)|)𝑝 |𝑑𝜁| ≤ 𝐶 ∫ 𝜔̄ 󸀠 (1 − |𝜑(𝜁𝑧)|)𝑝 |𝑑𝜁| 𝕋

𝕋

𝕋

× (1 − |𝑧|)−𝑝−2 ∫ {|𝜑(𝑤)| − |𝜑(𝜁𝑧)|}𝑝+ 𝑑𝐴(𝑤) 𝐷(𝑧𝜁)

= 𝐶 ∫ 𝜔̄ 󸀠 (1 − |𝜑(𝜁𝑧)|)𝑝 |𝑑𝜁| 𝕋

× (1 − |𝑧|)−𝑝−2 ∫ {|𝜑(𝜁𝑤)| − |𝜑(𝜁𝑧)|}𝑝+ 𝑑𝐴(𝑤) 𝐷(𝑧) −𝑝−2

= 𝐶 ∫ (1 − |𝑧|)

𝑑𝐴(𝑤)

𝐷(𝑧)

× ∫ 𝜔̄ 󸀠 (1 − |𝜑(𝜁𝑧)|)𝑝 {|𝜑(𝜁𝑤)| − |𝜑(𝜁𝑧)|}𝑝+ |𝑑𝜁| 𝕋

≤ 𝐶 ∫ (1 − |𝑧|)−𝑝−2 𝑑𝐴(𝑤) 𝐷(𝑧)

̄ − |𝜑(𝜁𝑧)|) − 𝜔(1 ̄ − |𝜑(𝜁𝑤)|)}𝑝+ |𝑑𝜁| × ∫{𝜔(1 𝕋

≤ 𝐶 ∫ (1 − |𝑧|)−𝑝−2 𝜔(1 − |𝑧|)𝑝 𝑑𝐴(𝑤) 𝐷(𝑧)

= (𝐶𝜋/4)Ψ󸀠 (1 − |𝑧|)𝑝 . This concludes the proof.

𝑝

8.4 Composition operators into 𝐻Λ 𝜔

| 259

Proof of “(C𝑝) ⇒ (A𝑝)”. By Lemma 8.5, it suffices to prove that (C𝑝) implies ‖{𝑔𝑤 − 𝑔𝑧 }+ ‖𝐿𝑝 (𝕋) ≤ 𝐶𝜔(1 − |𝑧|),

whenever |𝑤 − 𝑧| < (1 − |𝑧|)/2.

Let |𝑤 − 𝑧| < (1 − |𝑧|)/2, 𝜁 = 𝑧/𝑟, 𝑟 = |𝑧| ≠ 0, 𝜂 = 𝑤/𝜌, 𝜌 = |𝑤| ≠ 0. Then (D𝑝) implies ‖{𝑔𝑤 − 𝑔𝑧 }+ ‖𝑝 ≤ ‖𝑔𝑤 − 𝑔𝜂 ‖𝑝 + ‖𝑔𝜂 − 𝑔𝜁 ‖𝑝 + ‖𝑔𝜁 − 𝑔𝑧 ‖𝑝 ≤ 𝐶𝜔(1 − 𝜌) + 𝐶𝜔(|𝜂 − 𝜁|) + 𝐶𝜔(1 − 𝑟). Since the condition |𝑤 − 𝑧| < (1 − |𝑧|)/2 implies 1 − 𝜌 ≤ 3(1 − 𝑟)/2, we see that ‖{𝑔𝑤 − 𝑔𝑧 }+ ‖𝑝 ≤ 𝐶𝜔(1 − 𝑟) + 𝐶𝜔(|𝜂 − 𝜁|). On the other hand, 𝜔(|𝜂 − 𝜁|) ≤ 𝜔(|𝜂 − 𝑤|) + 𝜔(|𝑤 − 𝑧|) + 𝜔(|𝑧 − 𝜁|) ≤ 𝜔(1 − 𝜌) + 𝜔((1 − 𝑟)/2) + 𝜔(1 − 𝑟). The result follows. Thus we have proved Theorem 8.13. Proof of Theorem 8.14. Since obviously (C𝑝) implies (D𝑝), we have to prove (by The­ orem 8.13) that (D𝑝) implies (E𝑝) and that (E𝑝) implies (A𝑝). We need the following lemma. ̄ − |𝜑|) ∈ Λ𝑝𝜔 (𝕋), where 𝜔 is slow, then the Poisson integral of 𝑔 Lemma 8.6. If 𝑔 = 𝜔(1 𝑝 belongs to Λ 𝜔 (𝔻). Proof. As noted in Subsection 8.1.2, item (I), the 𝐿𝑝 -analog of Theorem 8.3 holds for 𝑝 ≥ 1. The lemma is a special case of this analog. Let us prove that (D𝑝) implies (E𝑝). Let ℎ = P[𝑔]. Assuming (D𝑝) we use the inequality 𝑔 ≤ ℎ to obtain 𝑔(𝑟𝜁) − ℎ(𝑟𝜁) = {𝑔(𝑟𝜁) − ℎ(𝑟𝜁)}+ ≤ {𝑔(𝑟𝜁) − 𝑔(𝜁)}+ + {𝑔(𝜁) − ℎ(𝑟𝜁)}+ = {𝑔(𝑟𝜁) − 𝑔(𝜁)}+ + |ℎ(𝜁) − ℎ(𝑟𝜁)|, whence ‖𝑔𝑟 − ℎ𝑟 ‖𝑝 ≤ ‖{𝑔𝑟 − 𝑔}+ ‖𝑝 + ‖ℎ − ℎ𝑟 ‖𝐿𝑝 (𝕋) . Hence, by Lemma 8.6, ‖𝑔𝑟 − ℎ𝑟 ‖𝑝 ≤ 𝐶𝜔(1 − 𝑟), which is another form of (E𝑝). In order to prove that (E𝑝) implies (A𝑝), let |𝑤 − 𝑧| ≤ (1 − |𝑧|)/2. Then {𝑔(𝑧𝜁) − 𝑔(𝑤𝜁)}+ ≤ {𝑔(𝑧𝜁) − ℎ(𝑤𝜁)}+ ≤ {𝑔(𝑧𝜁) − ℎ(𝑧𝜁)}+ + {ℎ(𝑧𝜁) − ℎ(𝑤𝜁)}+ .

260 | 8 Lipschitz spaces of first order Hence ‖{𝑔𝑧 − 𝑔𝑤 }+ ‖𝑝 ≤ ‖𝑔𝑧 − ℎ𝑧 ‖𝑝 + ‖ℎ𝑧 − ℎ𝑤 ‖𝑝 , and hence, by (E𝑝) and Lemma 8.6, ‖{𝑔𝑧 − 𝑔𝑤 }+ ‖𝑝 ≤ 𝐶𝜔(1 − |𝑧|). Now we use Lemma 8.5 to obtain (A𝑝). This completes the proof of Theorem 8.14.

8.5 Inner functions As a consequence of the above theorems we have the following result on division by inner functions. Theorem 8.15. Let 𝜔 be a regular majorant, 𝑝 ∈ [1, ⬦) ∪ {∞}. If 𝐼 is an inner function and 𝜑𝐼 belongs to C(𝜔,̄ 𝜔, 𝑝), then so does 𝜑. In other words, C(𝜔,̄ 𝜔, 𝑝) has the f -property. Proof. This follows from Theorem 8.13 and the inequality ̄ − |𝜑(𝑟𝜁)𝐼(𝑟𝜁)|) − 𝜔((1 ̄ − |𝜑(𝜁)|))} + ̄ − |𝜑(𝑟𝜁)𝐼(𝑟𝜁)|) − 𝜔((1 ̄ − |𝜑(𝜁)𝐼(𝜁)|)} + = {𝜔((1 {𝜔(1 ̄ − |𝜑(𝑟𝜁)|) − 𝜔(1 ̄ − |𝜑(𝜁)|)}+ , ≥ {𝜔(1 where 𝜁 ∈ 𝕋. Theorem 8.16. Let 𝑝 ≠ ⬦. For an inner function 𝐼, the following conditions are equiva­ lent: (a) 𝐼 ∈ C(𝜔,̄ 𝜔, 𝑝); (b) (∫𝕋 𝜔̄ 𝑝 (1 − |𝐼(𝑟𝜁)|) |𝑑𝜁|)1/𝑝 ≤ 𝐶𝜔(1 − 𝑟); 1/𝑝

(c) (∫𝕋 𝜔̄ 𝑝 ((1 − 𝑟)|𝐼󸀠 (𝑟𝜁)|) |𝑑𝜁|)

≤ 𝐶𝜔(1 − 𝑟).

For the proof we need a maximal lemma. Lemma 8.7. Let 𝑡0 ∈ (0, ∞), and 𝜙 : [0, 𝑡0 ] 󳨃→ [0, ∞) be a continuous function such that 𝜙(𝑡)/𝑡𝛼 is increasing for some 𝛼 > 0 and 𝜙(𝑡)/𝑡𝛽 is decreasing for some 𝛽 > 0. Let 𝑓 ∈ 𝐴(𝔻) be a function such that |𝑓(𝑧)| ≤ 𝑡0 for all 𝑧 ∈ 𝔻. Then ∫ 𝑀∗ [𝜙(|𝑓|)](𝜁) |𝑑𝜁| ≤ 𝐶𝛼,𝛽 ∫ 𝜙(|𝑓(𝜁)|) |𝑑𝜁|. 𝕋

(8.18)

𝕋

Proof. Assume that we have proved the lemma under the additional condition that the function 𝜙(𝑡) = √𝜙(𝑡2/𝛼 ) is convex. Then 2

∫ 𝑀∗ [𝜙(|𝑓|)](𝜁) |𝑑𝜁| = ∫ (𝑀∗ 𝜙[|𝑓|𝛼/2 ](𝜁)) |𝑑𝜁|. 𝕋

𝕋

8.5 Inner functions

| 261

The function 𝜙(|𝑓|𝛼/2 ) is subharmonic as a composition of a convex with a subhar­ monic function. This implies (8.18), by the subharmonic maximal theorem (Theo­ rem B.10). In the general case we define the function 𝜓 by 𝑡

√𝜓(𝑡2/𝛼 ) = ∫

√𝜙(𝑥2/𝛼 ) 𝑥

0

𝑑𝑥.

This function is defined correctly, possesses the above “good” property, and is equiv­ alent to 𝜙. This concludes the proof. Proof of Theorem 8.16. Let 𝑝 ≠ ∞. The equivalence (a) ⇔ (b) is a consequence of Theorem 8.13 (condition (C𝑝)). The implication (b) ⇒ (c) is a consequence of the Schwarz–Pick lemma: |𝐼󸀠 (𝑧)| ≤ (1 − |𝐼(𝑧)|2 )/(1 − |𝑧|2 ). In order to prove that (c) implies (b), we start from the inequality 1

1 − |𝐼(𝑟𝜁)| ≤ ∫ |𝐼󸀠 (𝜌𝜁)| 𝑑𝜌. 𝑟 −𝑗

Let 𝑟𝑗 = 1 − 2 . Then ∞

1 − |𝐼(𝑟𝑛 𝜁)| ≤ ∑ 2−𝑗 sup |𝐼󸀠 (𝜌𝜁)| 𝑗=𝑛

(8.19)

𝑟𝑗 0, while this can be proved by means of the formula 𝑖𝑗𝑡 ̂ − 1)𝑛 𝑒𝑖𝑗𝜃 Δ𝑛𝑡 𝑔(𝑒𝑖𝜃 ) = ∑ 𝑔(𝑗)(𝑒 |𝑗| 0,

and therefore ‖𝑢𝑘 ‖𝜔,𝑛 = 2𝑛 sup{| sin(𝑘𝑠/2)|𝑛 /𝜔(𝑡) : 𝑠 ≤ 𝑡 ≤ 1, 0 < 𝑠 ≤ 1}, Since 𝜔(𝑡) ≥ 𝜔(𝑠)/𝐶 for 0 < 𝑠 ≤ 𝑡 ≤ 1, we have ‖𝑢𝑘 ‖𝜔,𝑛 ≤ 𝐶 sup{| sin(𝑘𝑠/2)|𝑛 /𝜔(𝑠) : 0 < 𝑠 ≤ 1}, where 𝐶 is independent of 𝑘. If 1/𝑘 ≤ 𝑠 ≤ 1, then | sin(𝑘𝑠/2)|𝑛 /𝜔(𝑠) ≤ 𝐶/𝜔(1/𝑘) because the function 1/𝜔 is almost decreasing. If 0 < 𝑠 ≤ 1/𝑘, then | sin(𝑘𝑠/2)|𝑛 /𝜔(𝑠) ≤ 2−𝑛 𝑘𝑛 𝑠𝑛 /𝜔(𝑠) ≤ 𝐶 𝑘𝑛 (1/𝑘)𝑛 /𝜔(1/𝑘) because 𝑠𝑛 /𝜔(𝑠) is almost increasing. Thus ‖𝑢‖𝜔,𝑛 ≤ 𝐶/𝜔(1/𝑘). The proof of the reverse inequality is simpler. Now we are ready to prove the implication (b) ⇒ (c) in Theorem 9.1. Let 𝐻Λ𝜔,𝑛 = 𝐻∞ (𝜓)𝑛 . It follows from Lemma 9.4 and the closed graph theorem that ‖𝐷𝑛 𝑢‖𝜓 ≍ ‖𝑢𝑘 ‖𝜔,𝑛 , 𝑘 ≥ 1, where 𝑢𝑘 is as in Lemma 9.5. Hence, by Lemma 9.5, 𝜔(1/𝑘) ≍ (1/𝑘)𝑛 𝜓(𝑘), 𝑘 ≥ 1, which yields 0 < 𝑡 ≤ 1, (9.10) 𝜔(𝑡) ≍ 𝑡𝑛 𝜓(1/𝑡), and this is part of (b). In order to prove that (b) implies (9.2) for some 𝛽 < 𝑛, we need another lemma. Lemma 9.6. We have ∞

∑ 𝜓(𝑛)𝑟𝑛 ≤ 𝐶 𝑛=1

1 1 𝜓( ), 1−𝑟 1−𝑟

0 < 𝑟 < 1.

Proof. If 0 < 𝑟 < 1, choose an integer 𝑘 > 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 3.12; see [443]. Returning to the proof of the theorem, we consider the analytic functions ∞

𝑈𝑘 (𝑧) = 𝑘−𝑛 𝜓(𝑘)𝑧𝑘 + ∑ 𝑗−𝑛 (𝜓(𝑗) − 𝜓(𝑗 − 1)) 𝑧𝑗 ,

𝑧 ∈ 𝔻.

𝑗=𝑘+1

By summation by parts, we get ∞

∞

𝑗=𝑘+1

𝑗=𝑘

𝜓(𝑘)𝑟𝑘 + ∑ (𝜓(𝑗) − 𝜓(𝑗 − 1)) 𝑟𝑗 = (1 − 𝑟) ∑ 𝜓(𝑗)𝑟𝑗 .

(9.11)

9.2 Lipschitz spaces and spaces of harmonic functions

| 271

From this and (9.11) we obtain ∞

∞

𝑗=𝑘+1

𝑗=𝑘

𝑀(𝑟, 𝐷𝑛 𝑈𝑘 ) ≤ 𝜓(𝑘)𝑟𝑘 + ∑ (𝜓(𝑗) − 𝜓(𝑗 − 1)) 𝑟𝑗 = (1 − 𝑟) ∑ 𝜓(𝑗)𝑟𝑗 , and hence, by Lemma 9.6, 𝑀(𝑟, 𝐷𝑛 𝑈𝑘 ) ≤ 𝐶𝜓(1/(1 − 𝑟)). It follows that {𝑈𝑘 } is a norm bounded sequence in 𝐻∞ (𝜓)𝑛 . Now we use the inclusion 𝐻∞ (𝜓)𝑛 ⊂ 𝐻Λ𝜔,𝑛 to conclude that the functions 𝑈𝑘 are continuous on the closed disk and 𝜔𝑛 (𝑈𝑘∗ , 𝑡) ≤ 𝐶𝜔(𝑡),

(9.12)

0 < 𝑡 < 1,

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 we use the formula 𝑚

𝑗

𝑚

where 𝐵𝑗 = ∑ 𝑏𝜈 ,

∑ 𝑎𝑗 𝑏𝑗 = ∑ (𝑎𝑗 − 𝑎𝑗+1 )𝐵𝑗 + 𝑎𝑚+1 𝐵𝑚 , 𝑗=𝑘+1

𝑗=𝑘+1

𝜈=𝑘+1

to 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 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

𝑗=𝑘+1

𝑗=𝑘+1

𝑘−𝑛 𝜓(𝑘) + ∑ 𝑗−𝑛 (𝜓(𝑗) − 𝜓(𝑗 − 1)) ≥ ∑ (𝑗−𝑛 − (𝑗 + 1)−𝑛 )𝜓(𝑗) 𝑚

≥ 𝑛 ∑ 𝑗−𝑛−1 𝜓(𝑗) 𝑗=𝑘+1

(9.13)

272 | 9 Lipschitz spaces of higher order and, by (9.10), (9.12), and (9.13), ∞

∫ 𝑦−𝑛−1 𝜓(𝑦) 𝑑𝑦 ≤ 𝐶𝑘−𝑛 𝜓(𝑘),

𝑘 ∈ ℕ+ .

𝑘

It is easily verified 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 inci­ dentally of Theorem 9.1.

Proof of Proposition 9.1 In proving Propositions 9.1 and 9.2 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 |𝐷𝑛+1 𝑓(𝑧)| ≤ 𝐶(1 − |𝑧|)−1 ‖𝐷𝑛 𝑢‖∞ ,

|𝑓(𝑛+1) (𝑧)| ≤ 𝐶(1 − |𝑧|)−1 ‖𝐷𝑛 𝑢‖∞ ,

where 𝐶 is independent of 𝑓 and 𝑧. The proof is left to the reader as an exercise (al­ though these inequalities in a more general form were used in the proof of Theo­ rem 3.10; see also Exercise 3.1). In proving Proposition 9.1 we may assume that 𝑢 is real valued and harmonic in a neighborhood of the closed disk. For fixed 𝑟 < 1 let ℎ(𝜃) = 𝑢𝑟 (𝜃) = 𝑢(𝑟𝑒𝑖𝜃 ). Then (Δ𝑛𝑡 ℎ)(𝜃) = ∫ ℎ(𝑛) (𝜃 + 𝑥1 + ⋅ ⋅ ⋅ + 𝑥𝑛 ) 𝑑𝑥1 . . . 𝑑𝑥𝑛 , 𝑡𝐸

where 𝑡𝐸 is the 𝑛-dimensional cube [0, 𝑡]𝑛 . Hence ℎ(𝑛) (𝜃) = (𝐷𝑛 𝑢)(𝑟𝑒𝑖𝜃 )𝑡𝑛 = (Δ𝑛𝑡 𝑢𝑟 )(𝜃) − ∫ (ℎ(𝑛) (𝜃 + 𝑥1 + ⋅ ⋅ ⋅ + 𝑥𝑛 ) − ℎ(𝑛) (𝜃)) 𝑑𝑥1 . . . 𝑑𝑥𝑛 . 𝑡𝐸

Since

𝑥

󵄨󵄨 󵄨󵄨 |ℎ (𝜃 + 𝑥) − ℎ (𝜃)| = 󵄨󵄨󵄨󵄨 ∫ ℎ(𝑛+1) (𝜃 + 𝑦) 𝑑𝑦󵄨󵄨󵄨󵄨 ≤ 𝑀(𝑟, 𝐷𝑛+1 𝑢)𝑥, 󵄨 󵄨 (𝑛)

(𝑛)

0

where 𝑥 = 𝑥1 + ⋅ ⋅ ⋅ + 𝑥𝑛 , we get 𝑀(𝑟, 𝐷𝑛 𝑢)𝑡𝑛 ≤ ‖Δ𝑛𝑡 𝑢𝑟 ‖∞ + ∫ 𝑀(𝑟, 𝐷𝑛+1 𝑢)(𝑥1 + ⋅ ⋅ ⋅ + 𝑥𝑛 ) 𝑑𝑥1 . . . 𝑑𝑥𝑛 𝑡𝐸

= ‖Δ𝑛𝑡 𝑢𝑟 ‖∞ + (𝑛/2)𝑀(𝑟, 𝐷𝑛+1 𝑢)𝑡𝑛+1 ,

0 < 𝑟 < 1, 𝑡 > 0.

(9.15)

9.2 Lipschitz spaces and spaces of harmonic functions |

273

The function Δ𝑛𝑡 𝑢 defined by (Δ𝑛𝑡 𝑢)(𝑟𝑒𝑖𝜃 ) = (Δ𝑛𝑡 𝑢𝑟 )(𝜃) is harmonic on the closed disk and therefore ‖Δ𝑛𝑡 𝑢𝑟 ‖∞ ≤ ‖Δ𝑛𝑡 𝑢∗ ‖ ≤ 𝜔𝑛(𝑢∗ , 𝑡), 𝑡 > 0. These inequalities together with (9.14) yield (9.16) 𝑀(𝑟, 𝐷𝑛 𝑢) ≤ 𝑡−𝑛 𝜔𝑛(𝑢∗ , 𝑡) + 𝐾𝑡(1 − 𝑟)−1 𝑀((1 + 𝑟)/2, 𝐷𝑛 𝑢) (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 so 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

∫ 𝐴(𝑟) 𝑑𝑟 ≤ 𝑎

−𝑚

1

∫ 𝜔(𝑡) 𝑑𝑡 + (1/2)

𝜌

0 1−𝜌

≤𝑎

−𝑚

∫ 𝐴(𝑟) 𝑑𝑟 (1+𝜌)/2 1

∫ 𝜔(𝑡) 𝑑𝑡 + (1/2) ∫ 𝐴(𝑟) 𝑑𝑟. 𝜌

0 1

Hence, since the integral ∫𝜌 𝐴(𝑟) 𝑑𝑟 is finite, 1−𝜌

1

(1/2) ∫ 𝐴(𝑟) 𝑑𝑟 ≤ 𝑎

−𝑚

𝜌

∫ 𝜔(𝑡) 𝑑𝑡. 0

Now (9.5) follows from the inequalities 1−𝜌

∫ 𝜔(𝑡) 𝑑𝑡 ≤ (1 − 𝜌) 𝜔(1 − 𝜌), 0 1 𝑛+1

𝑀(𝑟, 𝐷

𝑛+1

𝑢)(1 − 𝜌)

≤ (𝑛 + 1) ∫ 𝐴(𝑟) 𝑑𝑟, 𝜌

which are valid because the functions 𝜔 and 𝑀 are increasing. Thus the proof of Propo­ sition 9.1 is finished.

Proof of Proposition 9.2 Let 𝑢 = Re 𝑓, where 𝑓 is analytic 𝑢 𝔻. Then 𝑛

1

𝑓(𝑘) (𝑟𝑧) 𝑘 1 𝑧 (1 − 𝑟)𝑘 + ∫(1 − 𝑠)𝑛 𝑧𝑛+1 𝑓(𝑛+1) (𝑠𝑧) 𝑑𝑠 𝑘! 𝑛! 𝑘=0

𝑓(𝑧) = ∑

𝑟

(9.17)

274 | 9 Lipschitz spaces of higher order (𝑧 ∈ 𝔻, 0 < 𝑟 < 1). Denoting the sum by 𝑓𝑟,𝑛 we have 1

|𝑓(𝑧) − 𝑓𝑟,𝑛 (𝑧)| ≤

1 ∫(1 − 𝑠)𝑛 𝑀(𝑠, 𝑓(𝑛+1) ) 𝑑𝑠. 𝑛! 𝑟

From this and (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 disk, we see that (9.6) implies the continuity of 𝑓, and consequently of 𝑢, on the closed disk. In order to prove (9.7) let 𝑢𝑟 (𝜃) = 𝑢(𝑟𝑒𝑖𝜃 ), 0 < 𝑟 ≤ 1. Then (9.7) is equivalent to 1

‖Δ𝑛𝑡 𝑢1 ‖∞

≤ 𝐶 ∫ (1 − 𝑠)𝑛−1 𝑀(𝑠, 𝐷𝑛 𝑢) 𝑑𝑠,

0 < 𝑡 < 1.

(9.18)

1−𝑡

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

‖Δ𝑛𝑡 𝑢𝑟 ‖ ≤ 𝑡𝑛 𝑀(𝑟, 𝐷𝑛 𝑢) ≤ 𝑛 ∫ (1 − 𝑠)𝑛−1 𝑀(𝑠, 𝐷𝑛 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 is enough to prove that 1

‖Δ𝑛𝑡 (𝑓1

− 𝑓𝑟 )‖ ≤ 𝐶 ∫ (1 − 𝑠)𝑛−1 𝑀(𝑠, 𝐷𝑛 𝑢) 𝑑𝑠. 1−𝑡

To prove this write (9.17) in the form 𝑛

𝑓1 (𝜃) − 𝑓𝑟 (𝜃) = 𝐻(𝜃) + ∑ ℎ𝑘 (𝜃)(1 − 𝑟)𝑘 /𝑘!,

where

𝑘=1 1

𝐻(𝜃) =

1 ∫(1 − 𝑠)𝑛 𝑒𝑖(𝑛+1)𝜃 𝑓(𝑛+1) (𝑠𝑒𝑖𝜃 ) 𝑑𝑠, 𝑛!

and ℎ𝑘 (𝜃) = 𝑓(𝑘) (𝑟𝑒𝑖𝜃 )𝑒𝑖𝑘𝜃 .

𝑟

We have 1

‖Δ𝑛𝑡 𝐻‖ ≤ 2𝑛 ‖𝐻‖ ≤

2𝑛 ∫(1 − 𝑠)𝑛 𝑀(𝑠, 𝑓(𝑛+1) ) 𝑑𝑠 𝑛! 𝑟

1

≤ 𝐶 ∫(1 − 𝑠)𝑛−1 𝑀((1 + 𝑠)/2, 𝐷𝑛 𝑢) 𝑑𝑠 𝑟 1 𝑛

= 2 𝐶 ∫ (1 − 𝑠)𝑛−1 𝑀(𝑠, 𝐷𝑛 𝑢) 𝑑𝑠, 1−𝑡

9.3 Conjugate functions |

275

where we have applied (9.14). In order to estimate ‖Δ𝑛𝑡 ℎ𝑘 ‖, let 𝑚 = 𝑛 − 𝑘 + 1 (1 ≤ 𝑘 ≤ 𝑛) and observe that (9.15) implies 𝑚 𝑘−1 ‖Δ𝑚𝑡 ℎ𝑘 ‖ ≤ 2𝑘−1 𝑡𝑚 ‖ℎ(𝑚) ‖Δ𝑛𝑡 ℎ𝑘 ‖ = ‖Δ𝑘−1 𝑡 Δ 𝑡 ℎ𝑘 ‖ ≤ 2 𝑘 ‖.

From this and the inequality ‖ℎ(𝑚) ‖ ≤ 𝐶(1 − 𝑟)−1 𝑀((1 + 𝑟)/2, 𝐷𝑛 𝑢) (see (9.14)) it follows 𝑘 that 1

‖Δ𝑛𝑡 ℎ𝑘 ‖

𝑛−𝑘

≤ 𝐶𝑡

𝑛

−𝑘

𝑀(1 − 𝑡, 𝐷 𝑢) ≤ 𝐶𝑡

∫ (1 − 𝑠)𝑛−1 𝑀(𝑠, 𝐷𝑛 𝑢) 𝑑𝑠, 1−𝑡

where 𝐶 is independent of 𝑡. Combining all the above 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 [405] says that if 𝜔 is a regular majorant (of order 1), then the Hilbert operator maps Λ 𝜔 (𝕋) into itself (see Theorem 8.4). This is a special 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 Λ 𝜔,𝑛 into Λ 𝜔,𝑛 if and only if 𝜔 is a slow majorant of order 𝑛. See Note 9.2. Since Λ 𝜔,𝑛 ≃ ℎ∞ (𝜓)𝑛 , where 𝜓(𝑥) = 𝑥𝑛 𝜔(1/𝑥), 𝑥 ≥ 1, we can consider equivalent question: when the space ℎ∞ (𝜓)𝑛 is self-conjugate, i.e. 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 |∇𝑢(𝑧)| ≤ 𝐶

1 1 ) 𝜓( 1−𝑟 1−𝑟

(𝑟 = |𝑧|).

Now “if part” follows from the formula |∇𝑢| = |∇𝑢̃ |. In proving “only if part” we use the fact that, by Proposition 8.3, there is a concave 𝛽 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 a positive integer. Let 𝑁 = 1 and ∞

𝑘𝜓 (𝑟𝑒𝑖𝜃 ) = 2 ∑ 𝜓(𝑘)𝑟𝑘 cos 𝑘𝜃. 𝑘=1

276 | 9 Lipschitz spaces of higher order Assume that we have proved 𝑀1 (𝑘𝜓 , 𝑟) ≤ 𝐶𝜓 ( and let

1 ), 1−𝑟

0 < 𝑟 < 1,

(9.19)

∞

𝑢(𝑟𝑒𝑖𝜃 ) = ∑ 𝑘−1 𝜓(𝑘)𝑟𝑘 sin 𝑘𝜃. 𝑘=1

Then

∞

̃ 𝑖𝜃 ) = − ∑ 𝑘−1 𝜓(𝑘)𝑟𝑘 cos 𝑘𝜃. 𝑢(𝑟𝑒 𝑘=1

We have

∞

𝑀∞ (𝑢,̃ 𝑟) ≥ ∑ 𝑘−1 𝜓(𝑘)𝑟𝑘

(9.20)

𝑘=1

and, by (9.19), 𝜃

2𝜋

|𝑢(𝑟𝑒𝑖𝜃 )| ≤ ∫ |𝐷𝑢(𝑟𝑒𝑖𝑡 )| 𝑑𝑡 ≤ ∫ |𝐷𝑢(𝑟𝑒𝑖𝑡 )| 𝑑𝑡 0

0 2𝜋

=

1 1 ) ∫ |𝑘𝜓 (𝑟𝑒𝑖𝑡 )| 𝑑𝑡 ≤ 𝐶𝜓 ( 2 1−𝑟

(𝜃 ∈ (0, 2𝜋)).

0

Hence 𝑢 ∈ ℎ∞ (𝜓) and therefore, by hypothesis, 𝑢̃ ∈ ℎ∞ (𝜓). From this and (9.20) we find that ∞ 1 ). ∑ 𝑘−1 𝜓(𝑘)𝑟𝑘 ≤ 𝐶𝜓 ( 1 − 𝑟 𝑘=1 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)𝑟𝑘 )(𝑘 + 1)𝐾𝑘 (𝜃), 𝑘=0

where 𝐾𝑘 (𝜃) is the Fejér kernel, 𝐾𝑘 (𝜃) =

1 𝑘 1 − cos(𝑘 + 1)𝜃 , ∑ 𝐷 (𝜃) = 𝑘 + 1 𝑗=0 𝑗 (𝑘 + 1)(1 − cos 𝜃)

9.3 Conjugate functions |

277

2𝜋

and Δ2 𝑎𝑘 = 𝑎𝑘 − 2𝑎𝑘+1 + 𝑎𝑘+2 . Since ∫0 |𝐾𝑘 (𝜃)| 𝑑𝜃 = 2𝜋, we have ∞

𝑀1 (𝑘𝜓 , 𝑟) ≤ ∑ |Δ2 (𝜓(𝑘)𝑟𝑘 )|(𝑘 + 1). 𝑘=0

On the other hand, (Δ2 (𝜓(𝑘)𝑟𝑘 ))(𝑘 + 1) = (𝜓(𝑘)𝑟𝑘 − 2𝜓(𝑘 + 1)𝑟𝑘+1 Now we use the formula Δ(𝑎𝑘 𝑏𝑘 ) = (Δ𝑎𝑘 )𝑏𝑘 + 𝑎𝑘+1 Δ𝑏𝑘 , where Δ𝑐𝑘 = 𝑐𝑘 − 𝑐𝑘+1 , to get Δ2 (𝑎𝑘 𝑏𝑘 ) = (Δ2 𝑎𝑘 )𝑏𝑘 + (Δ2 𝑏𝑘 )𝑎𝑘+2 + 2Δ1 𝑎𝑘+1 Δ1 𝑏𝑘 . Hence, taking 𝑎𝑘 = 𝑟𝑘 and 𝑏𝑘 = 𝜓(𝑘), we obtain Δ2 (𝜓(𝑘)𝑟𝑘 ) = 𝑟𝑘+2 Δ2 𝜓(𝑘) + 𝜓(𝑘)(1 − 𝑟)2 𝑟𝑘 + 2𝑟𝑘+1 (1 − 𝑟)Δ1 𝜓(𝑘). Since Δ2 𝜓(𝑘) ≤ 0 and |Δ1 𝜓(𝑘)| = 𝜓(𝑘 + 1) − 𝜓(𝑘) ≤

𝜓(𝑘 + 1) , 𝑘+1

we have ∞

𝑀1 (𝑘𝜓 , 𝑟) ≤ ∑ (−Δ2 𝜓(𝑘))(𝑘 + 1) 𝑟𝑘+2 𝑘=0 ∞

∞

𝑘=0

𝑘=0

+ (1 − 𝑟)2 ∑ 𝜓(𝑘)(𝑘 + 1) 𝑟𝑘 + 2(1 − 𝑟) ∑ 𝜓(𝑘) 𝑟𝑘+1 . The last two summands are ≤ 𝐶𝜓(1/(1 − 𝑟)), by Lemma 9.6. In order to estimate the first summand we use suitable changes of indices to get ∞

∞

𝑘=0

𝑘=2

∑ Δ2 𝜓(𝑘)(𝑘 + 1) 𝑟𝑘+2 = 𝑃(𝑟) + (1 − 𝑟)2 ∑ 𝑘𝜓(𝑘)𝑟𝑘 − (1 − 𝑟2 ) ∑ 𝜓(𝑘)𝑟𝑘 , 𝑘=2

where 𝑃 is a polynomial. In view of Lemma 9.6, this completes the proof of (9.19) in the case 𝑁 = 1. If 𝑁 = 2, i.e. if 𝜓 = 𝜓02 , where 𝜓0 is concave, we start from the formula 𝜋

∫ 𝑘𝜓0 (√𝑟𝑒𝑖𝑡 ) 𝑘𝜓0 (√𝑟𝑒𝑖(𝜃−𝑡) ) 𝑑𝑡, 𝑘𝜓 (𝑟𝑒𝑖𝜃 ) = − −𝜋

from which we get 𝑀1 (𝑘𝜓 , 𝑟) ≤ 𝑀1 (𝑘𝜓0 , √𝑟)2 ≤ 𝐶 (𝜓0 (

2 1 1 ). )) ≤ 𝐶𝜓 ( 1−𝑟 1 − √𝑟

The proof (for 𝑁 ≥ 3) is completed by induction on 𝑁.

278 | 9 Lipschitz spaces of higher order

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 ,

𝑡 > 0.

|𝑠| 0. Consequently, if 𝜔 is regular of order 𝑛, then 𝐻Λ 𝜔,𝑚 = 𝐻Λ 𝜔,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 𝛼 𝑝,𝑞 𝜔(𝑡) ≥ 𝑐𝑡 , which implies 𝑓 ∈ B𝛼 ⊂ 𝐻𝑝 so we have to prove that 𝑓 ∈ Λ𝑝,𝑞 𝜔,𝑛 . Let 𝑞 < 𝑝. Then 1

𝜔𝑛 (𝑓, 𝑡)𝑞𝑝

≤ 𝐶 ∫ (1 − 𝑟)𝑛𝑞−1 𝜑𝑞 (𝑟) 𝑑𝑟,

where 𝜑(𝑟) = 𝑀𝑝 (𝑟, 𝐷𝑛 𝑓).

(9.27)

1−𝑡

Multiplying this inequality by 𝜔(𝑡)−𝑞 𝑡−1 , then integrating and changing the order of integration, we obtain 1

1

1

𝑞

∫ (𝜔𝑛 (𝑓, 𝑡)𝑝 /𝜔(𝑡)) 𝑑𝑡/𝑡 ≤ 𝐶 ∫(1 − 𝑟)𝑛𝑞−1 𝜑(𝑟)𝑞 𝑑𝑟 ∫ 𝜔(𝑡)−𝑞 𝑡−1 𝑑𝑡. 0

0

1−𝑟

Now the result follows from the inequality 1

∫ 𝜔(𝑡)−𝑞 𝑡−1 𝑑𝑡 ≤ 𝐶𝜔(𝑥)−𝑞 , 𝑥

which holds because 𝜔 is fast. Assuming that 𝑞 ≥ 𝑝, we have by Jensen’s inequality 𝑞/𝑝

1 −𝛼𝑝

(𝑛−𝛼)𝑝

(𝛼𝑝𝑡

∫ (1 − 𝑟)

𝑝

𝛼𝑝−1

𝜑(𝑟) (1 − 𝑟)

𝑑𝑟)

1−𝑡 1 −𝛼𝑝

≤ 𝛼𝑝𝑡

∫ (1 − 𝑟)(𝑛−𝛼)𝑞 𝜑(𝑟)𝑞 (1 − 𝑟)𝛼𝑝−1 𝑑𝑟. 1−𝑡

From this and (9.27) it follows 1

𝜔𝑛(𝑓, 𝑡)𝑞𝑝 ≤ 𝐶𝑡𝜀 ∫ (1 − 𝑟)𝑛𝑞−𝜀−1 𝜑(𝑟)𝑞 𝑑𝑟, 1−𝑡

where 𝜀 = 𝛼(𝑞 − 𝑝). Hence 1

1 𝑞

1 𝑛𝑞−𝜀−1

∫ (𝜔𝑛(𝑓, 𝑡)/𝜔(𝑡)) 𝑑𝑡/𝑡 ≤ ∫(1 − 𝑟) 0

0

𝑞

𝜑(𝑟) 𝑑𝑟 ∫ 𝑡𝜀−1 𝜔(𝑡)−𝑞 𝑑𝑡. 1−𝑟

9.4 Integrated mean Lipschitz spaces | 283

Using the inequality 𝑡𝛼 /𝜔(𝑡) ≤ 𝐶(1 − 𝑟)𝛼 𝜔(1 − 𝑟), 𝑡 ≥ 1 − 𝑟, one shows that the inner integral is dominated by (1 − 𝑟)𝜀 /𝜔(1 − 𝑟)𝑞 . This, together with the inclusions (9.26) and 𝑝,𝑞 Λ̄ 𝑝,𝑞 𝜔,𝑛 ⊂ Λ 𝜔,𝑛 , completes the proof. The harmonic analogs of the previous theorems read: Theorem 9.11. Let 𝑝 ≥ 1 and 0 < 𝑞 ≤ ∞. If 𝜔 is a fast majorant satisfying (9.25), then 𝑝,𝑞 𝑝,𝑞 Λ̄ 𝑝,𝑞 𝜔,𝑛 ≃ Λ 𝜔,𝑛 ≃ T𝜔,𝑛 . Consequently, if 𝜔 is a regular majorant of order 𝑛, then 𝑝,𝑞 Λ𝑝,𝑞 𝜔,𝑚 = B𝜔 ,

for all 𝑚 ≥ 𝑛.

𝑝,∞

Exercise 9.1. Let Λ𝑝∗ = Λ 𝜔,2 , 𝜔(𝑡) = 𝑡; this space is called the 𝑝-Zygmund space. An old theorem of Timan and Timan [488] (𝑝 = 2) and Zygmund [535] (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 ∈ 𝐻𝑝 (𝕋), for 𝑝 < 1. (For the case of de­ creasing coefficients, see Aljančić [21].) This can be proved by means of (the harmonic version) Theorems 6.15 and 9.11. More generally, we can define the (𝑝, 𝑞)-Zygmund 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 space by Λ𝑝,𝑞 ∗ = Λ 1,2 . Then one can prove that if 𝑝 ≥ 1, then Λ ∗ ⊂ Λ 𝜔,1 , where 1/𝑠 𝑝,𝑞 𝑝 𝜔(𝑡) = 𝑡(log(2/𝑡)) , 𝑠 = min{𝑝, 2}; see Note 6.7. The inclusion 𝐻Λ ∗ := 𝐻 (𝕋) ∩ Λ𝑝,𝑞 ∗ ⊂ 𝑝,𝑞 Λ 𝜔,1 holds for all 𝑝 > 0. This is one of examples showing that considering the case of nonregular weights in Theorems 9.9 and 9.11 is indispensable.

Radial Lipschitz conditions 𝑝,𝑞 It is substantial that the definition of T𝜔,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

𝑝,𝑞

T𝜔,1 = {𝑢 ∈ ℎ(𝔻) : 𝐷1 𝑢 ∈ ℎ𝜙 } 𝑝,𝑞

𝑝,𝑞

we see from Corollary 3.6 that T𝜔,1 is self-conjugate, which implies that T𝜔,1 does not |𝑘| 𝑖𝑘𝜃 ̂ change when 𝐷1 𝑢 is replaced with R𝑢, where R𝑢 = 𝑟𝜕𝑢/𝜕𝑟 = ∑∞ 𝑒 . −∞ |𝑘|𝑢(𝑘)𝑟 Theorem 9.12. Let 𝑝 ≥ 1, 𝑞 > 0, 𝑢 = P[𝑔], 𝑔 ∈ 𝐿𝑝 (𝕋), and 𝜔 a regular majorant. Then ‖𝑢−𝑢𝑟 ‖𝑝 𝑝,𝑞 𝑞 𝑢 ∈ T𝜔,1 if and only if the function 𝜔(1−𝑟) , 0 < 𝑟 < 1, belongs to 𝐿 −1 . In fact we have Theorem 9.13. Let 𝑝 ≥ 1, 𝑢 = P[𝑔], 𝑔 ∈ 𝐿𝑝 (𝕋), and 𝜔 a fast majorant (of order 1) such 𝑞 that the function 𝑡/𝜔(𝑡), 0 < 𝑡 < 1, is in 𝐿 +0 . Then the function 𝑀𝑝 (𝑟, R𝑢)(1 − 𝑟) 𝜔(1 − 𝑟) 𝑞

,

0 < 𝑟 < 1, 𝑞

belongs to 𝐿 −1 if and only if the function ‖𝑢 − 𝑢𝑟 ‖𝑝 /𝜔(1 − 𝑟), 0 < 𝑟 < 1, belongs to 𝐿 −1 . In the simplest case 𝑝 = ⬦ and 𝑞 = ∞, this theorem says: A function 𝑔 ∈ 𝐶(𝕋) belongs to Λ 𝜔 (𝕋) if and only if ‖𝑢∗ − 𝑢𝑟 ‖∞ = O(𝜔(1 − 𝑟)). The first result of this kind is due to

284 | 9 Lipschitz spaces of higher order Hardy and Littlewood [191, Theorem 42]: 𝑔 belongs to Λ 𝛼 (𝕋) (0 < 𝛼 < 1) if and only if ‖𝑢𝜌 − 𝑢𝑟 ‖∞ ≤ 𝐶(𝜌 − 𝑟)𝛼 (0 < 𝑟 < 𝜌 < 1). Ravisankar [411, 412] proved that “transversally Lipschitz” harmonic functions defined on a domain with 𝐶2 -boundary are Lipschitz. 1 Theorem 9.13 is deduced from the inequality ‖𝑢 − 𝑢𝑟 ‖𝑝 ≤ 2 ∫𝑟 ‖R𝑢𝜌 ‖𝑝 𝑑𝜌 (𝑟 ≥ 1/2), and the following theorem, which is proved in a similar way as Theorem 9.6. Theorem 9.14. 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 ∫[(1 − 𝑟)−1 ‖𝑢 − 𝑢𝑟 ‖𝑝 ]𝑞 𝜓(1 − 𝑟) 𝑑𝑟, 0

(9.28)

0

where 𝐶 depends only on 𝑝, 𝑞, and 𝐾. This result can be further generalized by introducing the radial symmetric differences of order 𝑛 ≥ 2, e.g. 𝑛 𝑗 𝑛 Δ𝑛𝑟 𝑢(𝜁) = ∑ ( )𝑢 (𝜁 (1 − (1 − 𝑟))) , 𝑛 𝑗 𝑗=0

𝜁 ∈ 𝕋, 0 < 𝑟 < 1.

Inequality (9.28) should certainly remain true when R is replaced with R𝑛 , and (1 − 𝑟)−1 × ‖𝑢 − 𝑢𝑟 ‖𝑝 with (1 − 𝑟)−𝑛 ‖Δ𝑛𝑟 𝑢‖𝑝 , but the author has not written a proof.

9.5 Invariant Besov spaces 𝑝,𝑝

Membership of a function 𝑓 ∈ 𝐻(𝔻) in B𝜔,1 (0 < 𝑝 < ⬦), where 𝜔 is a fast majorant of order 1, can be expressed via an integral over the bicircle 𝕋2 ; namely: 𝑝,𝑝

Theorem 9.15. Under the above conditions, 𝑓 belongs to B𝜔,1 if and only if 𝑓 ∈ 𝐻𝑝 and ∫∫( 𝕋 𝕋

|𝑓(𝜁) − 𝑓(𝜂) 𝑝 |𝑑𝜁| |𝑑𝜂| ) < ∞. 𝜔(|𝜁 − 𝜂|) |𝜁 − 𝜂|

Proof. This can be obtained from the relation 1

∫ 0

𝜋

𝑝

‖Δ1 𝑓‖𝑝 𝜔(𝑡)𝑝 𝑡

𝑑𝑡 ≍ ∫ ∫ |𝑓(𝜁𝑒𝑖𝑡 ) − 𝑓(𝜁)|𝑝 −𝜋 𝕋

𝑑𝑡 |𝑑𝜁| . [𝜔(|𝑒𝑖𝑡 − 1|)]𝑝 |𝑒𝑖𝑡 − 1|

𝑝

Let 𝐵 (1 < 𝑝 < ⬦) denote the space of those 𝑓 ∈ 𝐻(𝔻) such that 1/𝑝 󸀠

𝑝

2 𝑝−2

∫ |𝑓 (𝑧)| (1 − |𝑧| ) 𝐵𝑝 (𝑓) := (−

𝑑𝐴(𝑧))

,

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 9.15 we have:

9.5 Invariant Besov spaces | 285

Theorem 9.16. Let 1 < 𝑝 < ⬦. A function 𝑓 ∈ 𝐻(𝔻) belongs to 𝐵𝑝 if and only if 𝑓 ∈ 𝐻𝑝 and 1/𝑝

|𝑓(𝜁) − 𝑓(𝜂)|𝑝 𝐵𝑝,1 (𝑓) := (− ∫∫ − |𝑑𝜁| |𝑑𝜂|) |𝜁 − 𝜂|2

< ∞.

𝕋 𝕋

The seminorm 𝐵𝑝,1 is also Möbius invariant, and the Douglas formula 𝐵2 = 𝐵2,1 holds [124]. When 𝑝 ≤ 1, we choose an integer 𝑁 so that 𝑁𝑝 − 2 > −1 and define the space 𝐵𝑝 by the requirement 𝑁−1

𝑝

|𝑓‖𝐵𝑝 = ∑ |𝑓(𝑗) (0)|𝑝 + ∫ |𝑓(𝑁) (𝑧)|𝑝 (1 − |𝑧|2 )𝑁𝑝−2 𝑑𝐴(𝑧) < ∞. 𝑗=0

(9.29)

𝔻 𝑝

It turns out that the norm of 𝐵 (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 [27] for 𝑝 = 1, and [527] for 𝑝 < 1. 𝑝,𝑝 𝑝,𝑝 The space 𝐵𝑝 (0 < 𝑝 < ⬦) coincides with the space B1/𝑝 ≃ 𝐻Λ 1/𝑝 and therefore it can be renormed in several ways. One of them is a special case of Theorem 5.15. Theorem 9.17. Let 0 < 𝑝 < ⬦. A function 𝑓 ∈ 𝐻(𝔻) belongs to 𝐵𝑝 if and only if 𝑓 ∈ 𝐻𝑝 and ∞

∑ 𝐸𝑛(𝑓)𝑝𝑝 < ∞. 𝑛=0

If 𝑝 = 2, then 𝐵 it coincides with the Dirichlet space D. The norm in D is given by ‖𝑓‖2D = |𝑓(0)|2 + D(𝑓), where 𝑝

∞

̂ 2. − |𝑓󸀠 |2 𝑑𝐴 = ∑ 𝑛|𝑓(𝑛)| D(𝑓)2 = ∫ 𝑛=1

𝔻

It is useful to check the validity of the preceding theorem in the case of D. Concerning deeper properties of D, see Carleson [87] (representation theorems), Nagel–Rudin–Shapiro [339] (tangential limits). Krotov [279] extended the results of [339] to the Hardy–Sobolev spaces on the ball; see also Twomey [494] and [94, Sec­ tion 2.6]. For further information and references we refer to the recent papers [28, 145]. Here we record Carleson’s representation formula in a simplified form. Let 𝑓 = 𝐵𝑆𝐹, where 𝐹 ∈ 𝐻2 , 𝐵 is the Blaschke product with the zeros 𝑧𝑛 , 𝑛 ≥ 1, and 𝑆 a singular inner function generated by the singular measure 𝑑𝜎. Then ∞

1 − |𝑧𝑛|2 𝑑𝜎(𝜉) + 2∫ ) |𝑑𝜁|. 2 |𝜁 − 𝜉|2 𝑛=1 |𝜁 − 𝑧𝑛 |

− |𝐹(𝜁)|2 ( ∑ D(𝑓)2 = D(𝐹)2 + ∫ 𝕋

(9.30)

𝕋

Richter and Sundberg [414] extended this formula to a class of weighted Dirich­ let spaces; see also PhD thesis of Chacon Perez [90]. Is there something similar for invariant Besov spaces?

286 | 9 Lipschitz spaces of higher order The expression inside the brackets in (9.30) occurs in the problem of the nontan­ gential derivative of an inner function. Generalizing and improving two old results of Riesz and Frostman, Ahern and Clark [9, 10] proved the following theorem. Theorem (Ahern–Clark). Let 𝜑(𝑧) = 𝐵(𝑧) exp (−− ∫ 𝕋

𝜁+𝑧 𝑑𝜈(𝜁)) , 𝜁−𝑧

𝑧 ∈ 𝔻,

(9.31)

where 𝐵 is Blaschke product with zeros 𝑧𝑛, 𝑛 ≥ 1, and 𝑑𝜈 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∫ 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 with ∫(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 some inner function 𝐼. By Theorem 9.4, 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 |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 9.20. In a similar way we prove that |𝑔∗ | ∈ Λ𝑝,𝑞 and 𝛼 P[|𝑔∗ |](𝑟𝜁) − |𝑔(𝑟𝜁)| ≤ P[|𝐹∗ |](𝑟𝜁) − |𝐹(𝑟𝜁)|. The latter can be expressed as 0 ≤ |𝐹(𝑟𝜁)| − |𝑔(𝑟𝜁)| ≤ P[|𝑓∗ |](𝑟𝜁) − P[|Ψ∗ |](𝑟𝜁),

where Ψ = 𝐼𝑔.

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 9.20. Since 𝑓 = 𝐼𝑔/(1/𝐺), it remains to prove that 1/𝐺 ∈ 𝐻Λ 𝜔 ; this 󸀠 󸀠 2 󸀠 holds because |(1/𝐺) | = |𝐺 |/|𝐺| ≤ |𝐺 |. The result follows. Theorem 9.20 gives in particular another equivalent quasinorm on 𝐵𝑝 .

290 | 9 Lipschitz spaces of higher order Theorem 9.21 (Böe [73]). A function 𝑓 ∈ 𝐻𝑝 belongs to 𝐵𝑝 (1 < 𝑝 < ⬦) if and only if 𝐵𝑝,1 (|𝑓∗ |) < ∞ and 1/𝑝 𝑝

𝐵𝑝,2 (𝑓) := (∫ (𝑃[|𝑓∗ |](𝑧) − |𝑓(𝑧)|) 𝑑𝜏(𝑧))

< ∞.

𝔻

We have 𝐵𝑝 (𝑓) ≍ 𝐵𝑝,1 (|𝑓∗ |) + 𝐵𝑝,2 (𝑓).

9.6.1 Division and multiplication by inner functions As a consequence of Theorem 9.18, we have essentially a special case of a fact which was already noted before. Corollary 9.4. If 𝜔 is a regular majorant, 𝑝 ≥ 1 and 𝑞 ≥ 1, then the space 𝐻Λ𝑝,𝑞 𝜔 has the f -property. 𝑝 𝑝 Proof. Since 𝑓𝐼 ∈ 𝐻Λ𝑝,𝑞 𝜔 ⊂ 𝐻 , we have 𝑓∗ ∈ 𝐿 (𝕋). By Smirnov’s theorem, this 𝑠 𝑝 and the hypothesis 𝑓 ∈ 𝐻 imply 𝑓 ∈ 𝐻 . Now the desired conclusion follows from Theorem 9.18 and the relation {|𝑓𝜁 𝐼𝜁 | − |𝐼𝑟 𝑓𝑟 |}+ = {|𝑓𝜁 | − |𝑓𝑟 𝐼𝑟 |}+ ≥ {|𝑓𝜁 | − |𝑓𝑟 |}+ .

Corollary 9.5. If 𝜔 is a regular majorant, 𝑝 ≥ 1, 𝑞 ≥ 1, 𝐼 an inner function, and 𝑓 ∈ 𝐻𝑝 𝑝,𝑞 be such that |𝑓∗ | ∈ Λ𝑝,𝑞 𝜔 (𝕋). Then 𝑓𝐼 ∈ 𝐻Λ 𝜔 if and only if 𝑞

the function 𝐹4 (𝑟) = ‖ |𝑓𝑟 |(1 − |𝐼𝑟 |)‖𝑝 /𝜔(1 − 𝑟) belongs to 𝐿 −1 (0, 1). Proof. This is an immediate consequence of Theorem 9.18, (a)⇔(c), and the relation |(𝑓𝐼)∗ | − |𝑓𝑟 𝐼𝑟 | = |𝑓∗ | − |𝑓𝑟 | + |𝑓𝑟 |(1 − |𝐼𝑟 |). In the case 𝑞 = ∞ and 𝑝 > 0, the relation (A𝑝) ⇔ (C𝑝) of Theorem 9.18 holds under 𝑝,𝑞 the hypothesis that 𝜔 is only a fast majorant such that B𝜔,1 ≠ 0, which can be proved by a modification of the proof of Theorem 8.13. This is true, doubtless, for all 𝑞, but the author did not persist in finding a proof. There is still an interesting special case in which this does hold. 𝑞

Theorem 9.22. If 𝑝 > 0, 𝑞 > 0, and 𝜔 a fast majorant such that 𝑡/𝜔(𝑡) ∈ 𝐿 +0 . Then, an 𝑝,𝑞 inner function 𝐼 belongs to 𝐻Λ 𝜔,1 if and only if the function ‖1 − |𝐼𝑟 | ‖𝑝 /𝜔(1 − 𝑟), 0 < 𝑟 < 1, 𝑞 belongs to 𝐿 −1 . In particular, taking 𝜔(𝑡) = 𝑡 and 𝑞 = ∞, we see that 𝐼 belongs to the 𝑝 Hardy–Sobolev space 𝑆1 (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−𝑟2 ). In the other direction, we use that 1−|𝐼(𝑟𝜁)| ≤ |𝐼(𝜁)−𝐼(𝑟𝜁)| and the analytic version of Theorem 9.13, which holds for all 𝑝 > 0.

9.6 BMO-type characterizations of Lipschitz spaces |

291

Further notes and results Theorem 9.1 solves a problem posed by Shields and Williams [443, Problem C] and provides more information. Theorem 9.5 is essentially due to them [444], although they proved it under the hypothesis that 𝜓 is slow and 𝜓 is convex, or 𝜓 is concave, or |Δ2 𝜓(𝑛)| ≤ −𝑐

Δ1 𝜓(𝑛) , 𝑛

𝑛 ≥ 1.

(SW)

However, the proofs are very similar. Section 9.2 follows the author’s paper [362]. Proposition 9.1 (for analytic functions) has recently been generalized by Kolomoit­ sev [271] 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 [364]. Some particular cases of Theorem 9.9 were previously obtained by Janson [225] and Blasco–de Souza [68, 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 3.11. 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 when 𝜔(𝑡) = 𝑡𝛼 (0 < 𝛼 < 1) all the results of Subsection 9.6.1, except Theorems 9.18 and 9.22, were proved by Böe [73]. However, the order of proving in [73] differs from our one: using Theorem 2.1 of [137], Böe proved his version of our Theo­ rem 9.20, and then deduced the other results from this one. The first results such as Theorem 9.19 were proved by Richter and Shields [413] (Dirichlet space), Aleman [19], and Dyakonov [138]; see also Walsch [499]. Ahern [4, Theorem 6] proved Theorem 9.22 when 𝜔(𝑡) = 𝑡𝛼 , 0 < 𝛼 < 1, and then used it and his Theorem 2.25 to prove Corollary 5.5. 9.1. For a slow majorant 𝜓 defined on [1, ∞), let 𝑡

̃ = 1+∫ 𝜓(𝑡) 1

𝜓(𝑥) 𝑑𝑥, 𝑥

𝑡 ≥ 1.

󸀠

Theorem 1 of [444] together with our proof Theorem 9.5 gives the following fact, which was proved in [444] under the hypothesis (SW) above. Theorem. Let 𝜓 be a slow majorant. (i) If 𝑢 ∈ ℎ𝑝 (𝜓), where 𝑝 ∈ {1, ∞}, then 𝑢̃ ∈ ℎ𝑝 (𝜓)̃ and (ii) there is a function 𝑢 ∈ ℎ𝑝 (𝜓) such that ̃( 𝑀𝑝 (𝑟, 𝑢)̃ ≥ 𝑐 𝜓

1 ). 1−𝑟

This shows again (take 𝜓(𝑡) ≡ 1) that the Riesz conjugate functions theorem does not hold for 𝑝 ∈ {1, ∞}.

292 | 9 Lipschitz spaces of higher order Assertion (i) is easy to prove and is extended to the case 𝑝 < 1: 1 1 ̃( )) 󳨐⇒ 𝑀𝑝𝑝 (𝑟, 𝑢)̃ = 𝑂 (𝜓 )) . 1−𝑟 1−𝑟 This generalizes the second inequality of Theorem 1.15. 𝑀𝑝𝑝 (𝑟, 𝑢) = 𝑂 (𝜓 (

9.2. Concerning Theorem 9.4, we note that in the case 𝑛 = 1 there is a stronger result [42, Theorem 7]: Theorem (Bary–Stechkin). Let 𝜔 be a majorant of order 1. Then the Hilbert operator maps Λ 𝜔 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 multipliers²: 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 𝜔 is “homogeneous”, the set of these multipliers, denoted by 𝑋∗ = (𝑋, 𝐻∞ ), is identical to 𝑋𝐴 provided 𝑞 ≤ ⬦. The proof of Theorem 5.11 can easily ∗ −𝑛 be modified to show that 𝑔 ∈ (B𝑝,𝑞 𝜔 ) if and only if the sequence 𝜔(2 )‖𝑉𝑛 ∗ 𝑓‖(𝐻𝑝 )∗ 󸀠

belongs to ℓ𝑞 . Since 󸀠

‖𝑉𝑛 ‖(𝐻𝑝 )∗

{2𝑛(1−1/𝑝 ) = 2𝑛/𝑝 , 𝑝 ≥ 1, ≍ { 𝑛(1/𝑝−1) 𝑛/𝑝 2 ‖𝑉𝑛 ‖∞ ≍ 2 , 𝑝 < 1, {

∞ −𝑛 𝑛/𝑝 (see Lemma 5.10) we see that B𝑝,𝑞 , 𝑛 ≥ 0, 𝜔 ⊂ 𝐻 if and only if the sequence 𝜔(2 )2 󸀠 𝑞 1/𝑝 belongs to ℓ . The latter can be stated in the form: the function 𝜔(𝑡)/𝑡 , 0 < 𝑡 < 1, 𝑞󸀠 ∞ belongs to 𝐿 +0 . In particular, if 𝑞 > 1, then 𝐻Λ𝑝,𝑞 𝛼 ⊂ 𝐻 (𝛼 > 0) if and only if 𝛼 > 1/𝑝, whilst 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 9.2, it should be known that there is a reach theory concern­ ing 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 [223]. “Jackson’s theorem” usually means the validity of the inequality 𝐸𝑘 (𝑔)𝑋 ≤ 𝐶𝑋,𝑛 𝜔𝑛 (𝑔, 1/𝑘)𝑋 ,

𝑘 ≥ 1,

(9.34)

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 were treated by Stechkin [457]. Further his­ torical and mathematical facts can be found and Bary and Stechkin’s paper [42] and, the recent monographs DeVore–Lorents [112] and Stepanets [467]. Storozhenko [476, 478] proved that (9.34) continues to be true for 𝑋 = 𝐻𝑝 , i.e. we have 𝐸𝑘 (𝑓)𝑝 ≤ 𝐶𝑝,𝑛 𝜔𝑛 (𝑓∗ , 1/𝑘)𝑝 ,

𝑛 ≥ 1, 𝑘 ≥ 1, 0 < 𝑝 < ⬦.

2 A theory of multipliers will be considered in Chapter 11.

(9.35)

9.6 BMO-type characterizations of Lipschitz spaces | 293

It is interesting that this inequality for 𝑛 = 1 together with the relations (9.23) only gives 𝐸𝑘 (𝐴 𝜆 )𝑝 ≤ 𝐶𝑝 𝜉𝑘,𝑝 (𝑘 ≥ 2), where 𝜉𝑘,𝑝 = 𝑘−1/2𝑝 for 𝑝 > 1/2, and 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 (𝐴 𝜆 )𝑝 ≤ 𝐶𝑝 𝑘−1/2𝑝 for all 𝑘 and 𝑝, which can also be deduced from (9.35) and (9.23) by taking sufficiently large 𝑛 for a fixed 𝑝. 9.5 (Several variables). The first results on Lipschitz (or Besov) spaces on the real sphere were obtained by Greenwald [173, 174], who considered some tangential moduli of continuity of first and second order. Lizorkin and Nikol’sk˘ı [306, 346] used moduli of continuity of arbitrary order and characterized these spaces via best approximation by spherical polynomials. In [230], the Besov spaces on the sphere are defined by us­ ing radial derivatives. Our Theorem 9.13 extends to several variables [358]. The theory of Lipschitz spaces on the complex ball is simpler; the first results were given by Kwon et al. [287]; further information and results can be found in [91, 286], and [234]³ and [381].

3 The author of the papers 11–13 in the list of references is Pavlović, not Oswald.

10 One-to-one mappings In this chapter, we prove some fundamental theorems of theory of univalent functions, due to Prawitz, Bieberbach, and Köbe, with applications to the problem of member­ ship of univalent functions to some function classes. In Section 10.2, 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 the most interesting cases is when 𝑋 = 𝐻𝑝 and 𝑌 = B𝑝,𝑝 , 𝑝 > 0 – a recent paper of Astala and Koskela enables a proof considerably shorter and clearer than the original one. Section 10.3 contains a characterization of the boundary behavior of a harmonic quasiconformal homeomor­ phism of the unit disk. A small piece of the Astala–Koskela theory on 𝐻𝑝 -classes of quasiconformal (harmonic) mappings is presented in the last section.

10.1 Integral means of univalent functions Recall that we defined U to be the set of all one-to-one analytic functions from 𝔻 into ℂ. Theorem 10.1 (Prawitz [404]). Let 𝑓 ∈ U, 𝑓(0) = 0. Then for every 𝑝 > 0 the function − |𝑓(𝑟𝜁)|−𝑝 |𝑑𝜁|, 𝐽𝑝 (𝑟) = 𝐽𝑝 (𝑟, 𝑓) = ∫

0 < 𝑟 < 1,

𝕋

is decreasing. What is interesting here is 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 2.2. Proof. We have 2𝜋𝐽 󸀠𝑝 (𝑟) = −𝑝 ∫ |𝑓(𝑟𝜁)|−𝑝−2 Re {𝑓(𝑟𝜁)𝑓󸀠 (𝑟𝜁)𝜁} |𝑑𝜁| 𝕋

= −(𝑝/𝑟) Im ∫ |𝑓(𝜁)|−𝑝−2 𝑓(𝜁)𝑓󸀠 (𝜁) 𝑑𝜁 |𝜁|=𝑟

= −(𝑝/𝑟) Im ∫ |𝑤|−𝑝−2 𝑤̄ 𝑑𝑤 𝛤𝑟

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)̄ = − |𝑤|−𝑝−2 , 𝜕𝑤̄ 2

10.1 Integral means of univalent functions

| 295

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 𝐽 󸀠𝑝 (𝑟) = −(𝑝/𝑟)𝑅−𝑝 − (𝑝2 /2𝜋𝑟) ∬ |𝑤|−𝑝−2 𝑑𝐴(𝑤). Ω𝑟,𝑅

Letting 𝑅 tend to ∞ we get 𝐽 󸀠𝑝 (𝑟) = −(𝑝2 /2𝜋𝑟) ∬ |𝑤|−𝑝−2 𝑑𝐴(𝑤), Ω𝑟

where Ω𝑟 is the “exterior” of the curve 𝛤𝑟 . This concludes the proof. Exercise 10.1. If 𝑓 is univalent in 𝔻, 𝑓(0) = 0, then 𝑝 𝑝 𝑀(𝑟)−𝑝 ≤ −𝐽𝑝󸀠 (𝑟) ≤ 𝑚(𝑟)−𝑝 , 𝑟 𝑟 where 𝑀(𝑟) = max|𝑧|=𝑟 |𝑓(𝑧)|, 𝑚(𝑟) = min|𝑧|=𝑟 |𝑓(𝑧)|.

10.1.1 Distortion theorems Theorem 10.2 (Bieberbach). If 𝑓 ∈ U, then |𝑓󸀠󸀠 (0)| ≤ 4|𝑓󸀠 (0)|. Proof [219]. 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 + 𝑧2 ℎ(𝑧)) , where ℎ is analytic in 𝔻. By Theorem 10.1, case 𝑝 = 1, the function 𝐽1 (𝑟, 𝑓) = 𝑟−1 (1 + |𝑓󸀠󸀠 (0)/4|2 𝑟2 + 𝑟4 𝑀22 (𝑟, ℎ)) is decreasing. Hence the function 𝑟−1 + |𝑓󸀠󸀠 (0)/4|2 𝑟 = 𝐽1 (𝑟, 𝑓) − 𝑟3 𝑀22 (𝑟, ℎ) is decreasing, i.e. (𝑑/𝑑𝑟)(𝑟−1 + |𝑓󸀠󸀠 (0)/4|2 𝑟) ≤ 0, and hence |𝑓󸀠󸀠 (0)/4| ≤ 1. Theorem 10.3 (Köbe 1/4-theorem). If 𝑓 ∈ U, then 𝑓(𝔻) contains the disk of radius |𝑓󸀠 (0)|/4 centered at 𝑓(0).

296 | 10 One-to-one mappings Proof [219]. 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 |𝑓󸀠󸀠 (0) + 2𝑓󸀠 (0)2 /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 10.4. If 𝑓 is a conformal mapping of 𝐷 ⊂ ℂ onto 𝐺, then |𝑓󸀠 (𝑧)| 𝛿𝐺 (𝑓(𝑧)) ≤ ≤ 4|𝑓󸀠 (𝑧)|, 4 𝛿𝐷 (𝑧)

(10.1)

𝑧 ∈ 𝐷.

Proof. The first inequality in (10.1) is just a reformulation of Theorem 10.3: the set 𝑓(𝔻) contains the disk of radius |𝑓󸀠 (𝑧)|𝛿𝐷 (𝑧)/4. Applying the first inequality to the inverse function we get the second inequality in (10.1). Theorem 10.5. If 𝑓 ∈ U, then there exists a point 𝑤 belonging to the disk of radius |𝑓󸀠 (0)| centered at 𝑓(0) and such that 𝑤 ∈ ̸ 𝑓(𝔻). Proof. Let 𝑓(0) = 0, 𝑓󸀠 (0) = 1. Assume that 𝑓(𝔻) contains the closed unit disk. 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, i.e. |𝑓󸀠 (0)| ≥ 𝑅 > 1, a contra­ diction. Exercise 10.2. Theorems 10.3 and 10.5 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 10.6 (Köbe distortion theorem). If 𝑓 ∈ U, then |𝑓󸀠 (𝑧)| 1+𝑟 1−𝑟 ≤ ≤ 󸀠 , 3 (1 + 𝑟) |𝑓 (0)| (1 − 𝑟)3

|𝑧| = 𝑟.

(10.2)

Proof. If we apply Bieberbach’s theorem to the function 𝑔(𝑤) = 𝑓(𝜎𝑧 (𝑤)), where 𝑧−𝑤 𝜎𝑧 (𝑤) = 1− , we get ̄ 𝑧𝑤 󵄨󵄨 󸀠󸀠 󵄨 2𝑟2 󵄨󵄨󵄨 󵄨󵄨 𝑓 (𝑧) 4𝑟 󵄨󵄨𝑧 󸀠 󵄨 − 󵄨󵄨 𝑓 (𝑧) 1 − 𝑟2 󵄨󵄨󵄨 ≤ 1 − 𝑟2 . 󵄨 󵄨 𝑓󸀠󸀠

𝜕 log |𝑓󸀠 | = Re(𝑧 𝑓󸀠 ) we see that Since 𝑟 𝜕𝑟

𝜕 2𝑟 + 4 2𝑟 − 4 log |𝑓󸀠 | ≤ ≤ . 2 1−𝑟 𝜕𝑟 1 − 𝑟2 Now the desired result is obtained by integration.

10.1 Integral means of univalent functions

| 297

As a consequence of the proof we have two assertions. Corollary 10.1. If 𝑓 ∈ U, then 󵄨󵄨 󸀠󸀠 󵄨󵄨 󵄨 𝑓 (𝑧) 󵄨 (1 − |𝑧|2 ) 󵄨󵄨󵄨󵄨 󸀠 󵄨󵄨󵄨󵄨 ≤ 6. 󵄨󵄨 𝑓 (𝑧) 󵄨󵄨 Corollary 10.2. If 𝑓 ∈ U, then 󵄨󵄨 𝑓(𝑧) − 𝑓(0) 󵄨󵄨 |𝑓󸀠 (0)| |𝑓󸀠 (0)| 󵄨󵄨 󵄨󵄨 ≤ ≤ 󵄨 󵄨 󵄨󵄨 (1 − |𝑧|)2 , (1 + |𝑧|)2 󵄨󵄨󵄨 𝑧 󵄨

𝑧 ∈ 𝔻.

(10.3)

Proof. The right-hand side inequality is obtained from (10.2) 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)| = ∫ |𝑑𝑤|. ℓ

Hence, by the change 𝑤 = 𝑓(𝜁), we get |𝑓(𝑧)| = ∫ |𝑓󸀠 (𝜁)| |𝑑𝜁| (where 𝐿 = 𝑓−1 (ℓ)) 𝐿

≥∫ 𝐿

|𝑧| 1 − |𝜁| 1 − |𝜁| |𝑑𝜁| ≥ ∫ 𝑑|𝜁| = . (1 + |𝜁|)3 (1 + |𝜁|)3 (1 + |𝑧|)2 𝐿

This completes the proof. Corollary 10.3. If 𝑓 ∈ U and 𝑓(𝑧) ≠ 0 for all 𝑧 ∈ 𝔻, then (1 − |𝑧|)2 /5 ≤

|𝑓(𝑧)| ≤ 5(1 − |𝑧|)−2 . |𝑓(0)|

(10.4)

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 disk 𝐷 of radius |𝑓󸀠 (0)|/4 centered at 𝑓(0). Since 0 ∈ ̸ 𝐷, we have |𝑓󸀠 (0)|/4 ≤ |𝑓(0)|, which along with the preced­ ing inequality proves part of (10.4). Now we applied this result to the function 𝑓 ∘ 𝜎𝑎 to get |𝑓(𝜎𝑎 (𝑧))| ≤ 5|𝑓(𝑎)|/(1 − |𝑧|)2 . Hence, taking 𝑧 = 𝑎, we get |𝑓(0)| ≤ 5|𝑓(𝑎)|/(1 − |𝑎|)2 , which completes the proof. Corollary 10.4. If 𝑓 ∈ U, then 1 − |𝑧| 󵄨󵄨󵄨󵄨 𝑓(𝑧) − 𝑓(0) 󵄨󵄨󵄨󵄨 1 + |𝑧| ≤󵄨 . 󵄨≤ 1 + |𝑧| 󵄨󵄨󵄨 𝑧𝑓󸀠 (𝑧) 󵄨󵄨󵄨 1 − |𝑧|

(10.5)

298 | 10 One-to-one mappings Proof. Let 𝑔(𝑤) = 𝑓(𝜎𝑧 (𝑤)), 𝑤, 𝑧 ∈ 𝔻. Applying Corollary 10.2 to 𝑔 we obtain |𝑓󸀠 (𝑧)|(1 − |𝑧|2 ) |𝑓(𝜎𝑧 (𝑤)) − 𝑓(𝜎𝑧 (0))| |𝑓󸀠 (𝑧)|(1 − |𝑧|2 ) ≤ ≤ . (1 + |𝑤|)2 |𝑤| (1 − |𝑤|)2 Now take 𝑤 = 𝑧 to conclude the proof. Remark 10.1. If we rewrite (10.5) as 󵄨 󵄨 1 − |𝑧| 󵄨󵄨󵄨 𝑧𝑓󸀠 (𝑧) 󵄨󵄨󵄨 1 + |𝑧| 󵄨󵄨 ≤ ≤ 󵄨󵄨󵄨 , 1 + |𝑧| 󵄨󵄨 𝑓(𝑧) − 𝑓(0) 󵄨󵄨󵄨 1 − |𝑧| and then multiply this inequality by (10.3) we return to (10.2).

10.2 Membership of univalent functions in some function classes 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 2.2). However, the proof of Theo­ rem 10.1 gives additional information on 𝐼𝑝 (𝑟), namely: If 𝑓 ∈ U, 𝑓(0) = 0 and 𝑓󸀠 (0) = 1, then 𝑝 𝑝2 ∬ |𝑤|𝑝−2 𝑑𝐴(𝑤). 𝐼𝑝󸀠 (𝑟) = 𝑅𝑝 − 𝑟 2𝜋𝑟 Ω𝑟,𝑅

This implies that 𝐼𝑝󸀠 (𝑟) ≤ (𝑝/𝑟)𝑀(𝑟)𝑝 . Combining this with Corollary 10.2 we get the following theorem [404]: Theorem 10.7 (Prawitz). If 𝑓 ∈ U, 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 Hardy 2 -theorem 2.16. 𝑀∞ 1

𝑝 (𝑟, 𝑓) 𝑑𝑟 Theorem 10.8 (Hardy–Prawitz). A function 𝑓 ∈ U is in 𝐻𝑝 if and only if ∫0 𝑀∞ < ∞. Moreover 1

‖𝑓‖𝑝𝑝

𝑝 ≍ ∫ 𝑀∞ (𝑟, 𝑓) 𝑑𝑟,

𝑓 ∈ U,

0

where the equivalence constants are independent of 𝑓.

10.2 Membership of univalent functions in some function classes

| 299

As a consequence of this fact and Corollary 10.2 we have Corollary 10.5. If 𝑓 ∈ U and 0 < 𝑝 < 1/2, then 𝑓 ∈ 𝐻𝑝 . Also 𝑀1/2 (𝑟, 𝑓) ≤ 𝐶 (log

2 2 ) 1−𝑟

and 𝑀𝑝 (𝑟, 𝑓) ≤ 𝐶(1 − 𝑟)1/𝑝−2 ,

𝑝 > 1/2,

where 𝐶 is independent of 𝑟. We have proved that part of the following theorem holds for every analytic function. Theorem 10.9 ([172, Theorem 2(a)]). Let 0 < 𝑝 < 𝑞 < ⬦, and 𝑓 ∈ U. Then 1

‖𝑓‖𝑝𝑝 ≍ ∫(1 − 𝑟)−𝑝/𝑞 𝑀𝑞𝑝 (𝑟, 𝑓) 𝑑𝑟. 0

Proof. By Theorems 10.8 and 4.4, 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 10.7. Taking 𝑞 = 2 in the preceding theorem we obtain the case 𝑝 ≤ 2 of the following result [215]; see (5.29) and (5.30). Theorem 10.10 (Holland–Twomey). For a function 𝑓 ∈ U, with 𝑓(0) = 0, the relation 1/𝑝

1 𝑝/2

‖𝑓‖𝑝 ≍ (∫ 𝐴(𝑟, 𝑓)

𝑑𝑟)

,

0 < 𝑝 < ⬦,

0

holds, where the equivalence constants depend only on 𝑝. Proof. We have only to prove the inclusion U ∩ 𝐻𝑝 ⊂ AR𝑝 for 2 < 𝑝 < ⬦. We come back to the proof of Prawitz’s theorem (with the notation used there) and then follow Spencer [456]. 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 2

≥ 𝑝 (∫ |𝑤| 𝑑(arg 𝑤)) 𝛤𝑟

1−𝑝/2

(∫ 𝑑(arg 𝑤)) 𝛤𝑟 𝑝/2

1−𝑝/2

= 𝑝(2𝜋)

(∫ Im(𝑤̄ 𝑑𝑤)) 𝛤𝑟

That is all.

= 𝑝(2𝜋)1−𝑝/2 2𝑝/2 𝐴(𝑟, 𝑓)𝑝/2 .

300 | 10 One-to-one mappings According to (5.28) and Lemma 3.8, we can rewrite Theorem 10.10 as follows. Theorem 10.11. For a function 𝑓 ∈ U the relation ‖𝑓‖𝑝𝑝

∞

𝑝/2 𝑛(𝑝/2−1)

≍ ∑2

̂ 2) ( ∑ |𝑓(𝑘)|

𝑛=0

,

0 < 𝑝 < ⬦,

𝑘∈𝐼𝑛

holds, where the equivalence constants depend only on 𝑝. Combining this theorem with the Hardy–Prawitz theorem we obtain another theorem from [215]. Let ∞

𝑛 ̂ . 𝑃(𝑟, 𝑓) = ∑ |𝑓(𝑛)|𝑟 𝑛=0

Theorem 10.12 (Holland–Twomey). For a function 𝑓 ∈ U the following relation holds: 1

‖𝑓‖𝑝𝑝 ≍ ∫ 𝑃(𝑟, 𝑓)𝑝 𝑑𝑟,

0 < 𝑝 < ⬦,

0

where the equivalence constants depend only on 𝑝. 𝑝

Proof. The inequality “‖𝑓‖𝑝 ≤ 𝐶 . . . ” follows from the Hardy–Prawitz theorem imme­ diately. On the other hand, we have 1

𝑝

∞

𝑝

−𝑛

∫ 𝑃(𝑟, 𝑓) 𝑑𝑟 ≍ ∑ 2 𝑛=0

0

̂ ( ∑ |𝑓(𝑘)|) 𝑘∈𝐼𝑛 𝑝/2

∞

−𝑛

≤ ∑2 𝑛=0 ∞

( ∑ 1) 𝑘∈𝐼𝑛

𝑝/2

̂ 2) ( ∑ |𝑓(𝑘)| 𝑘∈𝐼𝑛 𝑝/2

−𝑛 𝑛𝑝/2

≤ 𝐶∑2 2 𝑛=0

2

̂ ( ∑ |𝑓(𝑘)| )

.

𝑘∈𝐼𝑛

Now the result follows from Theorem 10.11. Exercise 10.3. Let 1 ≤ 𝑝 ≤ 2. A function 𝑓 ∈ U belongs to 𝐻𝑝 if and only if ∑(𝑛 + ̂ 𝑝 < ∞. The equivalence does not hold for large 𝑝 > 2; see [215]. 1)𝑝−2 |𝑓(𝑛)| Exercise 10.4. Let 0 < 𝑝 < 1. Then a function 𝑓 ∈ U belongs to 𝐻𝑝 if and only if ∞

̂ 𝑝 < ∞. ∑ 2𝑛(𝑝−1) sup |𝑓(𝑘)| 𝑛=0

𝑘∈𝐼𝑛

𝑠,𝑞

We know that 𝐻𝛼𝑝,𝑞 ⊂ 𝐻𝛽 , where 𝛽 = 𝛼 + 1/𝑝 − 1/𝑠 and 𝑝 < 𝑠 (Theorem 3.14). However: 𝑠,𝑞

Theorem 10.13. Let 𝛼 > 0. With the above notation we have 𝐻𝛼𝑝,𝑞 ∩ U = 𝐻𝛽 ∩ U.

10.2 Membership of univalent functions in some function classes

| 301

𝑠,𝑞

Proof. We have to prove that if 𝑓 ∈ 𝐻𝛽 ∩ U, then 𝑓 ∈ 𝐻𝛼𝑝,𝑞 . Let 𝑠 = ∞. Applying Prawitz’s theorem to the function 𝑓 − 𝑓(0) and using the maximum modulus principle 1 𝑝 𝑝 we get ‖𝑓‖𝑝 ≤ 𝐶 ∫0 𝑀∞ (𝜌, 𝑓) 𝑑𝜌 and then 1 𝑝 𝑀𝑝𝑝 (𝑟, 𝑓) ≤ 𝐶 ∫ 𝑀∞ (𝜌𝑟, 𝑓) 𝑑𝜌. 0

Now the result in the case 𝑠 = ∞ is obtained by the “fractional integration proposition” 𝑝 (Proposition 3.6); the subharmonicity of the function 𝑧 󳨃→ 𝑀∞ (𝜌|𝑧|, 𝑓) is to be used. 𝑠,𝑞 If 𝑝 < 𝑠 < ∞, then the result follows preceding case and the inclusions 𝐻𝛼𝑝,𝑞 ⊂ 𝐻𝛽 ⊂ ∞,𝑞 𝐻𝛾 , where 𝛽 = 𝛼 + 1/𝑝 − 1/𝑠 and 𝛾 = 𝛽 + 1/𝑠 = 𝛼 + 1/𝑝.

Littlewood–Paley theorem for univalent functions 𝑝

The Littlewood–Paley theorem states that 𝐻𝑝 ⊂ D𝑝−1 = B𝑝,𝑝 for 𝑝 ≥ 2, and that the reverse inclusion holds for 𝑝 < 2. However, as was proved in [41], we have: 𝑝

Theorem 10.14 (Baernstein–Girela–Peláez). If 0 < 𝑝 < ⬦, then U ∩ 𝐻𝑝 = U ∩ D𝑝−1, and moreover ‖𝑓‖𝑝 ≍ ‖𝑓‖D𝑝 , where the equivalence constants depend only on 𝑝. 𝑝−1

Before the proof, note an interesting consequence of this theorem, Corollary 10.5, and Theorem 5.8: Corollary 10.6 ([41]). If 𝑓(𝑧) = ∑ 𝑎𝑛 𝑧𝑘𝑛 is a univalent function with Hadamard gaps, then ∑ |𝑎𝑛 |𝑝 < ∞ for all 𝑝 > 0. The proof of Theorem 10.14 given in [41] is very complicated and depends on various techniques, which nevertheless are important because can be used in other situations, e.g. in proving an extension of a theorem of Pommerenke; see Note 10.1. We present a surprisingly simple proof, due to Astala and Koskela [32], that is based on a remarkable result of Jones (Lemma 10.2 below). 𝑝

Proof of Theorem 10.14. Let 𝑝 ≥ 2. The validity of the inclusion U ∩ 𝐻𝑝 ⊂ U ∩ D𝑝−1 follows Theorem 7.14 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

∫𝑟 0

1 −1

𝑝 𝑀∞ (𝑟, 𝑓) 𝑑𝑟

𝑝 ≤ 𝐶 ∫(1 − 𝑟)𝑝 𝑀∞ (𝑟, 𝑓󸀠 ) 𝑑𝑟, 0

and then the inequality 𝑝 (𝑟, 𝑓󸀠 ) ≤ 𝐶(1 − √𝑟)−1 𝑀𝑝𝑝 (𝑟, 𝑓) 𝑑𝑟. 𝑀∞

Combining these inequalities with Theorem 10.7, we conclude the proof for 𝑝 ≥ 2.

302 | 10 One-to-one mappings Let 0 < 𝑝 < 2. Assume first that 𝑓(𝑧) ≠ 0 for all 𝑧 ∈ 𝔻. Then, by Lemma 10.2 below, the measure 𝑑𝜇(𝑧) = |𝑓󸀠 (𝑧)|𝑝 |𝑓(𝑧)|−𝑝 (1 − |𝑧|2 )𝑝−1 𝑑𝐴(𝑧) (10.6) is Carleson and consequently ∫ |𝑓󸀠 (𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧) = ∫ |𝑓(𝑧)|𝑝 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 preceding result to the function 𝑓(𝑧) − 𝑤0 we find that ∫ |𝑓󸀠 (𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧) ≤ 𝐶𝑝 (‖𝑓‖𝑝𝑝 + |𝑓(0)|𝑝 + |𝑓󸀠 (0)|𝑝 ) ≤ 𝐶𝑝 ‖𝑓‖𝑝𝑝 , 𝔻

which proves the theorem.

Jones’ lemma The following fact is a very special case of Jones’ lemma [236]. Lemma (Jones). If 0 < 𝑝 < 2, 𝑓 ∈ U and 𝑓(𝑧) ≠ 0 for all 𝑧 ∈ 𝔻, then the measure 𝑑𝜇 defined by (10.6) is Carleson. Proof [32, Lemma 5.6]. We have ∫ |𝑓󸀠 (𝑧)|𝑝 |𝑓(𝑧)|−𝑝 (1 − |𝑧|2 )𝑝−1 𝑑𝐴(𝑧) 𝔻 𝑝/2 󸀠

2

−2

2 2𝜀/𝑝

≤ (∫ |𝑓 (𝑧)| |𝑓(𝑧)| (1 − |𝑧| )

𝑑𝐴(𝑧))

𝔻 (2−𝑝)/2 2 (𝑝−1−𝜀)2/(2−𝑝)

× (∫(1 − |𝑧| )

𝑑𝐴(𝑧))

𝔻 𝑝/2 󸀠

2

−2

2 2𝜀/𝑝

≤ 𝐶 (∫ |𝑓 (𝑧)| |𝑓(𝑧)| (1 − |𝑧| )

𝑑𝐴(𝑧))

=: 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 = 𝜌𝔻 ∩ 𝑓−1 (𝜌𝔻), and 𝐼2 (𝑓) over 𝐷2 = 𝜌𝔻 \ 𝑓−1 (𝜌𝔻), where 𝜌 = |𝑓(0)|. Then, by (10.4) 𝐼1 (𝑓) ≤ 𝐶1 ∫ |𝑓󸀠 (𝑧)|2 |𝑓(𝑧)|−2+𝜀/𝑝 |𝑓(0)|−𝜀/𝑝 𝑑𝐴(𝑧) 𝐷1

=

∫ 𝜌𝔻 ∩𝑓(𝜌𝔻)

|𝑤|−2+𝜀/𝑝 𝑑𝐴(𝑤) ≤ 𝐶2 ,

10.2 Membership of univalent functions in some function classes |

303

where 𝐶2 is independent of 𝑓. In a similar way we obtain 𝐼2 (𝑓) ≤ 𝐶3 , where 𝐶3 is independent of 𝑓. Consequently ∫ |𝑓󸀠 (𝑧)|𝑝 |𝑓(𝑧)|−𝑝 (1 − |𝑧|2 )𝑝−1 𝑑𝐴(𝑧) ≤ 𝐶𝑝 .

(10.7)

𝔻

We use this inequality to prove that 𝑑𝜇 is a Carleson measure, i.e. that ∫ |𝑓󸀠 (𝑧)|𝑝 |𝑓(𝑧)|−𝑝 (1 − |𝑧|2 )𝑝−1 |𝜎𝑎󸀠 (𝑧)| 𝑑𝐴(𝑧) ≤ 𝐶𝑝 .

(10.8)

𝔻

To do this we apply (10.7) to the function 𝑓 ∘ 𝜎𝑎 and after substitution 𝜎𝑎 (𝑧) = 𝑤 we get (10.8) with the same constant 𝐶𝑝 as in (10.7). (It is convenient to use the formula 1 − |𝑧|2 = (1 − |𝑤|2 )|𝜎𝑎󸀠 (𝑤)|.)

Univalent functions in the Bloch space and in BMOA As a consequence of Theorems 10.14 and 6.2 we have Theorem 10.15. Let 0 < 𝑝 < ⬦. A function 𝑓 ∈ U belongs to BMOA if and only if 𝐾𝑝 (𝑓) := sup ∫ |𝑓󸀠 (𝑧)|𝑝 (1 − |𝑧|2 )𝑝−1 |𝜎𝑎󸀠 (𝑧)| 𝑑𝐴(𝑧) < ∞, 𝑎∈𝔻

(10.9)

𝔻

1/𝑝

and we have 𝐾𝑝 (𝑓) ≍ ‖𝑓‖∗2 . The relation (10.9) 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 [402]. Corollary 10.7 (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 (10.1). Proof. Assuming that 𝑓 ∈ 𝐻(𝔻) we have ‖𝑓‖3∗2 ≤ 𝐶 sup ∫ |𝑓󸀠 (𝑧)|3 (1 − |𝑧|2 )2 |𝜎𝑎󸀠 (𝑧)| 𝑑𝐴(𝑧) 𝑎∈𝔻

𝔻

≤ 𝐶B(𝑓) sup ∫ |𝑓󸀠 (𝑧)|2 (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 10.15 (𝑝 < 2) and Jones’ lemma we have the following result from [38, 96].

304 | 10 One-to-one mappings Theorem 10.16 (Baernstein–Cima–Shober). If 𝑓 ∈ U and 𝑓(𝑧) ≠ 0 for all 𝑧 ∈ 𝔻, then log 𝑓 ∈ BMOA. Exercise 10.5. Let 𝑓 ∈ U, 0 < 𝑝 < ⬦, and 0 < 𝑞 ≤ 2. Then 𝑓 ∈ 𝐻𝑝 if and only if ∫ |𝑓|𝑝−𝑞 |𝑓󸀠 |𝑞 (1 − |𝑧|)𝑞−1 𝑑𝐴(𝑧) < ∞. 𝔻

For the case 𝑞 = 1, see [32, Corollary 5.4].

10.3 Quasiconformal harmonic mappings A mapping 𝑓 : 𝔻 󳨃→ ℂ is said to be quasiconformal (QC) if it is one-to-one, absolute continuous on straight lines, and satisfies the condition |𝑓󸀠 (𝑧)|2 ≤ 𝐾𝐽𝑓 (𝑧),

(10.10)

𝑧 ∈ 𝔻,

󸀠

where |𝑓 (𝑧)| = ‖𝑑𝑓(𝑧)‖, and 𝐽𝑓 is the Jacobian of 𝑓. The function 𝑓 is then called 𝐾quasiconformal (𝐾-QC); the smallest 𝐾 satisfying (10.10) is denoted by 𝐾𝑓 . Quasicon­ formal mappings are differentiable almost everywhere. If 𝑓 is in addition harmonic, then |𝑓󸀠 (𝑧)|2 = (|ℎ󸀠 (𝑧)| + |𝑔󸀠 (𝑧)|)2 , and 𝐽𝑓 (𝑧) = |ℎ󸀠 (𝑧)|2 − |𝑔󸀠 (𝑧)|2 , and 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.

10.3.1 Boundary behavior of QCH homeomorphisms of the disk Before stating the results, we note that if 𝑓 a QC-mapping from 𝔻 onto 𝔻, then it ex­ tends 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 [11]. Thus we have 𝑓(𝑒𝑖𝜃 ) = 𝑒𝑖𝜑(𝑡) ,

where 𝜑 is strictly increasing, continuous, and 𝜑(𝑡 + 2𝜋) − 𝜑(𝑡) ≡ 2𝜋.

(10.11)

10.3 Quasiconformal harmonic mappings |

305

Subsequently Ahlfors and Beurling [57] characterized the boundary mapping 𝑒𝑖𝜑 by the condition 𝜑(𝜃 + 2𝑡) − 𝜑(𝜃 + 𝑡) 1 ≤ ≤ 𝑀, 𝑡 > 0, 𝑀 𝜑(𝜃 + 𝑡) − 𝜑(𝜃) 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 (Mori [338]). 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),

(10.12)

where 𝐶 depends on |Φ(0)|; if Φ(0) = 0, then one can take 𝐶 = 16. A proof can be found in Ahlfors [13]. 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 shall use Mori’s theorem to characterize the boundary function of a QCH-map­ ping. As a consequence of the proof we show that a QCH-homeomorphism of 𝔻 satis­ fies the ordinary Lipschitz condition ((10.31)), which along with the Heinz inequality |𝑓󸀠 (𝑧)| ≥ 1/𝜋, valid for all 𝑓(0) = 0 [206] (see Theorem 10.20 below for a weaker vari­ ant), gives Theorem 10.17 (Pavlović [373]). If 𝑓 is a harmonic 𝐾-quasiconformal homeomorphism of 𝔻, then it is bi-Lipschitz, i.e. 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 We leave the deduction to the reader. It is known that a bi-Lipschitz mapping is qua­ siconformal. The QCH-analog of the Ahlfors–Beurling theorem reads Theorem 10.18 (Pavlović [373]). Let 𝑓 be a orientation preserving harmonic homeomor­ phism 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

𝐻(𝜑󸀠 )(𝜃) = −

1 𝜑󸀠 (𝜃 + 𝑡) − 𝜑󸀠 (𝜃 − 𝑡) 𝑑𝑡, ∫ 𝜋 2 tan(𝑡/2) +0

𝜃 ∈ ℝ,

306 | 10 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, 󸀠

ess sup 𝜑 < ∞, 󵄨 󵄨󵄨 𝜋 󸀠 󵄨󵄨 𝜑 (𝜃 + 𝑡) − 𝜑󸀠 (𝜃 − 𝑡) 󵄨󵄨󵄨 󵄨 󵄨󵄨 𝑑𝑡󵄨󵄨󵄨 < ∞. ess sup 󵄨󵄨 ∫ 󵄨󵄨 𝑡 𝜃∈ℝ 󵄨󵄨󵄨 󵄨󵄨 󵄨+0

(10.13) (10.14) (10.15)

The last condition is equivalent to the boundedness of 𝐻(𝜑󸀠 ) because 𝜑󸀠 ∈ 𝐿∞ and 1 1 − = O(𝑡2 ), 2 tan(𝑡/2) 𝑡

𝑡 → 0.

In [242], Kalaj considered harmonic mappings of 𝔻 onto a smooth convex do­ main 𝐺. Under the hypotheses of the Radó–Kneser–Choquet theorem 1.20, he ob­ tained the following result, which we state without proof. Theorem (Kalaj [242]). Let 𝛤 := 𝜕𝐺 ∈ 𝐶1,𝛼 for some 𝛼 ∈ (0, 1). Then the following condi­ tions are equivalent: (a) 𝑓 is quasiconformal. (b) 𝑓 is bi-Lipschitz in the Euclidean metric. (c) 𝛾 is absolutely continuous, 𝛾󸀠 ∈ 𝐿∞ , | ess inf 𝕋 𝛾󸀠 | > 0, and 𝐻(𝛾󸀠 ) ∈ 𝐿∞ . Although Kalaj also applies Mori’s theorem, his proof is not a complete imitation of that of Theorem 10.18. We note that “𝛤 ∈ 𝐶1,𝛼 ” means that the natural parametrization of 𝛤 is of class 𝐶1,𝛼 . It is not difficult to check that if 𝐺 = 𝔻, then the equivalence (a) ⇔ (c) reduces to Theorem 10.18 + 10.17. The author of Theorem 10.17 was motivated by the pioneering work of Martio [322]. Theorem (Martio [322]). The mapping 𝑓 = P[𝑒𝑖𝜑 ] is quasiconformal if 𝜑 ∈ 𝐶1 (ℝ), min 𝜑󸀠 > 0 and 𝜋

∫ 0

𝜔(𝑡) 𝑑𝑡 < ∞, 𝑡

(10.16)

where 𝜔(𝑡) = sup{ |𝜑󸀠 (𝑥) − 𝜑󸀠 (𝑦)| : |𝑥 − 𝑦| < 𝑡}. Condition (10.16), 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 which map 𝔻 into a relatively compact subset of the right halfplane. Having such an 𝐹, with Re 𝐹(0) = 1, we take 𝜑󸀠 (𝜃) = Re 𝐹∗ (𝑒𝑖𝜃 ), and since 𝜃

𝐻(𝜑󸀠 )(𝜃) = Im 𝐹∗ (𝑒𝑖𝜃 ), we see that the function 𝜑(𝜃) = ∫0 𝜑󸀠 (𝑡) 𝑑𝑡 satisfies the condition (b) of Theorem 10.18. 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.

10.3 Quasiconformal harmonic mappings

| 307

Before passing to the proof of Theorem 10.18, note that by a result of Lewy [292] (see Duren [131, Section 2.2]), the Jacobian of a univalent harmonic function is zero free in 𝔻, i.e. as we may assume, 2 ̄ 𝐽𝑓 (𝑧) = |𝜕𝑓(𝑧)|2 − |𝜕𝑓(𝑧)| > 0 (𝑧 ∈ 𝔻).

(10.17)

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 (10.17) as 󵄨󵄨󵄨 𝑔󸀠 (𝑧) 󵄨󵄨󵄨 󵄨󵄨󵄨 󸀠 󵄨󵄨󵄨 < 1 (𝑧 ∈ 𝔻). (10.18) 󵄨󵄨 ℎ (𝑧) 󵄨󵄨 󵄨 󵄨 Thus we consider those 𝑓 for which (10.18) can be improved to 󵄨󵄨 󸀠 󵄨󵄨 󵄨 𝑔 (𝑧) 󵄨 (10.19) 𝑘 = sup 󵄨󵄨󵄨󵄨 󸀠 󵄨󵄨󵄨󵄨 < 1. 𝑧∈𝔻 󵄨󵄨 ℎ (𝑧) 󵄨󵄨 This condition is very strong, as follows from Theorems 1.20 and 10.18.

Proof of Theorem 10.18 As mentioned after the formulation of the theorem, it suffices to consider conditions (10.13)–(10.15). 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 max­ imum 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, and shall, assume that 𝑓 = P[𝛾], where 𝛾 = 𝑒𝑖𝜑 and 𝜑 has the properties described in (10.11). In calculating the boundary values of the analytic functions ℎ󸀠 and 𝑔󸀠 it is useful to use the formulas 𝐷𝑓(𝑧) 1 ) (10.20) ℎ󸀠 (𝑧) = 𝜕𝑓(𝑧) = 𝑒−𝑖𝜃 (𝑅0 𝑓(𝑧) − 𝑖 2 𝑟 𝐷𝑓(𝑧) 1 ̄ ), (10.21) 𝑔󸀠 (𝑧) = 𝜕𝑓(𝑧) = 𝑒𝑖𝜃 (𝑅0 𝑓(𝑧) + 𝑖 2 𝑟 where 𝐷𝑓 = 𝜕𝑓/𝜕𝜃, 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 : 𝜋

1 𝐷𝑓(𝑟𝑒 ) = ∫ 𝑃(𝑟, 𝜃 − 𝑡) 𝑑𝛾(𝑡). 2𝜋 𝑖𝜃

−𝜋

(10.22)

308 | 10 One-to-one mappings 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

󵄨 󵄨 ∫ 󵄨󵄨󵄨󵄨𝑅0 𝑓(𝜌𝑒𝑖𝜃 )󵄨󵄨󵄨󵄨 𝑑𝜃 < ∞, sup − 𝜌 −1, which is possible because (𝛼−1)(2−𝛽) → 𝛼−1 > −1 as 𝛽 → 1− , to get max 𝜑󸀠 ≤ 𝐶2 , where 𝐶2 depends only on 𝐾 and |𝑓(0)|. From this and (10.28) we get 𝐴(𝜃) ≤ 2𝐾2 𝐶2 and hence, by (10.27) and (10.24), |ℎ󸀠 (𝑒𝑖𝜃 )| ≤ 𝐶3 . Since ℎ󸀠 ∈ 𝐻1 , we see, using Smirnov’s maximum principle, that |ℎ󸀠 (𝑧)| ≤ 𝐶3

(𝑧 ∈ 𝔻),

(10.31)

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 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 = 󵄨󵄨 󸀠 󵄨 󵄨󵄨 ℎ 󵄨󵄨 󵄨󵄨 (ℎ ∘ 𝑤𝑛 )𝑤󸀠 󵄨󵄨󵄨 ≤ 𝑘, 𝑛󵄨 󵄨 𝑛󵄨 󵄨 we can appeal to the preceding special case to conclude that |ℎ󸀠 (𝑤𝑛 (𝑧))| |𝑤𝑛󸀠 (𝑧)|/𝑟𝑛 ≤ 𝐶3 , where 𝐶3 is independent of 𝑛 and 𝑧. And since 𝐺𝑛 ⊂ 𝐺𝑛+1 and ∪𝐺𝑛 = 𝔻, we can ap­ ply Carathéodory’s convergence theorem (Theorem 10.21 below): 𝑤𝑛 (𝑧) 󴁂󴀱 𝑧, whence 𝑤𝑛󸀠 (𝑧) 󴁂󴀱 1 (𝑛 → ∞). Thus inequality (10.31) holds in the general case. Using this and (10.24) we get 𝜑󸀠 (𝜃) + |𝐵(𝜃)| ≤ 𝐶4 . Finally, it remains to apply (10.26).

Some consequences of the proof It was proved by Clunie and Sheil-Small [97, Theorem 5.7] 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, then 1/𝐶 ≤ |ℎ󸀠 | ≤ 𝐶. This means that Theorem 10.19. If 𝑓 is QCH, then ℎ is bi-Lipschitz.

10.3 Quasiconformal harmonic mappings | 311

The converse is not true. To see this, let 𝐷 = 1 + 𝔻, and 𝐹(𝑧) = 1 + 𝑧. Then let as before 𝜑󸀠 (𝜃) = Re 𝐹(𝑒𝑖𝜃 ) = 1 + cos 𝜃, which implies that 𝐻(𝜑󸀠 ) = Im 𝐹(𝑒𝑖𝜃 ) = sin 𝜃. Then we use the formula (10.24) to show that |ℎ󸀠 (𝑒𝑖𝜃 )| is bounded, which implies that ℎ󸀠 is bounded on 𝔻. On the other hand, from (10.24) and (10.29) it follows that |ℎ󸀠 (𝑒𝑖𝜃 )| ≥ 𝐴(𝜃)/2 ≥ (1 − |𝑓(0)|)/4, and hence |ℎ󸀠 (𝑧)| ≥ (1 − |𝑓(0)|)/4 (10.32) for all 𝑧 ∈ 𝔻. Thus ℎ is bi-Lipschitz, while 𝑓 is not quasiconformal because min𝕋 𝜑 = 0. Theorem 10.20. 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 (10.32) holds and hence |𝑓󸀠 (𝑧)| ≥ |ℎ󸀠 (𝑧)| ≥ (1 − |𝑓(0)|)/4.

Carathéodory convergence theorem Here we prove the simplest variant of Carathéodory’s theorem; for the general form see [171, Ch. II § 5] and [130]. Theorem 10.21. 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 is enough to prove that every 𝐻(𝔻)-convergent subsequence of {𝑓𝑛 } converges to the function 𝜑(𝑧) = 𝑧. There­ fore 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. Problem 10.1. Let QCH∗ = {𝑓∗ : 𝑓 is QCH}. Is QCH∗ a group with respect to composi­ tion? 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. [13]).

312 | 10 One-to-one mappings

10.4 𝐻𝑝 -classes of quasiconformal mappings The theory of univalent harmonic (UH) mappings [131] can be certainly used to ex­ tend some of the results of Section 10.2 to QCH-mappings. However, in view of the Astala–Koskela work [32], it seems that quasiconformality plays a primary role (which also could be seen from the proof of Theorem 10.18). This does not mean that some of the results on U, presented in Section 10.2, cannot be extended the class UH. For ex­ ample, Abu-Muhanna [2] extended Pommerenke’s theorem (stated as Corollary 10.7) to UH. Astala and Koskela proved in [32] that many theorems on conformal mappings continue to be true for arbitrary QC-mappings. Here we present some of their ideas within the framework of QCH-mappings. Theorem 10.22 (Astala–Koskela [32]). Let 0 < 𝑝 < ⬦. If 𝑓 is a QCH-mapping, then 1 𝑝 (𝑟, 𝑓) 𝑑𝑟, ‖𝑓‖𝑝𝑝 ≍ |𝑓(0)|𝑝 + ∫ |𝑓󸀠 (𝑧)|𝑝 (1 − |𝑧|)𝑝−1 𝑑𝐴(𝑧) ≍ ∫ 𝑀∞

(10.33)

0

𝔻

where the equivalence constants depend only on 𝐾 and 𝑝. Here ‖𝑓‖𝑝𝑝 = ∫ 𝑀∗ 𝑓(𝜁)𝑝 |𝑑𝜁| ≍ sup 𝑀𝑝𝑝 (𝑟, 𝑓). 𝕋

0 𝑝,

0

where the equivalence constants depend only on 𝑝, 𝑞, and 𝐾. Proof. In one direction, we have 1

1

∫ 𝑀𝑞𝑝 (𝑟, 𝑓)(1 − 𝑟)−𝑝/𝑞 𝑑𝑟 ≍ |𝑓(0)|𝑝 + ∫ 𝑀𝑞𝑝 (𝑟, 𝑓󸀠 )(1 − 𝑟)−𝑝/𝑞+𝑝 𝑑𝑟 0

0 1 𝑝

≤ 𝐶|𝑓(0)| + 𝐶 ∫ 𝑀𝑝𝑝 (𝑟, 𝑓󸀠 )(1 − 𝑟)𝑝−1 𝑑𝑟 ≤ 𝐶‖𝑓‖𝑝𝑝 . 0

In the reverse direction, we proceed as in the proof of Theorem 10.9. Corollary 10.8. For 𝑓 ∈ QCH each of the following three quantities is equivalent with 𝑝 ‖𝑓‖𝑝 , 0 < 𝑝 < ⬦: 1

∫ |𝑓(𝑟𝔻)| 0

We also have

𝑝/2

1 𝑝/2

𝑑𝑟;

󸀠 2

󸀠 2

∫ ( ∫ (|𝑔 | + |ℎ | ) 𝑑𝐴) 0

1

𝑑𝑟;

𝑟𝔻

∫ 𝑃(𝑟, 𝑓)𝑝 𝑑𝑟. 0

∞

̂ 𝑝 ‖𝑓‖𝑝𝑝 ≍ ∑ (|𝑛| + 1)𝑝−2 |𝑓(𝑛)|

(1 ≤ 𝑝 ≤ 2).

𝑛=−∞

The following assertion is a special case of [32, Theorem 5.1]. Theorem (Astala–Koskela). A function 𝑓 ∈ QCH belongs to 𝐻𝑝 (0 < 𝑝 < ⬦) if and only if the function 𝑀(𝜁) := sup0 𝑝), BMOA, B, etc. The multipliers from 𝐻𝑝 (𝑝 < 1) into ℓ𝑞 (𝑞 > 0) are calculated by means of a characterization of the “solid hull” of 𝐻𝑝 . We use 𝑝1 ,𝑞1 , and show the decomposition method to discuss the multipliers from B𝑝,𝑞 𝛼 to B𝛽 how it should be modified to characterize, in some cases, the multipliers between spaces of harmonic functions with subnormal weights; new spaces arise, which we call Jackson–Bernstein spaces because they are defined via the best approximations by polynomials. The interested reader will find a long list of multiplier theorems in the book [31], which will appear in 2014. However, this chapter is far from being a subset of [31].

11.1 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 {𝜇𝑛𝑎𝑛 } ∈ 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 but also in the case of admissible spaces of analytic or harmonic functions. For instance, we can identify 𝑓 ∈ 𝐻(𝔻), or ̂ and conversely: if a sequence {𝑎 } satisfies 𝑓 ∈ ℎ(𝔻), with the sequence 𝑎𝑛 = 𝑓(𝑛) 𝑛 𝑛 𝑛 the condition lim sup𝑛→∞ √|𝑎𝑛 | < ∞, 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. 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 𝐻𝑝 to 𝐻𝑞 , where 0 < 𝑝 < 𝑞 < ⬦. A less self-evident example is the following reformulation of a variant of the Little­ wood–Paley theorem, i.e. of the inclusion B𝑝,𝑝 ⊂ 𝐻𝑝 (𝑝 ≤ 2): The sequence (𝑛 + 1)−1/𝑝 is a multiplier from the Bergman space 𝐴𝑝 to 𝐻𝑝 for 0 < 𝑝 ≤ 2. We leave the proof as an exercise¹.

1 In the case 𝑝 ≥ 1, this was proved by MacGregor and Zhu [314] by using interpolation of linear operators.

11.1 Multipliers on abstract spaces | 319

The sequence 𝜎𝑛𝛼 𝑓 can be viewed as a sequence of multiplier transforms. As a corollary to 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 𝐻𝑝 . In order to unify the language, we shall formulate most assertions only for analytic functions. If 𝜇 is a sequence and 𝑓 ∈ 𝐻(𝔻), then we write ∞

𝑛 ̂ . (𝜇 ∗ 𝑓) = ∑ 𝜇𝑛 𝑓(𝑛)𝑧 𝑛=0

The following fact can be easily proved by using the closed graph theorem. Theorem 11.1. If 𝜇 is a multiplier from 𝑋 to 𝑌, where 𝑋 and 𝑌 are 𝐻-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 (1.17) 𝑛 𝑛 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 . Proposition 11.1 ([70]). If 𝑋 and 𝑌 are 𝐻-admissible spaces, then so is (𝑋, 𝑌). Proof. Let C(𝑧) = (1 − 𝑧)−1 . Take 𝑔 ∈ (𝑋, 𝑌) and observe that 𝑔 ∗ C𝑟 = 𝑔𝑟 and so 𝑀∞ (𝑟2 , 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 11.1, 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. This concludes the proof. Many results of Chapter 5 can be extended to “abstract” spaces (of analytic or har­ monic functions) which satisfy additional properties. Beside notions of an admissible space (Section 1.1) and a minimal space (Section 1.2) there are some other important notions. 11.1 (Homogeneous spaces and FP-spaces). Many classical spaces of analytic or har­ monic functions satisfies the following conditions: If 𝑓 ∈ 𝑋, then 𝑓𝜁 ∈ 𝑋 (|𝜁| ≤ 1) and sup ‖𝑓𝑤 ‖𝑋 = ‖𝑓‖𝑋 .

(11.1)

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

(11.2)

𝑤∈𝔻

𝑤∈𝔻

320 | 11 Coefficients multipliers Here 𝑓𝑤 (𝑧) = 𝑓(𝑤𝑧). If an admissible space 𝑋 satisfies condition (11.1), then it will be 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 (11.2). For example, the spaces ℎ𝑝,𝑞 𝛼 (𝑝 ≥ 1, 𝑝,𝑞 𝛼 > 0) and 𝐻𝛼 (𝑝 > 0, 𝛼 > 0) are homogeneous for all 𝑞; they have the Fatou property if and only if 𝑞 ≠ ⬦. The spaces BMOA, VMOA, 𝐻𝑝 (0 < 𝑝 ≤ ∞) are all homogeneous, but VMOA and 𝐻⬦ = 𝐴(𝔻) are not FP. 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 (11.2). We call a space an HFP-space if it is 𝐻-admissible, homogeneous, and FP. The following proposition explains the term “Fatou property”. Proposition 11.2. A homogeneous space 𝑋 is an FP-space if and only the following im­ plication 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 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. 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

𝑝

‖𝑓𝑤 ‖𝑋 ≤ ‖(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 𝑋𝐴 was defined on page 149. We also use the notation 𝑋∗ = (𝑋, 𝐻∞ ),

𝑋# = (𝑋, 𝐴(𝔻)).

(11.3)

Let A denote the class of all Abel summable sequences, i.e. of those sequences 𝑛 {𝑐𝑛 }∞ 0 such that the series 𝑠(𝑟) := ∑𝑛 𝑐𝑛 𝑟 , 0 < 𝑟 < 1, converges and the limit lim𝑟→1− 𝑠(𝑟) is finite. The Abel dual 𝑋𝐴 of 𝑋 coincides with (𝑋, A). However, a 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

11.1 Multipliers on abstract spaces |

321

is bounded in the sense of the theory of linear topological spaces. This means that for all 𝑟 and 𝜀 there is a sequence {𝑓𝑛 } ⊂ 𝑉𝑟,𝜀 and a seminorm 𝑝𝜌 such that lim𝑛 𝑝𝜌 (𝑓𝑛 ) = ∞. One 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 𝑋𝐴 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 }‖𝑋𝐴 ≤ 𝐶‖𝑓‖𝑋 . This gives, in 𝑛 particular, |𝑐𝑛 | ≤ 𝐶‖𝑒𝑛 ‖𝑋 , and hence, by (1.17), the function 𝑔(𝑧) = ∑∞ 𝑛=0 𝑐𝑛 𝑧 is analytic 𝐴 in 𝔻. Ergo, we can define 𝑋 as 𝑋𝐴 = {𝑔 ∈ 𝐻(𝔻) : lim− 𝑓 ∗ 𝑔(𝑟) exists and is finite}. 𝑟→1

The norm is given by ‖𝑔‖𝑋𝐴 = sup{|𝑓 ∗ 𝑔(𝑟)| : ‖𝑓‖𝑋 ≤ 1, 0 < 𝑟 < 1}. It is not too difficult to prove that then 𝑋𝐴 is complete, but we shall not do this because this fact is irrelevant for our aims. Here we collect some important although simple properties. Proposition 11.3. Let 𝑋 and 𝑌 be 𝐻-admissible spaces. Then the following assertions hold: (i) If 𝑋 is minimal and homogeneous, then 𝑋𝐴 ≅ 𝑋# ≅ 𝑋∗ and 𝑋𝐴 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. Further­ more, if 𝑋 is HFP, then 𝑋∗∗ ≅ 𝑋. (v) If 𝑋 and 𝑌 are HFP-spaces, then (𝑋, 𝑌) ≅ (𝑌∗ , 𝑋∗ ). The statement (i) says in particular that if 𝑓 ∈ 𝑋 and 𝑔 ∈ 𝑋𝐴 , then the function 𝑔 ∗ 𝑓 extends to a continuous function on 𝔻. This can be applied to 𝑋 = 𝐻𝑝 (𝑝 < 1) and 𝑌 = B1/𝑝−1 (by Theorem 5.13). Proof. The assertions (i), (ii), and (iii) are almost trivial, whilst (iv) can be easily de­ duced 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, assertion (i) does not hold for nonminimal spaces. For example, it is a delicate result of Piranian et al. [396] that (𝐻∞ )𝐴 = K𝑎 , where, we recall, K is the space of Cauchy transforms, and 𝐾𝑎 the “absolutely continuous part” of K. On the other hand (𝐻∞ )∗ = (𝐴(𝔻))∗ = K, by Proposition 11.3(iii).

322 | 11 Coefficients multipliers Preduals and the second duals It is known that a dual space can have different preduals; see, e.g. [170]. We will not consider this question. Instead, we show how to identify a predual of a minimal space in some important cases. Theorem 11.2. If 𝑋 is a minimal Banach HFP-space, then 𝑌𝐴 ≅ 𝑋, where 𝑌 = (𝑋𝐴 )P . Proof. Let 𝑓 ∈ 𝑌 and 𝑔 ∈ 𝑋. Since 𝑌 ⊂ 𝑋𝐴 , we see that ⟨𝑓, 𝑔⟩ exists and is finite, and that |⟨𝑓, 𝑔⟩| ≤ ‖𝑓‖𝑌 ‖𝑔‖𝑋 , which implies ‖𝑔‖𝑋 ≥ ‖𝑔‖𝑌𝐴 . Let 𝑔 ∈ 𝑌𝐴 . In view of Theorem 5.10 and the fact that 𝑋 is minimal, we have, for 0 < 𝜌 < 1, ‖𝑔𝜌 ‖𝑋 = sup{|⟨𝑓, 𝑔𝜌 ⟩| : 𝑓 ∈ 𝑋𝐴 , ‖𝑓‖𝑋𝐴 ≤ 1} = sup{|⟨𝑓𝜌 , 𝑔⟩| : 𝑓 ∈ 𝑋𝐴 , ‖𝑓‖𝑋𝐴 ≤ 1} ≤ sup{|⟨𝑓𝜌 , 𝑔⟩| : 𝑓 ∈ 𝑋𝐴 , ‖𝑓𝜌 ‖𝑋𝐴 ≤ 1} ≤ ‖𝑔‖𝑌𝑎 , where the relations 𝑓𝜌 ∈ (𝑋𝐴 )P and ‖𝑓𝜌 ‖𝑋𝐴 ≤ ‖𝑓‖𝑋𝐴 have been used. Now the 𝐹-property and the homogeneity of 𝑋 give 𝑔 ∈ 𝑋 and ‖𝑔‖𝑋 ≤ ‖𝑔‖𝑌𝐴 . This concludes the proof. Theorem 11.3. Let 𝑋 be a Banach HFP-space. Suppose that the space (𝑋P )𝐴 is mini­ mal. Then the second dual of 𝑋P is isometrically isomorphic to 𝑋. More precisely, we have ((𝑋P )𝐴 )𝐴 ≅ 𝑋. Proof. Let 𝑍 = 𝑌𝐴 , 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 5.10, ‖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 Corollary 11.1. If 𝑝 ≥ 1, then (B𝑝,⬦ ≅ B𝑝,∞ 𝛼 ) 𝛼 .

Abstract Besov spaces 𝑞 Let 𝑋 be a homogeneous 𝐻-admissible space. We define the “𝑋-Besov” space B𝑋,𝛼 = {𝑓 : . . . } by the requirement that the function (1 − 𝑟2 )𝑠−𝛼 ‖J𝑠 𝑓𝑟 ‖𝑋 , 0 < 𝑟 < 1, belongs 𝑞 to 𝐿 −1 for some 𝑠 > 𝛼. It follows from Lemmas 12.2 and 12.3 (see Remark 12.1) that ‖𝑉𝑛 ∗ 𝑓‖𝑋 ≤ 𝐶𝑋 ‖𝑓‖𝑋 , and 𝑛

𝑛

𝑐𝑟𝐴2 2𝑛𝑠 ‖𝑉𝑛 ∗ 𝑓‖𝑋 ≤ ‖𝑉𝑛 ∗ 𝑅𝑠 𝑓𝑟 ‖𝑋 ≤ 𝐶𝑟𝑎2 2𝑛𝑠 ‖𝑉𝑛 ∗ 𝑓‖𝑋 ,

11.1 Multipliers on abstract spaces | 323

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 𝑞

𝑞

B𝑋P ,𝛼 = B𝑋,𝛼 = 𝑉𝛼𝑞 [𝑋] = 𝑉𝛼𝑞 [𝑋P ].

(11.4)

The first and the last equality hold because ‖𝑓𝑟 ‖𝑋 = ‖𝑓𝑟 ‖𝑋P and ‖𝑉𝑛 ∗ 𝑓‖𝑋 = 𝑞 ‖𝑉𝑛 ∗ 𝑓‖𝑋P . The space B𝑋,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 5.10, the dual 𝑞 of B𝑋,𝛼 , 𝑞 ≤ ⬦, is isomorphic to its Abel dual. According to Theorem 5.11, the relations (11.4) and Proposition 11.3, the following theorem holds. 𝑞

𝑞󸀠

𝑞

𝑞󸀠

Theorem 11.4. If 𝑞 ≤ ⬦, then (B𝑋,𝛼 )𝐴 ≃ B(𝑋 )𝐴 ,−𝛼 , and also (B𝑋,𝛼 )∗ ≃ B(𝑋 )∗ ,−𝛼 for all P P 𝑞 ≤ ∞. This can be used to give a somewhat simpler proof of Theorem 5.12 in the case when = 𝑉𝛼𝑞 [𝑋], where 𝑋 = B𝑝,⬦ for 𝑝 < 1, and 𝑝 ≤ 1 or 𝑝 = ∞. Namely, we have B𝑝,𝑞 𝛼 𝑋 = B1,1 , for 𝑝 = 1, and 𝑋 = B∞,1 for 𝑝 = ∞. It is relatively easy to determine the Abel dual of these spaces (see the proof of Theorem 6.13).

11.1.1 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, i.e. if 𝑔∗ maps 𝐵(𝑋), the unit ball of 𝑋, into a totally bounded subset of 𝑌. Lemma 11.1. (a) A subset 𝑆 of an admissible space 𝑌 is totally bounded if (1) 𝑆 is bounded and, (2) sup ‖𝑔 − 𝑔𝑟 ‖𝑌 → 0 as 𝑟 → 1− .

(11.5)

𝑔∈𝑆

(b) If 𝑌 is minimal and 𝑆 ⊂ 𝑌 is totally bounded, then (11.5) holds. Proof. (a) Using Montel’s theorem and the definition of admissible spaces one proves that, for a fixed 𝑟 < 1, the set {𝑔𝑟 : 𝑔 ∈ 𝐵(𝑌)} is totally bounded. From this and (11.5) 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 (11.5)(2) does not hold. Denote by 𝜅(𝑋, 𝑌) the space of all compact multipliers from 𝑋 to 𝑌. The following the­ orem shows that describing 𝜅(𝑋, 𝑌) is practically equivalent to describing (𝑋, 𝑌). Theorem 11.5. If 𝑋 and 𝑌 be 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)

324 | 11 Coefficients multipliers By Lemma 11.1, 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 (†) ‖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 of 𝑌. This implies that (1/2)𝑗 𝑓𝑛̂ (𝑗)[𝑔(𝑗) − 𝑔(𝑗)𝑟𝑛𝑗 ] → (1/2)𝑗 ℎ(𝑗) 𝑗 ̂ follows that ℎ = 0 because (1/2) |𝑓𝑛 (𝑗)| ≤ 𝐶‖𝑓𝑛 ‖𝑋 ≤ 𝐶, where 𝐶 is independent of 𝑛. This contradicts (†) and proves the theorem.

11.2 Multipliers for Hardy and Bergman spaces In this section we apply the Coifman–Rochberg theorem (Theorem 11.6 below) to de­ scribe the set (𝐴𝑝 , 𝑌), 0 < 𝑝 ≤ 1, where 𝑌 a 𝑞-Banach space with 𝑞 ≥ 𝑝, and also the set (𝐻𝑝 , 𝑌) with 𝑞 > 𝑝, 𝑝 < 1. 𝑝 Let 0 < 𝑝 < ⬦ and 𝛽 > −1. The (already defined) Bergman space 𝐴 𝛽 is equal as a set to 𝐻𝛼𝑝,𝑝 , where 𝛼 = (𝛽 + 1)/𝑝. However, it is useful to introduce a different quasinorm, namely 𝛽+1 𝑝 𝑝 ∫ |𝑓(𝑧)|𝑝 (1 − |𝑧|2 )𝛽 𝑑𝐴(𝑧). ‖𝑓‖𝑝,𝛽 = ‖𝑓‖𝐴𝑝 = 𝜋 𝛽 𝔻

One of the reason is that lim𝛽→−1 ‖𝑓‖𝑝,𝛽 = ‖𝑓‖𝑝 . 𝑝 If 𝛽 = 0, then we write 𝐴𝑝 = 𝐴 𝛽 . Theorem 11.6 (On atomic decomposition). Let 0 < 𝑝 ≤ 1, 𝛽 > −1 and 𝛾 > 0. 𝑝 (a) There exists a sequence {𝑤𝑛} in 𝔻 and a constant 𝐶 such that every 𝑓 ∈ 𝐴 𝛽 can be represented as ∞ (1 − |𝑤𝑛 |2 )𝛾 (11.6) 𝑓(𝑧) = ∑ 𝑎𝑛 (1 − 𝑤𝑛𝑧)𝛾+(𝛽+2)/𝑝 𝑛=1 with ‖{𝑎𝑛 }‖ℓ𝑝 ≤ 𝐶‖𝑓‖𝑝,𝛽 . 𝑝 (b) Every function 𝑓 of the form (11.6) with {𝑎𝑛} ∈ ℓ𝑝 belongs to 𝐴 𝛽 and ‖𝑓‖𝑝,𝛽 ≤ 𝐶‖{𝑎𝑛 }‖ℓ𝑝 . A special case, Theorem 12.1, will be proved in the next chapter. 𝑞

Corollary 11.2. The 𝑞-Banach envelope of 𝐻𝑝 , where 𝑝 < 𝑞 ≤ 1 is equal to 𝐴 1/𝑞−2 . 𝑞

Proof. The space 𝑋 = 𝐻𝑝 is embedded into the 𝑞-Banach space 𝑌 = 𝐴 𝛽 ; see Corol­ lary 4.1. 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

11.2 Multipliers for Hardy and Bergman spaces | 325

and (∑ |𝑎𝑛 |𝑞 )1/𝑞 ≤ 𝐶‖𝑓‖𝑌 . Since 𝜋

∫ −𝜋

we have

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

𝑞

∞

𝑞

∑ ‖𝑓𝑛 ‖𝑋 ≤ 𝐶 ∑ |𝑎𝑛 |𝑞 ≤ 𝐶‖𝑓‖𝑌 . 𝑛=1

𝑛=1

Now the result follows from Proposition A.4. Proposition 11.4. Let 0 < 𝑝 ≤ 1, 𝛽 > −1, and let 𝑌 be a 𝑝-Banach space. Then 𝑝

(𝐴 𝛽 , 𝑌) = B∞ 𝑌,(𝛽+2)/𝑝−1

and 𝜅(𝐴𝑝 , 𝑌) = B⬦𝑌,(𝛽+2)/𝑝−1 .

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 Δ 𝑛 𝑓 if {𝑒𝑛} is a Schauder basis in 𝑌P ; examples: 𝐻𝑞 (1 < 𝑞 < ⬦), ℓ𝑞 (0 < 𝑞 ≤ ∞), 𝐴𝑞 (1 < 𝑞 < ⬦), etc. 𝑝

Proof. Let 𝑔 ∈ (𝐴 𝛽 , 𝑌). Let 𝑓(𝑧) = (1 − 𝑧)−𝑠−1 , where 𝑠 > 0 is sufficiently large. Then 𝑝

𝑝

𝑝

‖𝐷[𝑠] 𝑔𝑤 ‖𝑌 = ‖𝑔 ∗ 𝑓𝑤 ‖𝑌 ≤ 𝐶‖𝑓𝑤 ‖𝑝,𝛽 = (𝛽 + 1)𝑝∫ − 𝔻

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

where we have chosen 𝑠 so that (𝑠 + 1)𝑝 > 𝛽 + 2. (The operator 𝐷[𝑠] is defined by (5.38).) Using the formula 𝑛 𝐵2𝑘 1 1 1 + 𝑂 ( 2𝑛+1 ) , log 𝛤(𝑥) = (𝑥 − ) − 𝑥 + log 2𝜋 − ∑ 2𝑘−1 2 2 𝑥 𝑘=1 2𝑘(2𝑘 − 1)𝑥

where 𝐵2𝑘 are the Bernoulli numbers (see [342]) one proves that 𝐷[𝑠] can be replaced 𝑝 by R𝑠 . This proves the inclusion (𝐴 𝛽 , 𝑌) ⊂ B∞ 𝑌,(𝛽+2)/𝑝−1 . 𝑝

In order to prove the reverse inclusion, we use Theorem 11.6. Let 𝑓 ∈ 𝐴 𝛽 and 𝑔 ∈ B∞ 𝑌,(𝛽+2)/𝑝−1 . We have ∞

(𝑓 ∗ 𝑔)(𝑧) = ∑ 𝑎𝑛 (1 − |𝑤𝑛 |2 )1/𝑝 𝐷[𝑡] 𝑔(𝑤̄ 𝑛 𝑧),

𝑧 ∈ 𝔻,

𝑛=1

where 𝑡 = 1/𝑝 + (𝛽 + 2)/𝑝 − 1, and ‖{𝑎𝑛 }‖𝑝 ≤ 𝐶‖𝑓‖𝑝 . It follows that 𝑝

∞

𝑝

‖𝑓 ∗ 𝑔‖𝑌 ≤ ∑ |𝑎𝑛 |𝑝 (1 − |𝑤𝑛 |2 )‖𝐷[𝑡] 𝑔𝑤̄ 𝑛 ‖𝑌 𝑛=1 ∞

∞

𝑛=1

𝑛=1

≤ 𝐶 ∑ |𝑎𝑛 |𝑝 (1 − |𝑤𝑛 |2 )(1 − |𝑤𝑛 |2 )(𝛽+2)−𝑝−𝑡𝑝 = 𝐶 ∑ |𝑎𝑛 |𝑝 .

326 | 11 Coefficients multipliers This completes the proof of the first relation. The second relation follows from the first and Theorem 11.5. Using Corollary 11.2, Proposition A.3, and Proposition 11.4 we obtain a characterization of (𝐻𝑝 , 𝑌). Proposition 11.5. If 0 < 𝑝 < 𝑞 ≤ 1 and 𝑌 is a 𝑞-Banach space, then (𝐻𝑝 , 𝑌) = B∞ 𝑌,1/𝑝−1

and 𝜅(𝐻𝑝 , 𝑌) = B⬦𝑌,1/𝑝−1 .

It is useful to observe that this proposition is a formal special case (𝛽 = −1) of Propo­ 𝑞 sition 11.4 and the formal equality 𝐻𝑞 = 𝐴 −1 . 𝑝

Remark 11.1. As we see, the proof of the inclusion (𝐴 𝛽 , 𝑌) ⊂ B∞ 𝑌,(𝛽+2)/𝑝−1 depends neither of 𝑌 nor of the atomic decomposition. Thus this inclusion holds for all 𝑝 ∈ (0, ⬦), 𝛽 ≥ −1, and all admissible space 𝑌. A little more work is needed to prove that 𝑝 𝜅(𝐴 𝛽 , 𝑌) ⊂ B⬦𝑌,(𝛽+2)/𝑝−1 . As a first example we prove the following theorem. 𝑞,∞

Theorem 11.7. If 0 < 𝑝 ≤ 1 and 𝑞 ≥ 𝑝, then (𝐴𝑝 , 𝐴𝑞 ) ≃ B2/𝑝−1−1/𝑞 and 𝜅(𝐴𝑝 , 𝐴𝑞 ) ≃ 𝑞,⬦ B2/𝑝−1−1/𝑞 . 𝑝 𝑞 ⬦ Proof. By Proposition 11.4, we have (𝐴𝑝 , 𝐴𝑞 ) = B∞ 𝐴𝑞 ,2/𝑝−1 and 𝜅(𝐴 , 𝐴 ) = B𝐴𝑞 ,2/𝑝−1 . In

view of (11.4), this means that 𝑓 ∈ (𝐴𝑝 , 𝐴𝑞 ) if and only if ‖𝑉𝑛 ∗ 𝑓‖𝐴𝑞 ≤ 𝐶2𝑛(1−2/𝑝) . Now we use Lemma 2.2 to show that ‖𝑉𝑛 ∗ 𝑓‖𝐴𝑞 ≍ 2−𝑛/𝑞 ‖𝑉𝑛 ∗ 𝑓‖𝑞 . The result follows. 𝑝,∞

Corollary 11.3. If 0 < 𝑝 ≤ 1, then (B1/𝑝−1 , ∗) is a unital 𝑝-Banach algebra. This follows from the above theorem and the general fact that (𝑋, 𝑋) is a 𝑝-Banach algebra (with the function 1/(1 − 𝑧) as unit) for an arbitrary admissible space 𝑋. 2,∞ Corollary 11.4. If 0 < 𝑝 ≤ 1, then (𝐴𝑝 , 𝐴2 ) = B2,∞ = ℓ2/𝑝−1/2 . 2/𝑝−1/2

As an application of Proposition 11.5 and the canonical isomorphism between B∞ 𝐻𝑞,1/𝑝−1 𝑞 and the Lipschitz space 𝐻Λ 1/𝑝−1 we have the following theorem. 𝑞

Theorem 11.8. Let 0 < 𝑝 < 1 and 𝑝 < 𝑞 ≤ ∞. Then (𝐻𝑝 , 𝐻𝑞 ) ≃ 𝐻Λ 1/𝑝−1 and 𝑞 𝜅(𝐻𝑝 , 𝐻𝑞 ) = 𝐻𝜆 1/𝑝−1. Note that choosing 𝑞 = ∞ we get the Duren–Romberg–Shields theorem (Theo­ rem 5.13). Problem 11.1. What is (𝐻1 , 𝐻𝑝 ) for 0 < 𝑝 < 1? This question is interesting because (𝐻1 , 𝐻𝑝 ) is independent of 𝑝 < 1 and actually, by the Nikishin–Stein theorem B.17, (𝐻1 , 𝐻𝑝 ) = (𝐻1 , 𝐻1,⋆ ), where 𝐻1,⋆ is the weak 𝐻𝑝 -space; 𝐻1,⋆ can be defined in two equivalent ways: either by 𝐻1,⋆ = 𝐻𝑠 ∩ 𝐿1,⋆ , where 0 < 𝑠 < 1, or by the requirement sup𝑟 1 + 𝛿 be an integer, and apply Lemma 2.3 (𝑞 = ∞) to the polynomials 𝑒𝑖𝑀𝑛𝜃 𝑊(𝑒𝑖𝜃 ).

338 | 11 Coefficients multipliers Define the polynomials V𝑛,𝜆 : ∞

V0,𝜆 (𝑒𝑖𝜃 ) = ∑ 𝜓(𝑘/𝜆 0 )𝑒𝑖𝑘𝜃 𝑘=−∞ ∞

V𝑛,𝜆 (𝑒𝑖𝜃 ) = ∑ (𝜓 ( 𝑘=−∞

𝑘 𝑘 ) − 𝜓( )) 𝑒𝑖𝑘𝜃 , 𝜆𝑛 𝜆 𝑛−1

𝑛 ≥ 1.

We may and shall assume that 𝛿 ≤ 1. Then: Proposition 11.10. The polynomials V𝑛 = V𝑛,𝜆 , 𝑛 ≥ 0, possess the following properties: ̂) ⊂ [𝜆 , 𝜆 ] ∪ [−𝜆 , −𝜆 ] (𝜆 := 0). (i) supp (V 𝑛 𝑛−1 𝑛+1 𝑛+1 𝑛−1 −1 (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 ∈ 𝐻𝑝 , 0 < 𝑝 ≤ ∞). (v) ‖V𝑛 ∗ 𝑓‖𝑝 ≤ 𝐶𝑝 ‖𝑓‖𝑝 (𝑓 ∈ ℎ𝑝 , 1 ≤ 𝑝 ≤ ∞). Proof. Item (iv) is a consequence of (much stronger) Theorem 5.3. To prove (v) we use the inequality ‖𝑓 ∗ 𝑔‖𝑝 ≤ ‖𝑓‖𝑝 ‖𝑔‖1 (𝑝 ≥ 1) with 𝑔 = V𝑛 , and then (11.12) and the definition of V𝑛 . 𝑝,𝑞

In order to prove a decomposition theorem for the harmonic space ℎ[𝜙] (see Sub­ section 3.6.3) we need an analog of Lemma 2.2. 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) 𝑧̄𝑗 .

Observe that 𝑔(𝑧) = ∑∞ 𝑗=0 𝑔[𝑗] (𝑧), 𝑧 ∈ 𝔻. Lemma 11.3. If 𝑋 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’s theorem (‖𝜎𝑛 𝑔 ∗ 𝑓‖∞ ≤ ‖𝑔 ∗ 𝑓‖∞ ) and the inequality ‖𝑔 ∗ 𝑓‖∞ ≤ ‖𝑓‖𝑋∗ ‖𝑔‖𝑋 . Lemma 11.4. Let 𝑔 = ∑𝑛𝑚 𝑔[𝑗] ∈ 𝑋 (0 ≤ 𝑚 < 𝑛), where 𝑋 is a homogeneous Banach space. Then 3−1 𝑟2𝑛 ‖𝑔‖𝑋 ≤ ‖𝑔𝑟 ‖𝑋 ≤ 2𝑟𝑚/2 ‖𝑔‖𝑋 , 0 < 𝑟 < 1. Proof. Let 𝑚 ≥ 2. After two summations by parts we find ∞

∞

0

0

𝑔𝑟 = ∑ 𝑟𝑗 𝑔[𝑗] = (1 − 𝑟)2 ∑ 𝑟𝑗 (𝑗 + 1)𝜎𝑗1 𝑔.

11.5 Multipliers of spaces with subnormal weights |

339

Taking into account that 𝜎𝑗1 𝑔 = 0 for 𝑗 < 𝑚 and using the inequality ‖𝜎𝑗1 𝑔‖ ≤ ‖𝑔‖ (Lemma 11.3) we get ∞

‖𝑔𝑟 ‖ ≤ (1 − 𝑟)2 ∑ 𝑟𝑗 (𝑗 + 1)‖𝑔‖ = 𝑟𝑚 (1 + 𝑚(1 − 𝑟)). 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

0

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 11.3, 𝑛−1

‖𝑔‖ ≤ (𝑅 − 1)2 ∑ 𝑅𝑗 (𝑗 + 1)‖𝑓‖ + (𝑅 − 1)𝑅𝑛 𝑛‖𝑓‖ + 𝑅𝑛 ‖𝑓‖. 0

Finally, we use the inequalities 𝑛(𝑅 − 1) ≤ 𝑅𝑛 − 1 ≤ 𝑅𝑛 and 𝑛−1

𝑛−1

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. 𝑞

11.2. For an ℎ-admissible space 𝑋 we define the space ℎ𝑋,[𝜙] = {𝑓 ∈ ℎ(𝔻) : . . . } (𝜙 is 𝑞

subnormal) by the requirement ‖𝑓𝑟 ‖𝑋 ∈ 𝐿 (𝑑𝑚𝜙 ); recall that 𝑑𝑚𝜙 (𝑟) = 𝜙󸀠 (1−𝑟) 𝑑𝑟/𝜙(1−𝑟). 𝑝,𝑞 𝑞 Thus ℎ[𝜙] = ℎℎ𝑝 ,[𝜙] . 𝑝,𝑞

In order to formulate a duality theorem for the spaces ℎ[𝜙] , we introduce a class of generalized 𝑉𝛼𝑝,𝑞 spaces. Let C = {c𝑛 }∞ 0 be a sequence of positive real numbers such that c𝑛+1 c 0 < inf ≤ sup 𝑛+1 < ∞. (11.13) 𝑛 c𝑛

𝑛

c𝑛

Such a sequence will be called normal. Note the following: If 𝜙 is normal on [0, 1], then {𝜑(2−𝑛 )} is normal. Let V𝑛 be a sequence of polynomials satisfying the properties (i), (ii), and (iii) of 𝑞 Proposition 11.10. The space VC [𝑋], where 𝑋 is an ℎ-admissible space, consists of those 𝑓 ∈ 𝐻(𝔻) for which 𝑞 {c𝑛 ‖V𝑛 ∗ 𝑓‖𝑋 }∞ 0 ∈ ℓ ,

0 < 𝑞 ≤ ∞.

By means of Lemmas 11.4 and 3.12 one obtains the following decomposition theo­ rem. Theorem 11.22. Let 𝑋 be a Banach ℎ-homogeneous space, 𝜙 a subnormal function on 𝑞 𝑞 [0, 1], and 𝜆 𝑛 (𝜆 0 ≥ 1) a sequence such that C = {𝜙(1/𝜆 𝑛)} is normal. Then ℎ𝑋,𝜙 ≃ VC [𝑋].

340 | 11 Coefficients multipliers Remark 11.3. If {𝑒𝑛} (−∞ < 𝑛 < ∞) is a Schauder basis in the closure in 𝑌 of the harmonic polynomials, e.g. if 𝑌 = ℎ𝑝 , 1 < 𝑝 < ⬦, then the theorem remains true if we replace V𝑛,𝜆 with Δ 𝑛,𝜆 , where Δ 𝑛,𝜆 (𝑒𝑖𝜃 ) = ∑ 𝑒𝑖𝑘𝜃 , 𝑘∈𝐼𝑛,𝜆

where 𝐼0,𝜆 = (−𝜆 1 , 𝜆 1 ), and 𝐼𝑛,𝜆 = (−𝜆 𝑛+1 , −𝜆 𝑛] ∪ [𝜆 𝑛 , 𝜆 𝑛+1 ) for 𝑛 ≥ 1. This follows from [360, Theorem 4.1]; condition (0.5) in that paper should be “. . . |𝑗| ∈ ̸ [𝜆 𝑛−1 , 𝜆 𝑛+𝑁 ), where 𝑁 ≥ 0 . . . ”. This decomposition were recently rediscovered by Peláez and Rättyä in [389, Theorem 4]. They proved the analogous result for the analytic Bergman type 1 spaces 𝐴𝑝,𝑞 𝜑 (1 < 𝑝 < ⬦), where 𝜑 ∈ 𝐿 (0, 1) is a continuous function satisfying lim

𝑟→1−

(1 − 𝑟)𝜑(𝑟) 1

(11.14)

= 0.

∫𝑟 𝜑(𝑥) 𝑑𝑥

1

𝑝,𝑞

However, if we take 𝜙(𝑡)𝑞 = ∫1−𝑡 𝜑(𝑥) 𝑑𝑥, then 𝐴𝑝,𝑞 𝜑 = 𝐻[𝜙] , and condition (11.14) is equivalent to 𝑡𝜙󸀠 (𝑡) = 0. lim+ 𝑡→0 𝜙(𝑡) The set of such 𝜙 is a subset of the class of Karamata slowly varying functions [437]. Therefore Theorem 4 of [389] is a special case of [360, Theorem 4.1]. Theorem 11.23. If 0 < 𝑞 ≤ ⬦ and ‖V𝑛 ∗ 𝑓‖𝑋 ≤ 𝐶‖𝑓‖𝑋 , where 𝑋 is an ℎ-admissible quasi𝑞 Banach space, then the space 𝑉C [𝑋] is minimal, and we have 𝑞󸀠

𝐴

𝑞

(𝑉C [𝑋]) ≃ 𝑉1/C [(𝑋P )𝐴 ]. Proof. Imitate the proof of Theorem 5.11. 𝑝,𝑞

From the preceding two statements we can get a description of the dual of ℎ𝜙 in terms of “Jackson–Bernstein” spaces. For a subnormal function 𝜙 on [0, 1] extended to [0, ∞] so that it remains to be subnormal and satisfies the condition 𝜙(1/𝑥) ≍ 1/𝜙(𝑥), 𝑥 > 0, 𝑞 we define the Jackson–Bernstein space 𝐸𝑋,[𝜙] = {𝑓 ∈ 𝑋 : . . . } by the requirement ∞

𝑞

∑ [𝜙(𝑛 + 1)𝑞 − 𝜙(𝑛)𝑞 ]𝐸𝑛 (𝑓)𝑋 < ∞,

𝑞 < ⬦.

(11.15)

𝑛=1

The norm can be given, for instance, by “‖𝑓‖𝑋 +the sum”. If 𝑞 ∈ {⬦, ∞}, then we define 𝑞 𝐸𝑋,[𝜙] by 1 𝐸𝑛(𝑓)𝑋 = 𝑂 (𝜙 ( )) (𝑞 = ∞), 𝑛

1 𝐸𝑛 (𝑓)𝑋 = 𝑜 (𝜙 ( )) (𝑞 = ⬦), 𝑛 → ∞. 𝑛

We write 𝑝,𝑞

𝑞

𝐸[𝜙] = 𝐸ℎ𝑝 ,[𝜙] .

11.5 Multipliers of spaces with subnormal weights

| 341

Theorem 11.24. If 𝑞 ≤ ⬦, then, with the hypotheses of Theorem 11.22, we have 𝐴

𝑞

𝑞󸀠

(ℎ𝑋,[𝜙] ) ≃ 𝐸(𝑋

P)

𝐴 ,[𝜙]

.

Proof. Let 𝑌 = (𝑋P )𝐴 . We choose a lacunary sequence 𝜆 = {𝜆 𝑛} of positive integers such that 𝜙(𝜆 𝑛 ) is normal and satisfies 𝜙(𝜆 𝑛 ) ≍ 2𝑛 . Application of Theorem 11.23 shows 𝑝󸀠

𝑞

that (ℎ𝑋,[𝜙] )𝐴 ≃ V1/C [𝑌], where 1/C = {𝜙(𝜆 𝑛 )}. Then, proceeding just as in the proof of 𝑝󸀠

Theorem 5.15, we conclude that 𝑔 ∈ V1/C [𝑌] if and only if ∞

𝑞󸀠

( ∑ 𝜙(𝜆 𝑛) 𝑛=0

1/𝑞󸀠 𝑞󸀠 𝐸𝜆 𝑛 (𝑔)𝑌 )

(11.16)

< ∞.

In the case 𝑞󸀠 = ∞, this gives the conclusion at ones. Let 𝑞󸀠 < ⬦. Then summation by 󸀠 󸀠 parts together with the inequality ∑𝑛𝑘=0 𝜙(𝜆 𝑘 )𝑞 ≤ 𝐶𝜙(𝜆 𝑛 )𝑞 shows that (11.16) is equiva­ lent to ∞

𝑞󸀠

𝑞󸀠

󸀠

∑ [𝐸𝜆 𝑛 (𝑔)𝑌 − 𝐸𝜆 𝑛+1 (𝑔)𝑌 ]𝜙(𝜆 𝑛 )𝑞 < ∞.

𝑛=0

This sum is equal to ∞

󸀠

∑ 𝜙(𝜆 𝑛)𝑞 𝑛=0

𝜆 𝑛+1 −1

𝑞󸀠

𝑞󸀠

∞

𝑞󸀠

󸀠

𝑞󸀠

∑ [𝐸𝑘 (𝑔)𝑌 − 𝐸𝑘+1 (𝑔)𝑌 ] ≍ ∑ 𝜙(𝑘)𝑞 [𝐸𝑘 (𝑔)𝑌 − 𝐸𝑘+1 (𝑔)𝑌 ].

𝑘=𝜆 𝑛

𝑘=𝜆 0

Another summation by parts leads to the desired result. (The proof shows that the 𝑞 𝑞󸀠 norms in (ℎ𝑋,[𝜙] )𝐴 and 𝐸𝑌,[𝜙] are equivalent.) Remark 11.4. We note that the expression 𝜙(𝑛 + 1)𝑞 − 𝜙(𝑛)𝑞 in (11.15) can be replaced with 𝜙(𝑛 + 1) − 𝜙(𝑛) −Δ1 𝜙(𝑛) = 𝜙(𝑛)𝑞 , 𝜙(𝑛)𝑞 𝜙(𝑛) 𝜙(𝑛) 𝑝,𝑞

which resembles the definition of ℎ[𝜙] . When applied to the space 𝑋 = ℎ𝑝 , Theorem 11.24 gives the following. 𝑝,𝑞

𝑝󸀠 ,𝑞󸀠

Theorem 11.25. If 1 ≤ 𝑝 ≤ ∞ and 𝑞 ≤ ⬦, then (ℎ[𝜙] )𝐴 ≃ 𝐸[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 5.18.

More on duality In the “subnormal situation” the following theorem provides, maybe, a more natural 𝑝,𝑞 description of (ℎ[𝜙] )󸀠 .

342 | 11 Coefficients multipliers 𝑝,𝑞

𝑝󸀠 ,𝑞󸀠

Theorem 11.26. If 𝑝 ≥ 1 and 𝑞 ≤ ⬦, then (ℎ[𝜙] )󸀠 ≃ ℎ[𝜙] , with respect to the duality pairing (𝑓, 𝑔)𝜙 = ∫ 𝑓(𝑧)𝑔(𝑧)̄ 𝑑𝜈𝜙 (𝑧), where 𝑑𝜈𝜙 (𝑟𝑒𝑖𝜃 ) = (𝜙(1 − 𝑟))2 𝑑𝑚𝜙 𝑑𝜃.

(11.17)

𝔻

Recall that 𝑑𝑚𝜙 (𝑟) = 𝜙󸀠 (1 − 𝑟) 𝑑𝑟/𝜙(1 − 𝑟). In order to pass from the sum of the “blocks” to the integral we need a technical lemma. For a subnormal function Ψ on [1, ∞) define the operator JΨ : ℎ(𝔻) 󳨃→ ℎ(𝔻) by ∞

JΨ 𝑔(𝑧) = ∑ Ψ(𝑛 + 1)𝑔[𝑗] (𝑧). 𝑛=0 𝑗

𝑗

̂ ̂ Recall that 𝑔[𝑗] (𝑧) = 𝑔(𝑗)𝑧 + 𝑔(−𝑗) 𝑧̄ for 𝑗 ≥ 1 and 𝑔[0] (𝑧) = 𝑔(0). Lemma 11.5. Let Ψ be a subnormal function such that 1/Ψ is convex on [1, ∞), let {𝜆 𝑛}∞ 1 (𝜆 1 ≥ 1) be an increasing sequence of positive integers such that Ψ(𝜆 𝑛 ) ≍ Ψ(𝜆 𝑛+1 ), and 𝜆 𝑛+1 let 𝑔 = ∑𝑗=𝜆 𝑔[𝑗] . If 𝑋 is a homogeneous ℎ-admissible space, then 𝑛−1

‖JΨ 𝑔‖ ≍ Ψ(𝜆 𝑛 )‖𝑔‖,

𝑛 ≥ 1,

where the equivalence constants are independent of 𝑔, 𝑛, and 𝑋. Proof. Let 𝑘 = 𝜆 𝑛−1 and 𝑚 = 𝜆 𝑛+1 . Then 𝑚

JΨ 𝑔 = ∑ 𝐴 𝑗 𝑔[𝑗] ,

where 𝐴 𝑗 = Ψ(𝑗 + 1).

𝑗=𝑘

Applying the Abel identity twice, we get 𝑚

𝑚

𝑗=0

𝑗=0

1 ∑ 𝛼𝑗 𝛽𝑗 = 𝛼𝑚+1 𝑠𝑚 𝛽 + (𝑚 + 1)Δ1 𝛼𝑚+1 𝜎𝑚 𝛽 + ∑ (Δ2 𝛼𝑗 )(𝑗 + 1)𝜎𝑗1 𝛽,

𝑗

where 𝑠𝑗 𝛽 = ∑𝜈=0 𝛽𝑗 . Taking 𝛼𝑗 = 𝐴 𝑗 and 𝛽𝑗 = 𝑔[𝑗] we obtain 𝑚

𝑚

𝑗=𝑘

𝑗=𝑘

1 𝑔 + 𝐴 𝑚+1 𝑔, JΨ 𝑔 = ∑ 𝐴 𝑗 𝑔[𝑗] = ∑ Δ2 𝐴 𝑗 (𝑗 + 1)𝜎𝑗1 𝑔 + (𝐴 𝑚+1 − 𝐴 𝑚+2 )(𝑚 + 1)𝜎𝑚

(11.18)

where we have used the relations 𝑠𝑗 𝑔 = 𝜎𝑗1 𝑔 = 0 for 𝑗 < 𝑘, and 𝑠𝑚 𝑔 = 𝑔. Hence, by Lemma 11.3, 𝑚

‖JΨ 𝑔‖ ≤ ∑ |Δ2 𝐴 𝑗 |(𝑗 + 1)‖𝑔‖ + (𝐴 𝑚+2 − 𝐴 𝑚+1 )(𝑚 + 1)‖𝑔‖ + 𝐴 𝑚+1 ‖𝑔‖. 𝑗=𝑘

Letting 𝑎𝑗 = 1/𝐴 𝑗 we have Δ2 𝐴 𝑗 = −𝐴 𝑗 𝐴 𝑗+2 Δ2 𝑎𝑗 + 2𝐴 𝑗 (𝐴 𝑗+2 − 𝐴 𝑗+1 )(𝑎𝑗 − 𝑎𝑗+1 ).

11.5 Multipliers of spaces with subnormal weights |

343

Since 1/Ψ is convex, we see that, for 𝑥 > 1 fixed, the function 𝐹𝑥 (𝑡) :=

1 Ψ(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)

(11.19)

≤ 2/(𝑗 + 2)Ψ(1 + 𝑗/2) ≤ 𝐶𝑎𝑗 /(𝑗 + 1). Here we have used the relation Ψ(1 + 𝑗/2) ≍ Ψ(1 + 𝑗), which holds because Ψ is subnor­ mal. On the other hand, we have Δ2 𝑎𝑗 ≥ 0 because 1/Ψ is convex. From this, (11.19), and the identity 𝑚

∑ (Δ2 𝑎𝑗 )(𝑗 + 1) = (𝑎𝑘 − 𝑎𝑘+1 )𝑘 − (𝑎𝑚+1 − 𝑎𝑚+2 )(𝑚 + 1) + 𝑎𝑘 − 𝑎𝑚+1

(11.20)

𝑗=𝑘

we obtain 𝑚

𝑚

∑ |Δ2 𝐴 𝑗 |(𝑗 + 1) ≤ 𝐴 𝑚 𝐴 𝑚+2 ∑ (Δ2 𝑎𝑗 )(𝑗 + 1) 𝑗=𝑘

𝑗=𝑘 𝑚

+ 𝐶 ∑ 𝐴 𝑗 (𝐴 𝑗+2 − 𝐴 𝑗+1 )𝑎𝑗 𝑗=𝑘

≤ 𝐴 𝑚 𝐴 𝑚+2 [(𝑎𝑘 − 𝑎𝑘+1 )𝑘 + 𝑎𝑘 ] + 𝐶(𝐴 𝑚+2 − 𝐴 𝑘+1 ) ≤ 𝐴 𝑚 𝐴 𝑚+2 (𝐶𝑎𝑘 + 𝑎𝑘 ) + 𝐶𝐴 𝑚+2 ≤ 𝐶Ψ(𝜆 𝑛 ). In the same way we get (𝑚 + 1)(𝐴 𝑚+2 − 𝐴 𝑚+1 ) = 𝐴 𝑚+1 𝐴 𝑚+2 (𝑎𝑚+1 − 𝑎𝑚+2 )(𝑚 + 1) ≤ 𝐶𝐴 𝑚+1 𝐴 𝑚+2 𝑎𝑚+1 = 𝐶𝐴 𝑚+2 ≤ 𝐶Ψ(𝜆 𝑛 ). From these inequalities and (11.18), we obtain 𝑚

‖JΨ 𝑔‖ = ‖ ∑ 𝐴 𝑗 𝑔[𝑗] ‖ ≤ 𝐶Ψ(𝜆 𝑛 )‖𝑔‖. 𝑗=𝑘

In the other direction, let ℎ =

∑𝑚 𝑗=𝑘

𝐴 𝑗 𝑔[𝑗] . Then 𝑔 = ∑𝑚 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(𝜆 𝑛 )−1 ‖ℎ‖, where we have used (11.19) and (11.20). This completes the proof.

344 | 11 Coefficients multipliers Remark 11.5. The validity of (11.20) can be verified by induction on 𝑚. 𝑝󸀠 ,𝑞󸀠

𝑝,𝑞

Proof of Theorem 11.26. Let X = ℎ[𝜙] and Y = ℎ[𝜙] . We apply Hölder’s inequality to 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 (11.17) converges absolutely. To prove the converse, let 𝐿 ∈ X󸀠 . We extend 𝜙 to (0, ∞) by 1 −2

𝜙(𝑡)

1

= 𝑐∫𝑟

2(𝑡−1)

2

𝜙(1 − 𝑟) 𝑑𝑚𝜙 (𝑟) = 𝑐 ∫ 𝑟2(𝑡−1) 𝜙(1 − 𝑟)𝜙󸀠 (1 − 𝑟) 𝑑𝑟,

0

𝑡 > 1,

0

where 𝑐 is chosen so that 𝜙(1+) = 𝜙(1). Applying Lemma 11.5 with Ψ = 𝜙2 we see that 𝜙(1/𝜆 𝑛 ) ≍ 1/𝜙(𝜆 𝑛). Since the function 1/𝜙2 is convex we can apply Proposition 11.22 2

𝑞󸀠

󸀠

and Lemma 11.5 to conclude that there exists a unique ℎ such that J𝜙 ℎ ∈ VC [ℎ𝑝 ] = 󸀠

󸀠

2

𝑝 ,𝑞

ℎ[𝜙] , C = {𝜙(1/𝜆 𝑛)}, and 𝐿𝑓 = (2𝜋/𝑐)⟨𝑓, ℎ⟩. Letting 𝑔 = J𝜙 ℎ/𝑐 we have 1

̂ ̂ ∫ 𝑟2|𝑛| 𝜙(1 − 𝑟)2 𝑑𝑚𝜙 (𝑟) ℎ(𝑛) = 𝑔(𝑛) 0

and hence 1

∞

̂ 𝑔(𝑛) ̂ ∫ 𝑟2|𝑛| 𝜙(1 − 𝑟)𝜙󸀠 (1 − 𝑟) 𝑑𝑟 𝐿𝑓 = 2𝜋 ∑ 𝑓(𝑛) 𝑛=−∞

0

2𝜋 1

= ∫ ∫ 𝑓(𝑟𝑒𝑖𝜃 )𝑔(𝑟𝑒−𝑖𝜃 )𝜙(1 − 𝑟)𝜙󸀠 (1 − 𝑟) 𝑑𝜃 𝑑𝑟 0 0

= (𝑓, 𝑔)𝜙 𝑝,𝑞

for all harmonic polynomials 𝑓 ∈ ℎ[𝜙] . Since such polynomials are dense in this space, we have 𝐿𝑓 = (𝑓, 𝑔)𝜙 for all 𝑓 ∈

𝑝,𝑞 ℎ[𝜙] .

This completes the proof.

Some special cases deserve special attention. The space ℎ∞ (𝜓) = {𝑓 ∈ ℎ(𝔻) : . . . } was defined on page 266 by the requirement 1 )) , 𝑧 ∈ 𝔻, 1 − |𝑧| where 𝜓 > 0 is a subnormal majorant on [1, ∞) such that 𝜓(∞) = ∞. For a positive measure 𝜇 on 𝔻, let |𝑓(𝑧)| ≤ 𝐶 (𝜓 (

{ } 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]

11.5 Multipliers of spaces with subnormal weights |

345

Theorem 11.27. 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 . Example 11.2. Let 𝜓(𝑡) = (1 + log 𝑡)𝛾 , 𝛾 > 0. Then we can take 𝑑𝜇(𝑧) = (log

−𝛾−1 𝑒 𝑑𝐴(𝑧) ) , 1 − |𝑧| 1 − |𝑧|

(𝑓, 𝑔)𝜙 = ∫ 𝑓(𝑧)𝑔(𝑧)̄ (log 𝔻

−2𝛾−1 𝑑𝐴(𝑧) 𝑒 ) . 1 − |𝑧| 1 − |𝑧|

Another special case concerns weighted harmonic Bergman spaces. Theorem 11.28. Let 𝜑 be a Bergman weight such that (1 − 𝑟)𝜑(𝑟) is almost decreas­ ing, 𝜑(𝑟)/(1 − 𝑟)𝛾 is almost increasing in 𝑟 for some 𝛾 > −1. If 1 < 𝑝 < ⬦, then the 󸀠 dual of the Bergman space A𝜑𝑝 is isomorphic to A𝜑𝑝 under the duality pairing [𝑓, 𝑔]𝜑 = ̄ 𝑑𝐴(𝑧). ∫𝔻 𝑓(𝑧)𝑔(𝑧)𝜑(𝑧) 1

𝑝,𝑝

Proof. This is proved by taking 𝜙(𝑥) = (∫1−𝑥 𝜑(𝑟) 𝑑𝑟)1/𝑝 , so that A𝜑𝑝 = ℎ[𝜙] , then making 𝑝,𝑝

suitable substitutions and using the decomposition of ℎ[𝜙] into the blocks Δ 𝑛,𝜆 𝑓 = ∑ 𝑓[𝑗] 𝑗∈𝐼𝑛,𝜆

where 𝐼0,𝜆 = [0, 𝜆 0 )

and 𝐼𝑛,𝜆 = [𝜆 𝑛 , 𝜆 𝑛+1 )

for 𝑛 ≥ 1.

In this situation, the technique is much simpler because the sequence {𝑒𝑛 }∞ −∞ is a 𝑝,𝑞 Schauder basis in ℎ[𝜙] for 1 < 𝑝 < ⬦ and because if 1 < 𝑝 < ⬦, then Theorem 11.22 remains true if we replace 𝑉𝑛,𝜆 with Δ 𝑛,𝜆 .

Multipliers A careful examination of the proof of Theorem 11.18 shows that the “length” of the polynomials 𝑉𝑛 is irrelevant. So minor modifications of this proof lead to the following theorem. Theorem 11.29. Let C and A be two normal sequences, 𝑋 and 𝑌 ℎ-admissible spaces, and ‖V𝑛 ∗ 𝑓‖𝑋 ≤ 𝐶‖𝑓‖𝑋 . Then 𝑞

𝑞

𝑞⊖𝑞

(VC [𝑋], VA1 [𝑌]) = VA/C1 [(𝑋, 𝑌)]. From this Theorem 11.22 we obtain: Theorem 11.30. Let Λ = {𝜆 𝑛}∞ 0 (𝜆 0 ≥ 1), and let 𝜙 and 𝜓 be two subnormal functions such that the sequences C = {𝜙(1/𝜆 𝑛)} and A = {𝜓(1/𝜆 𝑛)} are normal. Let 𝑋 and 𝑌 be

346 | 11 Coefficients multipliers ℎ-admissible spaces such that ‖V𝑛 ∗ 𝑓‖𝑋 ≤ 𝐶‖𝑓‖𝑋 . Then 𝑞

𝑞

𝑞⊖𝑞

1 ) ≃ VA/C1 [(𝑋, 𝑌)]. (ℎ𝑋,[𝜙] , ℎ𝑌,[𝜓]

𝑝,𝑞

𝑝 ,𝑞

𝑞⊖𝑞

1 1 ) ≃ VA/C1 [(ℎ𝑝 , ℎ𝑝1 )] for 𝑝 ≥ 1, 𝑝1 ≥ 1. In particular (ℎ[𝜙] , ℎ[𝜓]

Since (ℎ1 , ℎ1 ) ≅ (ℎ∞ , ℎ∞ ) ≅ ℎ1 we have 1,𝑞

1,𝑞

∞,𝑞

∞,𝑞

(ℎ[𝜙] , ℎ[𝜙] ) ≃ (ℎ[𝜙] , ℎ[𝜙] ) ≃ V∞ [ℎ1 ],

(11.21)

where V∞ [ℎ1 ] consists of those 𝑢 ∈ ℎ(𝔻) for which (11.22)

sup ‖V𝑛,𝜆 ∗ 𝑢‖1 < ∞. 𝑛≥0

∞,𝑞

1,𝑞

Using (11.21) we can characterize those 𝜙 for which ℎ[𝜙] and ℎ[𝜙] are self-conjugate. Theorem 11.31. Let 0 < 𝑞 ≤ ∞, and 𝜙 subnormal. Then the following three conditions ∞,𝑞 1,𝑞 are equivalent: (a) ℎ[𝜙] is self-conjugate; (b) ℎ[𝜙] is self-conjugate; (c) The function 𝜙 is normal. 𝑝,𝑞

𝑝,𝑞

Proof. If 𝜙 is normal, then ℎ[𝜙] = ℎ𝜙 and hence these spaces are self-conjugate, by Corollary 3.6. If one of the spaces under consideration is self-conjugate, then the func­ tion 𝑢(𝑧) = (1 − 𝑧)−1 satisfies (11.22). This implies, via Hardy’s inequality, that ∞

̂ 𝑛,𝜆 (𝑘)/𝑘 ≤ 𝐶, ∑V 𝑘=1

where 𝐶 is independent of 𝑛. Since the sum on the left is greater or equal to a constant multiple log(𝜆 𝑛 /𝜆 𝑛−1 ), we conclude that {𝜆 𝑛} is normal, which leads to the conclusion that 𝜙 is normal. By means of Theorem 11.31 we can extend Privalov’s conjugate functions theorem, and moreover Theorem 9.4, in the following way. Theorem 11.32. Let 0 < 𝑞 ≤ ∞, and let 𝜔(𝑡), 0 ≤ 𝑡 ≤ 1, be a fast majorant of order 𝑛 satisfying (9.25). Then the following statements are equivalent: – The Hilbert operator is bounded on Λ∞,𝑞 𝜔,𝑛 . 1,𝑞 – The Hilbert operator is bounded on Λ 𝜔,𝑛 . – 𝜔 is a slow majorant of order 𝑛. Proof. Let 𝑞 < ⬦. If 𝜔 is regular of order 𝑛, then Λ𝑝,𝑞 𝜔,𝑛 can be identified, via the Poisson integral, with the Besov space B𝜔𝑝,𝑞 and is therefore self-conjugate, by Corollary 3.6. In the other direction, we appeal to Theorem 9.11 to reduce the implication to the follow­ 𝑝,𝑞 , 𝑝 ∈ {1, ∞} (see (9.24)), is self-conjugate, then 𝜔 is regular of order 𝑛. ing: if T𝜔,𝑛 Assume, as we may, that 𝜔 ∈ Δ[𝛼, 𝑛] for some 𝛼 > 0; see Proposition 3.9. We can 𝑝,𝑞 𝑝,𝑞 identify T𝜔,𝑛 with ℎ[𝜙] , where 𝑥

1

0

0

𝑡𝑛𝑞−1 𝑡𝑛𝑞−1 𝑛𝑞 𝜙(𝑥) = ∫ 𝑑𝑡 = 𝑥 ∫ 𝑑𝑡. 𝜔(𝑡)𝑞 𝜔(𝑥𝑡)𝑞 𝑞

11.5 Multipliers of spaces with subnormal weights

| 347

𝑝,𝑞 The function 𝜙 is subnormal because the majorant 𝑡𝛼 /𝜔(𝑡) decreases in 𝑡. Hence, if T𝜔,𝑛 𝑞 𝑞 󸀠 𝑞 is self-conjugate, then 𝜙 is normal, which implies (𝜙(𝑥) ) ≍ 𝑥𝜙(𝑥) , i.e. 1

𝜙(𝑥)𝑞 1 𝑡𝑛𝑞−1 =∫ 𝑑𝑡 ≍ , 𝑛𝑞 𝑥 𝜔(𝑥𝑡)𝑞 𝜔(𝑥)𝑞

0 < 𝑥 < 1.

0

Since 𝜙(𝑥)/𝑥𝛾 increases in 𝑥 for some 𝛾 > 0, we see that 𝜔(𝑥)/𝑥𝑛−𝛾 is decreasing. This proves the theorem in the case 𝑞 < ∞. We omit the (similar and simpler) proof in the remaining cases.

𝑝,𝑞

Multipliers of 𝐻[𝜙] The analytic analog of Theorem 11.22 holds not only for homogeneous Banach spaces but also for 𝑋 = 𝐻𝑝 , 𝑝 > 0, because of Proposition 11.10(iv) and Lemma 2.2. More­ over, this analog 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 the inequality 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 𝑘 󵄩 𝑚󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑟 󵄩󵄩 ∑ 𝑎𝑘 𝑒𝑘 󵄩󵄩 ≤ 󵄩󵄩 ∑ 𝑎𝑘 𝑟 𝑒𝑘 󵄩󵄩 ≤ 𝑟 󵄩󵄩 ∑ 𝑎𝑘 𝑒𝑘 󵄩󵄩󵄩󵄩 , 󵄩󵄩𝑗=𝑚 󵄩󵄩 󵄩󵄩𝑗=𝑚 󵄩󵄩 󵄩󵄩𝑗=𝑚 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑛

0 < 𝑟 < 1,

holds, which together with Lemmas 12.1 and 3.12 can be used to prove the analog of Theorem 11.22. If the quasinorm in an arbitrary quasi-Banach space is plurisubharmonic, then it is called a PL-convex space (see [105], where the term “PL-convex” was introduced, and [359, 363], 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. 𝑝,𝑞 If 1 < 𝑝 < ⬦, then the above theorems on ℎ[𝜙] extend to the analytic spaces in the 𝑝,𝑞

obvious way. If 𝑝 ∈ (0, 1]∪{⬦}, the structure of (𝐻[𝜙] )𝐴 is complicated. For instance, we 1,𝑞

𝑞󸀠

have (𝐻[𝜙] )𝐴 = 𝐸BMOA,[𝜙] , 𝑞 ≤ ⬦. This is a consequence of Fefferman’s duality theorem and the general formula 𝑞

𝑞󸀠

(𝐻𝑋,[𝜙] )𝐴 = 𝐸𝑌,[𝜙] ,

where 𝑌 = (𝑋P )𝐴 , and 𝑞 ≤ ⬦,

where 𝑋 is a PL-convex space and in particular 𝑋 = 𝐻𝑝 , 𝑝 > 0. Theorem 11.29 remains true if we assume that 𝑋 and 𝑌 are 𝐻-admissible spaces. We have in particular 𝑞 𝑝⊖𝑞 (V𝑝 [𝑋], VA [𝐻∞ ]) = VA [𝑋∗ ], 𝑝

where 𝑋 is a PL-convex admissible space, V𝑝 [𝑋] = VA [𝑋], and A = (1, 1, . . . ). Exercise 11.9. 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

348 | 11 Coefficients multipliers by the requirement ‖𝑓𝑟 ‖𝑌 = 𝑂 (𝜓 (

1 )) . 1−𝑟

󸀠

In particular, (𝐻𝑝 , 𝐻∞ (𝜓)) = 𝐻𝑝 (𝜓) for 1 < 𝑝 < ⬦.

11.6 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 only if 󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 𝑛−1 󸀠 ̂ 𝜑 (𝑧)󵄨󵄨󵄨 ≤ 𝐶‖𝑓‖𝑋 (1 − |𝑧|2 )−1 , 𝑧 ∈ 𝔻, 󵄨󵄨󵄨 ∑ 𝑓(𝑛)𝑛𝜑(𝑧) 󵄨󵄨 󵄨󵄨𝑛=0 󵄨 󵄨 where 𝐶 depends neither of 𝑓 nor of 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 𝑋𝐴 . If 𝑋 is homogeneous, then we can say somewhat more. Proposition 11.11. If 𝑋 is homogeneous, then 𝐶𝜑 maps 𝑋 into B if and only if the family of multiplier transforms defined by 𝑀|𝜑|,𝑧 (𝑛) = 𝑛|𝜑(𝑧)|𝑛−1 |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 se­ quence with special properties. Using this fact one can characterize composition op­ erators from 𝐻𝑝 to B for all 𝑝 < ⬦. For example, we have: Theorem 11.33. The operator C𝜑 maps 𝐻1 into B if and only if sup 𝑛2 |𝜑(𝑧)|𝑛−1 |𝜑󸀠 (𝑧)|(1 − |𝑧|2 ) < ∞. 𝑛≥1, 𝑧∈𝔻

Proof. By the above proposition we have to compute ‖𝑀|𝜑|,𝑧 ‖BMO . Since ̂ ‖𝑔‖BMO ≤ 𝐶 sup (𝑛 + 1)|𝑔(𝑛)|, 𝑛

we have ‖C𝜑‖ ≤ 𝐶‖𝑀|𝜑|,𝑧 ‖BMO ≤ 𝐶 sup 𝑛2 |𝜑(𝑧)|𝑛−1 |𝜑󸀠 (𝑧)|(1 − |𝑧|2 ). 𝑛

(11.23)

11.6 Some applications to composition operators

|

349

On the other hand, ‖𝑀|𝜑|,𝑧 ‖BMO ≥ 𝑐‖𝑀|𝜑|,𝑧 ‖B ≥ 𝑐 sup ‖𝑉𝑘 ∗ 𝑀|𝜑|,𝑧 ‖∞ 𝑘≥0

𝑘

≥ 𝑐 sup 2 |𝜑(𝑧)| 𝑘

2𝑘+1

|𝜑󸀠 (𝑧)|(1 − |𝑧|2 )‖𝑉𝑘 ‖∞

𝑘+1

≥ 𝑐 sup 22𝑘 |𝜑(𝑧)|2 |𝜑󸀠 (𝑧)|(1 − |𝑧|2 ). 𝑘

The result follows. Observe that condition (11.23) can be written as sup𝑛 𝑛‖𝜑𝑛 ‖B < ∞. More generally, C𝜑 maps 𝐻𝑝 into B (𝑝 ≤ 1) if and only if sup 𝑛1/𝑝 ‖𝜑𝑛 ‖B < ∞.

(11.24)

𝑛

In the case 𝑝 > 1, the corresponding condition reads (1 − |𝜑(𝑧)|)−1/𝑝−1 |𝜑󸀠 (𝑧)| ≤ 𝐶(1 − |𝑧|2 )−1 ,

(11.25)

which works for 𝑝 ≤ 1 as well [394]. However, a simple analysis shows that the last two conditions are equivalent. Therefore, the following holds: Theorem 11.34. The operator C𝜑 maps 𝐻𝑝 (0 < 𝑝 < ⬦) into B if and only if one of the conditions (11.24), (11.25) is satisfied. Exercise 11.10. It might be interesting to consider composition operators from B𝑝,𝑞 𝛼 into 𝐻∞ , B, or 𝐻Λ 𝛽 , 0 < 𝛽 < 1.

Further notes and results Various conditions on spaces of analytic functions were considered by Taylor [486]; a wide list can be found in [70]. Our requirements for a space to be homogeneous or FP are somewhat stronger than that in [70]. Proposition 11.5 is in fact a special case of Kalton’s theorem (to be stated in Chapter 12) on linear operators from 𝐻𝑝 (0 < 𝑝 < 1) to an arbitrary 𝑞-Banach space 𝑝 < 𝑞 ≤ 1. Blasco [61, 62] was the first who used Kalton’s ideas in considering coefficient multipliers, although he considered the case of Ba­ nach spaces, when the atomic decomposition need not be used. The reader should also consult his papers [63, 64] for various results on mixed norm spaces. 𝑞 The inclusion B1/𝑝−1 ⊂ (𝐻𝑝 , 𝐻𝑞 ) (of Theorem 11.8) when 𝑝 < 1 ≤ 𝑞 was stated without proof by Hardy and Littlewood [191, 193]. In [134], Duren and Shields proved the reverse inclusion for 𝑞 ≥ 1; the general case was published in [329]. Theorems 11.8 and 11.10 are taken from [329]. Theorem 11.9 is due to Nowak [348]. All these results were proved in a way which differs from that used in the text. Theorem 11.11 is only

350 | 11 Coefficients multipliers a reformulation of the relation (𝐻1 , 𝐻𝑝 ) = B𝑝,∞ , 2 ≤ 𝑝 < ⬦, due to Hardy and Little­ wood [194] (the inclusion (𝐻1 , 𝐻𝑝 ) ⊃ B𝑝,∞ ), and Stein and Zygmund [465] (the reverse inclusion). It is trivial that (𝐻𝑞 , 𝐻𝑝 ) = ℓ∞ for 𝑞 ≥ 2 ≥ 𝑝. As far as the author knows the set (𝐻𝑝 , 𝐻𝑞 ) (0 < 𝑝, 𝑞 < ⬦) in other case has not been determined yet. (Various sufficient conditions are known and are of great importance; see Stein [461, 462].) In particular, it is not known what is (𝐻𝑝 , 𝐻𝑞 ) for 2 > 𝑝 ≥ 𝑞 > 0. The notion of a solid space and the notations 𝑠(𝑋) and 𝑆(𝑋) were introduced by Anderson and Shields [26]. Theorems 11.14 and 11.15 were proved in [231]; see also [232] and Mastyło–Mleczko [323]. In the case 𝑞 ≥ 𝑝, Theorem 11.16 was proved by Duren and Shields [134], and in the general case – by Jevtić and Pavlović [228]. Theorem 11.19 is a reformulation of [360, Theorem 4.1]; in the case when 𝑝 ≤ 1 ≤ 𝑝1 and 𝑞 = 𝑞1 , it appears in [64, Theorem 5.4]. Theorem 11.21 is perhaps new. As far as the author knows, the only paper in which monotone multipliers are considered is that of Buckley et al. [78]. Also, there are only a few papers devoted to compact multipliers; we note Buckley–Ramanujan–Vukotić [79] and Mleczko [336]. Theorem 11.5 and some of its consequences are new. The decomposition method of finding multipliers between mixed norm spaces with subnormal weights were introduced in [360], but polynomials more complicated than 𝑉𝑛 were used. Besides, these papers contain many typos and several mistakes, which make them rather difficult to read; nevertheless, the results are formulated cor­ rectly. Subsequently part of [360], for the standard weights, was simplified in [229]. The ideas of Section 11.5 are from [360], but the proofs (for example of Theo­ rem 11.31) are simplified. Theorems 11.24, 11.26, and 11.32 are new. 11.1 (Tensor product [70]). Let 𝑋 and 𝑌 be normed 𝐻-admissible spaces. We define the space 𝑋 ⊗ 𝑌 to be the set of all ℎ ∈ 𝐻(𝔻) that can be represented in the form ℎ = ∑𝑛 𝑓𝑛 ∗ 𝑔𝑛 , 𝑓𝑛 ∈ 𝑋, 𝑔𝑛 ∈ 𝑌 so that the series converges in 𝐻(𝔻) and ∑ ‖𝑓𝑛 ‖𝑋 ‖𝑔𝑛 ‖𝑌 < ∞ 𝑛

The norm in 𝑋 ⊗ 𝑌 is given by ‖ℎ‖𝑋⊗𝑌 = inf ∑ ‖𝑓𝑛 ‖𝑋 ‖𝑔𝑛 ‖𝑌 , 𝑛

where the infimum is taken over all the above representations. The importance of this notion lies in the formula (𝑋, (𝑌, 𝑍)) = (𝑋⊗𝑌, 𝑍). Taking 𝑍 = 𝐻∞ or 𝑍 = 𝐴(𝔻) and imposing some additional hypotheses on 𝑋 and (or) 𝑌 one can describe some duality relations between tensor products and spaces of multipliers; see [70, Theorem 5.3]. 11.2. Theorem 11.27 is a generalization and improvement of a result of Shields and Williams [443], who used two rather strong additional hypotheses:

11.6 Some applications to composition operators

–

| 351

There is a positive finite Borel measure 𝑑𝜂 on [0, 1) such that 1

1 = ∫ 𝑟2𝑛 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.

11.3 ([360, 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 iso­ morphic via a multiplier transform. This is a solution to Problem B of [443]. 11.4. Using Theorem 11.6, one proves that if 0 < 𝑝, 𝑞 ≤ 1, 𝛼 > 0, and max{𝑝, 𝑞} ≤ 󰜚 ≤ 1, 󰜚,󰜚 󰜚 then the 󰜚-Banach envelope of 𝐻𝛼𝑝,𝑞 is equal to 𝐻𝛼+1/𝑝−1/󰜚 = 𝐴 𝛽 , where 𝛽 = 󰜚𝛼 + 󰜚/𝑝 − 2. Therefore, by Proposition 11.4, (𝐻𝛼𝑝,𝑞 , 𝑌) = B∞ 𝛼+1/𝑝−1 (𝑌), for a 𝜌-Banach space 𝑌. This was proved by Blasco [64, Theorem 3.2] for 󰜚 = 1. This paper and Yue’s paper [522] contain theorems on multipliers from 𝐻𝛼𝑝,𝑞 into 𝐻𝑠 or 𝐴𝑠 . Blasco proved his results for 𝑠 ≥ 1, while Yue requires only that 𝑠 ≥ max{𝑝, 𝑞}.

12 Toward a theory of vector-valued spaces In this chapter, we consider the simplest case of the Coifman–Rochberg theorem on the atomic decomposition of Bergman spaces (Theorem 12.1) and its applications to homogeneous admissible spaces and, more generally, to bounded vector-valued ana­ lytic functions. A theory of vector-valued Besov and 𝐻𝑝 -spaces (𝑝 < 1) is sketched. The Banach envelope and the dual of 𝐻𝑝 (𝑋) is determined for 𝑝 < 1. Some applications to composition operators are also presented.

12.1 Some properties of admissible spaces If 𝑋 is a homogeneous 𝐻-admissible space and 𝑓 ∈ 𝑋, then the vector-valued function ∞

𝑛 ̂ 𝐹(𝑧) = 𝐹𝑓 (𝑧) = ∑ 𝑓(𝑛)𝑒 𝑛𝑧 ,

𝑧 ∈ 𝔻,

𝑛=0

is analytic, bounded, i.e. belongs to 𝐻∞ (𝑋), and ‖𝐹‖∞,𝑋 =: sup𝑧∈𝔻 ‖𝐹(𝑧)‖𝑋 = ‖𝑓‖𝑋 . 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 ̂ such unit disk, it turns out that the analyticity implies the existence of vectors 𝐹(𝑛) that ∞ ̂ 𝑧𝑛 , for all 𝑧 ∈ 𝔻, 𝐹(𝑧) = ∑ 𝐹(𝑛) (12.1) 𝑛=0

with uniform convergence on compact subsets. However, the proof of this fact is rather difficult. In order to avoid this path, we can agree that 𝐹 is analytic in 𝔻 if there are vec­ ̂ such that (12.1) holds. These vectors are uniquely determined and are called tors 𝐹(𝑛) 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

(12.2)

𝑛→∞

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 func­ tions 𝐹 : 𝔻 󳨃→ 𝑋 analytic in 𝔻 can be identified with the set of the formal power series satisfying (12.2), i.e. with the set of the power series converging in 𝔻. We will denote

12.1 Some properties of admissible spaces |

353

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 spaces 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 [15], the (nonconstant) function 𝑣 is analytic in 𝔻, continuous on 𝔻, and vanishes on 𝕋. A characteristic property of infinite-dimensional spaces is the failure of Mon­ tel’s theorem. Namely, if 𝑋 is infinite-dimensional, then the unit sphere is not totally bounded, which implies the existence of a sequence {𝑓𝑛} on the unit sphere such that inf 𝑚=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 ∞ ̂ 𝑧𝑛 . ̂ 𝐹(𝑛) (𝜓 ∗ 𝐹)(𝑧) = ∑ 𝜓(𝑛) 𝑛=0

The proof that 𝜓 ∗ 𝐹 belongs to 𝐻(𝔻, 𝑋) is straightforward. Proposition 12.1. 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) = ∑𝑁 𝑗=1 (𝜓𝑗 ∗ 𝐹)(𝑧), and 𝑅𝑁 (𝑧) = ∑𝑘=𝑁+1 𝜓𝑘 (𝑧). Then ∞

‖𝐹𝑁 (𝑧) − 𝜓 ∗ 𝐹(𝑧)‖𝑝 = ‖𝑅𝑁 ∗ 𝐹(𝑧)‖𝑝 ≤ ∑ 𝐴 𝑁 (𝑗),

(12.3)

𝑗=0 𝑝 ̂ 𝑁 (𝑗)|𝑝 ‖𝐹(𝑗)‖ ̂ where 𝐴 𝑁 (𝑗) = |𝑅 |𝑧|𝑗𝑝 . The sequence 𝑅𝑁 (𝑧) is uniformly bounded on compact subsets and therefore for every 𝜌 < 1 there exists a constant 𝐶𝜌 such that ̂ 𝑁 (𝑗)| ≤ 𝐶𝜌 /𝜌𝑗 for all 𝑁 and 𝑗. From this and the inequality ‖𝐹(𝑗)‖ ̂ |𝑅 ≤ 𝐾𝜌 /𝜌𝑗 , we 2 𝑗𝑝 𝑝 𝑝 2 get 𝐴 𝑁 (𝑗) ≤ 𝑀(𝜌)(𝑟/𝜌 ) where 𝑀𝜌 = 𝐶󰜚 𝐾𝜌 . Thus by taking 𝜌 = √𝑟, we see that the sequence 𝐴 𝑁 has a summable majorant. On the other hand, from the hypothesis 𝑅𝑁 (𝑧) → 0, uniformly on compact subsets, it follows 𝐴 𝑁 (𝑗) → 0 (𝑁 → ∞), for every 𝑗. Therefore, we can apply the dominated convergence theorem: ∞

∞

lim ∑ 𝐴 𝑁 (𝑗) = ∑ lim 𝐴 𝑁 (𝑗) = 0.

𝑁→∞

𝑗=0

𝑗=0

𝑁→∞

Now the desired assertion follows from (12.3).

354 | 12 Toward a theory of vector-valued spaces The above proof is independent of the completness 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| −1. Then there exists a sequence {𝑤𝑛} in 𝔻 and a constant 𝐶 such that for every 𝑓 ∈ 𝐴𝑝 there exists a sequence {𝑎𝑛} ⊂ ℓ𝑝 with the properties ∞

𝑓(𝑧) = ∑ 𝑎𝑛 𝑛=1 ∞

𝑝

1 − |𝑤𝑛|2 , (1 − 𝑤̄ 𝑛 𝑧)2/𝑝+1

𝛽

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

(12.5)

𝔻

Postponing the proof to Section 12.4 we prove inequality (12.4) and some related in­ equalities. Theorem 12.2. Let 𝐹 ∈ 𝐻∞ (𝑋), where 𝑋 is a 𝑝-Banach space, 0 < 𝑝 ≤ 1. Then there exists a constant 𝐶𝑝 such that (12.4) holds. Proof. Let 𝜓 be a scalar-valued analytic function belonging to 𝐿𝑝 (𝔻). According to Theorem 12.1, we have ∞ 1 − |𝑤𝑘 |2 𝜓(𝑧) = ∑ 𝑎𝑘 , (1 − 𝑤𝑘 𝑧)2/𝑝+1 𝑘=1 where {𝑤𝑘 } is a sequence in 𝔻 and {𝑎𝑘 } is a sequence of complex numbers such that ‖{𝑎𝑘 }‖𝑝 ≤ 𝐶𝑝 ‖𝜓‖𝐿𝑝 (𝔻) , where 𝐶𝑝 is independent of 𝜓 (𝑤𝑘 are independent of 𝜓 as well). The series converges uniformly on compact subsets so we can apply Proposition 12.1; we get ∞

𝜓 ∗ 𝐺(𝑧) = ∑ 𝑎𝑘 (1 − |𝑤𝑘 |2 )𝐷[2/𝑝] 𝐺(𝑤𝑘 𝑧) 𝑘=1

(𝑧 ∈ 𝔻),

(12.6)

12.1 Some properties of admissible spaces |

355

where 𝐺 ∈ 𝐻(𝔻, 𝑋), while 𝐷[𝑠] is defined as in the scalar-valued case; see (5.38) and (5.39). Now let 𝐺 = 𝐷[2/𝑝] 𝐹 and put 𝜓(𝑧) = 𝑧𝑛 in (12.6) to obtain ∞

𝑛 ̂ = ∑ 𝑎𝑘 (1 − |𝑤𝑘 |2 )𝐹(𝑤𝑘 𝑧), 𝐵𝑛 𝐹(𝑛)𝑧 𝑘=1

where 𝐵𝑛 = 𝛤(2/𝑝 + 1)𝛤(𝑛 + 1)/𝛤(𝑛 + 1 + 2/𝑝). Hence ∞

𝑝 𝑛𝑝 ̂ 𝐵𝑝𝑛 ‖𝐹(𝑛)‖ ≤ ∑ |𝑎𝑘 |𝑝 (1 − |𝑧𝑘 |2 )𝑝 ‖𝐹(𝑤𝑘 𝑧)‖𝑝 𝑋 |𝑧| 𝑘=1 ∞

𝑝

≤ ∑ |𝑎𝑘 |𝑝 (1 − |𝑤𝑘 |2 )𝑝 ‖𝐹‖∞,𝑋 , 𝑘=1

Since, by Theorem 12.1, ∑ |𝑎𝑘 |𝑝 (1 − |𝑤𝑘 |2 )𝑝 ≤ 𝐶 ∫ |𝑧|𝑛𝑝 (1 − |𝑧|2 )𝑝 𝑑𝐴(𝑧) ≤ 𝐶𝑛−𝑝−1 , 𝑘

𝔻

and 𝐵𝑛 ≍ 𝑛−2/𝑝 , we have the desired conclusion. Define the space 𝐻𝑝 (𝑋) by the requirement ‖𝐹‖𝑝,𝑋 := sup0 0, that have this good prop­ erty. In the general case, one can use the following substitute for PL-convexity in order to develop a relatively reach theory of vector Besov spaces. Theorem 12.4. If 𝐹 ∈ 𝐻(𝔻, 𝑋), where 𝑋 is an arbitrary quasi-Banach space, then the function 𝑧 󳨃→ ‖𝐹(𝑧)‖𝑋 is quasi-nearly subharmonic. For the proof, we need the following substitute for the maximum modulus principle. Lemma 12.4 (Kalton’s maximum principle). If 𝐹 ∈ 𝐻(𝔻, 𝑋), and 0 < 𝑟 < 1, then there is a constant 𝐶𝑟 such that ‖𝐹(0)‖𝑋 ≤ 𝐶𝑟 sup𝑟 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 and ≤ 𝐶1 1 − |𝑤|

|𝐷𝑛| 𝜀2 ≤ ≤ 𝐶1 𝜀2 𝐶1 (1 − |𝑤|)2

(𝑤 ∈ 𝐷𝑛 ).

(12.19)

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

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

where 𝑎𝑛 =

1 ∫ 𝑓(𝑤) 𝑑𝜇𝑝 (𝑤). 1 − |𝑤𝑛 |2 𝐷𝑛

Proceeding as in the proof of Lemma 12.5, one proves that 𝑇 maps 𝐴𝑝 into 𝐴𝑝 . In order to conclude the proof it suffices to prove that 𝑇 is an isomorphism for 𝜀 small

12.4 Proof of the Coifman–Rochberg theorem | 373

enough and that (12.5) holds. The proof of the latter is easy and is independent of (12.19). 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 12.6, 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 12.5 shows that ||𝑇𝑓 − 𝑓|| ≤ 𝐶𝑝 𝜀||𝑓||. Finally we take 𝜀 = 1/2𝐶𝑝 and apply Proposition A.6. Exercise 12.2. The partition E can be used to reduce the proof of the following result of Duren [127] (see [129, Theorem 9.4]) to a Theorem of Hardy and Littlewood (Corollary 4.1): Let 𝑑𝜇 be a finite positive measure on 𝔻, and 0 < 𝑝 < 𝑞 < ⬦. In order that there is a constant 𝐶 such that ∫ |𝑓(𝑧)|𝑝 𝑑𝜇(𝑧) ≤ 𝐶‖𝑓‖𝑞𝑝 , (12.20) 𝔻

it is necessary and sufficient that 𝜇(𝑊(𝐼)) ≤ 𝐶1 |𝐼|𝑞/𝑝 for every arc 𝐼 ⊂ 𝕋. Recall that 𝑊(𝐼) denotes the Carleson window over 𝐼. In fact, it is easy to see that the last condition is sufficient for the validity of the inequality ∫ |𝑓(𝑧)|𝑝 𝑑𝜇(𝑧) ≤ 𝐶 ∫ |𝑓(𝑧)|𝑞 (1 − |𝑧|)𝑞/𝑝−2 𝑑𝐴(𝑧). 𝔻

𝔻

Even a weaker condition is sufficient for the validity of (12.20): 𝜇(𝐸𝑗,𝑘 ) ≤ 𝐶2−𝑘𝑞/𝑝 , where 𝐶 is independent of 𝑗, 𝑘. That Duren’s theorem can be reduced to the Hardy–Lit­ tlewood theorem was first observed by Blasco [65]. Exercise 12.3 ([35, 198]). Let 0 < 𝜀 < 1, 𝑝 > 0, and 𝛼 > −1. For a positive finite mea­ sure 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 disk 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|)−𝛼−2 , ∫ |1 − 𝑎𝑧| 𝔻

𝑎 ∈ 𝔻.

374 | 12 Toward a theory of vector-valued spaces

Further notes and results 𝑝

It is the idea of Coifman and Rochberg [99] to represent a member of 𝐴 𝛽 as a sum of “atoms” by replacing the integral in (12.17) with a Riemannian sum over a sufficiently fine partition of the disk. 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. [525, Theorem 4.4.9] (𝑝 = 1) and [374, Theorem 8.3.1] (𝑝 ≤ 1). For Corollary 11.2, see [15, 99, 501, 132, 133] (𝑞 = 1). Concerning the theory of analytic functions with values in a topological linear space we refer the reader to Turpin [493] and Etter [146]. The main property of such functions is that if 𝐹 : Ω 󳨃→ 𝑋 is analytic and Ω0 ⊂ Ω is open and relatively compact in Ω, then there is a Banach space 𝑌, an analytic function 𝑔 : Ω0 󳨃→ 𝑌 and a bounded linear operator 𝑇 : 𝑌 󳨃→ 𝑋 such that 𝐹(𝑧) = 𝑇(𝑔(𝑧)), 𝑧 ∈ Ω0 . For further informa­ tion and references see [250]; see also [252] for the theory of vector-valued harmonic functions. Two papers of Arregui and Blasco [30, 67] can serve as a good introduction into the theory of the Bergman spaces of functions with values in a Banach space. 12.1. If 𝑝 < 1, then there is no bounded projection from 𝐿𝑝 = 𝐿𝑝 (𝔻, 𝑑𝐴) onto 𝐴𝑝 because the dual of 𝐿𝑝 is trivial. In [331], 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 pseudo𝜀

hyperbolic disc of radius 𝜀, and 𝑑𝜏 the invariant measure on 𝔻. 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 0 < 𝑝 < ∞. The following statement holds [331, Theorem 3.1]: Let 0 < 𝑝 < 1. The operator 𝑇𝑠 (see (12.18)) 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 [307], and later were used in Li–Luecking [293].

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 and, in the context of 𝐹-spaces, in A.4. 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 examples are Lebesgue spaces 𝐿𝑝 . When 𝑝 < 1, the functional ‖ ⋅ ‖ = ‖ ⋅ ‖𝐿𝑝 is not a norm but satis­ fies (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 declar­ ing “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 quasi­ norm, 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 quasi-Banach space. A 𝑝-Banach space is a complete 𝑝-normed spaces. A linear operator 𝑇 : 𝑋 󳨃→ 𝑌, where 𝑋 and 𝑌 are quasi-Banach 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.

376 | A Quasi-Banach spaces Theorem A.1 (Aoki–Rolewicz [424]). If ‖ ⋅ ‖ is a quasinorm on 𝑋, then there is 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 |||𝑓||| = inf {(∑ ‖𝑓𝑗 ‖𝑝 ) { 𝑗=1

1/𝑝

𝑛 } : 𝑓 = ∑ 𝑓𝑗 } , 𝑗=1 }

where the infimum is taken over all finite sequences {𝑓𝑗 } ⊂ 𝑋. A.1. In order 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 𝐸 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 quasinormed space 𝑋 to a quasinormed 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.1. 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.2. Let {𝑓𝑗𝑘 } (𝑗, 𝑘 ≥ 1) be a double sequence in a 𝑝-Banach space 𝑋. If ∑𝑗,𝑘 ‖𝑓𝑗𝑘 ‖𝑝 < ∞, then the iterated series ∞

∞

∑ ( ∑ 𝑓𝑗𝑘 ) 𝑗=1

𝑘=1

∞

and

∞

∑ (∑ 𝑓𝑗𝑘 ) 𝑘=1

𝑗=1

converge and have the same sum.

A.2 𝑞-Banach envelopes 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

A.2 𝑞-Banach envelopes | 377

the functional 𝑁𝑞 (0 < 𝑞 ≤ 1) on 𝑋 in the following way: { 𝑁𝑞 (𝑓) = inf {(∑ ‖𝑓𝑗 ‖𝑞 ) { 𝑗

1/𝑞

} : ∑ 𝑓𝑗 = 𝑓} , 𝑗 }

where the infimum is taken over the set of finite sequences {𝑓𝑗 } ⊂ 𝑋. This functional is a “𝑞-seminorm,” i.e. satisfies the conditions {𝑁𝑞 (𝑓 + 𝑔)}𝑞 ≤ {𝑁𝑞 (𝑓)}𝑞 + {𝑁𝑞 (𝑔)}𝑞 ,

𝑁𝑞 (𝜆𝑓) = |𝜆| 𝑁𝑞 (𝑓).

The set {𝑓 ∈ 𝑋 : 𝑁𝑞 (𝑓) = 0} =: Ker 𝑁𝑞 is a closed subspace of 𝑋. If Ker 𝑁𝑞 = {0}, i.e. 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 a 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 envelops of the Hardy space 𝐻𝑝 (see Corollary 11.2). 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.3. 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.4. 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 ∞

𝑞

𝑞

∑ ‖𝑓𝑛 ‖𝑋 ≤ 𝐶‖𝑓‖𝑌 , 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 𝑞

𝑞

𝑞

‖𝑓‖𝑌 ≤ ∑ ‖𝑓𝑗 ‖𝑌 ≤ 𝐶𝑞 ∑ ‖𝑓𝑗 ‖𝑋 . 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 ∞

𝑞

𝑞

∑ ‖𝑓𝑛 ‖𝑋 ≤ 𝐶‖𝑓‖𝑌 . 𝑛=1

378 | A Quasi-Banach spaces 𝑞

𝑞 Since 𝑁𝑞 (𝑓𝑛 ) ≤ ‖𝑓𝑛 ‖𝑋 , we get ∑∞ 𝑛=1 𝑁𝑞 (𝑓𝑛 ) ≤ 𝐶‖𝑓‖𝑌 . Hence ∑ 𝑓𝑛 converges in [𝑋]𝑞 to 𝑓, and 𝑞 𝑁𝑞 (𝑓)𝑞 ≤ ∑ 𝑁𝑞 (𝑓𝑛 )𝑞 ≤ 𝐶‖𝑓‖𝑌 ,

which completes the proof. Proposition A.5. If 𝑋 is a quasi-Banach space such that Ker 𝑁𝑞 = {0} (0 < 𝑞 ≤ 1), then each 𝑓 ∈ [𝑋]𝑞 can be represented as ∞

(the series converges in [𝑋]𝑞 )

𝑓 = ∑ 𝑓𝑛 𝑛=1 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 so that 𝑘1

𝑞

and ∑ ‖𝑓𝑗 ‖𝑋 < (1 + 𝜀),

‖𝑓 − 𝑔1 ‖𝑌 < 𝜀,

𝑗=1

where 𝑘1

𝑔1 = ∑ 𝑓𝑗 . 𝑗=1

Then choose 𝑔2 =

𝑘2 ∑𝑗=𝑘 1 +1

𝑓𝑗 , 𝑓𝑗 ∈ 𝑋, so that

‖𝑓 − 𝑔1 − 𝑔2 ‖𝑌 < 𝜀2

𝑘2

and

𝑞

∑ ‖𝑓𝑗 ‖𝑋 ≤ (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 − ∑ 𝑓 󵄩󵄩󵄩 < 𝜀𝑛 󵄩󵄩 𝑗󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑌 𝑗=1 󵄩 󵄩

𝑘𝜈+1

and

𝑞

∑ ‖𝑓𝑗 ‖𝑋 ≤ (1 + 𝜀)𝜀𝜈

(𝑛 ≥ 1, 𝜈 ≥ 0).

𝑗=𝑘𝜈 +1

The series ∑∞ 𝑗=1 𝑓𝑗 is convergent in 𝑌 (and its sum is clearly equal to 𝑓) because ∞

𝑞

∞

𝑞

∑ ‖𝑓𝑗 ‖𝑌 ≤ ∑ ‖𝑓𝑗 ‖𝑋 ≤ 𝑗=1

𝑗=1

1+𝜀 < ∞. 1−𝜀

This incidentally proves the desired inequality. Exercise A.1. 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}.

A.3 Closed graph theorem |

379

Proposition A.6. Let 𝑋 be a 𝑝-Banach space and 𝑇 ∈ 𝐿(𝑋, 𝑋) such an operator that ‖𝐼 − 𝑇‖ < 1, where 𝐼 is the identity operator. Then 𝑇 is invertible and the inequality ‖𝑇−1 ‖𝑝 ≤ (1 − ‖𝐼 − 𝑇‖𝑝 )

−1

holds. 𝑘 Proof. Consider the series ∑∞ 𝑘=0 (𝐼 − 𝑇) . We have

󵄩󵄩𝑝 󵄩󵄩 𝑛 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.2. 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 [106].

A.3 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 member of 𝑌 can be approximated by members of 𝑋. This does not imply that members of a ball 𝐾1 ⊂ 𝑌 can be approximated by members of any fixed ball 𝐾2 ⊂ 𝑋, i.e. that 𝐾2 ⊃ 𝐾1 . (𝐾2 = the closure of 𝐾2 in the topology of 𝑌.) Namely, as the following theorem states, if 𝐾2 ⊃ 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. ̂ : 𝑋/ Ker 𝑇 󳨃→ 𝑌 is invertible. Then the mapping 𝑇 : 𝑋 󳨃→ 𝑌 is open and the operator 𝑇 A mapping is open if it maps open sets onto open sets. If 𝑇 ∈ 𝐿(𝑋, 𝑌), then the operator ̂ ∈ 𝐿(𝑋/ Ker 𝑇, 𝑌) is defined by 𝑇(𝑓 ̂ + Ker 𝑇) = 𝑇𝑓. The quasinorm in 𝑋/𝑍 is defined 𝑇 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 hy­ potheses 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

We will prove that 𝑉1 ⊂ 𝑇(Δ 2 ); then it will be easy to complete the proof.

(A.4)

380 | A Quasi-Banach spaces 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 ,

𝑓𝑘 ∈ 𝑈𝑘 .

𝑘=1 𝑝 It follows that 𝑔 = ∑∞ 𝑛=1 𝑇𝑓𝑛 . And 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 𝑇 ∈ ̂ acts as an isomorphism of 𝑋/ Ker 𝑇 onto 𝑌. 𝐿(𝑋, 𝑌). If 𝑇 is onto, then 𝑇 is open and 𝑇 In particular, 𝑇 is invertible if it is onto and one-to-one. Exercise A.3. 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 special case of the open mapping theorem we have: Theorem (The theorem on equivalent norms). Let ‖ ⋅ ‖1 and ‖ ⋅ ‖2 be quasinorms on a vector space 𝑋, and let ‖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.

Uniform boundedness principle Theorem (Banach–Steinhaus). Let 𝑋 and 𝑌 be quasi-Banach spaces, and let {𝐴 𝑠 } ⊂ 𝐿(𝑋, 𝑌) be a family of operators. If sup𝑠 ‖𝐴 𝑠 𝑓‖ < ∞, for all 𝑓 ∈ 𝑋, then sup𝑠 ‖𝐴 𝑠 ‖ < ∞. In particular, the limit of an everywhere convergent sequence of bounded operators is a bounded operator. Proof. Let ‖𝑓‖2 = ‖𝑓‖𝑋 + sup𝑠 ‖𝐴 𝑠 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 above proof of the Banach–Steinhaus theorem depends on the Baire category the­ orem. An elementary proof, essentially due to Hausdorff [200], can be read in [207]. Corollary A.1. 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 Closed graph theorem |

381

“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 {𝜆 𝑛(𝑓)} such that 𝑓 = ∑∞ 𝑛=1 𝜆 𝑛 (𝑓)𝑒𝑛 , the series converging in the topology of 𝑋. Proposition A.7. If {𝑒𝑛 : 𝑛 ≥ 1} is a Schauder basis of 𝑋, then the functionals 𝜆 𝑛 are con­ tinuous and the linear operators 𝑠𝑛 : 𝑋 󳨃→ 𝑋 defined by 𝑠𝑛𝑓 = ∑𝑛𝑘=1 𝜆 𝑘 (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 is enough 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 as 𝑗 → ∞,

(A.5)

𝑛≥1

and that for every 𝑘 the sequence {𝜆 𝑘 (𝑓𝑗 )}∞ 𝑗=1 converges; let 𝛾𝑘 = lim𝑗 𝜆 𝑘 (𝑓𝑗 ). Since the functional 𝜆 𝑘 is linear and the space 𝑠𝑛 (𝑋) is finite-dimensional, it follows that 𝜆 𝑘 (𝑔𝑛 ) = lim𝑗 𝜆 𝑘 (𝑠𝑛 𝑓𝑗 ) = lim𝑗 𝜆 𝑘 (𝑓𝑗 ) = 𝛾𝑘 for 𝑘 ≤ 𝑛, and 𝜆 𝑘 (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.4. Let {𝑒𝑗 : 𝑗 ≥ 1} be a Schauder basis in a Banach space 𝑋, and let {𝑎𝑗 : 𝑗 ≥ 1} be a sequence of complex numbers and {𝜆 𝑛} a monotone sequence of positive of real numbers. Then, for 𝑚 < 𝑛, 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 󵄨 𝜆 𝑚 /𝐶 ≤ 󵄨󵄨󵄨 ∑ 𝜆 𝑗 𝑎𝑗 𝑒𝑗 󵄨󵄨󵄨󵄨 ≤ 𝐶𝜆 𝑛, if {𝜆 𝑗 } is increasing, 󵄨󵄨 󵄨󵄨𝑗=𝑚 󵄨 󵄨 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 𝜆 𝑛/𝐶 ≤ 󵄨󵄨󵄨󵄨 ∑ 𝜆 𝑗 𝑎𝑗 𝑒𝑗 󵄨󵄨󵄨󵄨 ≤ 𝐶𝜆 𝑚 , if {𝜆 𝑗 } is decreasing, 󵄨󵄨 󵄨󵄨𝑗=𝑚 󵄨 󵄨 where 𝐶 is a constant independent of 𝑚, 𝑛, {𝑎𝑗 }, and {𝜆 𝑗 }.

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:

382 | A Quasi-Banach spaces For every sequence {𝑓𝑛 } ⊂ 𝑋 such that 𝑓𝑛 tends to 0 ∈ 𝑋 and 𝑇𝑓𝑛 tends to some 𝑔 ∈ 𝑌 we have 𝑔 = 0. 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 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); (c) 𝑁(𝜆𝑓) ≤ 𝑁(𝑓) for |𝜆| ≤ 1, and (A.6) lim 𝑁(𝜆𝑓) = 0. 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 𝑝 𝑁(𝑓) = ‖𝑓‖𝑋 . Besides, if 𝑋 is a locally convex space whose topology is given by a sequence of seminorms 𝑝𝑛 (𝑛 = 0, 1, 2, . . . ), then the formula ∞

2−𝑛 𝑝𝑛 (𝑓) 𝑛=0 1 + 𝑝𝑛 (𝑓)

𝑁(𝑓) = ∑

defines an 𝐹-norm on 𝑋 that induces the same topology.

A.4.1 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

∫ log (1 + |𝑓(𝑟𝑒𝑖𝜃 )|) 𝑑𝜃 < ∞. 𝑁(𝑓) := sup − 0 0,

𝑛=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. [326], where a more general result was proved). We extend Gurariy–Matsaev theorem in two directions. First, we consider the case where 𝑎𝑛 are members of a quasi-Banach space and, second, we 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)

386 | A Quasi-Banach spaces This class will be denoted by Δ 2 (𝑎, 𝑏; 𝛼, 𝛽). The 𝐹 ∈ Δ 2 means that 𝐹 belongs to Δ 2 (𝑎, 𝑏; 𝛼, 𝛽) for some values of the parameters. Theorem A.8. Let 𝑋 be a 𝑝-Banach space and 𝐹 ∈ Δ 2 (𝑎, 𝑏; 𝛼, 𝛽). Define L by (A.8) (𝜆 0 ≥ 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 < ∞

(A.11)

0

and

∞

∑ 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖) < ∞,

(A.12)

𝑛=0

are equivalent, and the corresponding quantities are equivalent, the equivalence con­ stants depending only on 𝑝, 𝛼, 𝛽, 𝑎, and 𝑏. We will give a detailed proof in the case where 𝑋 is a Banach space. If 𝑋 is 𝑝-normed we only have to replace the norm by the 𝑝th power of the 𝑝-norm and make minor modifications of the proof. We deduce the more difficult implication, (A.11) ⇒ (A.12), of Theorem A.8 from a weaker result, namely: Proposition A.8. If 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖) → 0 (𝑛 → ∞)¹, then we have 1

∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖) 𝑑𝑟 ≥ 𝑐 sup 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖),

(A.13)

𝑛≥0

0

where 𝑐 is a positive constant independent of {𝑎𝑛}. Proof. Let 𝐹 ∈ Δ 2 (𝑎, 𝑏; 𝛼, 𝛽). The integral in (A.13) is 1

≥ ∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖𝑟1/𝛽 ) 𝑑𝑟/𝑟. 0

The series L(𝑟)𝑟1/𝛽 is lacunary with the exponents 𝜆󸀠𝑛 = 𝜆 𝑛 + 1/𝛽, and since 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖) ≍ 𝐹(1/𝜆󸀠𝑛 , ‖𝑎𝑛 ‖), we see that it suffices to prove the inequality 1

∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖) 0

𝑑𝑟 ≥ 𝑐 sup 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖). 𝑟 𝑛≥0

(A.14)

In order to do this we need the following lemma.

1 It is easy to show that this condition implies the convergence of the series ∑ 𝑎𝑛 𝑟𝜆 𝑛 for 0 < 𝑟 < 1.

A.6 Lacunary series in quasi-Banach spaces |

387

Lemma A.3. Let 𝜆 𝑘+1 /𝜆 𝑘 ≥ 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 𝑟𝑁 + ⋅ ⋅ ⋅ + 𝑝𝑁+1 𝑟2𝑁 such that for 2−(1+𝜀) < 𝑟 < 2−1 ,

𝑃(𝑟) ≥ 1,

−𝑞(1−𝜀)

for 0 < 𝑟 < 2

0 < 𝑃(𝑟) ≤ 𝛿𝑟,

−(1+𝜀)/𝑞

0 < 𝑃(𝑟) ≤ 𝛿(1 − 𝑟), for 2

,

< 𝑟 < 1.

(A.15) (A.16) (A.17)

Proof. Let 𝑃(𝑟) = (

𝑟(1 − 𝑟) 𝑁 ) . 𝜌(1 − 𝜌)

where 𝑁 is an integer. Choose 𝜀 > 0 so that max{2−𝑞(1−𝜀) , 1 − 2−(1+𝜀)/𝑞 } < 2−(1+𝜀) ; this is possible because max{2−𝑞 , 1 − 21/𝑞 } < 2−1 . Let 𝜌 be an arbitrary number such that max{2−𝑞(1−𝜀) , 1 − 2−(1+𝜀)/𝑞 } < 𝜌 < 2−(1+𝜀) . If 2−(1+𝜀) < 𝑟 < 2−1 , then 𝑟(1 − 𝑟) > 2−(1+𝜀) (1 − 2−(1+𝜀) ) because the function 𝑟 󳨃→ 𝑟(1 − 𝑟) increases for 𝑟 < 1/2 and decreases for 𝑟 > 1/2.

(A.18)

Since 𝜌 < 2−(1+𝜀) < 1/2, we have, by the same reason 𝜌(1 − 𝜌) < 2−(1+𝜀) (1 − 2−(1+𝜀) ). This proves (A.15). In order to prove (A.16), it is enough to show that for every 𝛿 > 0 there exists 𝑁 such that 𝑟(1 − 𝑟) 𝑁−1 ) < 𝛿, 0 < 𝑟 < 2−𝑞(1−𝜀) . ( 𝜌(1 − 𝜌) In view of (A.18), we have 𝑟(1 − 𝑟) < 2−𝑞(1−𝜀) (1 − 2−𝑞(1−𝜀) ) =: 𝐵. By the same reasoning we have 𝜌(1 − 𝜌) > 2−𝑞(1−𝜀) (1 − 2−𝑞(1−𝜀) ) = 𝐵. because 2−𝑞(1−𝜀) < 𝜌 < 1/2. It follows that (

𝑁−1 𝐵 𝑟(1 − 𝑟) 𝑁−1 ) ) 2−(1+𝜀)/𝑞 (> 1/2). In view of (A.18), we have 𝑟(1 − 𝑟) < 2−(1+𝜀)/𝑞 (1 − 2−(1+𝜀)/𝑞 ). Since 1/2 > 𝜌 > 1 − 2−(1+𝜀)/𝑞 , we have, by (A.18), 𝜌(1 − 𝜌) > 2−(1+𝜀)/𝑞 (1 − 2−(1+𝜀)/𝑞 ), which proves (A.17), completing the proof. Remark A.1. The validity of (A.15) is independent of 𝛿.

Proof of (A.14) The value of 𝜆 0 is irrelevant so we assume that 𝜆 0 ≥ 1. Let 𝑚

𝑃(𝑟) = ∑ 𝑝𝑗 𝑟𝑗 𝑗=1

be an arbitrary polynomial. Since, by the properties of 𝐹, 1

1

𝑑𝑟 1 𝑑𝑟 = ∫(1 − 𝑟1/𝑠 )−1 𝐹(1 − 𝑟1/𝑠 , ‖L(𝑟)‖) ∫(1 − 𝑟) 𝐹(1 − 𝑟, ‖L(𝑟 )‖) 𝑟 𝑠 𝑟 −1

𝑠

0

0

1

≍ ∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖) 0

we have

1

󵄨∞

󵄨

󵄨󵄨 󵄨

󵄨󵄨 󵄨

𝑑𝑟 , 𝑟

𝑠 > 0,

󵄨󵄨 󵄨 𝑑𝑟 󵄨 −1 𝜆 󵄨󵄨 I(𝐹) := ∫(1 − 𝑟) 𝐹 (1 − 𝑟, 󵄨󵄨󵄨∑ 𝑎𝑘 𝑃(𝑟 𝑘 )󵄨󵄨󵄨) 0

0

𝑟

(A.19)

1

𝑑𝑟 , ≤ 𝐶 ∫(1 − 𝑟) 𝐹 (1 − 𝑟, ‖L(𝑟)‖) 𝑟 −1

0

where 𝐶 depends only of 𝑃 and (𝑎, 𝑏, 𝛼, 𝛽). Let 𝜈 be an integer and 𝐽𝜈 = (2−(1+𝜀)/𝜆 𝜈 , 2−(1−𝜀)/𝜆 𝜈 ) ,

0 < 𝜀 < 1.

Choose 𝜈 ≥ 0 so that 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖) ≥ 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖) for all 𝑘, and let 󵄨∞

󵄨

󵄨󵄨 󵄨

󵄨󵄨 󵄨

󵄨󵄨 󵄨 𝑑𝑟 󵄨 −1 𝜆 󵄨󵄨 , I𝜈 (𝐹) = ∫(1 − 𝑟) 𝐹 (1 − 𝑟, 󵄨󵄨󵄨∑ 𝑎𝑘 𝑃(𝑟 𝑘 )󵄨󵄨󵄨) 𝐽𝜈

0

𝑟

(A.20)

where 𝑃 is a polynomial satisfying (A.15)–(A.17), where 𝛿 will be chosen later on.

A.6 Lacunary series in quasi-Banach spaces | 389

Now we are going to estimate I𝜈 . Assume first that 𝐹 ∈ Δ 2 (𝑎, 𝑏; 1, 𝛽). This means that the function 𝑦 󳨃→ 𝐹(𝑥, 𝑦)/𝑦 (𝑦 > 0) decreases, which implies that 𝐹(𝑥, 𝑦1 + 𝑦2 ) ≤ 𝐹(𝑥, 𝑦1 ) + 𝐹(𝑥, 𝑦2 ), whence −1

𝜆

I𝜈 (𝐹) ≥ ∫(1 − 𝑟) 𝐹 (1 − 𝑟, 𝑃(𝑟 𝜈 )‖𝑎𝜈 ‖) 𝐽𝜈

𝑑𝑟 𝑟

− ∑ ∫(1 − 𝑟)−1 𝐹 (1 − 𝑟, 𝑃(𝑟𝜆 𝑘 )‖𝑎𝑘 ‖) 𝑘=𝜈+1 𝐽

𝑑𝑟 𝑟

𝜈

𝜈−1

− ∑ ∫(1 − 𝑟)−1 𝐹 (1 − 𝑟, 𝑃(𝑟𝜆 𝑘 )‖𝑎𝑘 ‖) 𝑘=0 𝐽

(A.21)

𝑑𝑟 𝑟

𝜈

=: 𝐼𝜈󸀠 − 𝐼𝜈󸀠󸀠 − 𝐼𝜈󸀠󸀠󸀠 (where 𝐼𝜈󸀠󸀠󸀠 = 0 if 𝜈 = 0). In estimating 𝐼󸀠 , 𝐼󸀠󸀠 , and 𝐼󸀠󸀠󸀠 we do note the hypothesis 𝛼 = 1 so we assume tem­ porarily that 𝛼 > 0 is arbitrary. By (A.15), we have 𝑃(𝑟𝜆 𝜈 ) ≥ 1 for 𝑟 ∈ (2−(1+𝜀)/𝜆 𝜈 , 2−1/𝜆 𝜈 ) := 𝐽𝜈󸀠 ⊂ 𝐽𝜈 . Hence 𝐼𝜈󸀠 ≥ ∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖𝑎𝜈 ‖) 𝐽𝜈󸀠

𝑑𝑟 𝑟

≥ (1 − 2−(1+𝜀)/𝜆 𝜈 )−1 𝐹(1 − 2−1/𝜆 𝜈 , ‖𝑎𝜈 ‖) ∫ 𝐽𝜈󸀠

= (1 − 2−(1+𝜀)/𝜆 𝜈 )−1 𝐹(1 − 2−1/𝜆 𝜈 , ‖𝑎𝜈 ‖)

𝑑𝑟 𝑟

𝜀 log 2. 𝜆𝜈

In order to estimate this quantity we use the Bernoulli inequalities: (1 + 𝑥)𝛾 ≥ 1 + 𝛾𝑥, 𝛾

(1 + 𝑥) ≤ 1 + 𝛾𝑥,

for 𝑥 > −1 and 𝛾 ∈ (−∞, 0) ∪ (1, ∞),

(A.22)

for 𝑥 > −1 and 0 ≤ 𝛾 ≤ 1.

(A.23)

It follows from (A.22) with 𝛾 = −(1 + 𝜀)/𝜆 𝜈 and 𝑥 = 1 that 2−(1+𝜀)/𝜆 𝜈 ≥ 1 −

1+𝜀 , 𝜆𝜈

and hence (1 − 2−(1+𝜀)/𝜆 𝜈 )−1 ≥

𝜆𝜈 . 1+𝜀

On the other hand, by (A.23), we have 1 1/𝜆 𝜈 1 1/𝜆 𝜈 = 1 − (1 − ) 1 − 2−1/𝜆 𝜈 = 1 − ( ) 2 2 1 1 ≥ 1 − (1 − )= , 2𝜆 𝜈 2𝜆 𝜈

390 | A Quasi-Banach spaces and hence 𝐹(1 − 2−1/𝜆 𝜈 , ‖𝑎𝜈 ‖) ≥ 𝐹 (

1 1 𝛼 , ‖𝑎𝜈 ‖) ≥ ( ) 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖). 2𝜆 𝜈 2

Combining the above inequalities we get 𝐼𝜈󸀠 ≥ 𝑐1 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖),

(A.24)

where

1 𝛼 𝜀 𝑐1 = ( ) log 2. 2 1+𝜀 Observe that 𝑐1 is independent of 𝛿. Now we pass to 𝐼𝜈󸀠󸀠 and 𝐼𝜈󸀠󸀠󸀠 . First step: we choose and fix 𝛿 < 1/2 so small that 𝐶2 𝛿𝛽 < 𝑐1 /4 and 𝐶3 𝛿𝛽 < 𝑐1 /4,

(A.25)

where 𝐶2 and 𝐶3 are positive constants independent of 𝛿 and are defined by (A.28) and (A.30) below. Second step: let 𝑃 be a polynomial from Lemma A.3; 𝑃 is now fixed. The following implications will be used: 𝑘 > 𝜈, 𝑟 ∈ 𝐽𝜈 ⇒ 𝑟𝜆 𝑘 ≤ 2−𝑞(1−𝜀) ,

(A.26)

𝑘 < 𝜈, 𝑟 ∈ 𝐽𝜈 ⇒ 𝑟𝜆 𝑘 ≥ 2−(1+𝜀)/𝑞 . Let 𝑘 > 𝜈. Then, by (A.26), (A.16), and Bernoulli’s inequalities, 𝐼𝑘,𝜈 := ∫(1 − 𝑟)−1 𝐹(1 − 𝑟, 𝑃(𝑟𝜆 𝑘 )‖𝑎𝑘 ‖) 𝐽𝜈

𝑑𝑟 𝑟

≤ (1 − 2−(1−𝜀)/𝜆 𝜈 )−1 ∫ 𝐹(1 − 2−(1+𝜀)/𝜆 𝜈 , 𝛿𝑟𝜆 𝑘 ‖𝑎𝑘 ‖) 𝐽𝜈

𝑑𝑟 𝑟

2𝜆 𝜈 𝑑𝑟 ∫ 𝐹((1 + 𝜀)/𝜆 𝜈 , 𝛿𝑟𝜆 𝑘 ‖𝑎𝑘 ‖) ≤ 1−𝜀 𝑟 𝐽𝜈

≤ 𝐾 𝐹(1/𝜆 𝜈 , 𝛿2−(1−𝜀)𝜆 𝑘 /𝜆 𝜈 ‖𝑎𝑘 ‖), where 𝐾 = 𝐾𝜀,𝑎 =

(1 + 𝜀)𝑎 4𝜀 log 2 . 1−𝜀

Hence, by the properties of 𝐹, 𝐼𝑘,𝜈 ≤ 𝐾(𝛿2−(1−𝜀)𝜆 𝑘 /𝜆 𝜈 )𝛽 𝐹(1/𝜆 𝜈 , ‖𝑎𝑘 ‖). Since 𝑡 := 𝜆 𝑘 /𝜆 𝜈 > 1, we have 𝐹(1/𝜆 𝜈 , ‖𝑎𝑘 ‖) = 𝐹(𝑡/𝜆 𝑘 , ‖𝑎𝑘 ‖) ≤ 𝑡𝑎 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖) = (

𝜆𝑘 𝑎 ) 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖) 𝜆𝜈

A.6 Lacunary series in quasi-Banach spaces | 391

and hence

∞

𝐼𝜈󸀠󸀠 ≤ 𝐾 ∑ (𝛿2−(1−𝜀)𝜆 𝑘 /𝜆 𝜈 )𝛽 ( 𝑘=𝜈+1

𝜆𝑘 𝑎 ) 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖). 𝜆𝜈

The function 𝑥 󳨃→ 𝑥𝑎 2−𝛽(1−𝜀)𝑥 , 𝑥 > 0, is decreasing for 𝑥 large enough, which together with the inequality 𝜆 𝑘 /𝜆 𝜈 ≥ 𝑞𝑘−𝜈 gives ∞

𝐼𝜈󸀠󸀠 ≤ 𝐾󸀠 𝛿𝛽 ∑ 𝑞(𝑘−𝜈)𝑎 2−𝛽(1−𝜀)𝑞

(𝑘−𝜈)

𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖)

𝑘=𝜈+1

(A.27)

∞

𝑘

≤ 𝐾󸀠 𝛿𝛽 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖) ∑ 𝑞𝑘𝑎 2−𝛽(1−𝜀)𝑞 = 𝐶2 𝛿𝛽 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖), 𝑘=1

where

∞

𝑘

𝐶2 = 𝐾𝜀,𝑎 ∑ 𝑞𝑘𝑎 2−𝛽(1−𝜀)𝑞 < ∞;

(A.28)

𝑘=1

the constant 𝐾󸀠 is independent of 𝛿 and so is 𝐶2 . (Here the inequality 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖) ≤ 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖) has been used.) Let 𝑘 < 𝜈 and 𝜈 ≥ 1. Then, by (A.26) and (A.17), 𝐼𝑘,𝜈 ≤ ∫(1 − 𝑟)−1 𝐹 (1 − 𝑟, 𝛿(1 − 𝑟𝜆 𝑘 )‖𝑎𝑘 ‖) 𝐽𝜈

≤

𝑑𝑟 𝑟

2𝜆 𝜈 𝑑𝑟 ∫ 𝐹 ((1 + 𝜀)/𝜆 𝜈 , 𝛿(1 − 𝑟𝜆 𝑘 ), ‖𝑎𝑘 ‖) 1−𝜀 𝑟 𝐽𝜈

2𝜆 𝜈 𝑑𝑟 ∫ 𝐹 ((1 + 𝜀)/𝜆 𝜈 , 𝛿(1 − 2−(1+𝜀)𝜆 𝑘 /𝜆 𝜈 )‖𝑎𝑘 ‖) ≤ 1−𝜀 𝑟 𝐽𝜈

4𝜀 𝐹 ((1 + 𝜀)/𝜆 𝜈 , 𝛿(1 + 𝜀)(𝜆 𝑘 /𝜆 𝜈 )‖𝑎𝑘 ‖) log 2 ≤ 1−𝜀 ≤ 𝐾󸀠 𝛿𝛽 (𝜆 𝑘 /𝜆 𝜈 )𝛽 𝐹(1/𝜆 𝜈 , ‖𝑎𝑘 ‖), where 󸀠󸀠 = 𝐾󸀠󸀠 = 𝐾𝜀,𝛽,𝑎

4𝜀(1 + 𝜀)𝛽+𝑎 log 2 . 1−𝜀

Since 𝜆 𝑘 /𝜆 𝜈 < 1, we have 𝐹(1/𝜆 𝜈 , ‖𝑎𝑘 ‖) ≤ (𝜆 𝑘 /𝜆 𝜈 )𝑏 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖). Hence 𝐼𝑘,𝜈 ≤ 𝐾󸀠󸀠 𝛿𝛽 (𝜆 𝑘 /𝜆 𝜈 )𝛽+𝑏 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖) ≤ 𝐾󸀠 𝛿𝛽 𝑞(𝑘−𝜈)(𝛽+𝑏) 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖) ≤ 𝐾󸀠 𝛿𝛽 𝑞(𝑘−𝜈)(𝛽+𝑏) 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖)

392 | A Quasi-Banach spaces and hence 𝜈

𝐼𝜈󸀠󸀠󸀠 ≤ 𝐾󸀠󸀠 𝛿𝛽 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖) ∑ 𝑞(𝑘−𝜈)(𝛽+𝑏) 𝑘=0 𝜈

󸀠󸀠 𝛽

(A.29) 𝛽+𝑏 𝑘

= 𝐾 𝛿 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖) ∑ (1/𝑞

𝛽

) ≤ 𝐶3 𝛿 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖),

𝑘=0

where

∞

󸀠󸀠 𝐶3 = 𝐾𝜀,𝛽,𝑎 ∑ (1/𝑞𝛽+𝑏 )𝑘 < ∞.

(A.30)

𝑘=0

Combining (A.24), (A.27), and (A.29), we get 𝛽

𝛽

I𝜈 (𝐹) ≥ 𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖)(𝑐1 − 𝐶2 𝛿 − 𝐶3 𝛿 ).

From this and (A.25) we obtain I(𝐹) ≥ I𝜈 (𝐹) ≥ (𝑐1 /2)𝐹(1/𝜆 𝜈 , ‖𝑎𝜈 ‖). This inequality to­ gether with (A.19) completes the proof of (A.14) and consequently of Proposition A.8, in the case 𝛼 = 1. Since Δ 2 (𝑎, 𝑏; 𝛼, 𝛽) ⊂ Δ 2 (𝑎, 𝑏; 1, 𝛽) for 𝛼 ≤ 1, we see that the propo­ sition also holds for 𝛼 < 1. Let 𝛼 > 1. The function 𝐹1/𝛼 is in Δ 2 (𝑎/𝛼, 𝑏/𝛼; 1, 𝛽/𝛼) and therefore ∞

∞

𝐹1/𝛼 (1 − 𝑟, ∑ ‖𝑎𝑘 ‖𝑃(𝑟𝜆 𝑘 ) ≥ 𝐹1/𝛼 (1 − 𝑟, ‖𝑎𝜈 ‖𝑃(𝑟𝜆 𝜈 )) − 𝐹1/𝛼 (1 − 𝑟, ∑ ‖𝑎𝑘 ‖𝑃(𝑟𝜆 𝑘 )) 𝑘=0

𝑘>𝜈 ∞

− 𝐹1/𝛼 (1 − 𝑟, ∑ ‖𝑎𝑘 ‖𝑃(𝑟𝜆 𝑘 )) . 𝑘 0, 𝜉 > 0, then 1

∫(1 − 𝜌)𝜂−1 (𝜌 − 𝑟)𝜉−1 𝑑𝜌 ≤ 𝐶𝜂,𝜉 (1 − 𝑟)𝜂+𝜉−1

(0 < 𝑟 < 1).

(A.31)

𝑟

Proof. Let 𝜂 < 1 and 𝜉 < 1. We split the interval (0, 1) by the point 𝜌 = √𝑟. If 𝑟 < 𝜌 < √𝑟, then (1 − 𝜌)𝜂−1 (𝜌 − 𝑟)𝜉−1 ≤ (1 − √𝑟)𝜂−1 (𝜌 − 𝑟)𝜉−1 . If √𝑟 < 𝜌 < 1, then (1 − 𝜌)𝜂−1 (𝜌 − 𝑟)𝜉−1 ≤ (1 − 𝜌)𝜂−1 (√𝑟 − 𝑟)𝜉−1 . The result follows by integration of these inequalities over the corresponding intervals. The other cases are similar.

A.6 Lacunary series in quasi-Banach spaces |

393

Lemma A.5. If 𝐺 is a positive measurable function on (0, 1), 𝛾 > 0, 𝑠 > 0, 𝑡 > 0, and 𝑠 + 𝑡 = 1, then 1

𝜌

1 𝛾−1

∫(1 − 𝑟)

𝑠𝛾−1

𝐺(𝑟) 𝑑𝑟 ≥ 𝑐 ∫(1 − 𝜌)

0

𝑑𝜌 ∫(𝜌 − 𝑟)𝑡𝛾−1 𝐺(𝑟) 𝑑𝑟,

0

0

where 𝑐 > 0 is independent of 𝐺. Proof. Apply Fubini’s theorem and inequality (A.31). Proof of Theorem A.8. First we prove that (A.11) implies (A.12). Put 𝐺(𝑟) = (1 − 𝑟)−𝑏 𝐹(1 − 𝑟, ‖L(𝑟)‖). According to Lemma A.5 we have 1

∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖) 𝑑𝑟 0

(A.32)

𝜌

1 𝑠𝑏−1

≥ 𝑐 ∫(1 − 𝜌) 0

𝑡𝑏−1

𝑑𝜌 ∫(𝜌 − 𝑟)

𝐺(𝑟) 𝑑𝑟.

0

On the other hand, the inner integral equals 1

𝜌𝑡𝑏 ∫(1 − 𝑟)𝑡𝑏−1 𝐺(𝜌𝑟) 𝑑𝑟 0 1 𝑡𝑏

= 𝜌 ∫(1 − 𝑟)𝑡𝑏−1 (1 − 𝜌𝑟)−𝑏 𝐹(1 − 𝜌𝑟, ‖L(𝜌𝑟)‖) 𝑑𝑟 0 1

≥ 𝜌𝑡𝑏 ∫(1 − 𝑟)−𝑠𝑏−1 𝐹(1 − 𝑟, ‖L(𝜌𝑟)‖) 𝑑𝑟. 0

(Here we used the fact that 𝑥−𝑏 𝐹(𝑥, 𝑦) increases with 𝑥, which follows from (A.10).) Further, the function 𝑥−𝑠𝑏 𝐹(𝑥, 𝑦) belongs to the class Δ 2 so we can apply Proposition A.8 to get 1 𝜆𝑛 ∫(1 − 𝑟)−𝑠𝑏−1 𝐹(1 − 𝑟, ‖L(𝜌𝑟)‖) 𝑑𝑟 ≥ 𝑐󸀠 𝜆𝑠𝑏 𝑛 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖𝜌 ). 0

Proposition A.8 can be applied because the coefficients of the function 𝑟 → L(𝜌𝑟) (0 < 𝜌 < 1) are equal to 𝑎𝑛 (𝜌) = 𝑎𝑛 𝜌𝜆 and 𝑎𝑛 (𝜌) → 0 (𝑛 → ∞), which implies that 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 (𝜌)‖) → 0.

394 | A Quasi-Banach spaces It follows that 𝜌 𝑡𝑏 𝛼𝜆 𝑛 ∫(𝜌 − 𝑟)𝑡𝑏−1 𝐺(𝑟) 𝑑𝑟 ≥ 𝑐󸀠 𝜆𝑠𝑏 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖). 𝑛𝜌 𝜌 0

for all 𝑛. Hence by (A.32) 1

1

∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖) 𝑑𝑟 ≥ 𝑐󸀠󸀠 ∫(1 − 𝜌)𝑠𝑏−1 𝑀(𝜌) 𝑑𝜌, 0

0

where 𝑡𝑏+𝛼𝜆 𝑛 𝑀(𝜌) = sup 𝜆𝑠𝑏 𝐹(1/𝜆 𝑛 , ‖𝑎𝑛 ‖), 𝑛𝜌 𝑛≥0

and 𝑠 is chosen so that 𝑠𝑏 − 1 < 0. Using this and the inequality ∞

𝜆𝑘 (1 − 𝜌)𝑠𝑏−1 ≥ 𝑐1󸀠 ∑ 𝜆1−𝑠𝑏 𝑘 𝜌 , 𝑘=0

valid because 1 − 𝑠𝑏 > 0, we get 1

1

∞

𝑡𝑏+𝛼𝜆 𝑘 𝑠𝑏 ∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖) 𝑑𝑟 ≥ 𝑐1󸀠󸀠 ∫ ∑ 𝜆1−𝑠𝑏 𝜆 𝑘 𝐹(1/𝜆 𝑘 , ‖𝑎𝑘 ‖) 𝑑𝜌. 𝑘 𝜌 0 𝑘=0

0

This proves that (A.11) implies (A.12), as desired. The proof of the implication (A.12) ⇒ (A.11) is much simpler. We need a simple lemma. Lemma A.6. If 𝐹 ∈ Δ 2 (𝑎, 𝑏; 𝛼, 𝛽), then 1

𝐹(1/𝑠, 𝑦)/𝐶 ≤ ∫(1 − 𝑟)−1 𝐹(1 − 𝑟, 𝑦)𝑟𝑠 𝑑𝑟 ≤ 𝐶𝐹(1/𝑠, 𝑦),

𝑠 ≥ 1, 𝑦 ≥ 0,

0

where 𝐶 is independent of 𝑦. Proof. For a fixed 𝑦 > 0, let 𝜑(𝑥) = 𝐹(𝑥, 𝑦). We have 1

1

∫(1 − 𝑟)−1 𝜑(1 − 𝑟)𝑟𝑠 𝑑𝑟 = ∫ 𝜑(𝑟)(1 − 𝑟)𝑠 0

0 𝑠

𝑠 𝑑𝑡

= ∫ 𝜑(𝑡/𝑠)(1 − 𝑡/𝑠) 0

𝑡

𝑑𝑟 𝑟

𝑠

≤ 𝜑(1/𝑠) ∫(𝑡𝑎 + 𝑡𝑏 )𝑒−𝑡 0

𝑑𝑡 ≤ 𝐶𝜑(1/𝑠). 𝑡

In the other direction, 1

1−1/2𝑠

∫(1 − 𝑟)−1 𝜑(1 − 𝑟)𝑟𝑠 𝑑𝑟 ≥ ∫ (1 − 𝑟)−1 𝜑(1 − 𝑟)𝑟𝑠 𝑑𝑟 0

1−1/𝑠

≥ (1/4)𝜑(1/2𝑠)𝑠(1/𝑠 − 1/2𝑠), for 𝑠 > 2, which completes the proof.

A.6 Lacunary series in quasi-Banach spaces |

395

As in the case of the implication (A.11) ⇒ (A.12) we first consider the case 𝛼 ≤ 1. Then the inequality 𝐹(𝑥, 𝑦1 + 𝑦2 ) ≤ 𝐹(𝑥, 𝑦1 ) + 𝐹(𝑥, 𝑦2 ) holds and hence 1

∞

1

∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖L(𝑟)‖) 𝑑𝑟 ≤ ∑ ∫(1 − 𝑟)−1 𝐹(1 − 𝑟, ‖𝑎𝑛 ‖)𝑟𝜆 𝑛 𝛽 𝑑𝑟. 𝑛=0

0

0

Now the desired result follows from (an obvious modification) of Lemma A.6. Let 𝛼 > 1, 𝛼󸀠 = 𝛼/(𝛼 − 1), and 𝛾 > 0. Then, applying first the inequality (𝑥1/𝛼 + 1/𝛼 𝛼 𝑥 ) ≥ 𝑥 + 𝑦 (𝑥, 𝑦 ≥ 0) and then Hölder’s inequality, we get 𝛼

∞

𝐹(1 − 𝑟, ‖L(𝑟)‖) ≤ ( ∑ 𝐹1/𝛼 (1 − 𝑟, ‖𝑎𝑛 ‖)𝑟𝜆 𝑛 𝛽 ) 𝑛=0 ∞

𝛼

󸀠

= ( ∑ 𝐹1/𝛼 (1 − 𝑟, ‖𝑎𝑛 ‖)𝑟𝜆 𝑛 𝛽(1/𝛼+1/𝛼 ) ) 𝑛=0 ∞

∞

𝑛=0 ∞

𝑛=0

𝛼−1

󸀠

𝜆𝑛𝛽 𝜆 𝑛𝛽 ≤ ∑ 𝜆−𝛾𝛼 ( ∑ 𝜆𝛾𝛼 ) 𝑛 𝑟 𝑛 𝐹(1 − 𝑟, ‖𝑎𝑛 ‖)𝑟 󸀠

𝜆𝑛𝛽 ≤ 𝐶 ∑ 𝜆−𝛾𝛼 (1 − 𝑟)−𝛾𝛼 (𝛼−1) 𝑛 𝐹(1 − 𝑟, ‖𝑎𝑛 ‖)𝑟 𝑛=0 ∞

𝜆𝑛𝛽 = 𝐶 ∑ 𝜆−𝛾𝛼 (1 − 𝑟)−𝛾𝛼 . 𝑛 𝐹(1 − 𝑟, ‖𝑎𝑛 ‖)𝑟 𝑛=0

Choose 𝛾 so small that the function 𝐹1 (𝑥, 𝑦) = 𝑥−𝛾𝛼 𝐹(𝑥, 𝑦) belongs to the class Δ 2 . Then integrate the preceding inequality and apply Lemma A.6 to the function 𝐹1 . The result follows, and the proof of Theorem A.8 is completed.

Further notes and results Subection A.6.1 is a slightly modified and narrowed version of the author’s paper [382]. 𝑝,𝑞 Let ℎ𝜙 (𝔹𝑁 ) denote the class of all complex-valued functions 𝑓 harmonic in the unit ball 𝔹𝑁 of ℝ𝑁 (abbreviated 𝑓 ∈ ℎ(𝔹𝑁 )) such that 1

∫ 𝑀𝑝𝑞 (𝑟, 𝑓) 0

𝜙𝑞 (1 − 𝑟) 𝑑𝑟 < ∞, 1−𝑟

(A.33)

where 𝜙 is a normal weight. Each 𝑓 ∈ ℎ(𝔹𝑁 ) can be represented in a unique way as ∞

𝑓(𝑟𝜁) = ∑ 𝑟𝑛𝑓𝑛 (𝜁), 𝑛=0

where 𝑓𝑛 is a spherical harmonic of degree 𝑛, i.e., the restriction to 𝑆𝑁 of a homoge­ neous harmonic polynomial of degree 𝑛. If ∞

𝑓(𝑟𝜁) = ∑ 𝑟𝜆 𝑛 𝑓𝜆 𝑛 (𝜁) 𝑛=0

(A.34)

396 | A Quasi-Banach spaces where {𝜆 𝑛} is lacunary, then 𝑓 is said to be a function with Hadamard gaps. We can associate to 𝑓 a function 𝑔 on (0, 1) with values in 𝐿𝑝 (𝕊𝑁 ) by 𝑔(𝑟)(𝜁) = 𝑓(𝑟𝜁). Consequently condition (A.33) can be rewritten as 1

∫ ‖𝑔(𝑟)‖𝑞𝑝 0

𝜙𝑞 (1 − 𝑟) 𝑑𝑟 < ∞. 1−𝑟

Here ‖ ⋅ ‖𝑝 = ‖ ⋅ ‖𝐿𝑝 (𝕊𝑁 ) . From this, by means of Theorem A.8 applied with 𝐹(𝑥, 𝑦) = 𝜙(𝑥)𝑞 𝑦𝑞 , we obtain immediately the following. Theorem A.9. Let 0 < 𝑝 ≤ ∞, 0 < 𝑞 < ⬦, and 𝜙 a normal weight. Let 𝑓 be a function 𝑝,𝑞 with Hadamard gaps given by (A.34). Then 𝑓 belongs to ℎ𝜙 (𝔹𝑁 ) if and only if ∞

∑ 𝜙(1/𝜆 𝑛 )𝑞 ‖𝑓𝜆 𝑛 ‖𝑞𝑝 < ∞.

𝑛=0

This theorem covers the case of analytic Bergman spaces on the ball of ℂ𝑛 . See [233], where a simple proof based partly on Paley’s theorem on lacunary trigonometric series is given; this method does not seem applicable in proving Theorem A.9. 𝑝,∞ Theorem A.9 holds for 𝑞 = ∞ with an appropriate interpretation of ℎ𝜙 . Two spaces arise: the space 𝑝,∞

ℎ𝜙 (𝔹𝑁 ) = {𝑓 ∈ ℎ(𝔹𝑁 ) : sup 𝜙(1 − 𝑟)𝑀𝑝 (𝑟, 𝑓) < ∞} 0 1 the space is locally convex, and for 𝑝 < 1 it is 𝑝-convex, i.e. there is an equivalent 𝑝-norm on it. For further information see Kalton [249]. See also [47] for the general theory of weak 𝐿𝑝 spaces.

Marcinkiewicz’s theorem Quasiilinear operators. Let 𝑇 be an operator acting from a vector space 𝑋 to the set of all nonnegative measurable functions defined on a measure space (Ω, 𝜇). Then 𝑇 is called a quasilinear operator if there exists a constant 𝐾 such that 𝑇(𝑓 + 𝑔) ≤ 𝐾(𝑇𝑓 + 𝑇𝑔)

(𝑓, 𝑔 ∈ 𝑋).

If 𝐾 = 1, then 𝑇 is said to be subadditive. If an operator 𝑆 with values in the set of finite measurable functions on Ω is linear, then the operator 𝑇𝑓 = |𝑆𝑓| is subadditive. Theorem (Marcinkiewicz’s interpolation theorem [320, 536]). Let 𝜇 and 𝜎 be sigmafinite measures on Ω and 𝑆, respectively, let 0 < 𝑝 < 𝑞 ≤ ∞ and let 𝑇 be a quasilin­ ear operator from 𝐿𝑝 (𝜎) + 𝐿𝑞 (𝜎) 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 ,

(B.3)

‖𝑇𝑓‖𝑞,⋆ ≤ 𝐶2 ‖𝑓‖𝑞 .

(B.4)

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

(B.5)

In the case 𝑞 = ∞ inequality (B.4) should be interpreted as ‖𝑇𝑓‖∞ ≤ 𝐶2 ‖𝑓‖∞ . Weak type and strong type If 𝑇 satisfies (B.3), i.e. if 𝑇 maps 𝐿𝑝 into 𝐿𝑝,⋆ and is continuous at zero, then we say that 𝑇 is of weak type (𝑝, 𝑝); if (B.5) holds, then 𝑇 is of strong type (𝑠, 𝑠).

B.2 Weak 𝐿𝑝 -spaces and Marcinkiewicz’s theorem | 401

Proof of Theorem. We consider the case where 𝐾 = 𝐶1 = 𝐶2 = 1 and 𝑞 < ∞, leaving the remaining cases to the reader. We have to deduce the inequality ∫ |𝑇𝑓|𝑠 𝑑𝜇 ≤ 𝐶 ∫ |𝑓|𝑠 𝑑𝜎 Ω

𝑆

from two “weak” inequalities: 𝜇(𝑇𝑓, 𝜆) ≤

1 ∫ |𝑓|𝑝 𝑑𝜎, 𝜆𝑝

(B.6)

1 ∫ |𝑓|𝑞 𝑑𝜎. 𝜆𝑞

(B.7)

𝑆

𝜇(𝑇𝑓, 𝜆) ≤

𝑆

To show this we represent the function 𝑓 in the form 𝑓 = 𝑔𝜆 + ℎ𝜆 , where {𝑓(𝜁), 𝑔𝜆 (𝜁) = { 0, {

if |𝑓(𝜁)| ≥ 𝜆, if |𝑓(𝜁)| < 𝜆.

Since 𝑇𝑓 ≤ 𝑇(𝑔𝜆 ) + 𝑇(ℎ𝜆 ), we have 𝜇(𝑇𝑓, 𝜆) ≤ 𝐺(𝜆) + 𝐻(𝜆), where 𝐺(𝜆) = 𝜇(𝑇𝑔𝜆 , 𝜆/2) and 𝐻(𝜆) = 𝜇(𝑇ℎ𝜆 , 𝜆/2). It follows from (B.6) and (B.7) that 𝐺(𝜆) ≤ (2/𝜆)𝑝 ∫ |𝑔𝜆 |𝑝 𝑑𝜎 = (2/𝜆)𝑝 ∫ |𝑓|𝑝 𝑑𝜎 𝑆

(B.8)

|𝑓|>𝜆

and 𝐻(𝜆) ≤ (2/𝜆)𝑞 ∫ |ℎ𝜆 |𝑞 𝑑𝜎 = (2/𝜆)𝑞 ∫ |𝑓|𝑞 𝑑𝜎. 𝑆

|𝑓|≤𝜆

Now we use the formula ∞

∞

𝑠

∫ |𝑇𝑓| 𝑑𝜇 = 𝑠 ∫ 𝜇(𝑇𝑓, 𝜆)𝜆 Ω

𝑠−1

𝑑𝜆 ≤ 𝑠 ∫ (𝐺(𝜆) + 𝐻(𝜆)) 𝜆𝑠−1 𝑑𝜆.

0

0

Multiplying inequality (B.8) by 𝑠𝜆𝑠−1 and then integrating over 𝜆 ∈ (0, ∞) we get ∞

∞

𝑠 ∫ 𝐺(𝜆)𝜆𝑠−1 𝑑𝜆 ≤ 𝑠2𝑝 ∫ (𝜆𝑠−𝑝−1 ∫ |𝑓|𝑝 𝑑𝜎) 𝑑𝜆 0

0

|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

402 | B Interpolation and maximal functions Paley’s theorem The implication ∞

󸀠

̂ 𝑝 𝜆} ≤ 𝜎{𝑛 : 𝐾𝑛 > 𝜆} ≤ ∑ 𝑛−2 ≤ 𝐶𝐾 min(1, 1/𝜆), 𝑛>𝜆/𝐾

which concludes the proof. Exercise B.3. That Theorem B.5 improves the implication (B.9) can be deduced from the implication ∞

∑ (𝑛 + 1)𝑝−2 (𝑐𝑛∗ )𝑝 < ∞ 󳨐⇒ 𝑐𝑛∗ = O((𝑛 + 1)1/𝑝−1 ) 𝑛=0

and the equality ∞

󸀠

∞

󸀠

∞

󸀠

̂ 𝑝 = ∑ (𝑐∗ )𝑝 = ∑ (𝑐∗ )𝑝 −𝑝 (𝑐∗ )𝑝 . ∑ |𝑓(𝑛)| 𝑛 𝑛 𝑛 𝑛=−∞

𝑛=0

𝑛=0

Exercise B.4 (Hardy–Littlewood [184]). If {𝑏𝑛 }∞ −∞ is a sequence such that, for some 𝑞 > 2, ∞

∑ (|𝑛| + 1)𝑞−2 |𝑏𝑛 |𝑞 < ∞, 𝑛=−∞

̂ = 𝑏𝑛 . then there exists a function 𝑔 ∈ 𝐿𝑞 (𝕋) such that 𝑔(𝑛)

B.3 Classical maximal functions |

403

Exercise B.5. If 𝑓 is a function harmonic in the unit ball 𝔹𝑛 ⊂ ℝ𝑛 , then 1/𝑝

1 𝑛−2

(∫(1 − 𝑟)

𝑝 𝑀∞ (𝑟, 𝑓) 𝑑𝑟)

≤ 𝐶𝑝,𝑛 ‖𝑓‖𝑝 ,

1 < 𝑝 ≤ ∞.

(B.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 their values in ℝ𝑛 . If 𝑓 is quasiconformal (not necessarily harmonic), then a result of Astala and Koskela [32, Theorem 3.3] asserts that inequality (B.10) and its reverse hold for all 𝑝 > 0.

B.3 Classical maximal functions The maximal function of a 2𝜋-periodic function 𝜙 ∈ 𝐿1 (−𝜋, 𝜋) is the (2𝜋-periodic) func­ tion M𝜙 defined as 𝜃+ℎ

1 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝜙(𝑡)󵄨󵄨󵄨 𝑑𝑡. (M𝜙)(𝜃) = sup 2ℎ 0 1, then M𝜙 ∈ 𝐿𝑝 (−𝜋, 𝜋) and ‖M𝜙‖𝑝 ≤ 𝐶𝑝 ‖𝜙‖𝑝 , where 𝐶𝑝 depends only of 𝑝. Proof. Assertion (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 | is 𝑖 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 ,

404 | B Interpolation and maximal functions 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 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝜙(𝑡 + 𝑥) − 𝜙(𝑥)󵄨󵄨󵄨 𝑑𝑡 = 0. ℎ→0 2ℎ lim

−ℎ

The set of all Lebesgue points of 𝜙 is called the Lebesgue set of 𝜙. Theorem B.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 B.1. The inequality |𝜙(𝜃)| ≤ (M𝜙)(𝜃) holds almost everywhere. Proof of Theorem. The operator ℎ

1 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝜙(𝑡 + 𝑥) − 𝜙(𝑥)󵄨󵄨󵄨 𝑑𝑡 𝑇𝜙(𝑥) = lim sup 2ℎ ℎ→0 −ℎ

satisfies: (a) 𝑇(𝜙1 + 𝜙2 ) ≤ 𝑇𝜙1 + 𝑇𝜙2 ; (b) 𝑇𝜙 ≤ |𝜙| + 𝑀𝜙 ; (c) 𝑇𝑔 = 0 if 𝑔 is continuous. Let 𝜙 ∈ 𝐿1 (−𝜋, 𝜋), 𝜆 > 0 and 𝜀 > 0. Choose a continuous function 𝑔 so that ‖𝜙 − 𝑔‖1 < 𝜀. From (a) we get 𝑇𝜙 ≤ 𝑇𝑔 + 𝑇(𝜙 − 𝑔) = 𝑇(𝜙 − 𝑔) and then, from (b), by Theorem B.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.

B.3 Classical maximal functions |

405

Radial maximal function Let 𝑢 be a complex-valued function defined 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 B.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 B.7 and Proposition B.1. The same holds in the case where 𝑝 = 1 and 𝑢 = P[𝜙]; if 𝑢 ∈ ℎ1 is arbitrary then, by Theorem B.7 and Proposition B.1, |{𝜁 ∈ 𝕋 : sup |𝑢(𝑟𝜁)| > 𝜆} ≤ 𝐶 0 1, then ̂ ̂ 𝑇𝑓(𝑛) = 𝑚𝑛 𝑓(𝑛)

(B.20)

for every integer 𝑛. Proof. Since 𝐿𝑝 is of type 𝑝 for 0 < 𝑝 ≤ 2, Nikishin–Stein theorem tells us that the operator 𝑇 with the above properties 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 𝜁𝑛 (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 ∫ − (𝑇𝑔)(𝑤)𝜙(𝑤) |𝑑𝑤| = ∫ − 𝜙(𝑤) |𝑑𝑤|− ∫ 𝜁−𝑛 (𝑇𝑔)(𝜁𝑤) |𝑑𝜁| 𝕋

𝕋

𝕋

=∫ − 𝜙(𝑤)𝑤𝑛 |𝑑𝑤|− ∫ 𝜁−𝑛 (𝑇𝑔)(𝜁) |𝑑𝜁|. 𝕋

𝕋

Hence (𝑇𝑔)(𝑤) = 𝑤𝑛∫ − 𝜁−𝑛 (𝑇𝑔)(𝜁) |𝑑𝜁| =: 𝑚𝑛𝑤𝑛

for a.e. 𝑤 ∈ 𝕋;

𝕋

this proves formula (B.19). The validity of (B.20) can then be deduced from the Weier­ strass 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 the­ orem (or by the Riesz–Thorin theorem), we can assume that 𝑞 > 2. Let 𝑓, 𝑔 be trigono­ metric polynomials. Then, in view of (B.19), we have 𝜋

𝜋 𝑖𝜃

−𝑖𝜃

∫ (𝑇𝑓)(𝑒 )𝑔(𝑒 − −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 ,

(B.21)

B.7 Banach’s principle and the theorem on a.e. convergence | 415

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 (B.21) for trigono­ metric polynomials implies (B.22) ‖𝑇𝑓𝑛 ‖𝑞 ≤ 𝐶𝑞 ‖𝑓‖𝑞 . 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 (B.22) give ‖𝑇𝑓‖𝑞 ≤ 𝐶𝑞 ‖𝑓‖𝑞 , which was to be proved.

B.7 Banach’s principle and the theorem on a.e. convergence The following fact, known as Banach’s principle, plays an important role in applica­ tions of Theorem B.17 to maximal operators. Theorem B.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)| < ∞,

for almost all 𝜔 ∈ Ω,

𝑛≥1

for every 𝑓 ∈ 𝑋. Then the operator 𝑇max : 𝑋 󳨃→ L0 (Ω, 𝜇) is continuous. Proof (cf. [158]). Let L0 (ℓ∞ ) denote the set of all functions 𝐹 = (𝑓1 , 𝑓2 , . . . ) : Ω 󳨃→ ℓ∞ with measurable coordinates. The following two facts imply the validity of the theo­ rem. (i) With the 𝐹-norm ‖𝐹(𝜔)‖∞ ∫ 𝑑𝜇(𝜔), 1 + ‖𝐹(𝜔)‖∞ 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 B.20 (Sawier [436]). 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, (b) 𝑇max 𝑓(𝜔) = ∞ a.e. on Ω1 for every 𝑓 from a subset of 𝑋 of the second Baire category.

416 | B Interpolation and maximal functions Proof (cf. [436]). 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, i.e. that 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,

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 ∪𝑁>1 𝐾𝑁 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. [460, 436]). The proof is similar to that of Theorem B.17. Theorem B.21 (Sawier–Stein). Let 𝑋 be as in Theorem B.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 B.22. Suppose the conditions of Theorem B.19 are satisfied. If the limit lim 𝑇𝑛 𝑓(𝜔) := 𝑇𝑓(𝜔)

𝑛→∞

a.e.

B.8 Vector-valued maximal theorem | 417

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 B.1, we have 𝜇(𝑆𝑓, 𝜀) ≤ 𝑐(𝜀/‖𝑓‖) (𝜀 > 0, 𝑓 ∈ 𝑋), (B.23) 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 (B.23) 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 above, the following alternative holds (cf. [460, 436]): Theorem B.23. Let 𝑋 be as in Theorem B.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 |𝑇𝑛𝑓(𝜔)| = ∞ a.e. in 𝜔 ∈ Ω for every 𝑓 from a set of the second category in 𝑋.

B.8 Vector-valued maximal theorem The following powerful theorem was proved by Fefferman and Stein in [148]. Theorem B.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 𝜋 𝑝/𝑞 𝑝/𝑞 ∞ ∞ 𝑞 󵄨 󵄨𝑞 ∫ ( ∑ (M𝑓𝑛 (𝑒𝑖𝑡 )) ) 𝑑𝑡 ≤ 𝐶 ∫ ( ∑ 󵄨󵄨󵄨󵄨𝑓𝑛 (𝑒𝑖𝑡 )󵄨󵄨󵄨󵄨 ) 𝑑𝑡, (B.24) −𝜋

𝑛=0

−𝜋

𝑛=0

where M denotes the Hardy–Littlewood maximal operator. A proof is also in [464], pp. 50–56. Actually Fefferman and Stein proved it in the case of functions defined on ℝ. The periodic version can then be deduced by consid­ ering the functions 𝜑𝑛 defined as 𝜑𝑛 (𝜃) = 𝑓𝑛 (𝑒𝑖𝜃 ) (|𝜃| < 2𝜋), and 𝜑𝑛 (𝜃) = 0 (|𝜃| > 2𝜋).

418 | B Interpolation and maximal functions This theorem can be expressed in terms of the space 𝐿𝑝 (ℓ𝑞 ). This space consists of those sequences {𝑓𝑛 }∞ 0 defined on 𝕋 for which 𝜋

‖{𝑓𝑛 }∞ 0 ‖𝐿𝑝 (ℓ𝑞 )

𝑝/𝑞

∞

:= − ∫ ( ∑ |𝑓𝑛 (𝑒𝑖𝑡 )|𝑞 𝑑𝑡) −𝜋

< ∞.

𝑛=0

Hence (B.24) can be written as ∞ ‖{M𝑓𝑛 }∞ 0 ‖𝐿𝑝 (ℓ𝑞 ) ≤ 𝐶‖{𝑓𝑛 }0 ‖𝐿𝑝 (ℓ𝑞 ) .

In the case 𝑝 = 1, Theorem B.24 does not hold, but we have (i) If {𝑓𝑛 } ∈ 𝐿1 (ℓ𝑞 ), then ‖{M𝑓𝑗 (𝜁)}‖ℓ𝑞 is finite for almost everywhere 𝜁 ∈ 𝕋. (ii) If {𝑓𝑛 } ∈ 𝐿1 (ℓ𝑞 ), then there is a constant 𝐶 > 0 such that 𝐶 ‖{𝑓𝑛 }‖𝐿1 (ℓ𝑞 ) , 𝜆

| {𝜁 ∈ 𝕋 : ‖{M𝑓𝑗 (𝜁)}‖ℓ𝑞 > 𝜆} | ≤

𝜆 > 0.

Further notes and results B.1. From Thorin’s proof of the Riesz–Thorin theorem (see [47] or [374]) it is seen that theorem holds for all 𝑝0 , 𝑝1 > 0. Calderón and Zygmund [83] proved that it holds for all 𝑝𝑗 > 0 and 𝑞𝑗 > 0 (𝑗 = 1, 2). B.2. Inequality (B.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 ‖𝑓 ̂ ‖𝐿𝑝󸀠 (ℝ) ≤ (𝑝1/𝑝 /𝑝󸀠 )

1/2

‖𝑓‖𝐿𝑝 (ℝ)

(see [45, 37]). B.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 (B.25) 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 all nonnegative 𝜇-measurable functions. If 𝑇 satisfies (B.3)(𝑝 = 1) and (B.4), then ∫ 𝑇𝑓 𝑑𝜇 ≤ 𝐾1 + 𝐾2 ∫ |𝑓| log+ |𝑓| 𝑑𝜎, Ω

where 𝐾1 and 𝐾2 are independent of 𝑓.

𝑆

(B.25)

B.8 Vector-valued maximal theorem | 419

The proof is similar to that of Theorem B.2. For the general Marcinkiewicz’s theo­ rem, see Zygmund [537, Ch. XII § 4]; the proof is much more difficult. B.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 the implication ∞

|𝑎𝑛 | 𝜆} ≤ ( ) . 𝑔(𝑤) 𝜆

See [502, 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) < ∞. B.10 (The case of ℝ𝑛 ). Let 0 < 𝑝 ≤ 2. Every continuous sublinear operator 𝑇 : 𝐿𝑝 (ℝ𝑛 ) 󳨃→ L0 (ℝ𝑛 ) which 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 [158, Corollary II.6.9].

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Index A Abel dual 149, 320 Aleman lemma 289 Approximation – by polynomials 13, 160 – of a singular inner function 165 – of the atomic function 165 – with inner functions 68 Area function 156 B Banach envelope 376 Banach principle 33, 134, 146, 415 Banach–Steinhauss principle 150 Bergman space – atomic decomposition 324, 354, 359 – Hardy–Stein characterization 117 – standard 103 – weighted 102 – weighted harmonic 103 Bergman weight 102 Besov spaces – abstract 322 – analytic 139 – best approximation 160 – decomposition 144, 163 – decomposition, 𝑝 > 1 139 – duality 139, 154 – f-property 171 – harmonic 138 – interpolation 146 – invariant 284 – isomorphism with B𝑝,𝑞 163 – K-property 172 – lacunary series 145 – membership of inner functions 167 – normal 162 – singular inner functions 166 – tangential 281 – vector-valued 361 Bi-Bloch lemma 199, 203, 255 Blaschke condition 49 Blaschke product 49, 52, 167, 170 – in Besov spaces 210 – in Hardy–Sobolev spaces 210

Bloch space 189 – compact composition operators 195 – Holland–Walsch characterization 209 – inner functions in b 208 – monotone coefficients 191 – Pavlović characterization 209 – predual – monotone coefficients 191 – weighted 197 – lacunary series 208 BMO 175 – B∞,2 ⊂ VMOA 186 𝑝 – B1/𝑝 ⊂ BMOA 186 – compact composition operators 196 – -f-property 180 – Fatou property 177 – Garsia norm 176 – higher order derivatives 207 – homogeneity 177 – inner functions in VMOA 185 – invariance of seminorms 178 – lacunary series 189 – Taylor coefficients 189 Bourgain’s lemma 119 C Carleson window 207 (𝐶, 1/𝑝 − 1)-convergence in 𝐻𝑝 132 Cesàro means of order 𝛼 > −1 131 Chebyshev inequality 400 Chebyshev lemma 54 Composition operators – from B to 𝐻𝑝 238 – from 𝐻𝑝 to B 349 – into mean Lipschitz spaces 254 – on Lipschitz spaces 251, 252 – derivative-free description 251 Composition with inner functions 64, 66 Convolution 4 Convolution lemma 336 D Decomposition – of abstract Besov spaces 323 – of Besov spaces, 0 < 𝑝 ≤ ∞ 144

444 | Index – of Besov spaces, 1 < 𝑝 < ⬦ 139 – of normal Besov spaces 163 – of polyharmonic spaces 116 – of spaces with subnormal weights 339 Density of polynomials in 𝐻𝑝 24 Diagonal spaces 102 Distortion function 103 Dual – of 𝐴(𝔻) 35 – of Bergman space with subnormal weights 345 – of Besov spaces, 0 < 𝑝 ≤ ⬦ 154 – of Besov spaces, 𝑝 > 1 139 – of 𝐻1 183 – of 𝐻𝑝 (𝑋), 𝑝 < 1 362 – of 𝐻𝑝 , 𝑝 < 1 155 – of 𝐻𝑝 , 𝑝 > 1 31 – of spaces with subnormal weights 341 𝑞 – of 𝑉𝑠 [𝑋] 151 F f-property 171, 260, 290 Fejér kernel 276 Formula – Carleson representation 285 – Cauchy integral 27 – Green 3, 47, 71, 150, 151, 179, 180, 182, 200 – Green–Poisson 3 – Jensen 48 – Parseval 11, 32 – Riesz representation 47, 48 Fractional derivative 164 Fractional integral 95, 101, 164 Fractional integration proposition 95 Function – almost monotone 99 – atomic 50, 63, 165 – derivatives 166 – moduli of smoothness 280 – conjugate 18, 19, 29 – convex of log 𝑟 42 – hyperbolic 𝑔-function 237 – increasing 23 – inner 50–52, 64, 66, 167, 170, 185, 260, 290 – inner in Besov spaces 167 – Köbe 61 – Littlewood–Paley 𝑔-funtion 213 – logarithmically convex 44 – log-subharmonic 41, 45, 53, 95

– Luzin area 213 – nearly convex 78, 79, 81 – normal 100, 103 – of bonded mean oscillation 175 – of class (H) 100, 157–159 – of class 𝑂𝐶2 115 – of vanishing mean oscillation 183 – outer 51, 62, 289 – polyharmonic 82, 115 – positive harmonic 7, 64 – quasi-nearly subharmonic 75, 78, 84, 88, 92, 94, 95, 101, 230, 256, 359 – several variables 117 – Rademacher 147, 409 – regularly oscillating 77, 98, 240, 255 – several variables 117 – regularly oscillating, vector 82 – semicontinuous 40, 67 – singular inner 50, 165, 166 – strictly increasing 23 – subharmonic 40, 43, 44, 48, 75, 95, 230 – discontinuous 40 – subnormal 100 – superharmonic 42, 253 – univalent 61 – Weierstrass 265 H Hadamard product 4, 353, 357 Hardy space – analytic 22 – and univalent functions 299 – atomic decomposition 73 – disk algebra 22 – equivalent ℓ𝑝 (𝐿2 )-norm 222 – harmonic 11 – harmonic, 𝑝 < 1 12, 15 – interpolation 118, 148 – on the sphere 72 – solid hull, 𝑝 < 1 331 – vanishing, 𝑜ℎ𝑝 16 Hardy–Bloch spaces 140 Hardy–Littlewood decomposition lemma 22, 49, 135 Hardy–Sobolev space 164, 281, 290 – membership of inner functions 170 Hardy–Stein identity 30, 59, 71, 72, 180, 181 – asymptotic form 90

Index |

Harmonic Schwarz lemma 69, 246 Hyperbolic Hardy class 235 I Inequality – Carleman 52, 54 – Clunie–MacGregor 192 – Fejér–Riesz 56 – Flett 156, 230 – Girela–Pavlović–Peláez 193 – Hardy 56, 130, 173 𝑝 – Hardy 𝑀∞ 57 – Hardy–Littlewood Σ 18, 118, 121, 131, 300, 314 – Hardy–Littlewood (1/𝑝 − 1) 15, 25, 118, 133, 280 – Hardy–Littlewood 𝑀𝑝2 59–61, 71, 155, 193 𝑝 𝑀𝑞

– Hardy–Littlewood 121, 135, 156 – Harnack 8, 79 – Hausdorff–Young 123, 189, 398, 402 – Heinz 305, 311 – Hilbert 56 – Hollenbeck–Verbitsky 29 – Huber 55 – isoperimetric 52 – John–Nirenberg 175 – strong 175 – Kahane 147 – Kalton (1/𝑝 − 1) 354 – Khintchin 409 – Kolmogorov–Smirnov 62 – Korenblum 193 – Littlewood–Paley 155, 193, 229–231 – Littlewood–Paley hyperbolic 237 – Luecking 234 – Makarov 192 – Mateljević–Pavlović 124 – Paley 419 – on lacunary series 60 – Pichorides 29 – Riesz–Zygmund 21, 56 – geometric interpretation 21 – Storozhenko 279, 280 – Wittman 137 Inner factor 51 Integrability of lacunary series 110 – with complex coefficients 112 – with positive coefficients 111 – with vector coefficients 386

Integrability of power series – with positive coefficients 110 Integral means 10 – convexity 44 – log-convexity 44, 45 – monotonicity 23 – monotonicity and convexity 42 – of polynomials 43 – of univalent functions 294 Isometry 𝐿𝑝 with ℎ𝑝 11 Isomorphism 𝐴𝑝 with ℓ𝑝 383 J Jones lemma 302 K Knese lemma 180 K-property 171 L Lacunary series 60, 61, 110, 204, 240 – and Peano curves 71 – in Besov spaces 145 – in BMOA 189 – in 𝐶[0, 1] 112 – in diagonal weighted spaces 102 – in quasi-Banach spaces 384 – in weighted 𝐿∞ -spaces 113 – in weighted mixed norm spaces 102 Laguerre polynomials 165 Laplacian 3 Lebesgue point 19, 20, 404 Lebesgue set 404 Lipschitz condition – for the modulus – of a harmonic function 262 – of an analytic function 249 – radial 283 Lipschitz space 241 – analytic 246 – and spaces of harmonic functions 267 – BMO-type characterizations 286 – composition operators 252 – generalized 280 – harmonic on 𝔻 245 – mean on 𝕋 247 – of higher order 265 – on the sphere 293 Littlewood’s conjecture 137

445

446 | Index Local-to-global estimates 86 Log-subharmonicity lemma 41, 215 M Majorant 242 – Δ 2 -condition 266 – concave 243 – Dini 242, 244, 249 – fast 242–244, 249, 266, 290 – fast of order 𝑛 283 – of order 𝑛 267 – on [1, ∞) 266 – regular 242, 250, 254, 260, 283, 286 – regular of order 𝑛 267, 281 – slow 242, 243, 245, 259, 266 – slow of order 𝑛 267 Makarov low of iterated logarithms 192 Maximal function 88, 92, 94, 95, 116, 142 – (𝐶, 𝛼) 132 – main 403 – non-tangential 132, 215, 407 – radial 72, 405 Maximal lemma 260 Maximal theorem – (𝐶, 𝛼) 133 – complex 26, 125, 134, 167, 169, 407 – main 134, 403, 406 – radial 405 – subharmonic 261, 406 – vector (𝐶, 𝛼) 212, 219, 221 – vector log-subharmonic 212, 215 – vector subharmonic 212 – 𝑊- 143 Maximum modulus principle 4, 12 – Kalton 359 – Smirnov 27, 32 Maximum principle 42, 177 Mean value property 4 Measure – Carleson 207, 234, 302 – Möbius invariant 2, 85 – Riesz 45, 230, 231 – local estimates 231 – of |𝑓|𝑝 58 Mixed norm spaces – analytic 83 – and Lipschitz spaces 99 – completeness 84 – harmonic 83, 92, 94, 116

– isomorphic classification 109 – minimal 85 – nonadmissible 85 – weighted 100, 101 – with normal weight 101 – with subnormal weights 105 Möbius dual 170 Möbius group 2 Moduli of continuity 241 Moduli of smoothness 264 Monotone coefficients in 𝐻1 128 Monotone coefficients in 𝐻𝑝 126 Multipliers 414 – between Besov spaces 334 – compact 323 – decomposition method 333 – monotone – between Besov spaces 335 – between Hardy spaces 336 – of Kellogg spaces 330 – of spaces with subnormal weights 345 – preduals 322 – second dual 322 N Nevanlinna class 382 Nontangential limits 10 – of an 𝐻𝑝 -function 24 Normal sequence 339 O Operator – composition 194 – compact 194 – from ℓ𝑝 to ℓ𝑞 384 – Hilbert 19, 30, 275, 346 – invertible 379 – maximal 403 – of strong type 400 – of weak type 400 – quasilinear 118, 119, 122, 400 – Riesz projection 29 – subadditive 400 – sublinear 403, 411 – Töplitz 171 – Wirtinger 3 Ortega–Fàbrega lemma 197 Oscillation 76, 77 Outer factor 51

Index | 447

P Poisson integral 66, 254, 259 – of a function 6 – semicontinuous 36, 68 – of a measure 6 – of log |𝑓∗ | 26 – of 𝑓∗ , 𝑓 ∈ 𝐻1 27 – of 𝑓∗ , 𝑓 ∈ ℎ𝑝 11 – of the conjugate function 32 – on the unit ball 38 Poisson kernel 3, 5, 16 – conjugate 18 Poisson–Stieltjes integral 8, 9, 28 Predual – of 𝐻∞ 35 – of 𝐻1 185 – of the Bloch space 190 Pseudohyperbolic metric 2 Q 𝑞-Banach envelope 376 – of 𝐻𝑝 324 Quasiconformal harmonic mappings 304 – bi-Lipschicity 305 – boundary correspondence 305 Quasiconformal mappings 304 R Radial derivative 97, 138, 139 Radial limits 9, 10 – and mean convergence 24 – of an ℎ𝑝 -function 11 – of conjugate function 19 – of Hardy–Bloch function 146 Radial Lipschitz condition – for the modulus – of a harmonic function 262 – of an analytic function 262 Representation – of outer functions 51 – of singular inner functions 50 Reproducing property 5 Riesz polynomials 173 S Schauder basis 31, 128, 132, 340, 381 – nonexistence 172 Schwarz modulus lemma 79, 249, 257 Smirnov class 383

Space – admissible 12 – Dirichlet 285 – 𝐹-space 382 – HFP 320 – homogeneous 319 – Jackson–Bernstein 340 – K𝑎 , K𝑠 35 – Kellog 122 – minimal 13 – of Borel measures 6 – of Cauchy transforms 35 – of type 𝑝 147, 411 – PL-convex 347 – 𝑝-normed 375 – 𝑝-Zygmund 283 – 𝑄𝑝 207 – quasinormed 375 – self-conjugate 92, 101, 105, 275, 346 – solid 329 – weak Lebesgue 118, 146, 399 – with Fatou property 320 – Zygmund 265 Stoltz angle 10, 211 Subharmonic behavior – and conformal mappings 80 – of |𝑢|, 𝑢 ∈ RO 77 – of |𝑢|𝑝 |∇𝑢|𝑞 79 – of ‖𝐹(𝑧)‖𝑋 359 – of |∇𝑢|, 𝑢 ∈ RO 77 – polyharmonic functions 82 Subordination principle 62, 63 – for BMOA 195 Symmetric difference 99, 217, 264 T Tangential derivative 98, 139 𝑝 Tauberian nature of B1/𝑝 188 Theorem – Ahern 63 – Ahern–Clark 286 – Ahern–Jevtić 167 – Aleksandrov 34 – on boundary decay 38, 39 – Almansi representation 82 – Aoki–Rolewicz 376 – Astala–Koskela 312, 314 – Baernstein–Cima–Shober 304 – Baernstein–Girela–Peláez 301

448 | Index – Banach–Steinhaus 380 – Bary–Stechkin 292 – Bieberbach 295, 296 – Blasco–de Souza 248 – Böe 290 – Bourdon–Shapiro–Sledd 186 – Burkholder–Gundy–Silverstein 72 – Calderón 213, 225 – Calderón–Zygmund 418 – Carathéodory convergence 310, 311 – Carleson – on Carleson measures 207 – Carleson–Hunt 36, 132 – Carleson–Jacobs–Havin – Shamoyan 262 – closed graph 381 – Coifman–Rochberg 354 – de Souza–Sampson 149 – Duren–Romberg–Shields 155 – Dyakonov 249, 250 – on non-BMO functions 210 – Fatou 9, 32, 51 – Fefferman duality 183 – Fefferman multiplier 330 – Fefferman–Stein 408 – on subharmonic behavior 13, 14, 74, 257 – Flett – for RO functions 240 – Garsia 179, 180 – Gurariy–Matsaev 102, 112 – Hardy–Littlewood 244 – fractional integration 97, 101 – harmonic conjugates 13, 85, 87, 89, 91 – on lacunary series 112 – projection 92 – Hardy–Littlewood fractional differentiation 145 – Hardy–Littlewood–Sobolev 124, 336 – Hardy–Littlewood–Sunouchi 133 – Hardy–Prawitz 298, 300 – Hardy–Rogosinski 134 – Herglotz–Plessner 7 – Hinkkanen 262 – Holland–Twomey 299, 300 – Hurwitz 8, 23 – hyperbolic Calderón 237 – increasing inclusion 92, 145 – Kalaj 306

– Kalton 37 – maximum modulus principle 359 – Karamata–Szász Tauberian 188 – Kislyakov–Xu 118, 121, 122 – Köbe distortion 81, 296 – Köbe one-quarter 81, 295, 297 – Kolmogorov – on conjugate functions 33 – Kolmogorov–Smirnov 32 – Konyagin 136 – Koskela–Manojlović 115 – Kwon 238 – Kwon–Pavlović 198 – Laitila–Nieminen–Saksman – Tylli 196 – Lindelöf 36 – Littlewood Tauberian 385 – Littlewood–Paley 240 – Littlewood–Paley 𝑔− 213, 222 – Lusky 109 – Madigan 251 – Marcinkiewicz 119, 400 – Marcinkiewicz 𝐿 log+ 𝐿 418 – Martio 306 – McGehee–Pigno–Smith 137 – mixed embedding 94, 145 – Montel 4 – Mori 305, 310 – Nikishin 411 – Nikishin–Stein 33, 146, 326, 412–414 – on a.e. convergence 416 – on Banach envelope of 𝐻𝑝 (𝑋) 362 – on equivalent norms 380 – on Lebesgue points 404 – on Riesz measure 45 – on strong convergence in 𝐻1 129 – open mapping 380 – Paley – on Fourier coefficients 402 – on lacunary series 61, 71 – Pavlović 104, 267, 278, 281 – on boundary decay 38 – Pavlović–Peláez 103 – Pommerenke 315 – univalent functions in BMOA 303 – Prawitz 294, 295, 298, 299, 301 – Privalov – absolute continuity of 𝑓∗ 29, 126

Index |

– conjugate functions 246, 275, 346 – uniqueness 26 – Privalov–Plessner 19, 20, 32 – Radó–Kneser–Choquet 22 – Ramey–Ulrich 197 – Riesz – conjugate functions 29 – projection 29 – radial limits 24 – Riesz brothers – absolute continuity of 𝑓∗ 28 – absolutely continuous measures 28 – Riesz factorization 50 – Riesz projection 183 – Riesz representation 47 – Riesz the Brothers 27 – Riesz–Herglotz 7, 8, 28, 50 – Riesz–Szegö – on log |𝑓∗ | 26 – uniqueness 26 – Riesz–Thorin 111, 397 – Rogosinski 70 – Sawier–Stein 134, 146, 416

– Shapiro 16, 34 – Shields–Williams 345, 350 – harmonic conjugates 275 – Sidon 72 – Smirnov – on conformal mappings 37 – Smirnov factorization 51 – Stein 413 – Stephenson 66 – tangential differentiation 98 – Tauber 385 – Zygmund 102, 269 – conjugate functions 33 – on (𝐶, 1/𝑝 − 1)-convergence 132 Transform(ation) – Cauchy 35, 171 – Hilbert 305, 306 – Hilbert matrix 70 – Kelvin 39 – Möbius 2, 39 U Uchiyama lemma 179 Uniform boundedness principle 380

449