Theory of Np Spaces (Frontiers in Mathematics) 3031397037, 9783031397035

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Frontiers in Mathematics

Le Hai Khoi Javad Mashreghi

Theory of 𝒩p Spaces

Frontiers in Mathematics Advisory Editors William Y. C. Chen, Nankai University, Tianjin, China Laurent Saloff-Coste, Cornell University, Ithaca, NY, USA Igor Shparlinski, The University of New South Wales, Sydney, NSW, Australia Wolfgang Sprößig, TU Bergakademie Freiberg, Freiberg, Germany

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Le Hai Khoi • Javad Mashreghi

Theory of N p Spaces

Le Hai Khoi Department of General Education University of Science and Technology of Hanoi, Vietnam Academy of Science and Technology Hanoi, Vietnam

Javad Mashreghi Department of Mathematics and Statistics Université Laval Québec, QC, Canada

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-031-39703-5 ISBN 978-3-031-39704-2 (eBook) https://doi.org/10.1007/978-3-031-39704-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

Banach and Hilbert spaces of analytic functions on the open unit disc .D and the operators acting on such spaces are currently active domains of research. The classical part of this subject was developed in the early days of the preceding century. However, due to numerous applications in science and engineering as well as its dominant role in other branches of mathematics, the topic has gained enormous momentum in recent years. A wide range of mathematicians and applied scientists are presently working on different aspects of the theory. A list of abstract analytic spaces is long, and there are several excellent books on each space. Since we are mainly dealing with Hardy, Dirichlet, and Bergman spaces in this note, a few references are mentioned below. A panoramic approach to function spaces is provided in [108]. Theory of Hardy Space is discussed in depth in [23, 38, 56, 66]. Some further spaces which live inside the Hardy space .H 2 are studied in [31, 32, 35, 36, 67]. Contrary to Hardy spaces, two close cousins, i.e., Bergman and Dirichlet spaces, are not completely settled yet and each topic contains numerous unsolved problems. Theory of Bergman spaces is available in [21,40,107], and Dirichlet spaces are discussed in two new texts [4, 25]. The above references also contain scattered information on Bloch, Blochtype, and Bergman-type spaces. The Bergman spaces and Bergman-type spaces have been recently the center of some intense studies. This stems from the fact that such spaces play a vital role in both function theoretic and operator theoretic development of function spaces and have a close connection to Bloch spaces, which appear as the images of the bounded functions under the Bergman projection. Bloch spaces by themselves are also important and play the role of the dual spaces of the Bergman spaces. Operators on these spaces, in particular, composition operators on .A−p (D), are concrete objects to study the geometric properties of these spaces. See [40] and references therein. In the family of Bergman and Bergman-type spaces, the class of .Qp -spaces, .p > 0, on .D have proved to be more important [7, 102]. The .Qp -space consists of functions in .Hol(D) such that  .

sup

a∈D D

|f  (z)|2 (1 − |σa (z)|2 )p dA(z) < ∞.

v

vi

Preface

Here .σa (z) = (a − z)/(1 − az) is the automorphism of .D that exchanges 0 and a, and dA is Lebesgue area measure on the plane, normalized so that .A(D) = 1. It is wellknown that .Qp -spaces coincide with the classical Bloch space .B for .p ∈ (1, ∞) and .Q1 is precisely the celebrated BMOA, the space of holomorphic functions on .D with bounded mean oscillation. However, for .p ∈ (0, 1), the .Qp -spaces are all totally new objects. The initial motivation of this work, the .Np -spaces, comes from the study of the various properties of .Qp -spaces. One notices that if, in the definition of the .Qp -space, .f  (z) is replaced by .f (z), then we have a new family, the so-called .Np -space on .D [73, 95]. More explicitly, the .Np -space consists of functions in .Hol(D) for which  .

sup

a∈D D

|f (z)|2 (1 − |σa (z)|2 )p dA(z) < ∞.

As a matter of fact, due to the similarities and the differences between .Qp -spaces and the Bergman-type spaces .B and .A−1 , it is natural to study the .Np -spaces and the operators acting on them. Some basic properties of .Np -spaces are as follows. For .p > 1, the .Np -space coincides with the Begrman-type space .A−1 consisting of holomorphic functions on the disc for which .sup |f (z)|(1 − |z|2 ) < ∞. However, for .p ∈ (0, 1], the .Np -spaces are all different function spaces with different topological and algebraic properties. Composition operators as well as weighted composition operators acting on .Np -spaces, or from .Np -spaces into Bergman-type spaces .A−q , have different features and form interesting objects to study as concrete operators. We also treat .Np -spaces and their operators in higher dimensions. Let .B denote the open unit ball in .Cn , with .S as its boundary. The space .Hol(B) consists of all holomorphic functions in .B equipped with the compact-open topology. Let .ϕ denote a non-constant holomorphic self-map of .B, and let .ψ be a holomorphic function on .B, which is not identically equal to zero. These functions induce the linear operator .Wψ,ϕ : Hol(B) → Hol(B), which is so-called weighted composition operator with symbols .ψ and .ϕ, and is defined by Wψ,ϕ (f )(z) = ψ(z) · (f ◦ ϕ(z)),

.

f ∈ Hol(B), z ∈ B.

Note that if .ψ is the constant function 1, then .Wψ,ϕ = Cϕ is the classical composition operator, while if .ϕ is the identity mapping, then .Wψ,ϕ = Mψ is precisely the multiplication operator .Wψ,ϕ = Mψ . The literature on composition operators and weighted composition operators acting on spaces of holomorphic functions on .D is simply beyond control. We just refer the readers to the monographs [17, 83] for further information. For composition operators on .A−p (D), see [40, 107] and references therein. This monograph consists of 12 chapters to describe .Np -spaces. At the beginning, we introduce some weighted spaces as well as the classical Hardy space .H 2 , Bergman space 2 .B , and Dirichlet space .D. These three families of Hilbert function spaces are the most

Preface

vii

senior function spaces that are needed in studying .Np -spaces. The definitions of these spaces can be generalized, which lead us to some classes of Banach spaces, such as Bloch p −p , where .p > 0. As a counterpart to these families .B spaces and Bergman-type spaces .A on the open unit disc .D, an important part of our studies is about entire functions in the complex plane .C, and even their several complex variable version in .Cn . A representative family is the Fock space of entire functions which is treated in depth. Hanoi, Vietnam Québec, QC, Canada

Le Hai Khoi Javad Mashreghi

Contents

1

Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Classical Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Weighted Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Special Families of Weighted Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Weighted Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Special Families of Weighted Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Hilbert Spaces of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Hilbert Spaces of Formal Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Some Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 9 11 14 17 22 25

2

The Counting Function and Its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Nevanlinna Counting Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Littlewood’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A Change of Variable Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Generalized Nevanlinna Counting Function . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Interrelation Between Nevanlinna Counting Functions . . . . . . . . . . . . . . . . . . . 2.6 The Littlewood–Paley Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 29 31 32 34 38

3

.

Np -Spaces in the Unit Disc .D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Comparison with Qp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Test Function kw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 3.3 An Equivalent Norm for Aγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Np Versus A−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Np Elements from the Hadamard Gap Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 41 41 43 46 53

4

The .α-Bloch Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Spaces Bα and Bα,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Some Operators on α-Bloch Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Pre-dual of α-Bloch Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 59 63

ix

x

Contents

4.4 The Space Hα∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 A Test Function for Hα∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68 70

5

Weighted Composition Operators on .D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Multiplication Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Carleson Measures for Np . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Boundedness of Wψ,ϕ : Hα∞ → Np . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Compactness of Wψ,ϕ : Hα∞ → Np . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Boundedness of Wψ,ϕ : Np → Hα∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The Compactness of Wψ,ϕ : Np → Hα∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72 72 75 78 85 86 90

6

Hadamard Gap Series in .Hμ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.1 Hadamard Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 α-Bloch Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.3 α-Bloch Spaces and Hadamard Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4 Normal Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ∞ Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Hμ∞ and Hμ,0 99 ∞ ∞ 6.6 Hμ and Hμ,0 Spaces and Hadamard Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.7 Lower Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Notes on Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7

Np Spaces in the Unit Ball .B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 On the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Automorphism a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Surjective Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Np (B) Is a Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

n+1

8

109 109 111 113 114

7.5 The Embedding Np (B) → A− 2 (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Embedding A−k (B) → Np (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Np (B) as a Banach Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 The Embedding B(B) → Np (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 117 118 120 120

Weighted Composition Operators on .B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 A Test Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Boundedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Compactness: Easy Reformulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Compactness: Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Estimation of |f (z) − f (w)| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Compactness of Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Essential Norm: Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Essential Norm: Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 125 127 131 135 146 149 151

Contents

xi

9

Structure of .Np -Spaces in the Unit Ball .B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Multipliers and M Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 An Upper Estimate for  · p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Space Np0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Closure of Polynomials in Np0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Carleson Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Np Spaces Are Not Homeomorphic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 154 156 158 160 164 171

10

Composition Operators Between .Np and .Nq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Boundedness of Cϕ : Np −→ Np , the Univalent Symbol . . . . . . . . . . . . . . . . 10.2 Boundedness of Cϕ : Np −→ Nq , the General Symbol . . . . . . . . . . . . . . . . . . 10.3 Characterizations of Compactness of Cϕ : Np −→ Nq . . . . . . . . . . . . . . . . . . . 10.4 A Sufficient Compactness Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 174 177 184 186

11

.

Np -Type Functions with Hadamard Gaps in the Unit Ball .B . . . . . . . . . . . . . . . . 11.1 Some Estimates for Volume and Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hadamard Gaps in Np -Spaces (p ≤ n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Hadamard Gaps in Np -Spaces (p > n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∞ ....................................... 11.4 A Characterization of Hμ∞ and Hμ,0 ∞ 11.5 The Growth Rate in Hμ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes on Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 190 197 201 205 213

12

.

N (p, q, s)-Type Spaces in the Unit Ball of .Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 N (p, q, s) as a Functional Banach Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Closure of Polynomials in N (p, q, s)-Type Spaces . . . . . . . . . . . . . . . . . . 12.3 The Space N∗ (p, q, s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Hadamard Gaps in N (p, q, s)-Type Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Carleson Measure and Embedding Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Embedding Relationship with Weighted Hardy Space . . . . . . . . . . . . . . . . . . . . Notes on Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 215 218 222 229 234 240 250

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

1

Function Spaces

1.1

Classical Function Spaces

Let .D be the open unit disc in the complex plane .C, let .Hol(D) be the space of holomorphic functions on .D, and let .dA = dxdy be the area Lebesgue measure. In studying .Np spaces, several classical function spaces enter our discussion. For reference, we just provide their definitions below. Then we treat some less well-known function spaces in further detail. The most classical function space is the Hardy space   H 2 := f ∈ Hol(D) : sup

2π

.

 |f (reiθ |2 dθ < ∞ .

0 1.

Hence, .b ∈ D . Therefore, we have .D  2β . Conversely, suppose that .D  2β . We show that .β ∗ ≥ 1 leads to a contradiction. If β ∗ = 1, then there is a subsequence .(βnk ) such that .limn→∞ βnk by

1/nk

.

an =

.

⎧ ⎨β −1

if n = nk , for some k,

⎩0

otherwise.

nk

−1/n

= 1. Define .a = (an )

Clearly, .lim supn→∞ |an |1/n = limk→∞ βnk k = 1, and thus .a ∈ D , but since .|ank |βnk = 1 = 0 for infinitely many .nk , we conclude .a ∈ / 2β , which contradicts .D  2β .

6

1 Function Spaces

If .β ∗ > 1, then there is a subsequence .(βnk ) such that .βnk .βnk > 1 which implies

1/nk

∞ .

> r > 1 for some r. Thus,

βn2 = +∞.

n=0

Now, consider .a = (an ), where .an ≡ 1. Then .a ∈ D . Since .D  2β , we must have ∞ 2 2 .a = n=0 βn < +∞, which is a again a contradiction. (iii) This part follows from (i) and (ii).  

1.3

Special Families of Weighted Sequence Spaces

For any positive sequence .β = (βn )n≥0 , we define the number βρ = lim inf

.

n→∞

log βn n log n

and the function .μβ : N → R+ by μβ (n) =

.

nβn−1 , βn

(n ≥ 1).

We also introduce the following family of sequences .β = (βn )n≥0 that are of great importance in the sequel: A = {β : β∗ = +∞},

.

B = {β : βρ > 0}, C = {β : βρ < +∞}, D = {β : βn /βn+1 ↓ 0}, G = {β : μβ is bounded}, H = {β : ∃ τ = τ (β) > 0 such that μβ (n) ↓ τ }. The interrelation between these classes are addressed in the following result. Proposition 1.3.1 The following assertions are true.

1.3 Special Families of Weighted Sequence Spaces

(i) (ii) (iii) (iv) (v)

7

D  A but .C  A. D \ G = ∅ and .G \ D = ∅. .G  B  A, so .B \ D = ∅ and .D \ B = ∅. .H  (A ∩ B ∩ C ∩ D ∩ G) = C ∩ D ∩ G. .(C ∩ G) \ (D ∩ G) = ∅ and .(D ∩ G) \ (C ∩ G) = ∅. . .

Proof (i) To show .D ⊆ A, let .β = (βn ) ∈ D. Since .βn+1 /βn ↑ +∞, there is some .N1 such that for .n > N1 , .βn+1 /βn > 2. So .βn − βn−1 > βn−1 ≥ βN1 (n > N1 ). Therefore, .βn ↑ +∞. Choose .N2 > N1 such that .βn > 1 for .n > N2 . For any .S > 0 arbitrarily large, choose .N3 such that whenever .n ≥ N3 , .βn+1 /βn > S 2 . Let .N = max{N2 , N3 }; then .β ∈ A because for .n > 2N, βn = βN

n−1 

βp+1 > 1 · S 2(n−N ) = S n+(n−2N ) > S n . βp

.

p=N

Therefore, .β∗ = +∞. To see why .D = A, choose the following sequence .β (1) , which is in .A but not .D: ⎧ ⎨nn if n is odd, (1) .βn = ⎩n2n if n is even. To show .C  A, let .βn ≡ 1 for all n. Then clearly .βρ = 0 < +∞, but .β∗ = 1 < +∞. √ √ (2) (2) (2) (ii) To verify .D \ G = ∅, let . βn = n!, then .βn /βn+1 = ( n + 1)−1 ↓ 0, but √ (2) (2) ∈ D \ G. .μ β (n) = n ↑ ∞. Equivalently, .β (3)

To show, choose .β (3) = (βn ): βn(3) =

.

⎧ ⎨n!

if n is odd,

⎩2n!

if n is even.

Then (3)

μβ (n) =

.

⎧ ⎨2

if n is odd,

⎩1

if n is even,

2

and

(3) βn (3) βn+1

=

⎧ ⎨

1 2(n+1) ⎩ 2 n+1

Therefore, .β (3) ∈ G but .β (3) ∈ / D, which implies .G \ D = ∅. (iii) Let .β ∈ B. Then there is .c > 0 such that

if n is odd, if n is even.

8

1 Function Spaces

.

log βn > c, n log n

n ≥ 1,

and thus lim

.

n→∞

 n

βn ≥ lim nc → ∞. n→∞

(4)

(4)

(4)

(4)

This observation shows .B ⊆ A. To see, take . β0 = β1 = β2 = 1 and .βn = (log n)n (4) (4) for .n ≥ 3. Then .β∗ = +∞, but .βρ = 0, so .β (4) ∈ A \ B. Note also that .β (4) ∈ D, so .D ⊆ B. To prove .G ⊆ B, let .β ∈ G. There is .S > 0 such that .μβ (n) < S for all n, so βn >

.

n(n − 1)βn−2 nβn−1 n!β0 > > ··· > n . S S S2

Since .n! ≥ nn/2 , we have .

log β0 + n2 log n − n log S log β0 + log n! − n log S 1 log βn > ≥ → n log n n log n n log n 2 (1)

as .n → ∞. Hence, .βρ > 0 as we need. Now choose .β = β (1) as above, clearly .βρ = 1,  4n2 2n but .μ(1) is not bounded, so .β ∈ / G and .G  B. This together with β (2n + 1) = 2n+1 .G ⊆ D also implies .B ⊆ D. Therefore, .G  B  A; .B \ D = ∅ and .D \ B = ∅. (iv) Let .β ∈ H. Since .(μβ (n)) is convergent, it is bounded and clearly .β ∈ G. From the ↓ τ > 0, we have hypothesis . nββn−1 n βn−1 βn (n + 1)βn > > nβn+1 βn+1 βn

.

and .

βn β0 nβn−1 < ··· < → 0, < (n + 1)βn (n + 1)β1 βn+1

so .β ∈ D. Also, as .n! ≤ nn , βn
0). Then it is clear that .β (5) ∈ C ∩ D ∩ G, (5) / H. Hence, .H  C ∩ D ∩ G. but .μβ (n) ↓ 0, so .β ∈ 2

(v) Consider .β = β (3) ; then .β ∈ (C ∩ G) \ (D ∩ G). Also, for .β = β (5) where .βn(5) = nn , then .β ∈ (D ∩ G) \ (C ∩ G).  

1.4

Weighted Hardy Spaces

Let  H(β) = f (z) =

∞

.

 an zn , (an ) ∈ 2β ∩ D .

n=0

More explicitly, the weighted Hardy space .H(β) consists of analytic functions f (z) =

∞

.

an zn ∈ Hol(D)

n=0

such that f H(β) :=

∞

.

1/2 βn2 |an |2

< +∞.

n=0

As in (1.3), with f, g =

∞

.

βn2 an bn ,

n=0

where .(an ) and .(bn ) are, respectively, the Taylor coefficients of f and g, we see that H(β) is in fact an inner product function space. If there is no ambiguity, we write .f  for 2 .f H(β) . Depending on .(βn ), the space . ∩ D may not be complete in norm (1.2), so β .H(β) is not necessarily complete and thus not a Hilbert space. Using Theorem 1.2.1, we obtain the following theorem which characterizes the completeness of .H(β) in terms of sequence .β. .

10

1 Function Spaces

Theorem 1.4.1 The space .H(β) is a Hilbert space of holomorphic functions on the open unit disc .D if and only if 1/n

β∗ = lim inf βn

.

n→∞

≥ 1.

(1.4)

The function space .H(β) induced by the weight .β that satisfies condition (1.4) is known as a weighted Hardy space. The name weighted Hardy space comes from the observation that if .βn ≡ 1, then .H(β) becomes the classical Hardy space .H 2 (D). Two further interesting cases are as follows. If 1 βn = √ , n+1

.

(n ≥ 0),

we obtain the Bergman space. If βn =

.

√ n + 1,

(n ≥ 0),

we end up with the Dirichlet space. A practically useful class of the weight sequences consists of those .β satisfying

.

∞ 1 < +∞. βn2

(1.5)

n=0

It is clear that condition (1.5) is stronger than condition (1.4). Any space .H(β) with .β satisfying condition (1.5) is called a small weighted Hardy space. An important property of these spaces is that functions in .H(β) extend continuously to the closed unit disc .D = {z ∈ C : |z| ≤ 1}. Here is yet another more extreme case which consists of entire functions. Theorem 1.4.2 The space .H(β) is a Hilbert space of entire functions if and only if .β∗ = +∞. In particular, since in this case .β∗ > 1 and (1.5) is satisfied, .H(β) is a small weighted Hardy space. If βn =

.

√

n!,

(n ≥ 0),

we obtain the classical Fock space .F 2 (C) of entire functions.

1.5 Special Families of Weighted Hardy Spaces

1.5

11

Special Families of Weighted Hardy Spaces

Using Proposition 1.3.1, we define some special types of Hilbert spaces of entire functions. Depending on the weights .β, we denote several subclasses .H(βE ) as follows: ⎧ ⎪ H(βρ ), if β ∈ B, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H(βρ+ ), if β ∈ B ∩ C, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ . H(βρ+ , T ), if β ∈ C ∩ G, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Eγ2 (T ), if β ∈ D ∩ G, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 Eγ (T , τ ), if β ∈ H. Note that .Eγ2 (T , τ ) is the smallest subclass of .H(βE ) we consider in this section. One of the main characteristics of an entire function is its growth order. Recall that an an zn has finite order .ρ if and entire function with power series representation .f (z) = only if ρ = lim sup

.

n→∞

n log n < +∞. − log |an |

We consider the following two properties: (.P1 ) Every function .f ∈ H(βE ) has a finite order. (.P2 ) There exists a function .g ∈ H(βE ) which has a nonzero order. In the next two theorems, we characterize the above two properties in .H(βE ) spaces. Theorem 1.5.1 A Hilbert space .H(βE ) has property .(P1 ) if and only if it is induced by β ∈ B.

.

Proof Necessity: Assume that every function in .H(βE ) has a finite order, but .βρ = 0. Thus, there exists an increasing sequence .(nk )k≥1 such that .

or equivalently

log βnk 1 < , nk log nk k

(k ≥ 1),

12

1 Function Spaces

.

log βnk
0, but there is some .f (z) = n=0 an z not having finite order in .H(βE ). We can thus choose an increasing sequence .(nk )k≥0 such that .|ank | < 1 and .

nk log nk > k, − log |ank |

(k ≥ 1),

or equivalently −nk /k

|ank | > nk

.

,

(k ≥ 1).

From the hypothesis .βρ > 0, there are numbers .c > 0 and .N ∈ N such that .

log βn > c, n log n

(n > N ).

Thus, βn > ncn ,

.

Take the integer K such that .1/K < c. Then

(n > N).

1.5 Special Families of Weighted Hardy Spaces cnk −nk /k

|ank |βnk > nk

.

n (c−1/k)

= nk k

13

> 1,

(k ≥ K).

This contradicts .f  < +∞.

 

Theorem 1.5.2 A Hilbert space .H(βE ) has property .(P2 ) if and only if it is induced by β ∈ A ∩ C.

.

Proof Since .β induces a .H(βE ) space if and only if .β ∈ A, we only need to prove that H(βE ) has property .(P2 ) if and only if .β ∈ C. Necessity: Suppose .βρ = +∞. For any .(an ) ∈ 2β that induces the power series of an element .f ∈ H(βE ), there is an integer N such that .|an |βn < 1 for all .n > N. Hence,

.

.

lim sup n→∞

n log n n log n ≤ lim sup = 0, − log |an | n→∞ log βn

which shows that all .f ∈ H(βE ) have order 0. Sufficiency Suppose .βρ < +∞. Then there exist .c > 0 and an increasing sequence (nk )k≥1 such that

.

.

log βnk < c. nk log nk

Choose .(an )n≥1 as in (1.6). Thus, .f ∈ H(βE ) and, moreover, .

lim sup n→∞

nk log nk n log n nk log nk 1 = lim sup ≥ lim sup = > 0, − log |an | log n β (cn + 1) log n c k nk k k k→∞ k→∞

so the power series induced by this sequence is of nonzero order.

 

As a consequence of Theorems 1.5.1 and 1.5.2, we have the following corollary. Corollary 1.5.3 For a Hilbert space .H(βE ) induced by .β, precisely one of the following alternative cases happens. (i) Every function .f ∈ H(βE ) has order 0 if and only if .βρ = +∞, i.e., .β ∈ B \ C. (ii) There exists a function .f ∈ H(βE ) that does not have finite order if and only if .βρ = 0, i.e., .β ∈ C \ B. (iii) Every .f ∈ H(βE ) has finite order and there exists .g ∈ H(βE ) having positive order if and only if .βρ ∈ (0, +∞), i.e., .β ∈ B ∩ C.

14

1 Function Spaces

By Corollary 1.5.3, a space .H(βρ ) is a Hilbert space of entire functions of finite orders, while a space .H(βρ+ ) is a space .H(βρ ) having at least one function of a positive order. Therefore, .H(βρ ) is a special case of .H(βE ), .H(βρ+ ) is a special case of .H(βρ ), and + + .H(βρ , T ) is a special case of .H(βρ ). 2 It is clear that any space .Eγ (T , τ ) is a space .H(βρ+ ) (Proposition 1.3.1 .(iv)) and any √ space .Fα2 is a space .H(βρ+ ) with .βn = α −n n!. Hence, both .Eγ2 (T , τ ) and .Fα2 belong to the same class .H(βρ+ ). However, while .Eγ2 (T , τ ) also belongs to the subclass .H(βρ+ , T ), √ it is not true for .Fα2 (as .μβ (n) = αn ↑ +∞).

1.6

Hilbert Spaces of Entire Functions

Denote by .mN the ordinary Lebesgue measure on .RN (i.e., the unit cube has measure 1). We denote the unit ball of .RN and its boundary, the closed unit sphere, respectively, by .BN and .SN −1 . Calculation of the volume of .BN and the surface area of .SN −1 is a standard exercise in measure theory. It is well-known that they are, respectively, given by VN =

.

π N/2

(1 + N/2)

(1.7)

2π N/2 .

(N/2)

(1.8)

and AN −1 =

.

The quantities (1.7) and (1.8) are used to define two normalized measures: the Lebesgue normalized measure on .RN , ηN :=

.

mN , VN

with respect to which the volume of .BN is one, and the area normalized measure on .SN −1 , σN −1 :=

.

mN , AN −1

with respect to which the surface area of .SN −1 is one. It is well-known that all these measures are rotation invariant. We just need the second normalized measure in our calculation. From now on, we assume that N is fixed. Hence, for simplicity of notations, we denote the unit ball and the closed unit sphere of .CN , respectively, by .B and .S. More explicitly, we write .B := B2N and .S := S2N −1 . Moreover, we put

1.6 Hilbert Spaces of Entire Functions

15

σ := σ2N −1 .

.

2 .L

The space of all square integrable functions on the closed unit sphere is denoted by = L2 (S, σ ) = L2 (S). This is a Hilbert space with inner product  f, g L2 =

f, g ∈ L2 (S).

f (z)g(z) 0. For βn = ω−n



.

(N − 1 + n)! (N − 1)!

1/2 ,

it is easy to verify that the condition (1.15) holds. Then the space .H(β) is identified with the Fock space .Fω2 discussed in Sect. 1.4. Example 1.7.3 Consider .N = 1. Let .β be any weight sequence for which the sequence is bounded and that of ratios . nββn−1 n .

lim inf n→∞

log βn < +∞. n log n

From Proposition 1.3.1, it follows that condition (1.15) is fulfilled. Every element f in the induced space .H(β) has finite growth order .ρf , and at least one element .g ∈ H(β) has positive growth order. Moreover, every translation operator .Tu : f (z) → f (z +u) (u ∈ C) is bounded on .H(β). This space is denoted by .H(βρ+ , T ). Let .H be a Hilbert space whose elements are functions from the set X to .C. Then .H is called a reproducing kernel Hilbert space (RKHS) if for every .y ∈ X, the evaluation functional .

δy : X −→ C f −  → f (y)

is bounded. Hence, by the Riesz representation theorem, there exists a unique element Ky ∈ H such that

.

f (y) = f, Ky H ,

.

(f ∈ H).

(1.17)

We call .Ky the reproducing kernel at the point y. The function .K : X × X → C defined by K(x, y) := Ky (x),

.

(x, y ∈ X),

is called the reproducing kernel for .H. Since .Ky ∈, the identity (1.17) immediately implies K(x, y) = Ky , Kx H ,

.

If .{ej : j ∈ J } is an orthonormal basis for .H, then

(x, y ∈ X),

1.7 Hilbert Spaces of Formal Power Series

21

Ky =



.

cj ej ,

j ∈J

where the coefficients .cj are given by cj = Ky , ej = ej , Ky = ej (y).

.

Hence, we have the representation Ky =



.

ej (y) ej ,

j ∈J

in which the series is convergent in the norm of .H. Furthermore, since the evaluation functionals .δx are continuous on .H, if we apply this functional to this representation, we see that K(x, y) =



.

ej (x)ej (y),

j ∈J

where the convergence is pointwise for each .x, y ∈ X. To tie the space introduced in this section to concepts of Sect. 1.4, we show that if all elements of .H(β) are entire functions, then .H(β) is a reproducing kernel Hilbert space. Theorem 1.7.4 Let .β = (βn ) be a positive sequence satisfying .

1/n

lim βn

n→∞

= ∞.

Then the space .H(β) is a reproducing kernel Hilbert space over .CN with the reproducing kernel .K : CN × CN → C given by K(z, w) =

.

∞ z, w n n=0

νn βn2

.

The convergence is uniform on compact subsets of .CN × CN . Proof From (1.16) in the proof of Lemma 1.7.1, we see that for any .z ∈ CN , there exists a constant .Mz > 0 such that |f (z)| ≤ Mz f H(β) ,

.

f ∈ H(β).

22

1 Function Spaces

Hence, each evaluation functional .δz is bounded, which shows that .H(β) is an RKHS. To   √ obtain an expression for K, recall that the set of monomials . (β|α| να )−1 zα : α ∈ NN form an orthonormal basis of .H(β). Using the multinomial expansion and (1.11), we get K(z, w) =

.

∞ ∞ zα w¯ α 1 n! α α z, w n z w ¯ = . = 2 α! ν β2 νn βn2 α να β|α| n=0 n=0 n n |α|=n

Note that since .limn→∞ (νn βn2 )−1/n = 0, the series converges uniformly on compact   subsets of .CN × CN . In the rest of the text, unless otherwise stated, whenever we write .β or .H(β), we implicitly assume that .β satisfies the condition (1.15) in Theorem 1.7.1, so that all elements of .H(β) are entire functions.

1.8

Some Estimations

We end this chapter with some integral estimations which will be used in studying operators on function spaces. The notation .A  B means that there are two constants .c1 and .c2 , independent of certain variables and parameters, such that c1 B ≤ A ≤ c2 B,

.

uniformly with respect to all variables and parameters involved. If there is ambiguity, we explicitly mention the independence status of .c1 and .c2 . Lemma 1.8.1 Let .a ∈ D, and let .b > −1. Then ⎧ ⎪ ⎪ ⎨1,



c < 0, (1 − |z|2 )b 2 −1 dA(z)  log(1 − |a| ) , c = 0, ⎪ |1 − az|1+b+c ⎪ ⎩ (1 − |a|2 )−c , c > 0.

.

D

For fixed b and c, the constants in . are uniforms with respect to .a ∈ D, in particular as − .|a| → 1 . Proof Write  Ib,c (a) =

.

D

(1 − |z|2 )b dA(z), |1 − az|2+b+c

(a ∈ D).

Since .b > −1, the integral .Ib,c (z) converges for every .a ∈ D. Put

1.8 Some Estimations

23

k=

.

1+b+c . 2

Note that whenever .k ≤ 0, we have .c < 0 and .Ib,c (a) is bounded. So, in the following, we consider the case where .k > 0. The power series representation ∞

.

(n + k) 1 (az)n = k (1 − az) n! (k) n=0

is a useful tool in the following. The measure .(1 − |z|2 )b dA(z) is orthogonal to all harmonics .einθ , .n = 0. Hence, by expanding .

1 1 1 , = k 2k (1 − az) (1 − az)k |1 − az|

we see that 

(1 − |z|2 )b dA(z) 2k D |1 − az|   ∞ 

(n + k) 2 2n = |a| (1 − |z|2 )b |z|2n dA(z) n! (k) D

Ib,c (a) =

.

n=0

=

 ∞ 

(n + k) 2 n! (k)

n=0

=

n! (k)

|a|

1

×2

(1 − r 2 )b r 2n rdr

0

 ∞ 

(n + k) 2 n=0

 2n

|a|2n ×

(b + 1) (n + 1)

(b + n + 2)

∞

=

(b + 1) (n + k)2 |a|2n . n! (b + n + 2)

(k)2 n=0

By Stirling’s formula,

.

(n + k)2 ∼ nc−1 , n! (b + n + 2)

(n → ∞).

Thus, Ib,c (a) 

∞

.

n=0

|a|2n . (n + 1)1−c

24

1 Function Spaces

Note that the constants involved in . depend on b and c, but uniformly hold with respect to a for a fixed b and c. There are now three cases to consider for c as follows. (i) .c < 0: the series ∞ .

n=0

|a|2n (n + 1)1−c

defines a bounded function on .D, and hence, .Ib,c (a) is bounded on .D. (ii) .c = 0: in this case, Ib,c (a) ∼

.

∞ 1 |a|2n ∼ log , n+1 1 − |a|2

(|a| → 1− ).

n=0

(iii) .c > 0: we have Ib,c (a) ∼

.

∞ (n + 1)c−1 |a|2n ∼ n=0

1 , 1 − |a|c

(|a| → 1− ),

since, again by Stirling’s formula, (n + 1)c−1 ∼

.

(n + c) n! (c)

and

.

∞

(n + c) n=0

n! (c)

|a|2n =

1 . 1 − |a|c  

Lemma 1.8.2 Let .a ∈ D. Then  .

0

2π

⎧ ⎪ ⎪1, ⎨

dθ  log(1 − |a|2 )−1 , ⎪ |1 − ae−iθ |1+c ⎪ ⎩ (1 − |a|2 )−c ,

c < 0, c = 0, c > 0.

For a fixed c, the constants in . are uniforms with respect to .a ∈ D, in particular as |a| → 1− .

.

Proof The proof is similar to the proof of Lemma 1.8.1, and hence, it is omitted.

 

1.8 Some Estimations

25

Notes on Chapter 1 Most of definitions and results of this chapter are taken from [19,20]. The sequence spaces 2β have many important applications in studying operators on function spaces; see, e.g., [17, 83]. The weighted Hardy spaces .H(β) are treated extensively in the past two decades. In particular, properties of composition operators on .H(β) spaces are investigated in [17], with an extension to the case of unit ball in several complex variables. See also [84]. For more details on entire functions and their growth, e.g., order as well as a type, we refer to [58]. The proof of Theorem 1.2.1 is quite analogous to the case of entire functions in [19], and part of it is shown in [76]. Lemma 1.6.1 is discussed in [79], while it was also stated in some other works [8,28,30]. Hilbert spaces .H(β) of formal power series are considered in [17, Chapter 2] as weighted Hardy spaces. The family of function spaces introduced in Example 1.7.3 and composition operators acting on them are studied in [19]. The theory of reproducing kernel Hilbert spaces was investigated in great details by Aronszajn in [5] in the 1950s. See also the new book [76]. Lemma 1.8.1 can be found in [40, Theorem 1.7] as well as in [108, Lemma 3.10], while Lemma 1.8.2 can be found in [40, Theorem 1.7]. Both lemmas are frequently used in estimations of operators on function spaces.

.

2

The Counting Function and Its Applications

2.1

The Nevanlinna Counting Function

Let .ϕ : D → D be a holomorphic self-map of the open unit disc .D. Then the Nevanlinna counting function of .ϕ is defined by 

Nϕ (w) =

.

z∈ϕ −1 {w}

log

1 , |z|

(w ∈ D \ {ϕ(0)}),

(2.1)

where .ϕ −1 {w} denotes the sequence of inverse images of w under .ϕ, each point being repeated according to its multiplicity. For a more detailed study of the counting function, let .w ∈ C \ {ϕ(0)}, and let .{zj (w) : j ≥ 1} denote the sequence of the preimages −1 {w}, arranged increasing order of absolute values, each point repeated according to .ϕ its multiplicity. Also, for .0 ≤ r < 1, let .n(r, w) = nϕ (r, w) denote the number of these points in the disc .rD. Then we define the partial counting functions of .ϕ by Nϕ (r, w) =

n(r,w) 

.

log

j =1

r . |zj |

(2.2)

With this notation, the original Nevanlinna counting function is precisely .Nϕ (1, w), i.e., Nϕ (w) = Nϕ (1, w) =



.

j

log

1 . |zj (w)|

With the convention that, for .0 ≤ r ≤ 1, .Nϕ (r, w) = 0, whenever .w ∈ / ϕ(rD), and so the counting functions can be regarded as defined on the entire complex plane. Note that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_2

27

28

2 The Counting Function and Its Applications

for each fixed complex number w, the partial counting function .Nϕ (r, w) increases with r. Hence, by the monotone convergence theorem, .

lim Nϕ (r, w) = Nϕ (w).

r→−1

Another useful reformulation of (2.2) is 

r

Nϕ (r, w) =

.

0

r log( ) dn(t, w), t

(2.3)

and in particular, (2.1) becomes  Nϕ (w) =

.

0

2.2

1

1 log( ) dn(t, w). t

(2.4)

Littlewood’s Inequality

We start with a simple, but very useful, estimation of .Nϕ (w). Theorem 2.2.1 (Littlewood) Let .ϕ : D → D be analytic, and let .Nϕ be the Nevanlinna counting function (2.1). Then, for each .w ∈ D \ {ϕ(0)}, we have    1 − ϕ(0)w    .Nϕ (w) ≤ log  .  ϕ(0) − w 

(2.5)

Regarding the case of equality, the following are equivalent. (i) Equality holds for some w. (ii) Equality holds for all .w ∈ D, except for an exceptional set of logarithmic capacity zero. (iii) .ϕ is an inner function. Proof Jensen identity is the main ingredient of proof. Let .f : D → C be analytic with f (0) = 0, and let .0 ≤ r < 1. Using the notations introduced in Sect. 2.1, the identity says

.

.

1 2π



2π 0

We rewrite the identity as

log |f (reiθ )| dθ = log |f (0)| +

n(r,w)  j =1

log

r . |zj |

2.3 A Change of Variable Formula

Nϕ (r, 0) =

.

29

1 2π



2π

log |f (reiθ )| dθ − log |f (0)|.

(2.6)

0

Put ϕw (z) =

.

w − ϕ(z) 1 − wϕ(z)

(z ∈ D).

Then .ϕw is a holomorphic self-map on .D, and .ϕ −1 {w} is precisely the zero set of .ϕw . Applying the representation (2.6) with .f = ϕw , where .w = ϕ(0), we obtain 1 .Nϕ (r, w) = Nϕw (r, 0) = 2π



2π

log |ϕw (reiθ )| dθ − log |ϕw (0)|.

(2.7)

0

Since .|ϕw | ≤ 1 on .D, the integral on the right-hand side of (2.7) is negative, and thus, for all .w ∈ D \ {ϕ(0)}, we have Nϕ (r, w) ≤ − log |ϕw (0)|.

.

Letting .r → 1, the Littlewood inequality (2.5) immediately follows. To prove the equivalence of .(i), .(ii), and .(iii), first note that, by (2.7), equality holds in (2.5) if and only if  .

lim

r→1 0

2π

log |ϕw (reiθ )| dθ = 0,

(2.8)

which is a characterization of Blaschke products. It is also elementary that .ϕ is inner if and only so is .ϕw . Thus, .(i) ⇒ (iii). Now, a classical theorem of Frostman says that if .ϕ is an inner function, then .ϕw is a Blaschke product for every .w ∈ D, except for a set of

logarithmic capacity zero, i.e., .(iii) ⇒ (ii). That .(ii) ⇒ (i) is trivial.

2.3

A Change of Variable Formula

We used dA to denote the normalized planar Lebesgue measure so that the area of the unit disc .D is 1. Set dA1 (z) = log(

.

1 ) dA(z), |z|

(z ∈ D).

Theorem 2.3.1 Let g be a positive measurable function on .D, and let .ϕ : D → D be analytic. Then

30

2 The Counting Function and Its Applications





2

.

D

(g ◦ ϕ)|ϕ | dA1 =

D

gNϕ dA,

where .Nϕ is the Nevanlinna counting function (2.1). Proof The singular points of .ϕ are the zeros of .ϕ , i.e., the set E := {z ∈ D : ϕ (z) = 0}

.

Let .D := D \ E. Then .ϕ is a local homeomorphism on the open set .D . Thus, there exists a countable collection .{Rj } of closed polar rectangles whose interiors are disjoint and their union is .D , and more importantly, .ϕ is one-to-one on each .Rj . Let us denote by .ψj the inverse of the restriction of .ϕ to .Rj , so that .ψj is a one-to-one map taking .ϕ(Rj ) back onto .Rj . By the usual change of variable formula applied on .Rj , with .z = ψj (w), we have  .





1 (g ◦ ϕ)(z) |ϕ (z)| dA1 (z) = g(w) log |ψ j (w)| Rj ϕ(Rj )

2

 dA(w).

(2.9)

Hence, if .χj denotes the characteristic function of the set .ϕ(Rj ), then  .

D

(g ◦ ϕ)|ϕ |2 dA1 =

 D

g(w)

⎧ ⎨ ⎩

j

⎫ ⎬ 1 χj (w) dA(w). |ψj (w)| ⎭ 

(2.10)

This is the desired formula, since, by definition, the term in curly brackets on the right

hand side of the equation above is precisely .Nϕ (w). In the above proof, we are slightly cheating. The points on the boundaries of .Rj are counting more than once when we add up the integrals in the left of (2.9) to obtain the left of (2.10). At the same token, to write  .

j



1 χj (w) |ψj (w)|

 = Nϕ (w),

there are w’s which are counted more than enough (images of the boundaries of .Rj ) and possibly even w’s which are not counted enough (image of singular points). However, this does not create any problem since the collection of all such exceptional points form sets of (two-dimensional Lebesgue) measure zero on both sides of (2.10).

2.4 The Generalized Nevanlinna Counting Function

2.4

31

The Generalized Nevanlinna Counting Function

For a holomorphic function .ϕ : D → D, .0 ≤ r < 1, and .γ ≥ 0, the generalized Nevanlinna partial counting function is defined by n(r,w) 

Nϕ,γ (r, w) :=

.

j =1



r log |zj (w)|

γ (w ∈ D \ {ϕ(0)}),

,

and thus, the generalized Nevanlinna counting function is given by Nϕ,γ (w) =



.

j ≥1

log

1 |zj (w)|

γ (w ∈ D \ {ϕ(0)}),

,

where .{zj (w)} denotes the sequence of .ϕ-preimages of w, multiplicity counted. In other words, .Nϕ,γ (w) = Nϕ,γ (1, w). Note also that .Nϕ,0 (r, w) is the multiplicity function .n(r, w) of .ϕ and .Nϕ,1 is the classical Nevanlinna counting function. The representation (2.3) and (2.4) can also be, respectively, generalized as 

 r γ log dn(t, w) t

r

Nϕ,γ (r, w) =

.

0

(2.11)

and  Nϕ,γ (w) =

.

0

1

1 log t

γ dn(t, w).

(2.12)

In the light of above formulas, we define the measure .dAγ on .D by 

1 γ dAγ (w) = log dA(w). |w|

.

The importance of .Nϕ,γ stems from the following result, which generalizes the change of variable formula. Theorem 2.4.1 Let g be a positive measurable function on .D, and let .ϕ : D → D be analytic. Then  .

D

where .c(γ ) =

2γ (γ +1) .

(g ◦ ϕ)|ϕ |2 dAγ = c(γ )

 D

gNϕ,γ dA,

32

2 The Counting Function and Its Applications

Proof The proof is almost identical with the original case .γ = 1, i.e., Theorem 2.3.1, and hence, it is omitted.

We also have the following extension of Littlewood’s inequality (Theorem 2.2.1). Theorem 2.4.2 Let .ϕ : D → D be analytic, and let .γ ≥ 1. Then we have    1 − ϕ(0)w  γ   .Nϕ,γ (w) ≤ log   ,  ϕ(0) − w  

(w ∈ D \ {ϕ(0)}).

Proof The case .γ = 1 is the original Littlewood’s inequality, which was studied in Sect. 2.2. For each .γ > 1 and for any positive sequence of numbers .(aj ), we have  .

γ

aj ≤



γ aj

.

Hence, we immediately deduce γ

Nϕ,γ (w) ≤ Nϕ,1 (w)

.

(w ∈ D).

Now, the desired result easily follows from the case .γ = 1.

2.5



Interrelation Between Nevanlinna Counting Functions

We have the following representation of the Nevanlinna counting function .Nϕ,γ in terms of .Nϕ = Nϕ,1 . Lemma 2.5.1 Let .ϕ : D → D be analytic, and let .γ ≥ 1. Then, for .w = ϕ(0), we have 

r

Nϕ,γ (r, w) = γ (γ − 1)

.

0

 r γ −2 dt. t −1 Nϕ (t, w) log t

Proof We integrate by parts twice. In each case, the integrated terms sum up to zero, because the condition .w = ϕ(0) guarantees that .n(t, w) and .Nϕ (t, w) both vanish for all t sufficiently close to zero. By (2.11), the first integration by parts yields 

 r γ log dn(t, w) t 0  r  r γ −1 t −1 n(t, w) log dt. =γ t 0

Nϕ,γ (r, w) =

.

r

In particular, in the special case .γ = 1, we obtain

2.5 Interrelation Between Nevanlinna Counting Functions



r

Nϕ (r, w) =

.

33

t −1 n(t, w) dt.

0

In other words, we can write t −1 n(t, w) dt = dNϕ (t, w).

.

Thus, applying the second integration by parts, we deduce 

r

t

.

0

−1

r γ −1 dt = (γ − 1) n(t, w) log t 



r 0

 r γ −2 dt, t −1 Nϕ (t, w) log t



from which the desired result follows.

The following result, which is very useful in applications, shows that the counting functions satisfy a specific mean value property. Lemma 2.5.2 Let .ϕ be a holomorphic self-map of .D, and let .γ > 0. If .ϕ(0) = 0 and .0 < r < |ϕ(0)|, then Nϕ,γ (0) ≤

.

1 r2

 Nϕ,γ (z) dA(z). rD

Proof First, we prove the lemma for the classical case .Nϕ (w). As usual, for .0 ≤ r < 1, we write .ϕr (z) = ϕ(rz), .z ∈ D. Set .dσ = dθ/2π , and let .μ be the Borel probability measure defined on .C (but supported only on .ϕr (∂D)) by .dμ = ϕr−1 dσ . Then for each fixed complex number .w = ϕ(0), Jensen’s formula applied to .f = ϕ − w is rewritten as  Nϕ (r, w) + log |ϕ(0) − w| =

.

log |ζ − w| dμ(ζ ).

The left-hand side of the above equation, being the logarithmic potential of a compactly supported probability measure, is subharmonic in the complex plane. Thus, .Nϕ (r, w) is a subharmonic function of w on .C \ {ϕ(0)}. Consequently, .Nϕ (r, w) is subharmonic, and since .Nϕ (r, w) ↑ Nϕ (w) as .r ↑ 1, the monotone convergence theorem insures that .Nϕ (w) inherits from .Nϕ (r, w) the subharmonicity and the submean inequality holds. Now, we return to the general case. By Lemma 2.5.1, .Nϕ,γ (r, w) inherits the continuity and subharmonicity of .Nϕ (r, w) on .D \ {ϕ(0)} for each .0 ≤ r < 1. By similar arguments as for the classical case and using again the monotone convergence theorem, the desired result follows.



34

2 The Counting Function and Its Applications

2.6

The Littlewood–Paley Formula

The norm in Hardy space .H p was defined as the supremum, even as the limit, of certain line integrals in Sect. 1.1. In the following result, we provide another formula in which the area integral is used. Theorem 2.6.1 (Littlewood–Paley) Let .f ∈ H p (D), .0 < p < ∞, with .f ≡ 0 and .f (0) = 0. Then we have 1 . 2π



2π

0

p2 |f (e )| dθ = 2 iθ



p

D

|f (z)|p−2 |f (z)|2 log(1/|z|) dA(z).

Proof Let .ρ be so small that on the closed disc .{ |z| ≤ ρ }, there are no zeros of f , except of course the origin. Let .1 > > ρ be such that on the circle .{ |z| = }, f has no zeros. Hence, on the annulus .{ ρ < |z| < }, there are a finite number of the zeros of f , say .z1 , z2 , · · · , zn . Let .ε be so small that all the discs .{ |z − zk | ≤ ε }, .1 ≤ k ≤ n, are entirely in the annulus .{ ρ < |z| < }. Finally, let  = { ρ < |z| < } \

n 

.

{ |z − zk | ≤ ε }.

k=1

In the following, we shall eventually let .ε → 0, and then .ρ → 0, and . → 1 through a sequence .( n )n≥1 , so that there is no zero on the circles .{ |z| = n }. ¯ Hence, The function .W (z) = |f (z)|p is infinitely differentiable in a neighborhood of .. by Green’s theorem, we have 



 log( /|z| ) ∇ W (z) dx dy = 2

.



∂

∂W ∂ log( /|z|) log( /|z| ) −W ∂n ∂n

 d.

First of all, a simple calculation shows that ∇ 2 W (z) = p2 |f (z)|p−2 |f (z)|2 .

.

Hence, the Green formula becomes 

log( /|z| ) |f (z)|p−2 |f (z)|2 dx dy = I − Iρ −

p2

.



n 

Ik ,

(2.13)

k=1

where the integrals on the right side are explained below. On the boundary .{ |z| = }, we have

2.6 The Littlewood–Paley Formula





2π

I =

35

log( 1 )

.

0



2π

=

∂W 1 ( eiθ ) + W ( eiθ ) ∂r



dθ

|f ( eiθ )|p dθ

0

By a well-known result in the theory of .H p spaces, we know that 

2π

I =

.

 |f ( e )| dθ −→ iθ

p

0

2π

|f (eiθ )|p dθ,

(2.14)

0

as . → 1. On the boundary .{ |z| = ρ }, we have 

 ∂W 1 (ρeiθ ) + W (ρeiθ ) ρ dθ ∂r ρ 0  2π  2π ∂|f |p = |f (ρeiθ )|p dθ + ρ log( /ρ) (ρeiθ ) dθ. ∂r 0 0 

Iρ =

.

2π

log( /ρ )

Since f is continuous at the origin, then 

2π

.

|f (ρeiθ )|p dθ −→ 2π |f (0)|p = 0,

0

as .ρ → 0. On the other hand, since .

∂|f |p = p (uur + vvr ) |f |p−2 , ∂r

by the Cauchy–Schwarz inequality,    

2π

.

0

  2π  ∂|f |p (ρeiθ ) dθ  ≤ p |f (ρeiθ )|p−1 |f (ρeiθ )| dθ. ∂r 0

If f has a zero of order .n0 ≥ 1 at the origin, then .|f (ρeiθ )|  ρ n0 , and .|f (ρeiθ )|  ρ n0 −1 , as .ρ → 0. Hence, 

2π

.

|f (ρeiθ )|p−1 |f (ρeiθ )| dθ  ρ pn0 −1 ,

0

which gives    . ρ log( /ρ) 

0

2π

  ∂|f |p iθ (ρe ) dθ  ≤ C ρ pn0 | log ρ|. ∂r

36

2 The Counting Function and Its Applications

Thus, we also have 

2π

ρ log( /ρ)

.

0

∂|f |p (ρeiθ ) dθ −→ 0, ∂r

as .ρ → 0. Therefore, Iρ → 0,

(2.15)

.

as .ρ → 0. Finally, on the boundary .{ |z − zk | = ε }, we have  Ik =

2π



∂|f |p (zk + εeiθ ) ∂n

log( /|zk + εeiθ | )

.

0

∂ log( /|z|) (zk + εeiθ ) −|f (zk + εe )| ∂n iθ

p

 ε dθ.

Since   ∂|f |p   ∂n

.

   ≤ | ∇ |f |p | ≤ p |f |p−1 |f |, 

if f has a zero of order .nk ≥ 1 at the .zk , then we have |Ik | ≤ C εpnk .

.

Note that the constant C depends on . . However, for a fixed . , we have Ik → 0,

(2.16)

.

as .ε → 0. Now, first let .ε → 0. Hence, by the monotone convergence theorem, and by (2.16), the Green formula (2.13) becomes  p

.

log( /|z| ) |f (z)|p−2 |f (z)|2 dx dy = I − Iρ .

2 ρ −1, define the measure .dAγ on .D by dAγ (w) = [log(1/|w|)]γ dA(w).

.

3 .Np -Spaces in the Unit Disc .D

42

p

Then, for .0 < p < ∞ and .γ > −1, we define the weighted Bergman space .Aγ by    Apγ = f ∈ O(D) : f Apγ := |f (w)|p dAγ (w) < ∞ .

.

D

Due to some observations, we will see that this definition can be extended to include p γ = −1. The formula in Corollary 2.6.4 leads to a very useful equivalent norm for .Aγ .

.

Lemma 3.3.1 Let .0 < p < ∞. Then p .f  p Aγ

 |f (0)| + p

D

|f (w)|p−2 |f  (w)|2 dAγ +2 (w).

Proof Without loss of generality, we may assume that .f (0) = 0. By Corollary 2.6.4, we have .

1 2π



2π

|f (reiθ |p dθ =

0

p2 2



|f (w)|p−2 |f  (w)|2 log rD

r dA(w). |w|

 γ We multiply both sides by .2r log 1r , integrate with respect to r from 0 to 1, and then apply Fubini’s theorem to obtain  .

D

|f (w)|p dAγ (w)

   r 1 γ χrD (w)|f (w)| |f (w)| log dr dA(w) 2r log |w| r D 0     1  p2 1 γ r p−2  2 = |f (w)| |f (w)| 2r log log dr dA(w). 2 D r |w| |w| p2 = 2



1 

p−2



2

For .1/2 ≤ |w| ≤ 1, the inner integral with respect to r is comparable to .(log 1/r)γ +2 . For .0 ≤ w ≤ 1/2, there are a finite number of singularities, and the integrand is integrable. Thus, the inner integral can be replaced by .(log 1/r)γ +2 without changing the space of functions, for which it is finite, and an equivalent norm results.   By Lemma 3.3.1 and formula in Corollary 2.6.3 with .ϕ(z) ≡ z, it is natural to extend p p the definition of .Aγ and define .A−1 to be the Hardy space .H p . Then we can write 

p

f Ap |f (0)|p +

.

γ

D

|f (w)|p−2 |f  (w)|2 dAγ +2 (w),

(γ ≥ −1),

(3.1)

and so we are able to provide a unified treatment of the Hardy spaces and the weighted Bergman spaces.

3.4 Np Versus A−1

43

Theorem 3.3.2 Let .ϕ be a holomorphic self-map of .D, and let f be holomorphic on .D. Then, for .γ ≥ −1, we have f

.

p ◦ ϕAp γ

 |f (ϕ(0)| + p

D

|f |p−2 |f  |2 Nϕ,γ +2 dA.

Proof This is an immediate consequence of the norm estimate (3.1) and the change of   variable formula, which was studied in Lemma 2.4.1.

3.4

Np Versus A−1

Some basic Banach space or Fréchet space properties of the .Np -spaces are discussed below. In particular, it will be showed that, for .p ∈ (0, 1), the .Np -spaces are all different topological vector spaces with independent interest. It will also be justified that the .Np spaces form a much bigger class of functions than the .Qp -spaces. Proposition 3.4.1 (i) For .p ∈ (0, ∞), we have . · A−1   · Np . That is, the inclusion −1 is well-defined and bounded. .Np → A (ii) For .p ∈ (1, ∞), we have . · A−1  · Np . That is, we have the Banach space equivalence .Np = A−1 . Proof To prove part .(i), first note that, for each .w ∈ D, f (w) = f ◦ σw (0).

.

Then, by subharmonicity of .|f ◦ σw (·)|2 , we have |f (w)|2 = |(f ◦ σw )(0)|2  ≤ const ·

.

√ |f {z:|σ0 (z)| 0, there exists a constant K depending only on c and p, such that

.

 1 ∞ ∞ 1 −nc p p 2 tn ≤ (1 − x)c−1 f (x)p dx ≤ K 2−nc tn . K 0 n=0

n=0

Theorem 3.5.1 can be directly proved by using Jensen’s inequality. Moreover, it is an immediate consequence of the following more general result. Theorem 3.5.2 Let F be a real-valued nonnegative function satisfying the following conditions. (i) F is defined on the interval .[0, ∞), and .F (0) = 0. F (t) F (t) (ii) For some .p ≥ q > 0, the function . p is nonincreasing, while the function . q is t t nondecreasing. Let .(ak )k≥1 be a sequence of nonnegative real numbers and .c > 0. Then 

1

F

.

0

∞

ak r (1 − r) k

c−1

dr

1, due to inequality (3.5), we can take .F (t) = t p . Put .c = min{1, c/p} ∞ and .s(r) = 2nc r 2n with .0 ≤ r < 1. By Jensen’s inequality, we have n=0

∞ .

∞

p ak r

≤

k

k=1

p tn r

2n

≤ s(r)p−1

n=0

∞

2nc r 2 2−ncp |tn |p . n

n=0

Combining inequality (3.7), the inequality r(1 − r)c−1 ≤ | log r|c−1 , 0 ≤ r < 1,

.

and the inequality 

1

.

| log r|c−1 r 2

n −1

dr = (c)2−nc ,

0

yields the desired result. Indeed, a direct computation shows that 

1

∞

.

0

p (1 − r)c−1 dr ≤ K2

ak r k

k=1

∞

⎛ ⎞p 2−nc ⎝ ak ⎠ ,

n=0

In

where (in the case .p ≥ c) .K2 = 2p−1 (c)p . - For case .p = 1, the proof is similar, but it is much simpler, because the function .s(r) is not needed. - For case .0 < p < 1, the proof is similar to .p = 1, with the use of inequality (3.6).   Theorem 3.5.3 Let .f (z) = Then .f ∈ Np if and only if

∞

k=0 bk z

k=0

be in the Hadamard gap class, and let .p ∈ (0, 1].

⎛

∞ .

nk

1 2k(1+p)



⎝

⎞ |bj |2 ⎠ < ∞.

2k ≤nj 1. Then: (i) .f ∈ Bα if and only if .(1 − |z|2 )α−1 f (z) is bounded on .D. (ii) .f ∈ Bα,0 if and only if .(1 − |z|2 )α−1 f (z) → 0 as .|z| → 1− . Proof (i) Suppose .f ∈ Bα . By Proposition 4.1.1 (iv), we have  f (z) = f (0) +

.

D

(1 − |w|2 )α f  (w) dA(w), w(1 − zw)α+1

(z ∈ D).

Hence,  |f (z) − f (0)| ≤ f α

.

D

dA(w , |w||1 − zw|α+1

(z ∈ D).

4.1 The Spaces Bα and Bα,0

57

Since the factor .|w| in the denominator does not change the growth rate of the integral for z near the boundary, by Lemma 1.8.1, there exists a constant .C > 0 such that |f (z) − f (0)| ≤ Cf α (1 − |z|2 )−(α−1) ,

.

(z ∈ D),

which shows that .(1 − |z|2 )α−1 f (z) is bounded on .D. Conversely, suppose that .(1 − |z|2 )α−1 f (z) ≤ M on .D for some constant .M > 0. By Proposition 4.1.1 (iii), we have  f (z) = α

.

D

(1 − |w|2 )α−1 f (w) dA(w), (1 − zw)α+1

(z ∈ D).

Differentiating under the integral sign, we obtain f  (z) = α(α + 1)



.

D

w(1 − |w|2 )α−1 f (w) dA(w), (1 − zw)α+2

(z ∈ D).

Again by Lemma 1.8.1, there exists a constant .C > 0 such that 



|f (z)| ≤ α(α + 1)M

.

D

dA(w) ≤ CM(1 − |z|2 )−α , (1 − zw)α+2

(z ∈ D),

which shows that .f ∈ Bα . (ii) Analyzing the proof of (i), we see that for .α > 1, the norm . · α on .Bα is actually equivalent to the norm f  = sup(1 − |z|2 )α−1 |f (z)|.

.

z∈D

Note that, on the one hand, .Bα,0 is the closure of the polynomials in .Bα under the norm . α and, on the other hand, the closure of the polynomials in .Bα under the norm 2 α−1 f (z) → 0 as .|z| → 1− . .  consists of holomorphic functions f with .(1 − |z| ) Then we can conclude that .f ∈ Bα,0 if and only if .(1−|z|2 )α−1 f (z) → 0 as .|z| → 1− . The proposition is proved completely.  Proposition 4.1.3 Let .α > 0 and .n > 1. Then: (i) A holomorphic function f on .D is in .Bα if and only if

58

4 The .α-Bloch Spaces

.

sup{(1 − |z|2 )α+n−1 |f (n) (z)|} < ∞. z∈D

(ii) A holomorphic function f on .D is in .Bα if and only is .

sup{(1 − |z|2 )α+n−1 |f (n) (z)|} < ∞. z∈D

Proof - Suppose .f ∈ Bα . Then by Proposition 4.1.1 (iv),  f (z) = f (0) +

.

D

(1 − |w|2 )α f  (w) dA(w), w(1 − zw)1+α

(z ∈ D).

Differentiating under the integral sign n times and applying Lemma 1.8.1, we obtain .

sup{1 − |z|2 )α+n−1 |f (n) (z)|} < ∞. z∈D

- Conversely, since subtracting a polynomial from f neither alters the assumption nor does the conclusion, without loss of generality, we may assume that the first .n + 1 Taylor coefficients of f all vanish. Then the function h(z) =

.

(α + 2)(1 − |z|2 )α+n−1 f (n) (z) (α + n) · (α + 2)zn

is bounded on .D. By Proposition 4.1.1 (iii),  f

.

(n)

(z) = (α + n)

D

(1 − |w|2 )α+n−1 f (n) (w) dA(w), (1 − zw)α+n+1

(z ∈ D).

Integrating from 0 to z yields  f (n−1) (z) − f (n−1) (0) =

.

D

  1 (1 − |w|2 )α+n−1 f (n) (w) − 1 dA(w). w (1 − zw)α+n

Since .f (n−1) (0) = 0 and  .

D

we obtain

(1 − |w|2 )α+n−1 f (n) (w) dA(w) = 0, w

4.2 Some Operators on α-Bloch Spaces

 f (n−1) (z) =

59

(1 − |w|2 )α+n−1 f (n) (w) dA(w), w(1 − zw)α+n

.

D

(z ∈ D).

Repeating the argument above .n − 1 times yields 

f  (z) =

h(w) dA(w), (1 − zw)α+2

.

D

(z ∈ D).

Now, by Lemma 1.8.1, there exists a constant .C > 0 such that |f  (z)| ≤ h∞



.

D

dA(w) ≤ Ch∞ (1 − |z|2 )−α , (1 − zw)α+2

(z ∈ D),

which implies that .f ∈ Bα . Now analyzing the proof above shows that the norm . · ∞ on .Bα is equivalent to the norm f  = |f (0)| + |f  (0)| + · + |f (n−1) (0)| + sup{(1 − |z|2 )α+n+1 |f (n) (z)|}.

.

z∈D

Then the closure of the polynomials in .Bα under the norm above consists of holomorphic functions f with .(1 − |z|2 )α+n+1 |f (n) (z)| → 0 as .|z| → 1− . This implies that a holomorphic function f is in .Bα,0 if and only if (1 − |z|2 )α+n+1 |f (n) (z)| → 0

.

as .|z| → 1− .

4.2



Some Operators on α-Bloch Spaces

Let .L1 (dA) denote the Bergman space of holomorphic functions f on .D such that  f  =

.

D

|f | dA < ∞.

As is well-known, the space .L1 (dA) is a Banach space with the norm above. Our purpose is to establish a duality relationship between .L1 (dA) and .Bα , .Bα,0 . To do so, in then sequel, we use the following operators:  Rα (f (z)) = α

.

D

f (w) dA(w), (1 − zw)1+α

60

4 The .α-Bloch Spaces

 Sα (f (z)) = 3(1 − |z| )

2 2 D

f (w) (1 − |w|2 )α−1 dA(w), (1 − zw)4

where .z ∈ D. Note that .Rα is not a projection, as it does not give holomorphic functions unless .α = 1. We use also the following subspaces of .L∞ (D): the space .C0 (D) of complex-valued continuous functions on .D which vanish on the boundary, the space .C(D) of complexvalued continuous functions on the closed unit disc .D, and the space .H C(D) of bounded complex-valued continuous functions on .D. The operators .Rα and .Sα help to find several useful relationships between spaces .L∞ (D), .C0 (D), .C(D), .H C(D), .Bα , and .Bα,0 . Proposition 4.2.1 For each .α > 0, we have: (i) .Rα is a bounded linear operator from .L∞ (D) onto .Bα ; .Rα also maps .H C(D) onto .Bα . (ii) .Rα is a bounded linear operator from .C(D) onto .Bα,0 ; .Rα also maps .C0 (D) onto .Bα,0 . (iii) There exists a constant .C > 0 such that C −1 f α ≤ inf{g∞ : f = Rα g, g ∈ L∞ (D)} ≤ Cf α

.

for all .f ∈ Bα and C −1 f α ≤ inf{g∞ : f = Rα g, g ∈ C0 (D)} ≤ Cf α

.

for all .f ∈ Bα,0 . Proof (i) First, take .g ∈ L∞ (D) and put .f = Rα g. We have  f (z) = α

.

D

g(w) dA(w), (1 − zw)1+α

and hence 



f (z) = α(α + 1)

.

D

wg(w) dA(w), (1 − zw)2+α

(z ∈ D).

By Lemma 1.8.1, there exists a constant .C > 0 such that 

|f (z)| ≤ α(α + 1)g∞

.

 D

dA(w) ≤ Cg∞ (1 − |z|2 )−α , (1 − zw)2+α

Also it is clear that .|f (0)| ≤ αg∞ . These estimates give

(z ∈ D).

4.2 Some Operators on α-Bloch Spaces

61

f α ≤ (C + α)g∞ ,

.

f ∈ L∞ (D),

which shows that .Rα maps .L∞ (D) boundedly into .Bα . Next, we prove that .Rα maps .H C(D) onto .Bα . We notice that for any nonnegative integer n,  α

.

D

(1 − |w|2 )2 w n 2α(α + 1) · · · (α + n) dA(w) = , 1+α (n + 3)! (1 − zw)

(z ∈ D),

which shows that for a polynomial p, then always there exists .g ∈ C0 (D) such that .p = Rα g. So if .f ∈ Bα , we can write f (z) = f (0) + f  (0)z +

.

f  (0) 2 z + f1 (z) 2!

with .f1 still in .Bα . Then we can find a function .g ∈ C0 (D) satisfying f (0) + f  (0)z +

.

f  (0) 2 z = Rα g(z). 2!

By Proposition 4.1.1 (iv), we also have .f1 (z) = Rα g1 (z), where g1 (z) =

.

(1 − |z|2 )α f1 (z) , αz

which is in .H C(D). Thus, .f = Rα (g + g1 ) and so .Rα maps .H C(D) onto .Bα . (ii) The “onto” part follows from the proof of part (i). So we prove that .Rα maps .C(D) into .Bα,0 . By the Stone–Weierstrass approximation theorem, each function from .C(D) can be uniformly approximated by a finite linear combination of functions of the form .zn z¯ m (m, n ≥ 0). Since, by part (i), .Rα maps .L∞ (D) boundedly into .Bα and n ¯ m into .B . .Bα,0 is closed in .Bα , it suffices to show that .Rα maps each function .z z α,0 But it is clear since an easy calculation using polar coordinates shows that .Rα maps each function of the form .zn z¯ m to a polynomial. (iii) This follows from the consideration of the quotient norm and the open mapping theorem.  Proposition 4.2.2 For each .α > 0, we have: (i) .Sα is a bounded linear operator from .Bα into .L∞ (D) and also is a bounded linear operator from .Bα,0 into .C0 (D).

62

4 The .α-Bloch Spaces

(ii) There exists a constant .C > 0 (depending only on .α) such that C −1 f α ≤ Sα f ∞ ≤ Cf α

.

for all .f ∈ Bα . (iii) In particular, for a holomorphic function f on .D, .f ∈ Bα if and only if .Sα f ∈ L∞ (D) and .f ∈ Bα,0 if and only if .Sα f ∈ C0 (D). Proof For .f ∈ Bα , there exists .g ∈ L∞ (D) such that .f = Rα g. By Proposition 4.1.1 (iii) and Fubini’s theorem, we have 

 (1 − |z|2 )2 g(u) dA(u) 2 α−1 (1 − |w| ) dA(w) 4 1+α D (1 − zw) D (1 − wu)   (1 − |w|2 )α−1 dA(w) = 3(1 − |z|2 )2 g(u) dA(u) α 4 1+α D D (1 − zw) (1 − wu)  g(u) dA(u) = 3(1 − |z|2 )2 . 4 D (1 − wu)

Sα f (z) = 3α

.

Hence,  |Sα f (z)| ≤ 3g∞ (1 − |z|2 )2

.

D

dA(u) = 3g∞ , |1 − zu|4

(z ∈ D).

This gives Sα f ∞ ≤ 3g∞

.

for all g ∈ L∞ (D) with Rα g = f.

By Proposition 4.2.1 (iii), there exists a constant .C > 0 such that Sα f ∞ ≤ 3Cf α ,

.

f ∈ Bα .

This shows that .Sα is a bounded linear operator from .Bα into .L∞ (D). Next, for .f ∈ Bα , again by Proposition 4.1.1 (iii) and Fubini’s theorem, we have 



(1 − |w|2 )2 (1 − |u|2 )α−1 f (u) dA(u) 4 D D (1 − wu)   (1 − |w|2 )2 dA(w) =α f (u)(1 − |u|2 )α−1 dA(u) · 3 4 α+1 D D (1 − wu) (1 − zw)  (1 − |u|2 )α−1 =α f (u) dA(u) = f (z). α+1 D (1 − zu)

Rα Sα f (z) = 3α

.

dA(w) (1 − zw)1+α

4.3 The Pre-dual of α-Bloch Space

63

This means that .Sα f ∈ L∞ (D) implies that .f ∈ Bα by Proposition 4.2.1 (i). Then there exists a constant .C > 0 such that f ∞ = Rα Sα f α ≤ CSα f ∞ ,

.

f ∈ Bα .

Also Proposition 4.2.1 (ii) shows that .Sα f ∈ C0 (D) implies that .f ∈ Bα,0 . Finally, we show that .Sα maps .Bα,0 into .C0 (D). Indeed, by the symmetry of the disc, the operator .Sα maps each polynomial to a polynomial times the function .(1 − |z|2 )2 . In particular, .Sα maps each polynomial to a function in .C0 (D). Moreover, since .Sα : Bα → L∞ (D) is bounded, .Bα,0 is the closure of the set of polynomials in .Bα , and .C0 (D) is closed  in .L∞ (D), we conclude that .Sα maps .Bα,0 into .C0 (D).

The Pre-dual of α-Bloch Space

4.3

We have now developed enough tools to completely characterize the pre-dual of .α-Bloch spaces. Theorem 4.3.1 (i) For each .α > 0, the dual of .L1 (dA) can be identified with .Bα (with equivalent norm) under the pairing  f, g α = lim

.

t→1− D

f (tz)g(tz)(1 − |z|2 )α−1 dA(z),

f ∈ L1 (dA), g ∈ Bα .

(ii) For each .α > 0, the dual of .Bα,0 can be identified with .L1 (dA) (with equivalent norm) under the pairing  f, g = lim

.

t→1− D

f (tz)g(tz)(1 − |z|2 )α−1 dA(z),

f ∈ Bα,0 , g ∈ L1 (dA).

Proof (i) Recall that  .

D

|g(z)|(1 − |z|2 )α−1 dA(z) < ∞,

g ∈ Bα .

First of all, we note that if .f ∈ L1 (dA) is bounded (i.e., .f ∈ H ∞ (D) and .g ∈ Bα ), then  2 α−1 dA(z) ≤ Cf L1 gα , . f (z)g(z)(1 − |z| ) D

64

4 The .α-Bloch Spaces

where .C > 0 is some constant independent of both f and g. Indeed, we write .g = Rα  for some . ∈ L∞ (D) and apply Fubini’s theorem to get  .

D

f (z)g(z)(1 − |z|2 )α−1 dA(z) 

 f (z)(1 − |z| )

2 α−1

α

D

 α

(w) dA(w)

D

dA(z)

D

(w) dA(w) (1 − wz)α+1

(1 − |z|2 )α−1 f (z)

(1 − wz)α+1 dA(z).

Furthermore, by Proposition 4.1.1 (iii), 



.

D

f (z)g(z)(1 − |z|2 )α−1 dA(z) =

D

(w) dA(w),

which gives .

 f (z)g(z)(1 − |z|2 )α−1 dA(z) ≤ f  1 ∞ . L D

Taking the infimum over . and applying Proposition 4.2.1 (i), we obtain a constant .C > 0 for which the desired result holds. Next, we prove that if .f ∈ L1 (dA) and .g ∈ Bα , then there exists  .

lim

t→1− D

f (tz)g(z)(1 − |z|2 )α−1 dA(z) = A,

with |A| ≤ Cf L1 gα ,

.

where .C > 0 is some constant independent of both f and g. Indeed, for .g ∈ Bα , as shown above,  Fg (f ) =

.

D

f (tz)g(z)(1 − |z|2 )α−1 dA(z),

f ∈ H ∞ (D),

can be extended to a bounded linear functional on .L1 with .Fg  ≤ Cgα . Therefore, we may assume that .Fg is defined on the whole space .L1 (but not via the above formula). We fix .f ∈ L1 (dA) and .g ∈ Bα and let .ft (z) = f (tz) for .0 < t < 1. Since .ft ∈ H ∞ and . lim ft − f L1 = 0, we have t→1−

4.3 The Pre-dual of α-Bloch Space

65

 .

lim

t→1− D

f (tz)g(z)(1 − |z|2 )α−1 dA(z) = lim Fg (ft ) = Fg (f ) t→1−

and |Fg (f )| ≤ Fg f L1 ≤ Cgα f L1 .

.

Furthermore, we show that if F is bounded linear functional on .L1 , then there exists a function .g ∈ Bα such that  F (f ) = lim

.

t→1− D

f (tz)g(z)(1 − |z|2 )α−1 dA(z),

f ∈ L1 (dA).

Indeed, by the Hahn–Banach extension theorem, F can be extended to a bounded linear functional on .L1 (D, dA) without increasing the norm. Since .(L1 )∗ = L∞ , there is a function . ∈ L∞ (D) such that  F (f ) =

f ∈ L1 .

f (tz)(z) dA(z),

.

D

By the first fact above, for each .f ∈ L1 , we have  F (f ) = lim

.

t→1− D

ft (z)(z) dA(z)



= lim

t→1− D

ft (z)Ra (z)(1 − |z|2 )α−1 dA(z).

Let .g = Ra ; then by Proposition 4.2.1 (i), .g ∈ Bα and  F (f ) = lim

.

t→1− D

f (tz)g(z)(1 − |z|2 )α−1 dA(z), f ∈ L1 .

Finally, by the rotational invariance of the measure .(1 − |z|2 )α−1 dA(z), we have 

 f (tz)g(z)(1 − |z| )

2 α−1

.

D

where .s =

dA(z) =

D

f (sz)g(sz)(1 − |z|2 )α−1 dA(z)

√ t. From this, it follows that 

 .

lim

t→1− D

f (tz)g(z)(1 − |z| )

2 α−1

dA(z) = lim

which completes the proof of the theorem.

t→1− D

f (tz)g(tz)(1 − |z|2 )α−1 dA(z),

66

4 The .α-Bloch Spaces

(ii) In view of part (i), we only need to show that each bounded linear functional F on 1 .Bα,0 arises from a function .g ∈ L in the following manner:  F (f ) = lim

.

f (tz)g(tz)(1 − |z|2 )α−1 dA(z),

t→1− D

f ∈ Bα,0 .

Indeed, by Proposition 4.2.1 (iii), the operator .Sα : Bα,0 → C0 (D) satisfies C −1 f α ≤ Sa f ∞ ≤ Cf α

.

 for some constant .C > 0 independent of f . We denote by .E = Sα Bα,0 , then E is a closed subspace of .C0 (D) and .F ◦ Sα−1 : E → C is a bounded linear functional. We extend it to the whole space .C0 (D) and apply the well-known Riesz representation theorem, to get a finite Borel measure .μ on .D so that F ◦ Sα−1 () =



.

D

(z)dμ(z),

 ∈ E.

From this, it follows that for each .f ∈ Bα,0 , we have  F (f ) =

.

D

(z) dμ(z).

Now by Fubini’s theorem, 

(1 − |w|2 )α−1 f (w) dA(w) (1 − zw)4 D D   (1 − |w|2 )α−1 f (w) 2 2 = 3 lim (1 − |z| ) dμ(z) dA(w) (1 − tzw)4 t→1− D D  = lim f (z)g(tz)(1 − |z|2 )α−1 dA(z)

F (f ) = 3

.



(1 − |z| ) dμ(z) 2 2

t→1− D



= lim

t→1− D

f (tz)g(tz)(1 − |z|2 )α−1 dA(z).

Here, g is a holomorphic function defined by  g(w) = 3

.

which belongs to .L1 , because

D

(1 − |z|2 )2 dμ(z) , (1 − wz)4

4.4 The Space Hα∞

67

 .

D

|g(w)| dA(w)

≤3



 D

(1 − |z|2 )2 d|μ|(z)

D

d(w) =3 |1 − zw|4

 D

d|μ|(z) = 3μ. 

The theorem is proved completely.

4.4

The Space Hα∞

Let .α ∈ (0, ∞). The weighted-type space .Hα∞ is the family of all .f ∈ Hol(D) such that f Hα∞ = sup(1 − |z|2 )α |f (z)| < ∞.

.

z∈D

Note that, based on our previous notation in Sect. 1.1, we have Hα∞ = A−α ,

.

the Bergman-type space. ∞ consists of elements .f ∈ H ∞ which satisfy the extra The closed subspace .Hα,0 α condition (1 − |z|2 )α |f (z)| → 0,

.

(|z| → 1).

∞ are related to .α-Bloch spaces as follows: The spaces .Hα∞ and .Hα,0

Hα∞ = Bα+1 ,

.

(α > 0),

(4.1)

and ∞ Hα,0 = Bα+1,0 ,

.

(α > 0).

We also have the inclusions B  Np  H1∞ ,

.

(0 < p ≤ 1),

and Np = H1∞ ,

.

(1 < p < ∞).

(4.2)

68

4 The .α-Bloch Spaces

The above relations are very similar to the corresponding ones between the .Qp -space and B. That is, .Qp  B if .p ∈ (0, 1] and .Qp = B if .p ∈ (1, ∞). We reformulate Theorem 4.3.1 in the previous section in terms of spaces .Hα∞ .

.

Theorem 4.4.1 Under the pairing  f, g α = lim

.

t→1− D

f (tz)g(tz)(1 − |z|2 )α dA(z),

we have

∞ ∗ Hα,0 ∼ = L1α (dA)

.

1 ∗ Lα (dA) ∼ = Hα∞ .

and

This result plays a crucial role in the study of weighted composition operators between spaces .Hα∞ and .Np in the sequel.

4.5

A Test Function for Hα∞

We introduce a test function for .Hα∞ which will be used later in the proofs of Theorem 5.3.1. Let fθ,r (z) =

∞

.

k

k

2kα (reiθ )2 z2 ,

(z ∈ D),

(4.3)

k=0

where .α ∈ (0, ∞), .θ ∈ [0, 2π ), and .r ∈ (0, 1]. The main properties of this test function are summarized below. Lemma 4.5.1 The function .fθ,r belongs to .Hα∞ and, moreover, .

sup fθ,r Hα∞ < ∞. r,θ

∞. If .r ∈ (0, 1), then .fθ,r ∈ Hα,0

Proof For each .z ∈ D \ {0}, we have |fθ,r (z)| ≤

∞

.

k+1 2kα ( r|z|)2

k=0

≤

∞  k=0 k

k+1

x 2kx ( r|z|)2 dx

4.5 A Test Function for Hα∞

69



∞

=

x 2kx ( r|z|)2 dx

o

1 α   1 log √r|z|



∞ 1 r|z|

log

s α−1 e−s ds

1 α .   1 log √r|z| Therefore, ⎛ (1 − |z|2 )α |fθ,r (z)|  ⎝

.

1 − |z|2 1 log √r|z|

⎞α ⎠ .

Since .log x1 ≥ 1 − x, this estimation implies (1 − |z|2 )α |fθ,r (z)|  1,

(z ∈ D).

.

∞ provided that .r = 1. Moreover, it shows .fθ,r ∈ Hα,0



For each .n ∈ N, consider the function n gθ,r (z) = zn fθ,r (z),

.

∞ ⊂ H ∞ , it follows from where .fθ,r is the test function defined in (4.3). Since .zn Hα,0 α,0 n n ∞ Lemma 4.5.1 that .{gθ,r }n∈N ⊂ Hθ,0 and the norm .gθ,r Hα∞ is uniformly bounded with n } respect to .θ, r and n. The following lemma says that .{gθ,r n∈N weakly converges to 0 in ∞ .Hα .

Lemma 4.5.2 For every . ∈ (Hα∞ )∗ , .

n sup |(gθ,r )| → 0,

(n → ∞).

θ,r

∞ )∗ . By Theorem 4.4.1, there exists Proof It suffices to prove the claim for every . ∈ (Hα,0 ∞ 1 .h ∈ La (dA) such that .(f ) = h, f α for .f ∈ H α,0 . By using the estimate in the proof of Lemma 4.5.1, we have

 .

n sup |(gθ,r )| ≤ sup | lim θ,r

θ,r

t→1 D

|tz|n |fθ,r (tz)| |h(tz)|(1 − |z|2 )α dA(z)

70

4 The .α-Bloch Spaces

  lim

t→1 D

|tz|n |h(tz)| dA(z)

 1 |z|n |h(z)| dA(z) t→1 t 2 D  = |z|n |h(z)| dA(z). ≤ lim

D

Since .|z|n |h(z)| → 0 as .n → ∞ for each fixed .z ∈ D, and .|zn h(z)| ≤ |h(z)| with 1 .h ∈ L (dA), the dominated convergence theorem ensures that  .

lim

n→∞ D

|z|n |h(z)| dA(z) = 0. 

Notes on Chapter 4 A paper by J. Anderson [2] is a good survey of the theory of classical Bloch functions. In this chapter, due to our needs, we considered more general spaces. The results provided in this chapter are justified by at least two considerations: firstly it gives a unified approach to some known duality results and secondly we developed several results along the way which are of independent interest. The results of this chapter are mostly from [112]. The last section is taken from [95]. Lemma 4.5.2 is similar to [60, Lemma 5]. Dualities of Bloch spaces’ some special cases have been obtained in [85] for .α > 1 and in [3] for .α = 1. For the cases .0 < α < 1, it can actually be deduced from the case .1 < α < 2 by considering .f  instead of f .

Weighted Composition Operators on D

5

.

Throughout this chapter, .ϕ denotes a non-constant holomorphic self-map of .D. Then the composition operator .Cϕ on .Hol(D) is defined by Cϕ f = f ◦ ϕ.

.

Let .ψ be a holomorphic function on .D, which is not identically equal to zero. Then the multiplication operator .Mψ on .Hol(D) is defined by Mψ f = ψf.

.

As a generalization of both concepts, the weighted composition operator .Wψ,ϕ on .Hol(D) is the mapping Wψ,ϕ f = ψ · f ◦ ϕ.

.

If .ψ(z) ≡ 1, then we obtain the standard composition operator .Cϕ , while the choice ϕ(z) = z gives us the standard multiplication operator .Mψ . We aim to study weighted composition operators .Wψ,ϕ acting between .Hα∞ and .Np spaces and relate operator theoretic properties, like boundedness and compactness, to function theoretic properties of the inducing functions .ψ and .ϕ. As consequences, we obtain characterizations for boundedness and compactness of the composition operators .Cϕ between these spaces. .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_5

71

5 Weighted Composition Operators on .D

72

5.1

The Multiplication Operator

There is a simple characterization of the multipliers on .Np spaces. Theorem 5.1.1 The multiplication operator .Mψ is well-defined and bounded on .Np if and only if .ψ ∈ H ∞ . Proof Suppose .ψ ∈ H ∞ . It is clear that Mψ f 2Np ≤ ψ2H ∞ f 2Np .

.

Conversely, using the test function .kw in Lemma 3.2.1, we have 1 ≥ kw Np

.

 Mψ kw Np  Mψ kw A−1 ≥

1 − |w|2 |ψ(w)|(1 − |w|2 ) = |ψ(w)| |1 − ww|2

for all .w ∈ D, i.e., .ψ ∈ H ∞ .

5.2



Carleson Measures for Np

For each arc I in the unit circle .T, the Carleson box based on I is the set of the form S(I ) = {z ∈ D : 1 − |I | ≤ |z| < 1, z/|z| ∈ I },

.

where .|I | denotes the normalized length of I . For each .p ∈ (0, ∞), a bounded positive Borel measure .μ on .D is called a p-Carleson measure if μ[S(I )] < ∞, p I ⊂T |I |

Cμ := sup

.

where the supremum is taken over all arcs .I ⊂ T; .μ is called a compact p-Carleson measure if .

μ[S(I )] = 0. |I |→0 |I |p lim

5.2 Carleson Measures for Np

73

Lemma 5.2.1 Let .μ be a positive measure on .D, and let .0 < p < ∞. Then the following assertions hold. (i) .μ is a p-Carleson measure if and only if   .

sup

D

w∈D

1 − |w|2 |1 − wz|2

p dμ(z) < ∞.

(5.1)

(ii) .μ is a compact p-Carleson measure if and only if   .

lim

|w|→1

1 − |w|2 |1 − wz|2

D

p dμ(z) = 0.

(5.2)

Proof (i) Suppose (5.1) is satisfied, and let M denote the supremum. Then the Carleson box S(I ) = {z ∈ D : 1 − h ≤ |z| ≤ 1, |θ − arg z| ≤ h}

.

considers the point .w = (1 − h)ei(θ+h/2) . Then we have p 1 − |w|2 dμ(z) .M ≥ 2 D |1 − wz| p   1 − |w|2 dμ(z) ≥ 2 S(I ) |1 − wz|  p   1 − |w|2 ≥ inf μ(S(I )) z∈S(I ) |1 − wz|2  



μ(S(I )) . |I |p

(5.3)

From this, it follows that μ(S(I ))  M < ∞. p I ⊂T |I |

Cμ = sup

.

Conversely, let .μ be a bounded p-Carleson measure on .D, that is, .Cμ < ∞. We have μ(D)  Cμ .

.

Now, we consider two cases. For .|w| ≤ 34 , we have the trivial estimate

(5.4)

5 Weighted Composition Operators on .D

74

  .

D

1 − |w|2 |1 − wz|2

p dμ(z)  μ(D)  Cμ .

For .|w| > 34 , let .w = r0 eiθ0 , and consider the sets En = {z ∈ D : |z − eiθ0 | < 2n (1 − r0 )}.

.

Then, since .En is inside a Carleson box, μ(En )  Cμ 2np (1 − |w|)p ,

(n ≥ 0).

.

We also have .

1 − |w|2 1 ,  2 1 − |w| |1 − wz|

z ∈ E1 ,

which implies that, with .E0 = ∅ and for all .n ≥ 1, .

1 − |w|2 1 ,  2n 2 |1 − wz| 2 (1 − |w|)

z ∈ En \ En−1 .

Consequently,   .

D

1 − |w|2 |1 − wz|2

p dμ(z) ≤

En \En−1

n=1





∞  

∞  n=1

1 − |w|2 |1 − wz|2

p dμ(z)

∞

 1 μ(En )  Cμ , np p 2 (1 − |w|) 2np

(5.5)

n=1

which means that (5.1) holds. (ii) Suppose that (5.2) is satisfied. Applying (5.3), we immediately see that .μ is a compact p-Carleson measure. Conversely, suppose that .μ is a compact p-Carleson measure. Then .μ must be bounded. For .t ∈ (0, 1), let .χD\Dt be the characteristic function of the set .D \ Dt , where .Dt = {z : |z| < t}. Let .dμt (z) = χD\Dt (z) dμ(z). Then from (5.4) and (5.5), it follows that   .

D

1 − |w|2 |1 − wz|2

p

p 1 − |w|2 dμ(z) |1 − wz|2 Dt D\Dt p p    1 − |w|2 1 − |w|2 ≤ dμt (z) μ(D) + 2 (1 − t)2 D |1 − wz|  

dμ(z) =



+



5.3 The Boundedness of Wψ,ϕ : Hα∞ → Np

   

75

1 − |w|2 (1 − t)2 1 − |w|2 (1 − t)2

p μ(D) + Cμt p Cμ + sup I

μ[(D \ Dt ) ∩ S(I )] . |I |p

This implies that (5.2) is true.

Theorem 5.2.2 Let .f ∈ Hol(D). Then .f ∈ Np if and only if dμf,p (z) = |f (z)|2 (1 − |z|2 )p dA(z)

.

is a p-Carleson measure. Furthermore, in this case, we have μf,p [S(I )] . |I |p I ⊂T

f 2Np  sup

.

Proof By the equality 1 − |σw (z)|2 =

.

(1 − |w|2 )(1 − |z|2 ) , |1 − wz|2

we have   f 2Np = sup

.

w∈D D

(1 − |w|2 ) |1 − wz|2

p dμf,p (z).

Combining this representation and Lemma 5.2.1 yields the desired result.

5.3



The Boundedness of Wψ,ϕ : Hα∞ → Np

We have developed all the required tools to study the boundedness of the weighted composition operator .Wψ,ϕ : Hα∞ → Np . Theorem 5.3.1 Let .ψ ∈ Hol(D) and .ϕ be an analytic self-map of .D. Then the following are equivalent: (i) .Wψ,ϕ : Hα∞ → Np is a well-defined bounded operator. (ii) .ψ and .ϕ satisfy

5 Weighted Composition Operators on .D

76

 .

sup

w∈D D

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z) < ∞. (1 − |ϕ(z)|2 )2α

(5.6)

(iii) .ψ and .ϕ satisfy

.

sup |I |

−p

I ⊂T

 S(I )

|ψ(z)|2 (1 − |z|2 )p dA(z) < ∞, (1 − |ϕ(z)|2 )2α

(5.7)

where the supremum is taken over all arcs .I ⊂ T. Proof .(i) ⇒ (ii): For each .θ ∈ [0, 2π ), we put .fθ = fθ,1 which is defined in (4.3). Let .w ∈ D. By Lemma 4.5.1 and Fubini’s theorem, we have 

2π

1

.

 ≥

0

D

Wψ,ϕ fθ 2Np

dθ 2π



2π

|ψ(z)|2 (1 − |σw (z)|2 )p

|fθ (ϕ(z))|2

0

dθ 2π

dA(z).

Furthermore, by Parseval’s formula, 

2π

.

0

dθ |fθ (ϕ(z))|2 = 2π =



2π 0

∞ 

∞

2



dθ k

 2

2kα eiθ ϕ(z)



2π k=0

 2k (2k )2α ϕ(z) .

(5.8)

k=0

In case .|ϕ(z)| > 12 , we have ∞  .

∞   2k  2k (2k )2α ϕ(z) = 2−2α (2k+1 )2α ϕ(z)

k=0

k=0

≥ 2−2α



∞

 2x (2x )2α ϕ(z) dx

0

2−2α



1 log = log 2 |ϕ(z)|2 

−2α 

1 , (1 − |ϕ(z)|2 )2α

where the last inequality follows from the simple fact that

∞

log

1 |ϕ(z)|2

s 2α−1 e−s ds

(5.9)

5.3 The Boundedness of Wψ,ϕ : Hα∞ → Np

.

log

1 ≤ (1 − x) log 4, x

77

(1/2 ≤ x ≤ 1).

Hence, we obtain  .

|ϕ(z)|> 21

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z)  1, (1 − |ϕ(z)|2 )2α

(5.10)

for all .w ∈ D. Furthermore, since .ψ ∈ Np , we also have  .

|ϕ(z)|≤ 21

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z)  ψ2Np , (1 − |ϕ(z)|2 )2α

(5.11)

for any .w ∈ D. The inequalities (5.10) and (5.11) together yield (5.6). ∞ .(ii) ⇒ (i): It is rather obvious. Indeed, for each .f ∈ Hα , we have  .

Wψ,ϕ f Np = sup 2

w∈D D

|ψ(z)|2 |f ◦ ϕ(z)|2 (1 − σw (z)|2 )p dA(z)

   |ψ(z)|2 2 p 2 2α 2 · (1 − |ϕ(z)| = sup dA(z) (1 − |σ (z)| ) ) |f ◦ ϕ(z)| w 2 2α w∈D D (1 − |ϕ(z)| )    |ψ(z)|2  sup (1 − |σw (z)|2 )p dA(z) · sup(1 − |ϕ(z)|2 )2α |f ◦ ϕ(z)|2 2 2α w∈D D (1 − |ϕ(z)| ) z∈D  |ψ(z)|2 ≤ sup (1 − |σw (z)|2 )p dA(z) · sup(1 − |z|2 )2α |f (z)|2 2 )2α (1 − |ϕ(z)| w∈D D z∈D 

 f 2Hα∞ , which shows that .Wψ,ϕ : Hα∞ → Np is bounded. .(i) ⇒ (iii): Suppose .Wψ,ϕ : Hα∞ → Np is bounded. We fix an arc .I ⊂ T and consider the test function .fθ , which was used in the proof of .(i) ⇒ (ii). By Theorem 5.2.2, Lemma 4.5.1, and Fubini’s theorem, we have |I |−p



.

S(I )∩|ϕ(z)|> √1

2

|ψ(z)|2 (1 − |z|2 )p dA(z)  1. (1 − |ϕ(z)|2 )2α

(5.12)

The boundedness of .Wψ,ϕ ensures that .ψ ∈ Np . Hence, by Theorem 5.2.2, .|ψ(z)|2 (1 − |z|2 )p dA(z) is a p-Carleson measure, and .

sup |I |

I ⊂T

−p

 S(I )

|ψ(z)|2 (1 − |z|2 )p dA(z)  ψ2Np .

5 Weighted Composition Operators on .D

78

Consequently, |I |

.



−p

S(I )∩|ϕ(z)|≤ √1

2

|ψ(z)|2 (1 − |z|2 )p dA(z)  ψ2Np . (1 − |ϕ(z)|2 )2α

(5.13)

Combining (5.12) and (5.13) yields (5.7). ∞ .(iii) ⇒ (i): For every .f ∈ Hα , we have .

sup |I |−p

I ⊂T

≤

 |ψ(z)|2 (1 − |z|2 )p |f ◦ ϕ(z)|2 dA(z) S(I )

f 2Hα∞

· sup |I |

−p

I ⊂T

 S(I )

|ψ(z)|2 (1 − |z|2 )p dA(z). (1 − |ϕ(z)|2 )2α

From this, together with condition (5.7), it follows that dμ(z) := |ψ(z)|2 |f ◦ ϕ(z)|2 (1 − |z|2 )p dA(z)

.

is a p-Carleson measure. Therefore, by Theorem 5.2.2, .Wψ,ϕ f ∈ Np , and moreover,  

p 1 − |w|2 .Wψ,ϕ f  |ψ(z)|2 |f ◦ ϕ(z)|2 (1 − |z|2 )p dA(z) Np = sup 2 w∈D D |1 − wz|  −p |ψ(z)|2 (1 − |z|2 )p |f ◦ ϕ(z)|2 dA(z)  sup |I | 2

I ⊂T



S(I )

f 2Hα∞ ,

which shows that .Wψ,ϕ : Hα∞ → Np is bounded.

5.4



The Compactness of Wψ,ϕ : Hα∞ → Np

We now consider the compactness of .Wψ,ϕ . As is usually the case in operator theory, if a “big-O” condition, like the one in Theorem 5.5.2, describes the bounded operators of a function space, the corresponding “little-o” condition describes the compact operators. Our weighted composition operator is not an exception. To verify this fact, we use a reformulation of weighted compactness for operators between .Np -spaces and the Bergman-type spaces .Hα∞ . More explicitly, a weighted composition operator .Wψ,ϕ : Hα∞ → Np is compact if and only if for every bounded sequence .(fn ) in .Hα∞ that converges to 0 uniformly on compact subsets of .D, we have

5.4 The Compactness of Wψ,ϕ : Hα∞ → Np .

79

lim Wψ,ϕ fn Np = 0.

n→∞

To do this, we need some preliminary results. Recall from Section 4.5 that n gθ,r (z) = zn fθ,r (z),

.

where .fθ,r is the test function defined in (4.3). By Lemmas 4.5.1 and 4.5.2, we see that for each fixed .θ ∈ [0, 2π ) and .r ∈ (0, 1) and any compact operator .T : Hα∞ → Np , we have n .T g θ,r → 0 in .Np as .n → ∞. However, for proving the main theorem of this section, we need a stronger result. Before stating and proving such a result, we recall that a bounded linear operator T from a Banach space X to a Banach space Y is called completely continuous if, for every weakly convergent sequence .(xn ) from X, the sequence .(T xn ) is norm-convergent in Y (see, e.g., [15, §VI.3]). Note that compact operators on Banach spaces are always completely continuous. The following result is a straightforward consequence of a complete continuity of a compact operator and Lemma 4.5.2. Lemma 5.4.1 For any compact operator .T : Hα∞ → Np , we have n lim sup T gθ,r Np = 0,

.

n→∞ θ,r

where the supremum is taken over all .θ ∈ [0, 2π ) and .r ∈ (0, 1). Proof Since T is a compact operator from a Banach space .Hα∞ to .Np , it is completely  n is weakly convergent in .Hα∞ . continuous. Furthermore, by Lemma 4.5.2, a sequence . gθ,r  n Hence, the sequence . T (gθ,r ) is norm-convergent in .Np , from which the desired result follows.

In the sequel, we study the essential norm of the weighted composition operators .Wψ,ϕ between spaces .Hα∞ and .Np . Recall, in a general setting, that for a bounded linear operator L acting from a Banach space X to a Banach space Y , the essential norm of L on X, denoted by .Le , is the distance from L to the set of all compact operators from X to Y in the operator norm, i.e., Le = inf{L − Kop },

.

where the infimum is taken over all compact operators .K : X → Y . Clearly, L is compact if and only if .Le = 0. Let us denote by .K = K(Hα∞ , Np , ) the set of all compact operators acting from .Hα∞ into .Np . Then the essential norm of .Wu,ϕ is

5 Weighted Composition Operators on .D

80

  Wψ,ϕ e = inf Wψ,ϕ − K .

.

K∈K

We have the following estimates of the essential norm of .Wψ,ϕ . Proposition 5.4.2 Let .ψ ∈ Hol(D), and let .ϕ be a holomorphic self-map of .D. Suppose that .Wψ,ϕ : Hα∞ → Np is bounded. Then  2 .Wψ,ϕ e

 lim sup sup r→1

w∈D |ϕ(z)|>r

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α

 k Proof In order to prove upper estimates, consider .Ck f (z) = f k+1 z for each positive integer k and .f ∈ Hol(D). It is clear that every .Ck is bounded on .Hα∞ . In fact, by the above reformulation of weighted compactness for operators between .Np -spaces and the k Bergman-type spaces .Hα∞ , applying to the case .ψ ≡ 1 and .ϕ(z) = k+1 z, .Ck is also ∞ compact on .Hα . Consequently, Wψ,ϕ e ≤ lim inf ψCϕ − ψCϕ Ck 

.

k→∞

= lim inf

sup

k→∞ f  ∞ ≤1 Hα

ψCϕ (I d − Ck )f Np .

(5.14)

Here, I denotes the identity operator on .Hα∞ . Fix a number .k ∈ N and .f ∈ Hα∞ with .f Hα∞ ≤ 1. For any .r ∈ (0, 1), we have ψCϕ (I − Ck )f 2Np

.

 2 



k ≤ sup |ψ(z)| f ◦ ϕ(z) − f ϕ(z)

(1 − σw (z)|2 )p dA(z) k+1 w∈D |ϕ(z)|≤r

 2  



k |ψ(z)|2

f ◦ ϕ(z) − f ϕ(z)

(1 − σw (z)|2 )p dA(z). + sup k+1 w∈D |ϕ(z)|>r 

2

Furthermore, by the growth estimate for .f ∈ Hα∞ , we also have







2f Hα∞ k

. f ◦ ϕ(z) − f . ϕ(z)

≤

k+1 (1 − |ϕ(z)|2 )α

(5.15)

Consequently, for any .r ∈ (0, 1) and any .k ∈ N, we get

 2 



k sup |ψ(z)| f ◦ ϕ(z) − f ϕ(z)

(1 − σw (z)|2 )p dA(z) k+1 w∈D |ϕ(z)|>r 

.

2

5.4 The Compactness of Wψ,ϕ : Hα∞ → Np

  sup

w∈D |ϕ(z)|>r

81

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α

(5.16)

Next we prove that

  2



k . sup sup |ψ(z)| f ◦ ϕ(z) − f ϕ(z)

(1 − σw (z)|2 )p dA(z) → 0 k+1 f ∞∞ w∈D |ϕ(z)|≤r 

2

Hα

(5.17) k as .k → ∞. Put .v = ϕ(z) and consider the radial segment by .[ k+1 v, v]. Integrating .f  along this radial segment gives

.

 



k

≤ 1 |v||f  (ξ(v))|,

f (v) − f v

k+1 k+1

(5.18)

k for some .ξ(v) ∈ [ k+1 v, v]. By Cauchy’s estimate for .f  on the circle centered at .ξ(v) of radius .R ∈ (0, 1 − r), we have

|f  (ξ(v))| ≤

.

1 max |f (ζ )|. R |ζ |=R+r

(5.19)

Combining (5.18) and (5.19) shows that the quantity

  2



k ϕ(z)

(1 − σw (z)|2 )p dA(z) . sup sup |ψ(z)| f ◦ ϕ(z) − f ∞ k + 1 f  ∞ w∈D |ϕ(z)|≤r 

2

Hα

is bounded above by

.

r2 1 · ψ2Np . · R 2 (k + 1)2 (1 − (R + r)2 )2α

Since .ψ ∈ Np , by the boundedness of .Wψ,ϕ : Hα∞ → Np , we obtain (5.17). Now by (5.14), (5.16), (5.18), and (5.19), we deduce  Wψ,ϕ 2e  sup

.

w∈D |ϕ(z)|>r

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z), (1 − |ϕ(z)|2 )2α

from which the upper estimate for .Wψ,ϕ 2e follows.



Proposition 5.4.3 Let .ψ ∈ Hol(D), and let .ϕ be a holomorphic self-map of .D. Suppose that .Wψ,ϕ : Hα∞ → Np is bounded. Then

5 Weighted Composition Operators on .D

82

 Wψ,ϕ 2e  lim sup sup

.

w∈D |ϕ(z)|>r

r→1

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α

n ) Proof Consider the functions .(gθ,t n∈N defined in Lemma 4.5.2. By Lemma 4.5.1, the n norm .gθ,t Hα∞ is uniformly bounded with respect to .θ, t, and n. So for any compact operator .K : Hα∞ → Np , we have n n n Wψ,ϕ − K  (Wψ,ϕ − K)gθ,t Np ≥ Wψ,ϕ gθ,t Np − Kgθ,t Np .

.

(5.20)

By Fatou’s lemma, for any .r ∈ (0, 1),  .

n sup Wψ,ϕ gθ,t Np ≥ lim inf t→1

θ,t

 ≥

D

|ϕ(z)|>r

n |ψ(z)|2 |gθ,t (ϕ(z))|2 (1 − |σw (z)|2 )p dA(z)

|ψ(z)|2 |ϕ(z)|2n |fθ (ϕ(z))|2 (1 − |σw (z)|2 )p dA(z).

Here, .fθ (w) denotes the function .fθ,1 (w). Integrating these inequalities with respect to .θ from 0 to .2π , by Fubini’s theorem, we get n sup Wψ,ϕ gθ,t Np

.

(5.21)

θ,t



 ≥

|ψ(z)| |ϕ(z)| (1 − |σw (z)| ) 2

|ϕ(z)|>r

2n

2 p

2π

|fθ (ϕ(z))|

0

Furthermore, by (5.8) and (5.9), for any .z ∈ D with .ϕ(z)| > 

2π

.

|fθ (ϕ(z))|2

0

√1 , 2

2 dθ

2π

dA(z).

we have

dθ 1 .  2π (1 − |ϕ(z)|2 )2α

(5.22)

Combining (5.21) and (5.22) yields  n . sup Wψ,ϕ gθ,t Np θ,t

r

2n

sup

w∈D |ϕ(z)|>r

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z), (1 − |ϕ(z)|2 )2α

for all .r ∈ ( √1 , 1). Letting .r → 1, we obtain 2

.

n sup Wψ,ϕ gθ,t Np θ,t

  lim sup sup r→1

w∈D |ϕ(z)|>r

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z) (1 − |ϕ(z)|2 )2α

(5.23)

5.4 The Compactness of Wψ,ϕ : Hα∞ → Np

83

n  Note that the last estimate does not depend on n. Since .supθ,t Kgθ,t Np → 0 as .n → ∞ ∞ for any compact operator .K : Hα → Np , by Lemma 5.4.1, (5.20) (5.23), we get

 Wψ,ϕ 2e  lim sup sup

.

w∈D |ϕ(z)|>r

r→1

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α



Combining Propositions 5.4.2 and 5.4.3 yields the estimate of the weighted composition operator .Wψ,ϕ acting from the space .Hα∞ to the space .Np . Theorem 5.4.4 Let .ψ ∈ Hol(D), and let .ϕ be a holomorphic self-map of .D. Suppose that Wψ,ϕ : Hα∞ → Np is bounded. Then

.

 Wψ,ϕ 2e  lim sup sup

.

w∈D |ϕ(z)|>r

r→1

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α

We provide another estimation which has the same spirit as Theorem 5.4.4. Theorem 5.4.5 Let .ψ ∈ Hol(D), and let .ϕ be a holomorphic self-map of .D. Suppose that Wψ,ϕ : Hα∞ → Np is bounded. Then

.

Wψ,ϕ 2e  lim sup sup |I |−p



.

I ⊂T

r→1

S(I )∩{|ϕ(z)|>r}

|ψ(z)|2 (1 − |z|2 )p (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α

Here, the supremum is taken over all arcs .I ⊂ T. Proof By Theorem 5.2.2, for each .r ∈ (0, 1), we have   Wψ,ϕ (I − Ck )f 2Np  sup |I |−p

+ sup |I |−p

.

I ⊂T



S(I )∩{|ϕ(z)|≤r}

I ⊂T



 S(I )∩{|ϕ(z)|>r}



  2



k 2 p 2

ϕ(z) (1 − |z| ) dA(z) . |ψ(z)| f ◦ ϕ(z) − f k+1

We write .J1 and .J2 for the last two integrals. On the one hand, from inequality (5.15), it follows that for any .r ∈ (0, 1) and .f ∈ Hα∞ with .f Hα∞ ≤ 1, J2  sup |I |

.

I ⊂T

−p



 sup

S(I )∩{|ϕ(z)|>r} w∈D |ϕ(z)|>r

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α

5 Weighted Composition Operators on .D

84

On the other hand, inequalities (5.18) and (5.19) and Theorem 5.2.2 imply that for .f ∈ Hα∞ with .f Hα∞ ≤ 1, J1 

.

r2 1 · · ψ2Np , R 2 (k + 1)2 (1 − (R + r)2 )2α

and the last expression tends to 0 as .k → ∞. Note that this estimation holds uniformly on the unit ball of .Hα∞ . Hence, by (5.14) and the estimations for .J1 and .J2 , and by letting .r → 1, we get Wψ,ϕ 2e  lim sup sup |I |−p



.

r→1

I ⊂T

S(I )∩{|ϕ(z)|>r}

|ψ(z)|2 (1 − |z|2 )p (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α

The upper estimate is proved. Also, by Theorem 5.2.2, n . sup Wψ,ϕ gθ,t Np θ,t

 |I |

−p

 S(I )

n |ψ(z)|2 |gθ,t (ϕ(z))|2 (1 − |z|2 )p dA(z),

for all arcs I . A similar argument in the proof of Theorem 5.4.4 implies 2 .Wψ,ϕ e

 lim sup sup |I | r→1

−p

 S(I )∩|ϕ(z)|>r

w∈T

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z). (1 − |ϕ(z)|2 )2α



The lower estimate is proved. Corollary 5.4.6 Let .ψ ∈ Hol(D), and let .ϕ be a holomorphic self-map of .D. (i) .Wψ,ϕ : Hα∞ → Np is a well-defined compact operator. (ii) .ψ and .ϕ satisfy  .

lim sup sup r→1

w∈D |ϕ(z)|>r

|ψ(z)|2 (1 − |σw (z)|2 )p dA(z) = 0. (1 − |ϕ(z)|2 )2α

(iii) .ψ and .ϕ satisfy

.

lim sup sup |I |−p r→1

I ⊂T

 S(I )∩{|ϕ(z)|>r}

|ψ(z)|2 (1 − |z|2 )p (1 − |σw (z)|2 )p dA(z) = 0, (1 − |ϕ(z)|2 )2α

where the supremum is taken over all arcs .I ⊂ T.

5.5 The Boundedness of Wψ,ϕ : Np → Hα∞

85

The final remark of this section is that taking a special case .ψ(z) ≡ 1 in theorems above, we can obtain the corresponding results for essential norm and compactness of composition operators .Cϕ acting from the space .Hα∞ to the space .Np .

The Boundedness of Wψ,ϕ : Np → Hα∞

5.5

In this section, we consider the weighted composition operator .Wψ,ϕ acting from the space Np to the space .Hα∞ . As it is proved in Proposition 3.4.2, the .Np -spaces are functional Banach spaces. Thus, the weighted composition operator .Wψ,ϕ : Np → Hα∞ is well-defined and bounded if and only if it maps .Np into .Hα∞ . Therefore, our first concern is boundedness of .Wψ,ϕ : Np → Hα∞ , and we have the following result.

.

Theorem 5.5.1 Let .ψ ∈ Hol(D) and .ϕ be a holomorphic self-map of .D. Then Wψ,ϕ : Np → Hα∞ is a well-defined bounded operator if and only if .ψ and .ϕ satisfy

.

.

|ψ(z)|(1 − |z|2 )α < ∞. 1 − |ϕ(z)|2 z∈D

sup

(5.24)

Proof First suppose .Wψ,ϕ : Np → Hα∞ is bounded. Consider the function kw (z) =

.

1 − |w|2 , (1 − wz)2

(z ∈ D),

with .w = ϕ(z0 ), where .z0 ∈ D is fixed. Since .kw ∈ Np and .kw Np ≤ 1, the boundedness of .Wψ,ϕ implies 1  Wψ,ϕ kw Hα∞ ≥ (1 − |z0 |2 )α |ψ(z0 )||kw (ϕ(z0 ))| =

.

|ψ(z0 )|(1 − |z0 |2 )α , 1 − |ϕ(z0 )|2

from which (5.24) follows. Conversely, suppose that (5.24) is satisfied. Recall that .Np → H1∞ . Then for each .z ∈ D and .f ∈ Np , we have (1 − |z|2 )α |Wψ,ϕ f (z)| ≤

.

|ψ(z)|(1 − |z|2 )α |ψ(z)|(1 − |z|2 )α ∞ f   f Np . H1 1 − |ϕ(z)|2 1 − |ϕ(z)|2

From this, it follows that Wψ,ϕ f Hα∞  f Np ,

.

5 Weighted Composition Operators on .D

86

which shows a boundedness of .Wψ,ϕ : Np → Hα∞ .



As an immediate consequence of Theorem 5.5.1, we have the following characterization of bounded composition operator .Cϕ : Np → Hα∞ . Corollary 5.5.2 The composition operator .Cϕ : Np → Hα∞ is well-defined and bounded if and only if

.

(1 − |z|2 )α < ∞. 2 z∈D 1 − |ϕ(z)|

sup

(5.25)

The Compactness of Wψ,ϕ : Np → Hα∞

5.6

In this section, we give a complete characterization of the compactness of .Wψ,ϕ : Np → Hα∞ . The characterization is purely based on the function theoretic properties of .ψ and .ϕ. Recall that the notation . · e denotes the essential norm of an operator. Proposition 5.6.1 Let .ψ ∈ Hol(D) and .ϕ be a holomorphic self-map of .D. Suppose that Wψ,ϕ : Np → Hα∞ is bounded. Then

.

|ψ(z)|(1 − |z|2 )α . 1 − |ϕ(z)|2 |ϕ(z)|→1

Wψ,ϕ e  lim sup

.

(5.26)

Proof Consider the operator .Ck as in the proof of Theorem 5.4.4. For each positive integer k, we have  2  



k 2 p



.Ck f  Np = sup

f k + 1 z (1 − |σw (z)| ) dA(z) w∈D D 2

 ≤ sup

w∈D D



f 2H ∞ 1

1−

k 2 k+1 z)

(1 − |σw (z)|2 )p dA(z)

(k + 1)4 f 2Np . (2k + 1)2

This shows that .Ck is bounded on .Np . Moreover, again by a reformulation of weighted compactness for operators between .Np -spaces and the Bergman-type spaces .Hα∞ , .Ck is compact on .Np , and hence, .Wψ,ϕ Ck is compact from .Np to .Hα∞ . Hence, we have Wψ,ϕ e ≤ Wψ,ϕ − Wψ,ϕ Ck  =

.

sup

f Np ≤1

Wψ,ϕ (I − Ck )f Hα∞ ,

(5.27)

5.6 The Compactness of Wψ,ϕ : Np → Hα∞

87

where I is the identity operator on .Np . Fix a positive number k and .f ∈ Np with .f Np ≤ 1. For any .r ∈ (0, 1), we have Wψ,ϕ (I − Ck )f Hα∞

.

 



k

≤ sup (1 − |z| )|ψ(z)| f ◦ ϕ(z) − f ϕ(z)

k+1 |ϕ(z)|≤r

 



k + sup (1 − |z|2 )|ψ(z)|

f ◦ ϕ(z) − f ϕ(z)

k+1 |ϕ(z)|>r 2

Furthermore,

 



2f ||Np k

ϕ(z)

 , . f ◦ ϕ(z) − f

k+1 1 − |ϕ(z)|2 and thus, for each .r ∈ (0, 1), .

 



k |ψ(z)|(1 − |z|2 )α ϕ(z)

 sup|ϕ(z)|>r . sup (1 − |z|2 )|ψ(z)|

f ◦ ϕ(z) − f k+1 1 − |ϕ(z)|2 |ϕ(z)|>r

Moreover, by the same argument in the proof of Theorem 5.4.4, inequalities (5.18) and (5.19) show that for .R ∈ (0, 1 − r), we have .

 



k ϕ(z)

sup (1 − |z| )|ψ(z)|

f ◦ ϕ(z) − f k+1 |ϕ(z)|≤r 2



ψHα∞ r , R(k + 1) 1 − (R + r)2

(5.28)

which tends to 0 as .k → ∞. By (5.26), (5.27), and (5.28), we have for any .r ∈ (0, 1) Wψ,ϕ e  sup

.

|ϕ(z)|>r

|ψ(z)|(1 − |z|2 )α . 1 − |ϕ(z)|2

Letting .r → 1, we get the desired result.



Proposition 5.6.2 Let .ψ ∈ Hol(D) and .ϕ be a holomorphic self-map of .D. Suppose that Wψ,ϕ : Np → Hα∞ is bounded. Then

.

|ψ(z)|(1 − |z|2 )α . 1 − |ϕ(z)|2 |ϕ(z)|→1

Wψ,ϕ e  lim sup

.

(5.29)

Proof Take any sequence .(zn ) ∈ D with .|ϕ(zn )| → 1 as .j → ∞. Put .wn = ϕ(zn ) and fn (z) = kwn (z), where .kw is the function defined in the proof of Theorem 5.5.1. Then .(fn )

.

5 Weighted Composition Operators on .D

88

forms a bounded sequence in .Np which converges uniformly to 0 on compact subsets of D. More precisely, inequality

.

fn H1∞  fn Np ≤ 1

.

∞ . Note that .(f ) → 0 weakly in .N . Then shows that .(fn ) is a bounded sequence in .H1,0 n p ∞ .Kfn H ∞ → 0 as .j → ∞ for any compact operator .K : Np → Hα . α We also have

Wψ,ϕ Hα∞ ≥ (1 − |zn |2 )α ||Wψ,ϕ f (zn )| =

.

|ψ(zn )|(1 − |zn |2 )α (1 − |ϕ(zn )|2 ) , (1 − |ϕ(zn )|2 )2

and hence, for any .j ∈ N, Wψ,ϕ Hα∞ ≥

.

|ψ(zn )|(1 − |zn |2 )α . 1 − |ϕ(zn )|2

Combining the above arguments and the following inequalities Wψ,ϕ − K  (Wψ,ϕ − K)fn Hα∞ ≥ Wψ,ϕ fn Hα∞ − Kfn Hα∞ ,

.

for any compact operator .K : Np → Hα∞ , yields Wψ,ϕ e  lim sup

.

j →∞

|ψ(zn )|(1 − |zn |2 )α . 1 − |ϕ(zn )|2

Since .(zn ) ⊂ D with .|ϕ(zn )| → 1 as .j → ∞ is arbitrary, this implies |ψ(z)|(1 − |z|2 )α . 1 − |ϕ(z)|2 |ϕ(z)|→1

Wψ,ϕ e  lim sup

.

Combining Propositions 5.6.1 and 5.6.2 yields the following result. Theorem 5.6.3 Let .ψ ∈ Hol(D) and .ϕ be a holomorphic self-map of .D. Suppose that Wψ,ϕ : Np → Hα∞ is bounded. Then

.

|ψ(z)|(1 − |z|2 )α . 1 − |ϕ(z)|2 |ϕ(z)|→1

Wψ,ϕ e  lim sup

.

(5.30)

5.6 The Compactness of Wψ,ϕ : Np → Hα∞

89

From Theorem 5.6.3, we obtain a complete characterization of boundedness of weighted composition operator .Wψ,ϕ acting from the space .Np to the space .Hα∞ . Theorem 5.6.4 Let .ψ ∈ Hol(D) and .ϕ be a holomorphic self-map of .D. The weighted composition operator .Wψ,ϕ : Np → Hα∞ is a well-defined compact operator if and only if .ψ and .ϕ satisfy

.

|ψ(z)|(1 − |z|2 )α = 0. |ϕ(z)|→1 1 − |ϕ(z)|2 lim

As an immediate consequence of Theorem 5.5.1, we have the following characterization of bounded composition operator .Cϕ : Np → Hα∞ . Corollary 5.6.5 The composition operator .Cϕ : Np → Hα∞ is compact if and only if .

lim sup

r→1 |ϕ(z)|>r

(1 − |z|2 )α = 0. 1 − |ϕ(z)|2

(5.31)

Corollary 5.6.5 can also be proved directly in another way. Below we provide such a proof which may have its own interest. Proof Suppose .Cϕ : Np → Hα∞ is compact. Assume, on the contrary, that there exist .ε0 > 0 and a sequence .(zn ) ⊂ D such that .

(1 − |zn |2 )α ≥ ε0 1 − |ϕ(zn )|2

whenever |ϕ(zn )| > 1 −

1 . n

By passing to a subsequence if needed, we may assume that .wn = ϕ(zn ) tends to .w0 ∈ T as .n → ∞. Let us consider the sequence .kwn , which clearly satisfies .kwn → kw0 with respect to the compact-open topology. Put .fn = kwn − kw0 . Then, by Lemma 3.2.1, .fn Np ≤ 1 and .fn → 0 uniformly on compact subsets of .D. Hence, .fn ◦ ϕ → 0 in the norm of .Hα∞ . However, for n large enough, we have Cϕ fn Hα∞ ≥ |kwn (ϕ(zn )) − kw0 (ϕ(zn ))|(1 − |zn |2 )α



(1 − |w0 |2 )(1 − |wn |2 )

(1 − |zn |2 )α

1 − =

≥ ε0 , 1 − |ϕ(zn )|2

|1 − w0 wn |2

.

which is a contradiction. Conversely, assume that (5.31) holds. Let .(fn ) be a bounded sequence of functions in .Np which converges to 0 on compact subsets of .D. We may surely assume that .ϕ(z)| > δ. Then we have

5 Weighted Composition Operators on .D

90

(1 − |z|2 )α |f (ϕ(z))|(1 − |ϕ(z)|2 ). 2 n z∈D 1 − |ϕ(z)|

Cϕ fn Hα∞ = sup

.

By Proposition 3.4.1, we obtain Cϕ fn Hα∞ ≤ εfn H1∞  εfn Np ≤ ε.

.

Hence, .Cϕ fn → 0 is the norm topology of .Hα∞ , and this fact implies that .Cϕ is compact.



Notes on Chapter 5 The results of this chapter are mostly from [95]. Lemma 5.2.1 is proved in [6, Lemma 2.1]. The counterpart of Theorem 5.1.1 for .Qp -spaces is still unknown. In [101,102], Xiao conjectured that .Mψ is bounded on .Qp if and only if ψ ∈ H∞

.

 and

sup log2 (1 − |a|) a∈D

D

|ψ  (z)|2 (1 − |σa (z)|2 )p dA(z) < ∞.

In [102, p. 22], Xiao also stated as an open problem to characterize the bounded composition operators on .Qp . The problem was completely solved by Pau and Peláez [75]. We state here as an open problem to give a full characterization of when .Cϕ is bounded on .Np . By doing so, one should be able to combine this with Theorem 5.1.1 and thereby to give a full description of when .Wϕ,ψ is bounded on .Np . Having done that, the open problem about bounded composition operators on .Qp should be solvable. A similar interplay between weighted composition operators (on Bergman-type spaces) and composition operators (on Bloch-type spaces) has been done in [14] and also in [74]. Using the derivative operator .f → f  , .Qp -spaces are closely related to .Np -spaces and Bloch-type spaces .Bα related to .Hα∞ . Hence, the results of this chapter also cover the corresponding results for .Cϕ (with .ψ = ϕ  ) acting between .Bα and .Qp -spaces which are presented in [102]. In [100], Xiao characterized the boundedness and compactness of .Cϕ : B → Qp by using a p-Carleson measure. For the case .Wψ,ϕ , an argument based on a p-Carleson measure is adopted. In [6], the authors characterized the .Qp -space in terms of a p-Carleson measure. For the .Np -space, an analogous characterization holds. The well-known result on the compactness of the composition operator on the Hardy spaces is provided in [17, Proposition 3.11].

Hadamard Gap Series in Hμ∞

6

.

In this chapter, the Hadamard gap series and the growth rate of the functions in the space Hμ∞ in the unit disc are studied.

.

6.1

Hadamard Gaps

An analytic function .f (z) on the open unit disc .D is said to have Hadamard gaps if its Taylor series expansion has the form f (z) =

∞ 

.

ak znk ,

k=1

where .nk+1 /nk ≥ q > 1, for all .k ≥ 1. Our goal is to find a necessary and sufficient condition on .μ which guarantees that f belongs to the weighted space .Hμ∞ or to the ∞ . We recall two classical results about lacunary corresponding little weighted space .Hμ,0 series. Theorem 6.1.1 (Hardy-Littlewood) Suppose that .f (z) =

∞ 

ck znk has a Hadamard

k=1

gap. Assume that the radial limit .

lim f (reiθ0 ) = f (eiθ0 )

r→1

exists. Then the series .

∞ 

ck eink θ0 is convergent.

k=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_6

91

6 Hadamard Gap Series in .Hμ∞

92

Theorem 6.1.2 (Zygmund) Suppose that the trigonometric series ∞  .

ak cos(nk θ ) + bk sin(nk θ ),

k=1

in which .nk+1 /nk ≥ q > 1, converges on a set of positive measure. Then the numerical series

.

∞  (ak2 + bk2 ) k=1

converges.

6.2

α-Bloch Spaces

Recall that for .α ∈ (0, ∞), the .α-Bloch space .Bα consists of all functions .f ∈ Hol(D) which satisfy the condition f Bα = sup(1 − |z|)α |f  (z)| < ∞.

.

z∈D

The space .B 1 is the Bloch space .B, while for .α ∈ (0, 1), .Bα is precisely the classical .(1 − α)-Lipschitz classes. It is also well-known that .Bα is a Banach space with respect to the norm .|f (0)| + f Bα . The little .α-Bloch space .Bα,0 consists of functions .f ∈ Hol(D) with .

lim (1 − |z|)α |f  (z)| = 0.

|z|→1

For .α-Bloch spaces, the following result holds. Lemma 6.2.1 Let .g(z) =

∞ 

bn zn ∈ Hol(D). If .g ∈ Bα , then

n=0 .

lim sup |bn |n1−α < ∞. n→∞

Similarly, if .g ∈ Bα,0 , then .

lim |bn |n1−α = 0.

n→∞

Proof Suppose .g ∈ Bα . By the Cauchy formula, we have

6.3 α-Bloch Spaces and Hadamard Gaps

93

   2π  1  1 g  (reiθ ) r 1−n ei(1−n)θ dθ   (1 − r)−α r 1−n , |bn | =  2π in 0 n

.

for all .0 < r < 1 and .n ≥ 1. Hence, for .n > 1 and the optimal choice .r = 1 − n1 , we get |bn |  n

α−1

.

  1 1−n , 1− n

which implies that .

lim sup |bn |n1−α < ∞. n→∞

The proof for the case .g ∈ Bα,0 is similar.

6.3



α-Bloch Spaces and Hadamard Gaps

We continue to complete Lemma 6.2.1 and provide a characterization of analytic functions with Hadamard gaps in .α-Bloch spaces. Theorem 6.3.1 Let f be a holomorphic function on .D with Hadamard gaps. Then the following assertions hold. (i) .f ∈ Bα if and only if .lim supk→∞ |ak |n1−α < ∞. k (ii) .f ∈ Bα,0 if and only if . lim |ak |n1−α = 0. k k→∞

Proof By Lemma 6.2.1, it suffices to prove the sufficiency. First, consider the case .

lim sup |ak |n1−α < ∞. k

(6.1)

k→∞

Recall the expansion ∞

.

 1 = An |z|n , 1+α (1 − |z|)

An ∼ (1 + α)−1 nα .

n=0

Thus, we have

.

∞  (n + 1)α |z|n ≤ n=0

C2 , (1 − |z|)1+α

(z ∈ D).

(6.2)

6 Hadamard Gap Series in .Hμ∞

94

It then follows from (6.1) that ∞  ∞     nk  .|zf (z)| =  ak nk z   nαk |z|nk .   

k=1

k=1

Hence, making use of the Cauchy product, we obtain ∞

|zf  (z)|  .  1 − |z|



n=1

 nk ≤n

nαk

|z|n .

Let .K = max{k : nk ≤ n}. We have n−α



.

nk ≤n

 α  

n α n1 nK−1 α K + ··· + 1+ n nK nK

nαk =

≤ 1 + q −α + q −2α + · · · =

qα . −1

qα

(6.3)

Therefore, by (6.2), we get

.

∞ |zf  (z)|  α n n |z|  1 − |z| n=1



∞ 

(n + 1)α |z|n

n=0



|z| , (1 − |z|)α+1

(z ∈ D),

which implies that .f ∈ Bα . Next, suppose that .

lim |ak |n1−α = 0. k

k→∞

For any .ε > 0, there exists .k0 ≥ 2 such that |ak |n1−α < ε, k

.

(k ≥ k0 ).

Put k0 −1 1  .P (z) = |ak |nk |z|nk , |z| k=1

(6.4)

6.3 α-Bloch Spaces and Hadamard Gaps

95

which is bounded in .D. Then there exists .r ∈ (0, 1) such that (1 − |z|)α P (z) < ε,

.

(r < |z| < 1).

(6.5)

Then we have |zf  (z)| ≤

∞ 

.

|ak |nk |z|nk ≤ |z|P (z) + ε

∞ 

nαk |z|nk ,

k=k0

k=1

which implies that

.

∞    α n |zf  (z)| |z|P (z)| nk |z| . ≤ +ε 1 − |z| 1 − |z| n ≤n n=1

k k≥k0

By (6.2) and (6.3), we have .

ε|z| |z|P (z)| |zf  (z)| +  . 1 − |z| 1 − |z| (1 − |z|)1+α

(6.6)

Combining (6.5) and (6.6) yields (1 − |z|α )|f  (z)|  ε,

.

(r < |z| < 1),

which shows that .f ∈ Bα,0 .



We recall a celebrated theorem of Fatou, which says that a bounded holomorphic function in the unit disc has radial limits almost everywhere on the unit circle [27]. Moreover, we need the notion of Fatou’s set whose elements are called Fatou’s points. Namely, let .f (z) be a non-constant holomorphic function from the Riemann sphere onto itself. Such functions are precisely the non-constant complex rational functions, that is, .f (z) = p(z)/q(z), where .p(z) and .q(z) are complex polynomials. We may assume that p and q have no common roots and at least one has degree larger than 1. Then there is a finite number of open sets .F1 , . . . , Fk , which satisfy the following conditions. (i) They are invariant under f . (ii) The union of the sets .Fj is dense in the plane. (iii) .f (z) behaves in a regular and equal way on each of the sets .Fj . The last statement means that either the termini of the sequences of iterations generated by the points of .Fj are precisely the same set, which is then a finite cycle, or they are finite cycles of circular- or annular-shaped sets that are lying concentrically. In the first case, the

6 Hadamard Gap Series in .Hμ∞

96

cycle is called attracting and in the second, neutral. The sets .Fj are the Fatou domains of f (z), and their union is called the Fatou set .F (f ) of f . Each of Fatou’s domains contains at least one critical point of f , that is, a finite point z satisfying .f  (z) = 0 or .f (z) = ∞ if the degree of the numerator p is at least twice larger than the degree of the denominator q or if .f = 1/q + c for some c and a rational function g. As a corollary of Theorem 6.3.1, we have the following result for Fatou’s points.

.

Corollary 6.3.2 Let .f ∈ Hol(D) with the Hadamard gap series expansion. If ∞  .

|ak |2 = ∞

lim |ak | = 0,

and

k→∞

k=1

then .f ∈ B1,0 , and f has no finite radial limit at almost every point of the unit circle .T. Proof We first note that (6.4) holds whenever .α = 1, because .|ak | → 0 as .k → ∞. Hence, by the Hardy–Littlewood and Zygmund Theorems 6.1.1 and 6.1.2, f has no finite radial limit at almost every point of .T. Let g be meromorphic in .D, .F (g) the set of all Fatou’s points of g, and .F ∗ (g) the set of .ξ ∈ F (g) where g has a finite angular limit. Then .F (g) − F ∗ (g) is of Lebesgue measure zero. Since g is pole-free in a terminal part of each angular domain at .ξ ∈ F ∗ (g), it follows from the Cauchy formula for .g  that .F ∗ (g) ⊂ F  (g), where .F  (g) is the set of  .ξ ∈ T where .(1 − |z|)|f (z)| has zero as the angular limit. Now considering the function f in the corollary, we see that .F  (f ) = T and .F ∗ (f ) is of measure zero. In other words, the set .F  (f ) − F ∗ (f ) has full measure .2π ..  Applying Theorem 6.3.1, we now prove the following result. Corollary 6.3.3 Let .α ∈ (0, ∞). Then there exist two functions .f1 , f2 ∈ Bα and a positive constant C, such that |f1 (z)| + |f2 (z)| ≥

.

C , (1 − |z|2 )α

(z ∈ D).

Equivalently, there exist two functions .g1 , g2 ∈ Hα∞ , such that |g1 (z)| + |g2 (z)| ≥

.

C , (1 − |z|2 )α

(z ∈ D).

Proof For a large integer q, consider the gap series fα (z) =

∞ 

.

j =0

j

q j (α−1) zq ,

(z ∈ D).

6.3 α-Bloch Spaces and Hadamard Gaps

97

By Theorem 6.3.1, with .bj = q j (α−1) and .nj = q j , we see that (1 − |z|2 )α |fα (z)|  1,

(z ∈ D).

.

Now we show that (1 − |z|2 )α |fα (z)|  1,

1 − q −k ≤ |z| ≤ 1 − q −(k+1/2) ,

.

(k ≥ 1).

(6.7)

Indeed, for each .z ∈ D, we have |fα (z)| ≥ q kα |z|q − α

k−1 

.

∞ 

j

q j α |z|q −

j =0

j

q j α |z|q = S1 − S2 − S3 .

j =k+1

For the part .S1 , we fix z with .|z| ∈ [1 − q −k , 1 − q −(k+1/2) ], .k ≥ 1, and put .τ = |z|q . Then k

 −1/2

k k+1/2 q (1 − q −k )q ≤ τ ≤ (1 − q −(k+1/2) )q .

.

With q large enough, .k ≥ 1, we have .

and hence .S1 ≥

q kα 3 .

 1 q −1/2 1 ≤τ ≤ , 3 2

(6.8)

Next, it is easy to check that S2 ≤

k−1 

.

qjα ≤

j =0

q kα . qα − 1

Lastly, note that |z|q

.

n (q−1)

≤ |z|q

k+1 (q−1)

,

(n ≥ k + 1).

In the sum .S3 , the quotient of two successive terms is not bigger than the ratio of the first two terms. Hence, the series of .S3 is controlled by the geometric series which has the same first two terms. Thus, by (6.8), we get S3 ≤ q (k+1)α |z|q

.

k+1

∞

j  k+2 k+1 q α |z|(q −q ) j =0

6 Hadamard Gap Series in .Hμ∞

98

= = ≤

q (k+1)α |z|q 1 − q α |z|(q

k+1

k+2 −q k+1 )

q kα q α τ q 1 − q α τ (q

2 −q)

q kα q α 2−q 1 − q α 2−(q

1/2

3/2 −q 1/2 )

.

Therefore, we obtain |fα (z)| ≥

.

q kα 1 (q α )k+1/2 ≥ α/2 , = α/2 4 4q 4q (1 − |z||)α 

which gives (6.7).

6.4

Normal Weights

A positive continuous function .μ on .[0, 1) is called normal if there exist positive constants α and .β with .α < β, and .δ ∈ (0, 1), such that

.

.

μ(r) μ(r) is decreasing on [δ, 1), lim = 0, α r→1 (1 − r)α (1 − r)

and .

μ(r) μ(r) is increasing on [δ, 1), lim = ∞. r→1 (1 − r)β (1 − r)β

The normal function itself is decreasing in a neighborhood of 1 and satisfies . lim μ(r) = r→1−

0. For example, μ(r) = 1/log log e2 (1 − r 2 )−1

.

and

 μ(r) = (1 − r 2 )a logb eb/a (1 − r 2 )−1 ,

.

(a > 0, b ≥ 0),

are normal functions.  Lemma 6.4.1 Suppose .μ is normal and .0 < α < min constant C, independent of .α, such that

1 e,1 − δ

 . Then there is a positive

∞ Spaces 6.5 Hμ∞ and Hμ,0

99



∞

e−αt

.

μ(1 −

e

1 t)

dt ≤

C . αμ(1 − α)

Proof We have  .

e

∞



e−αt μ(1 − 1t )



e−αt

1/α

dt =

μ(1 − 1t )

e

dt +

∞

1/α

e−αt μ(1 − 1t )

dt = A + B.

On the one hand, since .μ is essentially decreasing on .[0, 1), 

1/α

A≤

.

dt μ(1 −

e

1 t)

≤

C μ(1 − α)



1/α

dt ≤

e

C . αμ(1 − α)

On the other hand, since .μ is normal,  B≤

∞

.

1/α

(αt)b e−αt μ(1 −

1 t)

∞ dt =

s b e−s ds C ≤ . αμ(1 − α) αμ(1 − α) 1



Combining these inequalities yields the desired result.

In the sequel, positive constants are denoted by C, and they may have different values at different places.

6.5

∞ Hμ∞ and Hμ,0 Spaces

The weighted space .Hμ∞ is defined as 



Hμ∞ = f ∈ Hol(D) : f  = sup |f (z)|μ(|z|) < ∞ ,

.

z∈D

where .μ is normal on .[0, 1). It is straightforward to verify that .Hμ∞ is a Banach space. ∞ is the closed subspace of .H ∞ that consists of .f ∈ H ∞ The little weighted space .Hμ,0 μ μ satisfying .

lim |f (z)|μ(|z|) = 0.

|z|→1−

In the special but important case .μ(|z|) = (1 − |z|2 )α , α > 0, the induced spaces .Hμ∞ and ∞ ∞ ∞ .H μ,0 are denoted by .Hα and .Hα,0 , respectively.

6 Hadamard Gap Series in .Hμ∞

100 ∞ 

Lemma 6.5.1 Suppose .f (z) =

an zn ∈ Hμ∞ . Then

n=0

.

 1 lim sup |an |μ 1 − < ∞. n n→∞

(6.9)

Proof By the Cauchy integral formula, we have    1  .|an | =  2π 1 ≤ 2π



2π

iθ

f (re )r

−n −inθ

e

0 2π

  dθ 

|f (reiθ )|r −n dθ ≤ f Hμ∞

0

For .n ≥ 2, we choose the optimal value .r = 1 −

1 n

r −n . μ(r)

(6.10)

to get the estimation

     1  −1 1 −n |an | ≤ f Hμ∞ μ 1 − . 1− n n

.

Hence, .

  1 lim sup |an | 1 − ≤ f Hμ∞ · e < ∞. n n→∞ 

∞ , we have the following corresponding result. For the space .Hμ,0 ∞ 

Lemma 6.5.2 Suppose .f (z) =

∞ . Then an zn ∈ Hμ,0

n=0

.

 1 lim |an |μ 1 − = 0. n

n→∞

(6.11)

∞ , for every .ε > 0, there exists .δ ∈ (0, 1), such that whenever Proof Since .f ∈ Hμ,0 .δ < |z| < 1,

μ(|z|)|f (z)| < ε.

.

Let .n0 be the least integer satisfying .1 − have

1 n0

(6.12)

> δ. By (6.10) and (6.12), for .r ∈ (δ, 1), we

∞ Spaces and Hadamard Gaps 6.6 Hμ∞ and Hμ,0

|an | ≤

.

Substituting .r = 1 −

1 n



ε 2π

2π 0

101

r −n r −n dθ = ε . μ(r) μ(r)

into the last inequality, we obtain   1 μ 1− |an | < εe, n

.

(n ≥ n0 ), 

from which the result follows.

6.6

∞ Hμ∞ and Hμ,0 Spaces and Hadamard Gaps

These results of Sect. 6.3 give the motivation to investigate the situation for the more general weighted space .Hμ∞ . This is logical since the spaces with normal weights are the most natural generalization of .Hα∞ . Theorem 6.6.1 Let f (z) =

∞ 

.

ak znk ∈ Hol(D),

k=0

where .nk is a sequence of integers with .nk+1 /nk ≥ λ > 1, and let .μ be a normal function on the interval .[0, 1). Then .f ∈ Hμ∞ if and only if .

  1 |ak | < ∞. sup μ 1 − nk k≥0

(6.13)

Proof By Lemma 6.5.1, it suffices to prove the sufficiency. Since

 .μ is normal, without 1 is decreasing. Then loss of generality, we may assume that the sequence . μ(1 − nk ) k≥0

we have  ∞ ∞    |z|nk  nk  , .|f (z)| =  ak z  ≤ C   μ(1 − n1k ) k=0 k=0 and hence ∞

 |f (z)| ≤C . 1 − |z| n=1







1

nk ≤n μ(1 −

1 nk )

|z|n

(6.14)

6 Hadamard Gap Series in .Hμ∞

102

for some positive constant C. Write .K = K(n) = max{k : nk ≤ n}. Then, by the definition of the normal weight, we have  1 1  μ 1− n n ≤n μ(1 −

.

k

=

1 nk )

≤

 μ(1 − μ(1 − n1 ) K−1 μ(1 − K−1  n=0

≤

K−1  n=0

≤

1 nK ) n=0

μ(1 −

μ(1 − μ(1 −

1 nK ) 1 nK −n )

nK − n α nK

λα < ∞. −1

λα

Consequently,  .

nk ≤n

1 μ(1 −

≤

1 nk )

C μ(1 − n1 )

and, with (6.14), ∞

.

 |f (z)| |z| ≤C . 1 − |z| μ(1 − n1 ) n=1

Note that the function gx (t) =

.

1

xt μ(1 − 1t )

=

e−t log x

μ(1 − 1t )

is decreasing in t, for t large enough and each .x ∈ (0, 1), and 

∞ 

|z|

n=1

μ(1 − n1 )

.

∞

∼

1

e−t log |z|

e

μ(1 − 1t )

log

1 |z| μ

dt.

By Lemma 6.4.1, we get  .

e

∞

1

e−t log |z| μ(1 −

Using the asymptotic estimate

1 t)

dt ≤ C

1 .  1 1 − log |z|

1 nK ) 1 nK −n )

∞ Spaces and Hadamard Gaps 6.6 Hμ∞ and Hμ,0

.

103

1 ∼ 1 − |z|, |z|

log

as |z| → 1− ,

and the normality of .μ, we thus obtain ∞ 

|z|

n=1

μ(1 − n1 )

.

≤

C , (1 − |z|)μ(|z|) 

from which the desired result follows.

The following result has the same flavor as Theorem 6.6.1. Hence, its proof is considerably shortened. Theorem 6.6.2 Let f (z) =

∞ 

.

ak znk ∈ Hol(D),

k=0

where .nk is a sequence of integers with .nk+1 /nk ≥ λ > 1, and let .μ be a normal function ∞ if and only if on the interval .[0, 1). Then .f ∈ Hμ,0 .

  1 = 0. lim |ak |μ 1 − k→∞ nk

(6.15)

Proof By Lemma 6.5.2, it suffices to prove the sufficiency. By (6.15), for every .ε > 0, there is an integer .k0 ≥ 1 such that 

1 .|ak |μ 1 − nk

 < ε,

(k ≥ k0 ).

Then |f (z)| ≤

k 0 −1

.

k=0

|z|nk μ(1 −

1 nk )

+ε

∞  k=k0

|z|nk μ(1 −

1 nk )

As in the proof of Theorem 6.6.1, we obtain .

|f (z)| εC Pk (z) , + ≤ 0 1 1 1 − |z| log |z| 1 − |z| μ(1 − log |z| )

.

6 Hadamard Gap Series in .Hμ∞

104

where .Pk0 is a polynomial of order .nk0 , which is bounded on .D. From normality of .μ, (6.15), and the condition . lim μ(|z|) = 0, the desired result follows.  |z|→1−

6.7

Lower Estimations

For each .α ∈ (0, ∞), there are two functions .f1 , f2 ∈ Bα and a constant .C > 0, such that |f1 (z)| + |f2 (z)| ≥

.

C , (1 − |z|2 )α

z ∈ D.

(6.16)

In other words, there exist two functions .g1 , g2 ∈ Hα∞ such that |g1 (z)| + |g2 (z)| ≥

.

C , (1 − |z|2 )α

z ∈ D.

(6.17)

These results are still valid for a more general context of weighted space .Hμ∞ . It is quite logical, as the spaces with normal weights are the most natural generalization of the space ∞ .Hα . Applying Theorem 6.6.1, we deduce the generalization below. Theorem 6.7.1 Let .μ : [0, 1) → [0, ∞) be nonincreasing radial weight function and normal on the interval .[0, 1). Then there exist two functions .f1 , f2 ∈ Hμ∞ such that |f1 (z)| + |f2 (z)| ≥

.

C , μ(|z|)

(z ∈ D).

Proof Denote f (z) =

∞ 

.

j =1

zq

j

μ(1 −

1 qj

)

(z ∈ D),

,

where .q ∈ N large enough. By Theorem 6.6.1, with .aj = μ(1 − we easily see that .f ∈ Hμ∞ . We show that |f (z)| ≥

.

C , μ(|z|)

1 qj

)

−1

1

if 1 − q −k ≤ |z| ≤ 1 − q −(k+ 2 ) , k ≥ 0.

For .z ∈ D, we have |f (z)| ≥

.

|z|q μ(1 −

k

1 ) qk

−

k−1 

|z|q

j =1

μ(1 −

j

1 qj

)

−

∞ 

|z|q

k+1

μ(1 −

j

1 qj

)

and .nj = q j ,

6.7 Lower Estimations

105

= A − B − C. First, since

(1 − q −k )q ≤ |z|q k

k

.

 q − 21 1 k+   1 q 2 ≤ 1 − q −(k+ 2 ) ,

we have 1 k ≤ |z|q ≤ . 3

 q − 21 1 . 2

Then for q large enough, A≥

.

1 3μ(1 −

1 ) qk

.

Next, by the definition of the normal function .μ, for q large enough, and for each .j ∈ N, we have



α b 1 − (1 − q1j ) 1 − (1 − q1j ) μ(1 − q1j ) α ≤ ≤

.

b , μ(1 − q j1+1 ) 1 − (1 − q j1+1 ) 1 − (1 − q j1+1 ) and also by the simple computation, we have 1 < qα ≤

.

μ(1 − μ(1 −

1 qj 1

)

) q j +1

≤ qb.

(6.18)

Then B≤

.

≤

Last, we have

k−1 μ(1 − 

1 μ(1 −

1 ) qk

1 ) q j +1 1 μ(1 − q j ) j =1

1

k−1 

μ(1 −

1 ) q k j =1

1 q α(k−j )

≤

·

μ(1 − μ(1 − 1

μ(1 −

1 ) qk

1 ) q j +2 1 ) q j +1

·

qα

···

μ(1 − μ(1 −

1 . −1

1 ) qk 1

q k−1

)

6 Hadamard Gap Series in .Hμ∞

106

C≤

.

≤

|z|q

μ(1 − |z|q

∞ μ(1 − 

k+1

1 ) q k j =k+1 ∞ 

k+1

μ(1 −

μ(1 −

1 ) q k j =k+1

1 ) j k+1 qk |z|(q −q ) 1 ) qj

μ(1 − μ(1 −

1 ) qk 1

·

μ(1 −

1 ) q k+1 1 ) q k+2

) μ(1 − q k+1

···

μ(1 −

1 ) q j −1 1 μ(1 − q j )

· |z|(q

j −q k+1 )

Then by (6.18), we obtain C≤

.

≤

=

|z|q

∞ 

k+1

μ(1 −

1 ) q k j =k+1

q b · (q b )(j −(k+1)) |z|(q

j −q k+1 )

k

s (|z|q )q qb b b (q k+2 −q k+1 ) q · q |z| = · k 2 1 1 μ(1 − q k ) s=0 μ(1 − q k ) 1 − q b (|z|q )(q −q)

|z|q

∞ 

k+1

1 μ(1 −

1 ) qk

·

1

qb 1 − q b (|z|q )(q k

2 −q)

≤

1 μ(1 −

1 ) qk

·

q b 2−q 2 3

1

.

1 − q b 2−(q 2 −q 2 )

For q large enough and each .k ∈ N, we have ⎞ 1 b 2−q 2 1 q C 1 ⎠≥ ⎝ − .|f (z)| ≥ − 3 1 α 1 μ(1 − q k ) 3 q − 1 1 − q b 2−(q 2 −q 2 ) μ(1 − ⎛

1

By the normality of .μ and the inequality 1

1 − q −k ≤ |z| ≤ 1 − q −(k+ 2 ) ,

.

we obtain |f (z)| ≥

.

C μ(1 −

1 ) qk

≥

C  Cq μ 1 −

≥

1 q

k+ 12

Similarly, we can easily prove that for g(z) =

∞ 

.

j =1

z nj μ(1 − 1

1 q

j + 12

where .nj is the largest integer closest to .q j + 2 , we have

)

,

C . μ(|z|)

1 ) qk

.

.

6.7 Lower Estimations

|g(z)| ≥

.

C , μ(|z|)

107

1

whenever 1 − q −(k+ 2 ) ≤ |z| ≤ 1 − q −(k+1) , k ∈ N.

Finally, the result of the theorem follows by taking f1 (z) = r + sf (z) and

.

f2 (z) = tg(z), z ∈ D,

for some appropriate positive constants .r, s, and t.



Notes on Chapter 6 The concept of normal weights is taken from [85]. The Hardy–Littlewood Theorem 6.1.1 is proved in [39] and Zygmund Theorem 6.1.2 is available in [114]. Theorem 6.3.1 and its Corollary 6.3.2 are proved in [103], while the case .α = 1 is proved in [70]. Basing on Theorem 6.3.1, Corollary 6.3.3 is proved in [102]. However, this latter result had already been proved earlier in [78] for the basic case .α = 1. Other results are proved in [104]. Fatou’s points are introduced in [13]. The lower estimations (6.16) and (6.17) were studied in [102].

Np Spaces in the Unit Ball B

.

7

.

The aim of this chapter is to characterize the .Np -spaces in the unit ball as well as the behavior of the weighted composition operators acting on these spaces. We study different properties of the weighted composition operators acting on these spaces.

7.1

On the Unit Ball

Let .B be the open unit ball in the complex vector space .Cn ; let .Hol(B) denote the space of functions that are holomorphic in .B, with the compact-open topology; and let .H ∞ (B) denote the Banach space of bounded holomorphic functions on .B with the norm f ∞ = sup |f (z)|.

.

z∈B

If .z = (z1 , z2 , . . . , zn ) ∈ Cn and .ζ = (ζ1 , ζ2 , . . . , ζn ) ∈ Cn , we define the inner product z, ζ  = z1 ζ¯1 + · · · + zn ζ¯n

.

and correspondingly the Euclidean norm |z| = (|z1 |2 + · · · + |zn |2 )1/2 .

.

If X and Y are two topological vector spaces, then the symbol .X → Y indicates the continuous embedding of X into Y . The Beurling-type space, also called the Bergman-type space, .A−p (B), .p > 0, in the unit ball is defined as

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_7

109

7 .Np Spaces in the Unit Ball .B

110





A−p (B) = f ∈ Hol(B) : |f |p = sup |f (z)|(1 − |z|2 )p < ∞ .

.

z∈B

Weighted composition operators for function spaces defined on the open unit disc .D were defined in Sect. 5. In the present context, the definitions are similar. For the sake of completeness, we mention them here. For .ϕ a holomorphic self-map of .B and a holomorphic function .u : B → C, the linear operator .Wu,ϕ : Hol(B) → Hol(B) defined by Wu,ϕ (f )(z) = u(z) · (f ◦ ϕ(z)),

.

f ∈ Hol(B),

(z ∈ B),

is called the weighted composition operator with symbols u and .ϕ. We see that .Wu,ϕ = Mu Cϕ , where .Mu (f ) = uf is the multiplication operator with symbol u and .Cϕ (f ) = f ◦ ϕ is the composition operator with symbol .ϕ. If u is the constant function 1, then .Wu,ϕ = Cϕ , and if .ϕ is the identity, then .Wu,ϕ = Mu . Let .a ∈ B. Then the orthogonal projection of .Cn on the (at most) one-dimensional subspace .[a] generated by a is given by Pa z =

.

⎧ ⎨0,

if a = 0,

⎩ z,a a, a,a

if a = 0.

We write .Qa = I − Pa for the projection on the orthogonal complement of .[a] and, for the simplicity of notations, let .sa = (1 − |a|2 )1/2 . Then, as we will see in Sect. 7.2, the mapping a (z) =

.

a − Pa z − sa Qa z 1 − z, a

(7.1)

is an automorphism of the unit ball .B, i.e., .a ∈ Aut(B). For .p ∈ (0, ∞) and .f ∈ Hol(B), we define  f p := sup

.

a∈B

1/2 |f (z)| (1 − |a (z)| ) dV (z) 2

B

2 p

Then .Np is the space

 Np (B) = f ∈ Hol(B) : f p < ∞ ,

.

where dV is the Lebesgue normalized volume measure on .B, i.e., .V (B) = 1. When there is no ambiguity, we write .Np for .Np (B).

7.2 The Automorphism a

7.2

111

The Automorphism a

Recall that .a is defined by (7.1). It is clear that for .n = 1, we have .Pa = I and .Qa = 0, and hence .a (z) becomes the automorphism .σa of the open unit disc .D. We now show that .a satisfies the same property on the ball .B. It is easy to directly verify that a (0) = a

.

and a (a) = 0.

.

In other words, .a exchanges the points 0 and a. Moreover,  a (0) = −sa2 Pa − sa Qa

.

and  a (a) = −

.

Qa Pa − . 2 sa sa

Using the definition of inner product, we see that 1 − a (z), a (w) =

.

(1 − a, a)(1 − z, w) (1 − z, a)(1 − a, w)

(7.2)

holds for all .z ∈ B, w ∈ B. In particular, when .w = z, it leads to the identity 1 − |a (z)|2 =

.

(1 − |a|2 )(1 − |z|2 ) , |1 − z, a|2

(z ∈ B).

(7.3)

Hence, .a maps .B into itself. But, .a is also an involution, i.e., a (a (z)) = z,

.

(z ∈ B).

From this, we deduce that .a is a homeomorphism of the closed unit ball .B onto itself and an automorphism of the open unit ball .B. The pseudo-hyperbolic metric in the ball is defined by ρ(z, w) = |w (z)|,

.

(z, w ∈ B).

(7.4)

It is easy to verify that .ρ(0, w) = |w| and that .ρ(w (z), w) = |z|. Clearly, .ρ(z, w) ≥ 0 and .ρ(z, w) = 0 if and only if .z = w, because .w (z) = 0 only for .z = w. The symmetry

7 .Np Spaces in the Unit Ball .B

112

property .ρ(z, w) = ρ(w, z) follows from (7.3). The triangle inequality is less trivial, and we deduce it below from a stronger result. Theorem 7.2.1 The pseudo-hyperbolic metric .ρ has the following properties. (i) Rotation invariance:

ρ U (z), U (w) = ρ(z, w)

.

for all .z, w ∈ B and all unitary matrices U . (ii) Möbius invariance:

ρ a (z), a (w) = ρ(z, w)

.

for all .z, w ∈ B. (iii) Strong triangle inequality: .

|ρ(z, a) − ρ(a, w)| ρ(z, a) + ρ(a, w) ≤ ρ(z, w) ≤ 1 − ρ(z, a)ρ(a, w) 1 + ρ(z, a)ρ(a, w)

for all .z, w, a ∈ B. Proof (i): Since unitary transformations preserve inner products, the identity

2 1 − ρ U (z), U (w) = 1 − ρ(z, w)2

.

follows directly from (7.3). (ii): Due to (7.2) and (7.3), we have 

2

2 1 − ρ U (z), U (w) = 1 − a (w) a (z) 

.

=

(1 − |a (w)|2 )(1 − |a (z)|2 ) |1 − a (z), a (w)|2

=

(1 − |w|2 )(1 − |z|2 ) |1 − z, w|2

= 1 − |w (z)|2 = 1 − ρ(z, w)2 . Combining (i) and (ii), we see that .ρ is invariant under every automorphism of the ball.

7.3 Surjective Isometries

113

(iii): Since by (ii) .ρ is Möbius-invariant, we may assume that .a = 0. Hence, we need to show that   |z| − |w| |z| + |w| ≤ |w (z)| ≤ . . (7.5) 1 − |z||w| 1 + |z||w| Applying the Cauchy–Schwarz inequality, we have 1 − |z||w| ≤ |1 − z, w| ≤ 1 + |z||w|.

.

Then (7.5) follows by using the basic identity (7.3). 

7.3

Surjective Isometries

By using weighted composition operators with particular symbols, we obtain an alternate description of the norm in .Np . Furthermore, for any . ∈ Aut(B), as is well-known, there exists a unitary operator U such that . = U a , where .a = −1 (0). This shows that .|(z)| = |a (z)|, for all .z ∈ B. Consequently, we obtain f p = sup



.

a∈B

=

B



sup ∈Aut(B)

=

1/2 |f (z)|2 (1 − |a (z)|2 )p dV (z)

sup ∈Aut(B)



1/2 |f (z)|2 (1 − |(z)|2 )p dV (z)

B

1/2 |f (z)|2 (1 − |−1 (z)|2 )p dV (z) .

B

For . ∈ Aut(B), let .a = −1 (0). For simplicity, write W := Wka , .

.

By the change of variables .z = (w), we obtain  .

B

|f (z)|2 (1 − |−1 (z)|2 )p dV (z)

 =

B

|f ((w))|2 (1 − |w|2 )p



1 − |a|2 n+1 dV (w) |1 − w, a|2

(7.6)

7 .Np Spaces in the Unit Ball .B

114

 =

B

|f ((w))|2 |ka (w)|2 (1 − |w|2 )p dV (w)

= W f 2A2 .

(7.7)

p

Here, .A2p is the weighted Bergman space over .B defined by   A2p = f ∈ Hol(B) : f A2p < ∞ ,

.

where f A2p =



.

B

1/2 |f (z)|2 (1 − |z|2 )p dV (z)

According to the formulas (7.7), we have   f p = sup W f A2p :  ∈ Aut(B) .

.

(7.8)

Theorem 7.3.1 For any automorphism . of the unit ball .B, the weighted composition operator .W is a surjective isometry on .Np . Proof By a direct calculation that for any two automorphisms . and . in .Aut(B), there exists a complex number .λ with modulus one such that .W W = λW◦ . Consequently, by (7.8),   W f p = sup W W f A2p :  ∈ Aut(B)   = sup W◦ f A2p :  ∈ Aut(B) = f p .

.



7.4

Np (B) Is a Chain

As in many function spaces, we show that .Np (B) is a chain with respect to the parameter p. Theorem 7.4.1 For .p > q > 0, we have H ∞ (B) → Nq (B) → Np (B).

.

Proof Let .f ∈ H ∞ (B). Then we have

7.5 The Embedding Np (B) → A−

n+1 2 (B)

115

 2 .f q

= sup

|f (z)|2 (1 − |a (z)|2 )q dV (z)

a∈B B

2



≤

sup |f (z)| z∈B

 sup

a∈B B

(1 − |a (z)|2 )q dV (z)

≤ f 2∞ , which implies the first embedding. Next, let .f ∈ Nq (B). Then  f 2p = sup

.

a∈B B

|f (z)|2 (1 − |a (z)|2 )p dV (z)



= sup

a∈B B

|f (z)|2 (1 − |a (z)|2 )q · (1 − |a (z)|2 )p−q dV (z)



≤ sup

a∈B B

|f (z)|2 (1 − |a (z)|2 )q dV (z) = f 2q , 

which shows the second embedding.

7.5

The Embedding Np (B) → A−

n+1 2

(B)

Recall that n is the dimension of ambient space, i.e., .B ⊂ Cn . Theorem 7.5.1 For .p > 0, we have Np (B) → A−

.

n+1 2

(B).

Proof Write .B1/2 = {z : |z| < 1/2}. For each .f ∈ Np (B), we have  f 2p = sup

.

a∈B B

|f (z)|2 (1 − |a (z)|2 )p dV (z)



≥  =  ≥

B

B

|f (z)|2 (1 − |0 (z)|2 )p dV (z) |f (z)|2 (1 − |z|2 )p dV (z)

B1/2

|f (z)|2 (1 − |z|2 )p dV (z)

 p  3 ≥ |f (z)|2 dV (z), 4 B1/2

7 .Np Spaces in the Unit Ball .B

116

that is,  p  4 2 . f p ≥ |f (z)|2 dV (z). 3 B1/2

(7.9)

Since .f ∈ Hol(B), the function .|f |2 is subharmonic in .Cn and hence subharmonic in .R2n . Then   1 2 .|f (0)| ≤ |f (z)|2 dV (z) = 4n |f (z)|2 dV (z). (7.10) V (B1/2 ) B1/2 B1/2 Combining (7.9) and (7.10) yields |f (0)|2 ≤

.

4p+n f 2p , f ∈ Np (B). 3p

(7.11)

For every fixed .z ∈ B, we put n+1

(1 − |z|2 ) 2 .Fz,f (w) = (f ◦ z (w)) · , (1 − w, z)n+1

(w ∈ B),

which is clearly a holomorphic function in .B. We prove that .Fz,f (w) ∈ Np (B). First, by change of variables .t = z (w), we have 

(1 − |z|2 )n+1 (1 − |a (w)|2 )p dV (w) |1 − w, z|2n+2 a∈B B   p = sup |f (t)|2 1 − |a (z (t))|2 dV (t),

Fz,f 2p = sup

.

|f ◦ z (w)|2 ·

a∈B B

due to the fact that  .

1 − |z|2 |1 − w, z|2

n+1

is the real Jacobian determinant of the automorphism .z (w). Furthermore, by the transitivity of .Aut(B), .a ◦ z belongs to .Aut(B), and so .a ◦ z = U ◦ ba , where .ba = a (z), U is a unitary transformation of .B, and .ba is the involution. Then  .

sup

a∈B B



= sup

a∈B B

 p |f (t)|2 1 − |a (z (t))|2 dV (t)  p |f (t)|2 1 − |U (ba (t))|2 dV (t)

7.6 The Embedding A−k (B) → Np (B)

 = sup

a∈B B

117

 p |f (t)|2 1 − |ba (t)|2 dV (t)

( |U (z)| = |z|, z ∈ B)



≤ sup

c∈B B

|f (t)|2 (1 − |c (t)|2 )p dV (t) = f 2p .

Consequently, Fz,f 2p ≤ f 2p ,

.

which shows that .Fz,f ∈ Np (B). Then by (7.11), we have |f (z)|2 (1 − |z|2 )n+1 = |Fz,f (0)|2 ≤

.

4p+n 4p+n 2 F  ≤ f 2p , z,f p 3p 3p

(z ∈ B),

which implies that |f | n+1 = sup |f (z)|(1 − |z|2 )

.

2

n+1 2

≤

z∈B

That is, .Np (B) → A−

n+1 2

2p+n f p , 3p/2

(f ∈ Np (B)).



(B).

The Embedding A−k (B) → Np (B)

7.6

The following result can be considered as a dual to Theorem 7.5.1. Theorem 7.6.1 Let .k ∈ (0, (n + 1)/2]. Then, for .p > max{0, 2k − 1}, we have A−k (B) → Np (B).

.

In particular, when .p > n, .Np (B) = A−

n+1 2

(B).

Proof Let .f ∈ A−k (B). Then  f 2p = sup

.

a∈B B

|f (z)|2 (1 − |a (z)|2 )p dV (z)



= sup

a∈B B

1 (1 − |a (z)|2 )p dV (z) (1 − |z|2 )2k  (1 − |a (z)|2 )p · sup dV (z) (1 − |z|2 )2k a∈B B

|f (z)|2 (1 − |z|2 )2k ·

≤ sup |f (z)|2 (1 − |z|2 )2k z∈B

(7.12)

7 .Np Spaces in the Unit Ball .B

118

 = |f |2k sup

a∈B B



(1 − |a (z)|2 )p dV (z) (1 − |z|2 )2k

(1 − |a|2 )p (1 − |z|2 )p−2k dV (z) (by (7.3)) |1 − z, a|2p a∈B B  (1 − |z|2 )p−2k 2 2 p dV (z). = |f |k sup(1 − |a| ) 2p B |1 − z, a| a∈B

= |f |2k sup

Furthermore, similar to the estimate in Lemma 1.8.1, we have  .

B

(1 − |z|2 )p−2k dV (z) = |1 − z, a|2p 

 B

(1 − |z|2 )p−2k dV (z) |1 − z, a|n+1+(p−2k)+p−n−1+2k

⎧ ⎪ ⎪bounded in B, ⎨

p < n + 1 − 2k,

1 log 1−|a| 2, ⎪ ⎪ ⎩ (1 − |a|2 )n+1−p−2k ,

p = n + 1 − 2k,

(7.13)

p > n + 1 − 2k.

In any case, there exists a positive constant C such that  (1 − |a|2 )p

.

B

(1 − |z|2 )p−2k dV (z) ≤ C, |1 − z, a|2p

(a ∈ B),

which implies f p ≤

√

.

C|f |k ,

That is, .A−k (B) → Np (B). In particular, for all .p > n, by taking .k = A−

.

n+1 2

(f ∈ Np (B)).

n+1 2 ,

we have .p > 2k − 1 and hence

(B) → Np (B).

By Theorem 7.5.1, we always have .Np (B) → A− Np (B).

7.7

n+1 2

(B). Hence, in fact, .A−

n+1 2

(B) = 

Np (B) as a Banach Space

Theorem 7.5.1 and estimations obtained in its proof enable us to show that .Np (B) is a Banach space in which the point evaluations are continuous linear functionals.

7.7 Np (B) as a Banach Space

119

Theorem 7.7.1 .Np (B) is a functional Banach space with the norm . · p , and moreover, its norm topology is stronger than the compact-open topology. Proof We first verify the triangle inequality for . · p . For .f, g ∈ Np (B) and .a ∈ B, and by the Cauchy–Schwarz inequality,  .

 ≤

B

B

1/2 |f (z) + g(z)|2 (1 − |a (z)|2 )p dV (z) 1/2  1/2 |f (z)|2 (1 − |a (z)|2 )p dV (z) + |g(z)|2 (1 − |a (z)|2 )p dV (z) . B

Taking the supremum of both sides with respect to .a ∈ B gives the triangle inequality f + gp ≤ f p + gp . The other properties of a norm are easy to verify. To establish the completeness of .Np (B), let .(fm ) be a Cauchy sequence in .Np (B) with respect to the norm . · p . From this, it follows that .(fm ) is a Cauchy sequence in the space .Hol(B), and hence it converges to some .f ∈ Hol(B). It remains to show that .f ∈ Np (B). Indeed, there exists a . 0 ∈ N such that .fm − f p ≤ 1 for all .m, ≥ 0 . Fix an arbitrary .a ∈ B. Then, by Fatou’s lemma, we have .

 .

B

|f (z) − f 0 (z)|2 (1 − |a (z)|2 )p dV (z) 

≤ lim inf

→∞

B

|f (z) − f 0 (z)|2 (1 − |a (z)|2 )p dV (z)

≤ lim inf f − f o 2p ≤ 1,

→∞

which implies  f − f 0 p = sup

.

a∈B B

|f (z) − f 0 (z)|2 (1 − |a (z)|2 )p dV (z) ≤ 1,

and hence .f p ≤ 1 + f 0 p < ∞. In other words, .f ∈ Np (B). In the estimations above, if we replace 1 by .ε (which of course changes . 0 to another index, say . ε ), we see that the sequence .(fm ) actually converges to f in the .Np (B)-norm. Finally, we prove that .Np (B) is a functional Banach space, i.e., we show that for each .z ∈ B, the point evaluation .f −→ f (z) is continuous on .Np (B). Indeed, from (7.12), we have |f (z)| ≤

.

2p+n 3p/2 (1 − |z|2 )

n+1 2

f p ,

(f ∈ Np (B)),

which means that the point evaluation is bounded on .Np (B), and hence it is continuous.

7 .Np Spaces in the Unit Ball .B

120 n+1

Since the norm topology of .A− 2 (B) is clearly stronger than the compact-open topology, the second statement follows from (7.12). 

The Embedding B(B) → Np (B)

7.8

In the following, .B(B) is the Bloch space in .B. Theorem 7.8.1 For .0 < p < ∞, .B(B) → Np (B). Proof Let .f ∈ B(B). We have |f (z) − f (w)| ≤

.

1 1 + |z (w)| f B log , 2 1 − |z (w)|

(z, w ∈ B, z = w),

(7.14)

where .f B = |f (0)| + supz∈B |∇f (z)|(1 − |z|2 ). From (7.14), it follows that 1 4 3 4 |f (z)| ≤ |f (0)| + f B log ≤ f B log . 2 2 1 − |z|2 1 − |z|2

.

Consequently, for all .f ∈ B(B), we have  2 .f p

= sup a∈B

|f (z)| (1 − |a (z)| ) dV (z) 2

B

2 p

2   3 4 (1 − |a (z)|2 )p dV (z) f 2B sup log 2 1 − |z|2 a∈B B 2   4 3 2 dV (z) = Cf 2B , log ≤ f B · 2 1 − |z|2 B ≤

which shows that .B(B) → Np (B).



Notes on Chapter 7 Composition operators and weighted composition operators acting on spaces of holomorphic functions in the unit disc .D of the complex plane have been extensively studied. We refer the readers to the monographs [17, 83] for detailed information. In the onedimensional case, composition operators acting on the .Np -space in the unit disc .D were considered in [73]. Some results on boundedness and compactness of these operators were obtained. Moreover, weighted composition operators acting on the .Np -space were considered in [95]. The results of this chapter are mostly the generalization of corresponding

7.8 The Embedding B(B) → Np (B)

121

results in [73, 95] and are from [42]. The automorphism .a ∈ Aut(B) and its properties are provided in [79, pages 25–27]. Information about pseudo-hyperbolic metric for the ball is given in [22]. For properties of pseudo-hyperbolic metric in Theorem 7.2.1, proofs of parts (i) and (ii) are given in [22], while the proof of part (iii) is implicitly given in [64, Lemma 3]. The integral formula (7.13) is provided in [79, Proposition 1.4.10]. Formula (7.14) is provided in [108, Theorem 3.6]. When .n = 1, we obtain Proposition 3.1 of [73] as special cases of Theorems 7.5.1, 7.6.1, 7.7.1, and 7.8.1.

Weighted Composition Operators on B

8

.

In this chapter, we consider the weighted composition operators acting from .Np (B) to −q (B). .A

8.1

A Test Function

The following family of functions in .Np -spaces is important in applications. We usually exploit them as a test function to verify the necessity of conditions. Lemma 8.1.1 For each .w ∈ B, put  kw (z) =

.

1 − |w|2 (1 − z, w)2

 n+1 2 .

Then .kw ∈ Np (B) and, moreover, kw p ≤ 1,

.

w ∈ B.

Proof That .kw ∈ Hol(B) is trivial. Then 2 .kw p

n+1    1 − |w|2   = sup   (1 − |a (z)|2 )p dV (z) 2   (1 − z, w) a∈B B   ≤ sup

a∈B B

1 − |w|2 |1 − z, w|2

n+1 dV (z) = 1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_8

123

8 Weighted Composition Operators on .B

124

 n+1 1−|w|2 The last equality follows from the fact that . |1−z,w| is the real Jacobian determinant 2 of .w (z).  

8.2

Boundedness

First, we study the boundedness of the weighted composition operator .Wu,ϕ . As the first observation, the norm topology in both spaces .Np (B) and .A−q (B) is stronger than the compact-open topology, and hence, it is stronger than the pointwise convergence topology. Hence, if the weighted composition operator .Wu,ϕ maps .Np (B) into .A−q (B), an application of the closed graph theorem shows that .Wu,ϕ is automatically bounded from −q (B). .Np (B) into .A Theorem 8.2.1 Let .ϕ : B → B and .u : B → C be holomorphic mappings, and let .p, q > 0. Then the weighted composition operator .Wu,ϕ : Np (B) → A−q (B) is bounded if and only if

.

sup |u(z)| z∈B

(1 − |z|2 )q (1 − |ϕ(z)|2 )

< ∞.

n+1 2

(8.1)

Proof Necessity Suppose .Wu,ϕ is a bounded operator acting from .Np (B) into .A−q (B). Then there exists a positive constant M such that .

  Wu,ϕ (f ) ≤ M f p , q

(f ∈ Np ).

Fix .z0 ∈ B, and consider the test function .kw0 , where .w0 = ϕ(z0 ). By Lemma 8.1.1, we have .

    Wu,ϕ (kw ) ≤ M kw  ≤ M. 0 q 0 p

Moreover,      n+1  2 2     1 − |w0 |  (1 − |z|2 )q . Wu,ϕ (kw0 ) = sup u(z)   q 2 (1 − ϕ(z), w ) 0 z∈B   

1 − |w0 |2 ≥ |u(z0 )| |1 − ϕ(z0 ), w0 |2 = |u(z0 )|

(1 − |z0 |2 )q (1 − |ϕ(z0 )|2 )

n+1 2

.

 n+1 2

(1 − |z0 |2 )q

8.3 Compactness: Easy Reformulations

125

Thus, for every .z0 ∈ B, (1 − |z0 |2 )q

|u(z0 )|

.

(1 − |ϕ(z0 )|2 )

n+1 2

≤ M,

from which (8.1) follows. Sufficiency Suppose (8.1) holds. By Theorem 7.5.1, for some positive constant .c1 , we have .

sup |f (ϕ(z))|(1 − |ϕ(z)|2 )

n+1 2

≤ |f | n+1 ≤ c1 f p , 2

z∈B

(f ∈ Np (B)).

Then, for all .f ∈ Np (B), we have .

  Wu,ϕ (f ) = sup |u(z) · f (ϕ(z))| (1 − |z|2 )q q z∈B

≤ sup |u(z)| z∈B

≤ sup |u(z)| z∈B

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

· sup |f (ϕ(z))|(1 − |ϕ(z)|2 ) z∈B

· c1 f p = c2 f p ,

which shows that .Wu,ϕ is bounded from .Np (B) into .A−q (B).

8.3

n+1 2

 

Compactness: Easy Reformulations

To study the compactness of .Wu,ϕ , we start with the following useful test for compactness of weighted composition operators. Lemma 8.3.1 Let .ϕ : B → B and .u : B → C be holomorphic mappings, and let .p, q > 0. Then the weighted composition operator .Wu,ϕ : Np (B) → A−q (B) is compact if and only if |Wu,ϕ (fm )|q → 0,

.

(m → ∞),

for each bounded sequence .(fm ) in .Np (B) such that .(fm ) uniformly converges to 0 on compact subsets of .B. Proof Necessity Suppose that .Wu,ϕ : Np (B) → A−q (B) is compact. Take any bounded sequence .(fm ) in .Np (B) such that .(fm ) uniformly converges to 0 on compact subsets of .B. We need to prove that .|Wu,ϕ (fm )|q → 0 as .m → ∞. Assume that this is not true. Then

8 Weighted Composition Operators on .B

126

there exist an .ε0 > 0 and a subsequence .(fmk ) of .(fm ), which is a fortiori bounded in Np (B), such that

.

|Wu,ϕ (fmk )|q ≥ ε0 ,

(k ≥ 1).

.

(8.2)

On the one hand, since .Wu,ϕ is compact, .{Wu,ϕ (fmk )} has a subsequence, denoted by {Wu,ϕ (gj )} that converges to some .g ∈ A−q (B). Since the norm topology of .A−q (B) is stronger than the compact-open topology, which in turn is stronger than the pointwise topology, this implies that .{Wu,ϕ (gj )} converges pointwise to g in .B. On the other hand, .fmk → 0 uniformly on every compact subset of .B, in particular, on the singleton .{ϕ(z)}, for each fixed .z ∈ B. This means that .{Wu,ϕ (gj ) = u · gj ◦ ϕ} converges pointwise to 0 in .B. Thus, .g(z) = 0, .z ∈ B, and this contradicts (8.2). Sufficiency We show that for any sequence .(gm ) from the unit ball in .Np (B), the sequence .{Wu,ϕ (gm )} contains a Cauchy subsequence with respect to the norm .| · |q . First, we prove that .(gm ) is a normal sequence in .Hol(B). Indeed, by Theorem 7.5.1, .

|gm | n+1 ≤

.

2

2p+n 2p+n gm p ≤ p/2 , p/2 3 3

(m ∈ N).

From this, it follows that for an arbitrary closed ball .Bδ = {z ∈ B : |z| ≤ δ}, δ ∈ (0, 1), and any .m ∈ N, we have .

sup |gm (z)| = sup |gm (z)|(1 − |z|2 ) z∈Bδ

n+1 2

z∈Bδ

≤ sup |gm (z)|(1 − |z|2 )

n+1 2

z∈B

1 (1 − |z|2 )

· sup z∈Bδ

2

z∈Bδ

(1 − |z|2 )

2p+n 3p/2 (1 − δ 2 )

n+1 2

1 (1 − |z|2 )

n+1 2

1

= |gm | n+1 · sup ≤

·

n+1 2

n+1 2

,

which means that .(gm ) is a locally bounded sequence in .Hol(B). By Montel’s theorem, .(gm ) is a normal sequence. Next, we show that .g ∈ Np (B). From the normal sequence .(gm ), we can extract a subsequence .(gmk ) that converges uniformly on compact subsets of .B to some function g. That is, .(gmk ) converges to some g in the space .Hol(B). By Fatou’s lemma, for each .a ∈ B, we have 

 |g(z)| (1 − |a (z)| ) dV (z) = 2

.

B

2 p

lim inf |gmk (z)|2 (1 − |a (z)|2 )p dV (z)

B k→∞

8.4 Compactness: Characterization

127

 ≤ lim inf k→∞

B



≤ sup

k∈N B

|gmk (z)|2 (1 − |a (z)|2 )p dV (z)

|gmk (z)|2 (1 − |a (z)|2 )p dV (z)

= sup gmk p ≤ 1. k∈N

Finally, since .g ∈ Np (B), .(gmk − g) is a bounded sequence in .Np (B) that converges uniformly to zero on every compact subset of .B. By the assumption, .|Wu,ϕ (gmk − g)|q → 0, as .k → ∞. From this, it follows that .{Wu,ϕ (gmk )}k∈N is a Cauchy subsequence of .{Wu,ϕ (gm )}m∈N , and hence .Wu,ϕ is compact.   We also need the following sufficient condition for compactness. Lemma 8.3.2 Suppose .ϕ is a self-map of .B such that .ϕ∞ < 1, and let .u ∈ Np . Then the weighted composition operator .Wu,ϕ : Np −→ Np is compact. Proof Let .r = ϕ∞ . Take any .f ∈ Np . We then have Wu,ϕ f p = u · (f ◦ ϕ)p ≤ f ◦ ϕ∞ up ≤

.



sup |f (z)| up < ∞.

{z:|z|≤r}

This shows that .Wu,ϕ maps .Np into itself. Now suppose that .(fm ) is a bounded sequence in .Np that converges to zero uniformly on every compact subset of .B. Applying the above estimate to .f = fm , we obtain Wu,ϕ fm p ≤

.



sup |fm (z)| up .

{z:|z|≤r}

Since the set .{z : |z| ≤ r} is compact, the right-hand side of the last quantity converges to 0 as .m → ∞. Hence, so does the sequence .{Wu,ϕ fm p }. This means that .Wu,ϕ is compact.  

8.4

Compactness: Characterization

Using the tools provided in Sect. 8.3, we now formulate the following criteria for compactness of weighted composition operators. Theorem 8.4.1 Let .ϕ : B → B and .u : B → C be holomorphic mappings, and let .p, q > 0. Then the weighted composition operator .Wu,ϕ : Np (B) → A−q (B) is compact if and only if

8 Weighted Composition Operators on .B

128

.

lim

sup |u(z)|

r→1− |ϕ(z)|>r

(1 − |z|2 )q (1 − |ϕ(z)|2 )

= 0.

n+1 2

(8.3)

Proof Necessity Let .Wu,ϕ : Np (B) → A−q (B) be compact. By Theorem 8.2.1, M := sup |u(z)|

.

z∈B

(1 − |z|2 )q (1 − |ϕ(z)|2 )

< ∞.

n+1 2

Put F (r) := sup |u(z)|

.

|ϕ(z)|>r

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

,

which is clearly bounded and decreasing on .(0, 1). Hence, . lim F (r) exists. We show r→1−

that this limit is necessarily zero. Assume that . lim F (r) = L > 0. Then there exists a r→1−

r0 ∈ (0, 1) such that for all .r ∈ (r0 , 1), we have .F (r) > L/2. We are going to construct a sequence .(zm ) ⊂ B, by the standard diagonal process, to get a contradiction. Take .r1 ∈ (r0 , 1). Since

.

F (r1 ) =

.

sup |u(z)|

|ϕ(z)|>r1

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

>

L , 2

there exists a point .z1 ∈ {z : |ϕ(z)| > r1 } such that |u(z1 )|

.

(1 − |z1 |2 )q (1 − |ϕ(z1 )|2 )

n+1 2

> L/4.

 Next we take .r2 > max |ϕ(z1 )|, 34 . Since

.

 3 max |ϕ(z1 )|, ≥ |ϕ(z1 )| > r1 > r0 , 4

we have F (r2 ) =

.

sup |u(z)|

|ϕ(z)|>r2

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

>

L , 2

which implies the existence of another point .z2 ∈ {z : |ϕ(z)| > r2 } such that |u(z2 )|

.

(1 − |z2 |2 )q (1 − |ϕ(z2 )|2 )

n+1 2

> L/4.

8.4 Compactness: Characterization

129

Note that since .|ϕ(z2 )| > r2 > |ϕ(z1 )|, we have .z2 = z1 . Continuing this process, for any 1 .m ≥ 2, we take .rm > max |ϕ(zm−1 )|, 1 − m . Since 2

.



1 max |ϕ(zm−1 )|, 1 − m ≥ |ϕ(zm−1 )| > rm−1 > · · · > r1 > r0 , 2

we have F (rm ) =

.

sup

|ϕ(z)|>rm

|u(z)|

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

>

L , 2

and so there exists .zm ∈ {z : |ϕ(z)| > rm } such that |u(zm )|

.

(1 − |zm |2 )q (1 − |ϕ(zm )|2 )

n+1 2

> L/4

and .zm = z1 , . . . , zm−1 . Note that for each .m ≥ 2, we have .|ϕ(zm )| > 1 − 21m , which shows that .|ϕ(zm )| → 1 as .m → ∞. Consider the test functions .kwm , where .wm = ϕ(zm ), defined in Lemma 8.1.1. It is easy to see that .kwm → 0 uniformly on every compact subset of .B. Moreover, for each .m ≥ 1, .kwm p ≤ 1. Since .Wu,ϕ is compact, by Lemma 8.3.1, .|Wu,ϕ (kwm )|q → 0 as .m → ∞. However, for each .m ≥ 1, |Wu,ϕ (kwm )|q = sup |u(z)| · |kwm (ϕ(z))|(1 − |z|2 )q

.

z∈B

≥ |u(zm )| · |kwm (ϕ(zm ))|(1 − |zm |2 )q = |u(zm )| ·

(1 − |zm |2 )q (1 − |ϕ(zm

)|2 )

n+1 2

≥

L , 4

which gives a contradiction. Sufficiency Suppose that (8.3) holds. Let .(fm ) be a bounded sequence in .Np (B) which converges to zero uniformly on every compact subset of .B. We have .fm p ≤ M, .m ≥ 1, for some .M > 0. Since . lim F (r) = 0, for each .ε > 0, there exists .r0 ∈ (0, 1) such that r→1−

for all .r ∈ (r0 , 1), .

sup |u(z)|

|ϕ(z)|>r

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2


r

and C2,m = sup |u(z)||fm (ϕ(z))|(1 − |z|2 )q .

.

|ϕ(z)|≤r

Note that, for any .m ≥ 1, we have C1,m = sup |u(z)|

.

|ϕ(z)|>r

≤ ≤

sup |u(z)|

|ϕ(z)|>r

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

· |fm (ϕ(z))|(1 − |ϕ(z)|2 )

n+1 2

· sup |fm (ϕ(z))|(1 − |ϕ(z)|2 )

n+1 2

z∈B

n+1 ε sup |fm (ϕ(z))|(1 − |ϕ(z)|2 ) 2 2c1 M z∈B

ε ε |fm | n+1 ≤ · c1 fm p 2 2c1 M 2c1 M ε ε ≤ · c1 M = . 2c1 M 2 ≤

On the other hand, according to the condition, it is easy to verify that

.

sup |u(z)| · z∈B

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

< M1 .

Then C2,m = sup |u(z)|

.

|ϕ(z)|≤r

≤ sup |u(z)| |ϕ(z)|≤r

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

· |fm (ϕ(z))|(1 − |ϕ(z)|2 )

· sup |fm (ϕ(z))|(1 − |ϕ(z)|2 ) |ϕ(z)|≤r

≤ M1 sup |fm (ϕ(z))|(1 − |ϕ(z)|2 ) |ϕ(z)|≤r

n+1 2

n+1 2

.

n+1 2

8.5 Estimation of |f (z) − f (w)|

131

Since .(fm ) converges to zero on every compact subset of .B, it implies that there exists mε such that for .m > mε ,

.

.

sup |fm (ϕ(z))|(1 − |ϕ(z)|2 )

n+1 2

|ϕ(z)|≤δ

ε . 2M1


mε . Combining these two parts, we conclude that for any .ε > 0, there exists an .mε , such that for all .m > mε , .|Wu,ϕ (fm )|q ≤ ε, which means .|Wu,ϕ (fm )|q → 0 as .m → ∞. By   Lemma 8.3.1, we conclude that .Wu,ϕ : Np (B) → A−q (B) is compact. As a special case, we immediately obtain the following useful characterization for the composition operator. Corollary 8.4.2 Let .ϕ : B → B be a holomorphic mapping, and let .p, q > 0. Then the composition operator .Cϕ acting from .Np (B) → A−q (B) is: (i) Bounded if and only if

.

sup z∈B

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

r

(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

=0

Estimation of |f (z) − f (w)|

Recall that the pseudo-hyperbolic metric in the ball was defined by ρ(z, w) = |w (z)|,

.

(z, w ∈ B).

See (7.4). The following technical lemmas play an important role in the proof of our main result. Lemma 8.5.1 For .z, w ∈ B, if .ρ(z, w) ≤ 12 , then .

1 1 − |z|2 ≤ 6. ≤ 6 1 − |w|2

8 Weighted Composition Operators on .B

132

Proof Let .z, w ∈ B. For simplicity, denote .r = ρ(z, w). We have .

|ρ(w (z), 0) − ρ(0, w)| ρ(w (z), 0) + ρ(0, w) ≤ ρ(w (z), w) ≤ 1 − ρ(w (z), 0)ρ(0, w) 1 + ρ(w (z), 0)ρ(0, w)

(8.4)

or, equivalently, .

|r − |w|| r + |w| ≤ |z| ≤ . 1 − r|w| 1 + r|w|

As a matter of fact, for the left inequality, we need only a weaker version. Namely, .

|w| − r |w| + r ≤ |z| ≤ . 1 − r|w| 1 + r|w|

(8.5)

Furthermore, since .r ∈ (0, 12 ], from the left inequality of (8.5), it follows that

.

|w|−r 1 − 1−r|w| 1 − |z| 1+r ≤ = ≤ 3, 1 − |w| 1 − |w| 1 − r|w|

while the right inequality of (8.5) gives

.

|w|+r 1 − 1+r|w| 1 − |z| 1 − |r| 1 ≥ = ≥ . 1 − |w| 1 − |w| 1 + r|w| 3

We also have .

1 1 1 + |z| 2 ≤ ≤ ≤ ≤ 2. 2 1 + |w| 1 + |w| 1 + |w|

.

1 1 − |z| 1 + |z| 1 − |z|2 = ≤ · ≤ 6. 2 6 1 − |w| 1 + |w| 1 − |w|

Therefore, we get

  Lemma 8.5.2 For .f ∈ Np (B) and .z, w ∈ B, we have  |f (z) − f (w)| ≤ c f p max

.

where .A =

6

√ √ n+1 2 ·2p+n+1 (3+2 3) n p/2 3

.

1 (1 − |z|2 )

n+1 2

,



1 (1 − |w|2 )

n+1 2

ρ(z, w),

8.5 Estimation of |f (z) − f (w)|

133

Proof We consider two cases. Cases 1 .ρ(z, w) ≥ 14 . Since .|f (z) − f (w)| ≤ |f (z)| + |f (w)|, by Theorem 7.5.1, we have .

 n+1 n+1 min (1 − |z|2 ) 2 , (1 − |w|2 ) 2 |f (z) − f (w)| ≤ (1 − |z|2 )

n+1 2

≤ 2|f | n+1 ≤

|f (z)| + (1 − |w|2 )

2p+n+1 3p/2

2

f p ≤

n+1 2

2p+n+3 3p/2

|f (w)| f p ρ(z, w),

which implies   1 2p+n+3 1 f p max .|f (z) − f (w)| ≤ , ρ(z, w). n+1 n+1 3p/2 (1 − |z|2 ) 2 (1 − |w|2 ) 2 Cases 2 .ρ(z, w) < 14 . Fix .w ∈ B. From .ρ(w (z), w) = |z|, it follows that if .z ∈ B1/2 , then .ρ(w (z), w) ≤ 12 . In this case, by Theorem 7.5.1 and Lemma 8.5.1, we have |f (w (z))| ≤

.

≤

= ≤

|f | n+1 2

(1 − |w (z)|2 )

n+1 2

2p+n f p 3p/2 (1 − |w (z)|2 )



2p+n f p 3p/2 (1 − |w|2 ) 6

n+1 2

n+1 2

· 2p+n f p

3p/2 (1 − |w|2 )

n+1 2

n+1 2

1 − |w|2 · 1 − |w (z)|2

 n+1 2

.

Now, we follow the standard scheme to estimate a quantity .|f (z) − f (w)|. Set .gw = f ◦ w ; then |f (z) − f (w)| = |f (w (w (z)) − f (w (0))| = |gw (w (z)) − gw (0)|.

.

For each .z ∈ B with .ρ(z, w) = |w (z)| < 14 , we have |f (z) − f (w)| = |gw (w (z)) − gw (0)| ≤ |∇gw (t)| · |w (z)| = |∇gw (t)|ρ(z, w),

.

where .t = (t1 , t2 , . . . , tn ) is some point in .B with .|t| ≤ |w (z)| ≤ 14 . Furthermore,

8 Weighted Composition Operators on .B

134

   ∂gw   .|∇gw (t)|ρ(z, w) ≤ nρ(z, w) max  (t) 1≤k≤n ∂zk    1   √ gw (t1 , t2 , . . . , ξk , . . . , tn )   dξ ≤ nρ(z, w) max  √ k 2  1≤k≤n  2π i |ξk |= 3 (ξ − t ) k k 4   √   gw (t1 , t2 , . . . , ξk , . . . , tn )  nρ(z, w)   |dξk |. max ≤ √   2 3 1≤k≤n |ξk |= 2π (ξ − t ) k k 4 √

Note that for .(t1 , t2 , . . . , ξk , . . . , tn ) with .|t| ≤ .

1 4

and .|ξk | =

√

3 4 ,

we have

ρ (w (t1 , t2 , . . . , ξk , . . . , tn ), w) = ρ((t1 , t2 , . . . , ξk , . . . , tn ), 0) = |(t1 , t2 , . . . , ξk , . . . , tn )|   √ 2     1 2 3 1  2 2 ≤ |t| + |ξk | ≤ + = , 4 4 2

and thus n+1



6 2 · 2p+n f p .|gw (t1 , t2 , . . . , ξj , . . . , tn )| = |f w (t1 , t2 , . . . , ξk , . . . , tn ) | ≤ . n+1 3p/2 (1 − |w|2 ) 2 Also,  .

max

1≤k≤n

√ |ξk |= 43

|dξk | ≤ max |ξk − tk |2 1≤k≤n

 ≤ max

1≤k≤n |ξk |=

√ 3 4

 √ |ξk |= 43

|dξk |

√ ( 43

− |tk |)2 √  2 √ |dξk | 3 4 √ ≤ 2π = 4π(3 + 2 3). · √ 4 3−1 ( 43 − 14 )2

Consequently, √ n+1 √ nρ(z, w) 6 2 · 2p+n f p · 4π(3 + 2 3) · n+1 2π 3p/2 (1 − |w|2 ) 2 √ √ n+1 1 6 2 · 2p+n+1 (3 + 2 3) n · · f p · ρ(z, w). = n+1 3p/2 (1 − |w|2 ) 2

|f (z) − f (w)| ≤

.

Combining the results of the two cases yields

8.6 Compactness of Difference

135

 |f (z) − f (w)| ≤ Af p max

.

where .A =

6

√ √ n+1 2 ·2p+n+1 (3+2 3) n p/2 3

1 (1 − |z|2 )

n+1 2

,



1 (1 − |w|2 )

n+1 2

ρ(z, w),

 

.

Lemma 8.5.2 gives the estimate for the distance between .f (z) and .f (w) for .f ∈ Np and .z, w ∈ B, in which the constant A is involved. In the special case where z and w are multiple of each other, the above estimate can be simplified. This is explained below. Lemma 8.5.3 Let f be in .Np . For any .a ∈ B and any .κ ∈ (0, 1), we have |f (a) − f (κa)| ≤

.

f p Af p A(1 − κ)|a| · ≤ . n+1 n+1 2 1 − κ|a| (1 − |a|2 ) 2 (1 − |a|2 ) 2

(8.6)

Consequently, for any .0 < r < 1, we have .

sup |f (a) − f (κa)| ≤

Ar(1 − κ)f p (1 − r 2 )

|a|≤r

n+3 2

(8.7)

.

Here, A is the constant from Lemma 8.5.2. Proof Lemma 8.5.2 shows that  |f (a) − f (κa)| ≤ Af p max

.

1 (1 − |a|2 )

n+1 2

,



1 (1 − |κa|2 )

n+1 2

ρ(a, κa).

By the definition of .ρ, |ρ(a, κa)| = |a (κa)| =

.

(1 − κ)|a| ≤ 1. 1 − κ|a|2

On the other hand, .(1 − |κa|2 )−(n+1)/2 ≤ (1 − |a|2 )−(n+1)/2 . The inequalities in (8.6) now follow. If .|a| ≤ r, then .1 − κ|a|2 ≥ 1 − r 2 . Taking supremum of (8.6) in a yields (8.7).  

8.6

Compactness of Difference

In this section, we study the compactness of the difference of two bounded weighted composition operators acting from .Np (B) into .A−q (B). Let .ϕ1 , ϕ2 be two holomorphic self-mappings on .B, let .u1 , u2 : B → C be two holomorphic mappings, and let .p, q > 0.

8 Weighted Composition Operators on .B

136

Consider .Wu1 ,ϕ1 and .Wu2 ,ϕ2 the two corresponding weighted composition operators acting from .Np (B) into .A−q (B). We seek conditions under which the difference .Wu1 ,ϕ1 − Wu2 ,ϕ2 is compact. Inspiring by the pseudo-hyperbolic metric in the unit ball, for two holomorphic mappings .ϕ, ψ : B → B, we define   ρϕ,ψ (z) = ϕ(z) (ψ(z)) ,

(z ∈ B).

.

Evidently, .ρϕ,ψ = ρψ,ϕ . Theorem 8.6.1 Let .ϕ1 , ϕ2 be two self-mapping of .B, let .u1 , u2 : B → C be two holomorphic mappings, and let .p, q > 0. Let, further, .Wu1 ,ϕ1 and .Wu2 ,ϕ2 be two weighted composition operators acting from .Np (B) into .A−q (B). Then .Wu1 ,ϕ1 − Wu2 ,ϕ2 is compact if and only if the following two conditions are satisfied: (i)  .

lim

sup

r→1− |ϕk (z)|>r

|uk (z)|(1 − |z|2 )q (1 − |ϕk (z)|2 )

n+1 2

 ρϕ1 ,ϕ2 (z) = 0,

(k = 1, 2).

(ii) As .r → 1− ,  .



sup |u1 (z) − u2 (z)| min

(1 − |z|2 )q (1 − |ϕ1 (z)|2 )

n+1 2

,



(1 − |z|2 )q (1 − |ϕ2 (z)|2 )

n+1 2

→ 0.

The supremum is taken over the set .{min{|ϕ1 (z)|, |ϕ2 (z)|} > r}. Proof It is well-known that the operator .Wu1 ,ϕ1 − Wu2 ,ϕ2 : Np (B) → A−q (B) is compact if and only if |(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(fm )|q → 0,

.

(m → ∞),

(8.8)

for any bounded sequence .(fm ) in .Np (B) such that .(fm ) converges to zero uniformly on every compact subset of .B. This is used twice below. Necessity Suppose .Wu1 ,ϕ1 − Wu2 ,ϕ2 is a compact operator. Proof of (i). It suffices to prove for .k = 1. Since .Wu1 ,ϕ1 is bounded, by Theorem 8.2.1 and the fact that .ρϕ1 ,ϕ2 (z) ≤ 1, for all .z ∈ B , we have  .

sup z∈B

Set

|u1 (z)|(1 − |z|2 )q (1 − |ϕ1 (z)|2 )

n+1 2

 ρϕ1 ,ϕ2 (z) ≤ sup z∈B

|u1 (z)|(1 − |z|2 )q (1 − |ϕ1 (z)|2 )

n+1 2

< ∞.

8.6 Compactness of Difference

 G(r) =

sup

.

137

|u1 (z)|(1 − |z|2 )q (1 − |ϕ1 (z)|2 )

|ϕ1 (z))|>r

n+1 2

 ρϕ1 ,ϕ2 (z) ,

(0 < r < 1).

It is clear that G is bounded and decreasing on .(0, 1), and hence . lim G(r) exists. r→1−

.

To get a contradiction, assume that .(i) is not true. Then there exists an .L > 0, such that lim G(r) > L. By the standard diagonal process, we can choose a sequence .(zm ) ⊂ B,

r→1−

such that .|ϕ1 (zm )| → 1, as .m → ∞, and .

|u1 (zm )|(1 − |zm |2 )q (1 − |ϕ1 (zm )|2 )

n+1 2

ρϕ1 ,ϕ2 (zm ) >

L , 4

(m ≥ 1).

(8.9)

Consider the functions gm (z) = ϕ2 (zm ) (z) · kwm (z),

.

(z ∈ B),

where .wm = ϕ1 (zm ), .m ≥ 1. Obviously, .gm ∈ Hol(B). Moreover, since .|ϕ2 (zm ) (z)| ≤ 1, .z ∈ B, we have gm (z)p ≤ kwm (z)p ≤ 1,

.

which shows that .gm (z) ∈ Np (B) for all .m ≥ 1 and that the sequence .{gm } is bounded in Np (B). Furthermore, by the fact that .kwm converges to zero uniformly on every compact subset of .B, and .|gm (z)| ≤ |kwm (z)|, .z ∈ B, we see that .gm also converges to zero uniformly on every compact subset of .B. By (8.8),

.

|(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(gm )|q → 0,

.

(m → ∞).

(8.10)

Note that for each .m ≥ 1, .ϕ2 (zm ) (ϕ2 (zm )) = 0, which implies .gm (ϕ2 (zm )) = 0. Then, .

|(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(gm )|q  = sup |u1 (z)gm (ϕ1 (z)) − u2 (z)gm (ϕ2 (z))| · (1 − |z|2 )q z∈B

≥ |u1 (zm )gm (ϕ1 (zm )) − u2 (zm )gm (ϕ2 (zm ))| · (1 − |zm |2 )q = |u1 (zm )gm (ϕ1 (zm ))| · (1 − |zm |2 )q =

|u1 (zm )|(1 − |zm |2 )q (1 − |ϕ1 (zm )|2 )

n+1 2

· |ϕ2 (zm ) (ϕ1 (zm ))|

8 Weighted Composition Operators on .B

138

=

|u1 (zm )|(1 − |zm |2 )q (1 − |ϕ1 (zm )|2 )

n+1 2

ρϕ1 ,ϕ2 (zm ) >

L , 4

by Lemma 8.1.1, which contradicts (8.10). Thus, we must have  .

lim

sup

r→1− |ϕ1 (z)|>r

|u1 (z)|(1 − |z|2 )q (1 − |ϕ1 (z)|2 )

n+1 2

 ρϕ1 ,ϕ2 (z) = 0,

and .(i) is proved. Proof of (ii). Since both .Wu1 ,ϕ1 and .Wu2 ,ϕ2 are bounded, by Theorem 8.2.1, we have 

 .

sup |u1 (z) − u2 (z)| min z∈B





n+1



+ sup |u2 (z)| min

(1 − |ϕ1 (z)|2 ) 2  (1 − |z|2 )q

z∈B

≤ sup z∈B

(1 − |ϕ1 (z)|2 )

|u1 (z)|(1 − |z|2 )q (1 − |ϕ1 (z)|2 )

(1 − |ϕ1 (z)|2 )

(1 − |z|2 )q

≤ sup |u1 (z)| min z∈B

(1 − |z|2 )q

n+1 2

+ sup z∈B

,

n+1 2

,

(1 − |ϕ2 (z)|2 )  (1 − |z|2 )q

(1 − |ϕ2 (z)|2 )

n+1 2

(1 − |ϕ2 (z)|2 )

|u2 (z)|(1 − |z|2 )q (1 − |ϕ2 (z)|2 )

n+1 2

n+1 2

n+1 2



(1 − |z|2 )q

,



(1 − |z|2 )q

n+1 2

< ∞.

For each .r ∈ (0, 1), set 



H (r) = sup |u1 (z) − u2 (z)| min

.

(1 − |z|2 )q (1 − |ϕ1 (z)|2 )

n+1 2

,



(1 − |z|2 )q (1 − |ϕ2 (z)|2 )

,

n+1 2

where the supremum is taken over the set .{min{|ϕ1 (z)|, |ϕ2 (z)|} > r}. The function .H (r) is bounded and decreasing on .(0, 1), and hence, . lim H (r) exists. We prove that this limit r→1−

is zero. We follow the same scheme of proving .(i), but it requires more delicate arguments. Assume that . lim H (r) = L > 0. Again by the standard diagonal process, we can choose r→1−

a sequence .(zm ) ⊂ B, such that .min{|ϕ1 (zm )|, |ϕ2 (zm )|} → 1 as .m → ∞ and that, for each .m ≥ 1,  .

|u1 (zm ) − u2 (zm )| min

(1 − |zm |2 )q (1 − |ϕ1 (zm )|2 )

n+1 2

,



(1 − |zm |2 )q (1 − |ϕ2 (zm )|2 )

n+1 2

>

L . 4

For each .m ≥ 1, we have either .|ϕ1 (zm )| ≥ |ϕ2 (zm )| or .|ϕ1 (zm )| ≤ |ϕ2 (zm )|. Choose a subsequence .(zmk ) of .(zm ), such that for each .k ≥ 1,

8.6 Compactness of Difference

139

|ϕ1 (zmk )| ≥ |ϕ2 (zmk )|.

.

Otherwise, there are only finitely many indexes m, such that .|ϕ1 (zm )| ≥ |ϕ2 (zm )|, and in this case, we choose a subsequence .(zmk ) of .(zm ), such that for each .k ≥ 1, .|ϕ1 (zmk )| ≤ |ϕ2 (zmk )|. We only consider the first case, since the second case can be proved by interchanging the role of .ϕ1 and .ϕ2 , and without loss of generality, we write .(zm ) for .(zmk ). Since .|ϕ1 (zm )| ≥ |ϕ2 (zm )| for each .m ≥ 1, we have 

(1 − |zm |2 )q

|u1 (zm ) − u2 (zm )| min

.

(1 − |ϕ1 (zm )|2 )

(1 − |zm |2 )q

= |u1 (zm ) − u2 (zm )|

(1 − |ϕ2 (zm )|2 )

n+1 2

,

n+1 2

>



(1 − |zm |2 )q (1 − |ϕ2 (zm )|2 )

n+1 2

L . 4

Since the sequence .{ρϕ1 ,ϕ2 (zm )} is bounded, it contains a convergent subsequence. Without loss of generality, we can assume that .

lim ρϕ1 ,ϕ2 (zm ) = ≥ 0.

m→∞

There are two cases for . to consider. Case 1: . > 0. In this case, there exists an .m0 ≥ 1 such that .ρϕ1 ,ϕ2 (zm ) > , .m > m0 . In 2 this case, we have (1 − |zm |2 )q |u1 (zm ) − u2 (zm )|

.

(1 − |ϕ2 (zm )|2 ) ≤

(1 − |zm |2 )q |u1 (zm )| n+1 2

n+1 2

+

(1 − |zm |2 )q |u2 (zm )| n+1

(1 − |ϕ2 (zm )|2 ) (1 − |ϕ2 (zm )|2 ) 2   (1 − |zm |2 )q |u1 (zm )| (1 − |zm |2 )q |u2 (zm )| 2 + ≤ ρϕ1 ,ϕ2 (zm ) n+1 n+1 (1 − |ϕ2 (zm )|2 ) 2 (1 − |ϕ2 (zm )|2 ) 2   (1 − |zm |2 )q |u1 (zm )| (1 − |zm |2 )q |u2 (zm )| 2 ≤ ρϕ1 ,ϕ2 (zm ) + , n+1 n+1 (1 − |ϕ1 (zm )|2 ) 2 (1 − |ϕ2 (zm )|2 ) 2

which gives  ρϕ1 ,ϕ2 (zm )

.

(1 − |zm |2 )q |u1 (zm )| (1 − |ϕ1 (zm

)|2 )

n+1 2

+

(1 − |zm |2 )q |u2 (zm )| (1 − |ϕ2 (zm

)|2 )

However, since .|ϕ2 (zm )| ≤ |ϕ1 (zm )| ≤ 1, .m ∈ N, from

n+1 2

 ≥

L , 8

(m > m0 ).

8 Weighted Composition Operators on .B

140

.

lim min{|ϕ1 (zm )|, |ϕ2 (zm )|} = 1

m→∞

it follows that .

lim |ϕ1 (zm )| = lim |ϕ2 (zm )| = 1.

m→∞

m→∞

Hence, by .(i),

.

lim ρϕ1 ,ϕ2 (zm )

m→∞

(1 − |zm |2 )q |uk (zm )| (1 − |ϕk (zm )|2 )

n+1 2

= 0,

(k = 1, 2),

and thus, as .m → ∞,  ρϕ1 ,ϕ2 (zm )

.

(1 − |zm |2 )q |u1 (zm )| (1 − |ϕ1 (zm )|2 )

n+1 2

+

(1 − |zm |2 )q |u2 (zm )| (1 − |ϕ2 (zm )|2 )



n+1 2

→ 0,

which is impossible. Cases 2: . = 0. We claim, for the test functions .kwm , where .wm = ϕ1 (zm ), that .

  n+1   1 − (1 − |ϕ1 (zm )|2 ) 2 kwm (ϕ2 (zm )) → 0,

(as m → ∞).

(8.11)

Indeed, by Lemma 8.5.2, we have   n+1   1 − (1 − |ϕ1 (zm )|2 ) 2 kwm (ϕ2 (zm ))

.

= (1 − |ϕ1 (zm )|2 )

n+1 2

  · kwm (ϕ1 (zm )) − kwm (ϕ2 (zm ))

n+1

≤ (1 − |ϕ1 (zm )|2 ) 2 · ckwm p · ρϕ1 ,ϕ2 (zm ) ·   1 1 , max n+1 n+1 (1 − |ϕ1 (zm )|2 ) 2 (1 − |ϕ2 (zm )|2 ) 2 ≤

(1 − |ϕ1 (zm )|2 )

n+1 2

cρϕ1 ,ϕ2 (zm )

(1 − |ϕ1 (zm )|2 )

n+1 2

= cρϕ1 ,ϕ2 (zm ).

The last expression converges to . = 0 as .m → ∞, and (8.11) follows. Here, c is the constant defined in Lemma 8.5.2. Furthermore, since . lim ρϕ1 ,ϕ2 (zm ) = 0, there exists m→∞

m1 ≥ 1, such that .ρϕ1 ,ϕ2 (zm ) < 12 , .m ≥ m1 . Then by Lemma 8.5.1,

.

.

1 − |ϕ2 (zm )|2 ≤ 6. 1 − |ϕ1 (zm )|2

8.6 Compactness of Difference

141

Also, since .Wu2 ,ϕ2 is bounded from .Np (B) into .A−q (B), by Theorem 8.2.1, there exists some positive number .K > 0, such that

.

sup z∈B

(1 − |z|2 )q |u2 (z)| (1 − |ϕ2 (z)|2 )

n+1 2

< K.

Again by . lim ρϕ1 ,ϕ2 (zm ) = 0, there exists .m2 ≥ 1, such that .ρϕ1 ,ϕ2 (zm ) < m→∞

m > m2 . Consequently, for .m > max{m1 , m2 },

L 8·6

n+1 2 cK

.

.

|(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(kwm )|q   = sup(1 − |z|2 )q u1 (z)kwm (ϕ1 (z)) − u2 (z)kwm (ϕ2 (z)) z∈B

  ≥ (1 − |zm |2 )q u1 (zm )kwm (ϕ1 (zm )) − u2 (zm )kwm (ϕ2 (zm ))     u1 (zm )  2 q = (1 − |zm | )  − u (z )k (ϕ (z )) 2 m wm 2 m   (1 − |ϕ (z )|2 ) n+1  2 1 m     u1 (zm ) u2 (zm )   ≥ (1 − |zm |2 )q  − n+1 n+1   (1 − |ϕ (z )|2 ) 2 2 (1 − |ϕ1 (zm )| ) 2  1 m     u (z ) 2 m   −(1 − |zm |2 )q  − u2 (zm )kwm (ϕ2 (zm )) n+1  (1 − |ϕ (z )|2 ) 2  1

= (1 − |zm |2 )q

m

|u1 (zm ) − u2 (zm )| (1 − |ϕ1 (zm )|2 )

−(1 − |zm |2 )q

n+1 2

|u2 (zm )| (1 − |ϕ1 (zm )|2 )

n+1 2

  n+1   1 − (1 − |ϕ1 (zm )|2 ) 2 kwm (ϕ2 (zm )) .

Moreover, we also have (1 − |zm |2 )q

.

|u1 (zm ) − u2 (zm )| (1 − |ϕ1 (zm )|2 )

n+1 2

≥

(1 − |zm |2 )q |u1 (zm ) − u2 (zm )| (1 − |ϕ2 (zm )|2 )

n+1 2

>

and .

(1 − |zm |2 )q = (1 − |zm |2 )q

Therefore, we arrive at

|u2 (zm )| (1 − |ϕ1 (zm )|2 )

n+1 2

|u2 (zm )| (1 − |ϕ2 (zm

)|2 )

n+1 2

(1 − |ϕ2 (zm )|2 )

n+1 2

(1 − |ϕ1 (zm

n+1 2

)|2 )

≤K ·6

n+1 2

.

L 4

,

8 Weighted Composition Operators on .B

142

|(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(kwm )|q ≥

.

n+1 L L L L − 6 2 Kcρϕ1 ,ϕ2 (zm ) ≥ − = . 4 4 8 8

(8.12)

However, since .Wu1 ,ϕ1 − Wu2 ,ϕ2 is compact, and .kwm converges to zero uniformly on every compact subset of .B, we must have .

lim |(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(kwm )|q = 0,

(8.13)

m→∞

which contradicts (8.12). Sufficiency Assume that the conditions .(i) and .(ii) hold. Take an arbitrary bounded sequence .(fm ) in .Np (B) that converges to zero uniformly on compact subsets of .B. By (8.8), it is enough to show that |(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(fm )|q → 0,

(m → ∞).

.

To find a contradiction, assume that there are an .ε0 > 0 and a subsequence .(fmk ) of .(fm ) such that |(Wu1 ,ϕ1 − Wu2 ,ϕ2 )(fmk )|q ≥ ε0 ,

.

(k ∈ N).

(8.14)

Indeed, without loss of generality, by relabeling the sequence, we may assume .(fm ) itself has the above property. From this, it follows that there exists a sequence .(zm ) ⊂ B, such that Hm = (1 − |zm |2 )q |u1 (zm )fm (ϕ1 (zm )) − u2 (zm )fm (ϕ2 (zm ))| ≥

.

ε0 , 2

(8.15)

for all .m ≥ 1. Here, .zm ’s are not necessarily distinct. We may also assume that both sequences .{ϕ1 (zm )} and .{ϕ2 (zm )} converge (as otherwise, we can consider some convergent subsequences). Note that since both .Wu1 ,ϕ1 and .Wu2 ,ϕ2 are bounded, by Theorem 8.2.1, there exists .K > 0, such that sup |uk (z)|(1 − |z|2 )q ≤ sup

.

z∈B

z∈B

|uk (z)|(1 − |z|2 )q (1 − |ϕk (z)|2 )

n+1 2

≤ K, k = 1, 2.

Now we consider the sequence .max{|ϕ1 (zm )|, |ϕ2 (zm )|}. It is clear that .

lim max{|ϕ1 (zm )|, |ϕ2 (zm )|} = q ≤ 1.

m→∞

As a matter of fact, .q = 1. To verify this, assume that .q < 1. Then, by (8.15), .

  ε0 ≤ Hm ≤ (1 − |zm |2 )q |u1 (zm )fm (ϕ1 (zm ))| + |u2 (zm )fm (ϕ2 (zm )| 2

(8.16)

8.6 Compactness of Difference

143

  ≤ K |fm (ϕ1 (zm ))| + |fm (ϕ2 (zm ))| for all .m ≥ 1. Furthermore, there exists .m3 ≥ 1 such that .

max{|ϕ1 (zm )|, |ϕ2 (zm )|} ≤

1+q , 2

(m > m3 ).

  , k = 1, 2. Since . z : |z| ≤ 1+q In particular, for all .m > m3 , .ϕk (zm ) ∈ z : |z| ≤ 1+q 2 2 is a compact set of .B, the sequence .{fm (z)} converges uniformly to zero on this set, and hence, both sequences .{fm (ϕk (zm ))} , k = 1, 2, converge to zero, as .m → ∞, which shows that Hm ≤ K (|fm (ϕ1 (zm ))| + |fm (ϕ2 (zm ))|) → 0,

.

(m → ∞).

ε0 But, this contradicts the fact that .Hm ≥ , .m ≥ 1. 2 Thus, we have .

max{|ϕ1 (zm )|, |ϕ2 (zm )|} → 1,

(m → ∞).

(8.17)

Then at least one of the limits . lim |ϕk (zm )| (k = 1, 2) must be 1. So we may assume that m→∞

⎧ ⎨ lim ϕ1 (zm ) = P , with |P | = 1, .

m→∞

⎩ lim ϕ2 (zm ) = Q, with |Q| ≤ 1.

(8.18)

m→∞

Furthermore, we may also assume that there exists the limit .

lim ρϕ1 ,ϕ2 (zm ) = ≥ 0,

m→∞

since otherwise, we consider a convergent subsequence. As a matter of fact, . = 0. To verify this, assume in contrary that . > 0. Consider two cases of .|Q| ≤ 1. Case 1: .|Q| = 1. In this case, from .(i) and (8.18), it follows that .

lim

m→∞

|uk (zm )|(1 − |zm |2 )q (1 − |ϕk (zm )|2 )

n+1 2

= 0 (k = 1, 2).

Then, by Theorem 7.5.1 and (8.15), we have Hm ≤

.

(1 − |zm |2 )q |u1 (zm )| (1 − |ϕ1 (zm )|2 )

n+1 2

(1 − |ϕ1 (zm )|2 )

n+1 2

|fm (ϕ1 (zm ))|

(8.19)

8 Weighted Composition Operators on .B

144

+ 

(1 − |zm |2 )q |u2 (zm )| (1 − |ϕ2 (zm )|2 )

n+1 2

(1 − |ϕ2 (zm )|2 )

(1 − |zm |2 )q |u1 (zm )|

n+1 2

|fm (ϕ2 (zm ))|

(1 − |zm |2 )q |u2 (zm )|



+ |fm | n+1 n+1 n+1 2 (1 − |ϕ1 (zm )|2 ) 2 (1 − |ϕ2 (zm )|2 ) 2   2p+n (1 − |zm |2 )q |u1 (zm )| (1 − |zm |2 )q |u2 (zm )| ≤ p/2 + fm p . n+1 n+1 3 (1 − |ϕ1 (zm )|2 ) 2 (1 − |ϕ2 (zm )|2 ) 2 ≤

In the last inequality, letting .m → ∞, by (8.19) as well as boundedness of .(fm ) in .Np (B), we get .

lim Hm = 0,

m→∞

which is impossible, because it contradicts (8.15).  , Case 2: .|Q| < 1. In this case, the second limit in (8.18) gives .ϕ2 (zm ) ∈ z : |z| ≤ 1+|Q| 2 for all m large enough, say .m > m4 . Then by (8.16) and Theorem 7.5.1, for all .m > m4 , we have Hm ≤ (1 − |zm |2 )q |u1 (zm )||fm (ϕ1 (zm ))|

.

+(1 − |zm |2 )q |u2 (zm )||fm (ϕ2 (zm ))| =

(1 − |zm |2 )q |u1 (zm )| (1 − |ϕ1 (zm

)|2 )

n+1 2

(1 − |ϕ1 (zm )|2 )

n+1 2

|fm (ϕ1 (zm ))|

+(1 − |zm |2 )q |u2 (zm )||fm (ϕ2 (zm ))| ≤ ≤

(1 − |zm |2 )q |u1 (zm )| (1 − |ϕ1 (zm )|2 )

n+1 2

|fm | n+1 + K|fm (ϕ2 (zm ))| 2

2p+n (1 − |zm |2 )q |u1 (zm )| · fm p + K|fm (ϕ2 (zm ))|. n+1 3p/2 (1 − |ϕ1 (zm )|2 ) 2

Letting .m → ∞, from  (8.19) and the fact that .fm (ϕ2 (zm )) converges to zero uniformly 1+|Q| on the compact set . z : |z| ≤ 2 , it follows that right-hand side of the last inequality tends to 0 as .m → ∞, which again contradicts (8.15). Thus, we have .

lim ρϕ1 ,ϕ2 (zm ) = 0.

m→∞

(8.20)

We also have .|Q| = 1, that is, . lim |ϕ2 (zm )| = 1. Indeed, for each .m ≥ 1, we have m→∞

1 − ρϕ21 ,ϕ2 (zm ) = 1 − |ϕ2 (zm ) (ϕ1 (zm )|2

.

8.6 Compactness of Difference

145

= 1−

(1 − |ϕ1 (zm )|2 )(1 − |ϕ2 (zm )|2 ) . |1 − ϕ1 (zm ), ϕ2 (zm )|2

Since . lim |ϕ1 (zm )| = 1, if . lim |ϕ2 (zm )| = |Q| < 1, then we would have for all m large m→∞ m→∞ enough |1 − ϕ1 (zm ), ϕ2 (zm )| ≥ 1 − |ϕ1 (zm ), ϕ2 (zm )|

.

≥ 1 − |ϕ1 (zm )||ϕ2 (zm )| ≥ 1 − |Q| > 0. But, this implies that . lim ρϕ1 ,ϕ2 (zm ) = 1, which contradicts (8.20). m→∞ Now, by the same reasoning as in the necessity part, we may assume that |ϕ1 (zm )| ≥ |ϕ2 (zm )|,

(m ≥ 1).

.

(8.21)

Then, from (8.15) and (8.16) and Lemma 8.5.2, it follows that for each .m ≥ 1, Hm = (1 − |zm |2 )q |u1 (zm )fm (ϕ1 (zm )) − u2 (zm )fm (ϕ2 (zm ))|

.

≤ (1 − |zm |2 )q |u1 (zm )fm (ϕ1 (zm )) − u1 (zm )fm (ϕ2 (zm ))| +(1 − |zm |2 )q |u1 (zm )fm (ϕ2 (zm )) − u2 (zm )fm (ϕ2 (zm ))| =

(1 − |zm |2 )q |u1 (zm )| (1 − |ϕ1 (zm )|2 ) +

· (1 − |ϕ1 (zm )|2 )

(1 − |zm |2 )q |fm (ϕ2 (zm ))| (1 − |ϕ2 (zm 

)|2 )

≤ cK · fm p max +

n+1 2

n+1 2

n+1 2

|fm (ϕ1 (zm )) − fm (ϕ2 (zm ))|

· (1 − |ϕ2 (zm )|2 )

(1 − |ϕ1 (zm )|2 )

n+1 2

(1 − |ϕ1 (zm )|2 )

n+1 2

,

n+1 2

|u1 (zm ) − u2 (zm )|

(1 − |ϕ1 (zm )|2 )

n+1 2

(1 − |ϕ2 (zm )|2 )

n+1 2

(1 − |zm |2 )q |u1 (zm ) − u2 (zm )| 2p+n . f  · m p n+1 3p/2 (1 − |ϕ2 (zm )|2 ) 2

However, by (8.21),  .

max

(1 − |ϕ1 (zm )|2 )

n+1 2

(1 − |ϕ1 (zm )|2 )

n+1 2

, n+1 n+1 (1 − |ϕ1 (zm )|2 ) 2 (1 − |ϕ2 (zm )|2 ) 2   n+1 (1 − |ϕ1 (zm )|2 ) 2 = max 1, = 1, n+1 (1 − |ϕ2 (zm )|2 ) 2



 ρϕ1 ,ϕ2 (zm )

8 Weighted Composition Operators on .B

146

and hence, Hm ≤ cKfm p · ρϕ1 ,ϕ2 (zm )

.

+

(1 − |zm |2 )q |u1 (zm ) − u2 (zm )| 2p+n f  · m p n+1 3p/2 (1 − |ϕ2 (zm )|2 ) 2

= cKfm p · ρϕ1 ,ϕ2 (zm ) 2p+n +fm p · p/2 |u1 (zm ) − u2 (zm )| · 3   (1 − |zm |2 )q (1 − |zm |2 )q , . min n+1 n+1 (1 − |ϕ2 (zm )|2 ) 2 (1 − |ϕ1 (zm )|2 ) 2 In the inequality above, since .(fm p ) is bounded, by (8.20), lim fm p · ρϕ1 ,ϕ2 (zm ) = 0.

.

m→∞

Furthermore, by .(ii) and . lim |ϕ1 (zm )| = lim |ϕ2 (zm )| = 1, we get m→∞

m→∞

 .

lim fm p |u1 (zm ) − u2 (zm )| · min

m→∞

(1 − |zm |2 )q (1 − |ϕ2 (zm )|2 )

n+1 2

,



(1 − |zm |2 )q (1 − |ϕ1 (zm )|2 )

n+1 2

= 0.

These equalities imply that . lim Hm = 0, but this contradicts (8.15). m→∞

8.7

 

Essential Norm: Upper Bound

In this section and the next, we study the essential norm of the weighted composition operator .Wu,ϕ : Np → A−q . Let us denote by .K = K(Np , A−q ) the set of all compact operators acting from .Np into .A−q . Recall that the essential norm of .Wu,ϕ is defined by Wu,ϕ e = inf Wu,ϕ − K.

.

K∈K

Trivially, the essential norm of a compact operator is zero. Theorem 8.7.1 Let p and q be two positive numbers. Let .ϕ : B → B be a holomorphic self-map of .B, and let .u : B → C be a holomorphic function. Suppose that .Wu,ϕ is a bounded operator acting from .Np to .A−q . Then

8.7 Essential Norm: Upper Bound

147

Wu,ϕ e ≤ A lim

.

sup

r→1− |ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

,

where A is the constant from Lemma 8.5.2. Proof Since .Wu,ϕ is bounded, we see that u belongs to .A−q and Theorem 8.2.1 shows that .

lim

sup

r→1− |ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

exists and is a finite real number. First, we prove that, for any .r ∈ [0, 1), Wu,ϕ e ≤ A sup

|u(z)|(1 − |z|2 )q

.

|ϕ(z)|>r

(1 − |ϕ(z)|2 )

n+1 2

.

(8.22)

For each .k ≥ 1, set .ϕk (z) = kz/(k + 1), .z ∈ B. By Lemma 8.3.2, .Cϕk is compact on Np , and hence, .Wu,ϕ ◦ Cϕk is compact acting from .Np into .A−q . We then have, for each .k ≥ 1, .

Wu,ϕ e ≤ Wu,ϕ − Wu,ϕ ◦ Cϕk  = sup | Wu,ϕ − Wu,ϕ ◦ Cϕk (f )|q ,

.

f p ≤1

which implies Wu,ϕ e ≤ inf

.

k∈N





sup | Wu,ϕ − Wu,ϕ ◦ Cϕk (f )|q .

f p ≤1

For .f ∈ Np , we estimate .

| Wu,ϕ − Wu,ϕ ◦ Cϕk (f )|q 

  k   2 q   ϕ(z)  (1 − |z| ) = sup |u(z)| f (ϕ(z)) − f k+1 z∈B  

 k    ϕ(z)  (1 − |z|2 )q ≤ sup |u(z)| f (ϕ(z)) − f k+1 |ϕ(z)|>r



  k   |u(z)| f (ϕ(z)) − f ϕ(z)  (1 − |z|2 )q . k+1 |ϕ(z)|≤r

+ sup

On the one hand, by (8.6),

(8.23)

8 Weighted Composition Operators on .B

148

sup

.

|ϕ(z)|>r

    k  2 q   ϕ(z)  (1 − |z| ) |u(z)| f (ϕ(z)) − f k+1

≤



A|u(z)|(1 − |z|2 )q 

sup

(1 − |ϕ(z)|2 )

|ϕ(z)|>r

n+1 2

f p .

On the other hand, by (8.7),     k  ϕ(z)  (1 − |z|2 )q |u(z)| f (ϕ(z)) − f k+1 |ϕ(z)|≤r     k  ≤ sup f (ϕ(z)) − f ϕ(z)  · sup |u(z)|(1 − |z|2 )q k+1 |ϕ(z)|≤r z∈B   Ar|u|q f p . ≤ n+3 (k + 1)(1 − r 2 ) 2

sup

.

Therefore, if .f p ≤ 1, then .

| Wu,ϕ − Wu,ϕ ◦ Cϕk (f )|q ≤A



|u(z)|(1 − |z|2 )q 

sup

|ϕ(z)|>r (1 − |ϕ(z)|2 )

n+1 2

It then follows that  .

inf

k∈N

sup | Wu,ϕ − Wu,ϕ ◦ Cϕk (f )|q

+

Ar|u|q (k + 1)(1 − r 2 )

n+3 2

.



f p ≤1



≤ inf

k∈N



A

= A sup

|ϕ(z)|>r

sup

|ϕ(z)|>r

|u(z)|(1 − |z|2 )q  (1 − |ϕ(z)|2 )

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

n+1 2

+



Ar|u|q (k + 1)(1 − r 2 )

n+3 2

.

Combining this and (8.23), we obtain (8.22). Now, letting .r → 1− in (8.22), we arrive at the desired inequality Wu,ϕ e ≤ A lim

.

sup

r→1− |ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

.

 

8.8 Essential Norm: Lower Bound

8.8

149

Essential Norm: Lower Bound

We now discuss the estimation for the lower bound of .Wu,ϕ e . We will make use of weakly convergent sequences in the Bergman space .A2 . The following lemma plays an important role. Lemma 8.8.1 Suppose .(fm )m≥1 ⊂ A2 is a sequence that converges weakly to zero in .A2 . Then .(fm )m≥1 converges weakly to zero in .Np as well. Proof Let . be a bounded linear functional on .Np . Then  (A2 ) = sup

.

f ∈A2

| (f )| | (f )| | (f )| ≤ sup ≤ sup =  (Np ) , f A2 f ∈Np f p f ∈A2 f p

which implies . is also a bounded linear functional on .A2 . The second and third inequalities follow from the fact that .f p ≤ f A2 for any .f ∈ A2 . This means that, as sets, .(Np ) is a subset of .(A2 ) . Thus, if .fm → 0 weakly in .A2 , then we necessarily have .fm → 0 weakly in .Np as well.   Corollary 8.8.2 Let .{wm }m∈N ⊂ B, and assume that .|wm | → 1 as .m → ∞. Then kwm → 0 weakly in .Np .

.

Proof It is well-known that .kwm → 0 weakly in .A2 as .m → ∞. Indeed, for any .f ∈ A2 , using the reproducing property, we have f, kwm  = (1 − |wm |2 )(n+1)/2 f (wm ),

.

which converges to zero as .m → ∞. The result now follows immediately from   Lemma 8.8.1. Theorem 8.8.3 Let p and q be two positive numbers, let .ϕ : B → B be a holomorphic self-map of .B, and let .u : B → C be a holomorphic function. Suppose that .Wu,ϕ is a bounded operator acting from .Np to .A−q . Then Wu,ϕ e ≥ lim

.

sup

r→1− |ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

.

Proof The case .ϕ∞ < 1 is obvious since the right-hand side is zero. Now assume that ϕ∞ = 1. For any .r ∈ (0, 1), the set .Sr = {z ∈ B : |ϕ(z)| > r} is not empty. For each .z ∈ B, consider the test function .kϕ(z) in Lemma 8.1.1. Then, for any compact operator .Q ∈ K, we have .

8 Weighted Composition Operators on .B

150

Wu,ϕ − Q = sup |(Wu,ϕ − Q)(f )|q

.

f p ≤1

≥ |(Wu,ϕ − Q)(kϕ(z) )|q ≥ |Wu,ϕ (kϕ(z) )|q − |Q(kϕ(z) )|q ≥

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

− |Q(kϕ(z) )|q ,

n+1 2

which is equivalent to |u(z)|(1 − |z|2 )q

Wu,ϕ − Q + |Q(kϕ(z) )|q ≥

.

(1 − |ϕ(z)|2 )

(8.24)

.

n+1 2

Taking the supremum on z over the set .Sr on both sides of (8.24) yields Wu,ϕ − Q + sup |Q(kϕ(z) )|q ≥ sup

.

z∈Sr

z∈Sr

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

,

which is Wu,ϕ − Q + sup |Q(kϕ(z) )|q ≥ sup

.

|ϕ(z)|>r

|ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

.

(8.25)

Denote .H (r) = sup|ϕ(z)|>r |Q(kϕ(z) )|q . Since .H (r) decreases as r increases, . lim H (r) r→1−

exists. We claim that this limit is necessarily zero. For the purpose of obtaining a contradiction, assume that . lim H (r) = L > 0. Then there is a sequence .{zm } ⊂ B r→1−

satisfying .|ϕ(zm )| → 1 as .m → ∞, and for each .m ≥ 1, |Q(kϕ(zm ) )|q >

.

L . 2

(8.26)

By Corollary 8.8.2, .{kϕ(zm ) } converges weakly to zero in .Np . Since Q is compact, we have .{|Q(kϕ(zm ) )|q } converges to zero as .m → ∞, which contradicts (8.26). Therefore, . lim sup|ϕ(z)|>r |Q(kϕ(z) )|q = 0. r→1−

Letting .r → 1− on both sides of (8.25), we conclude that for any compact operator .Q ∈ K, Wu,ϕ − Q ≥ lim

.

From this, it follows that

sup

r→1− |ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

.

8.8 Essential Norm: Lower Bound

151

Wu,ϕ e = inf {|Wu,ϕ − Q} ≥ lim

.

sup

|u(z)|(1 − |z|2 )q

r→1− |ϕ(z)|>r

Q∈K

(1 − |ϕ(z)|2 )

n+1 2

.

 

In conclusion, combining Theorems 8.7.1 and 8.8.3, we obtain a full description of the essential norm of .Wu,ϕ . Theorem 8.8.4 Let p and q be two positive numbers, let .ϕ : B → B be a holomorphic self-map of .B, and let .u : B → C be a holomorphic function. Suppose that .Wu,ϕ is a bounded operator acting from .Np to .A−q . Then Wu,ϕ e  lim

.

sup

r→1− |ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

.

Theorem 8.8.4 provides us a characterization of compact weighted composition operators from .Np to .A−q , as does Theorem 8.2.1 for the boundedness. Corollary 8.8.5 Suppose that .Wu,ϕ is a bounded operator acting from .Np to .A−q as in Theorem 8.8.4. Then .Wu,ϕ is compact if and only if

.

lim

sup

r→1− |ϕ(z)|>r

|u(z)|(1 − |z|2 )q (1 − |ϕ(z)|2 )

n+1 2

= 0.

Notes on Chapter 8 The main results in this chapter are from papers [43, 45]. When .n = 1, Theorem 8.2.1 reduces to Theorem 3 of [95] as a special case. Moreover, in this special case, Theorem 8.4.1 contains [95, Corollary 2], and Corollary 8.4.2 contains Theorems 4.1 and 4.3 of [73]. The fact that formula (7.4) is a true metric is proved in [22]. Formula (8.4) is provided in [22, Theorem 1(c)]. Lemma 8.5.3 is taken from [43].

Structure of Np -Spaces in the Unit Ball B .

9

.

Multipliers and M Invariance

9.1

Recall that a function .u : B → C is a multiplier of .Np if uf belongs to .Np for all .f ∈ Np . An application of the closed graph theorem shows that for any .u ∈ Mult(Np ), the multiplication operator .Mu is bounded on .Np . We first describe the space .Mult(Np ) of multipliers of .Np . This result shows that the situation is similar to Hardy spaces .H p . Theorem 9.1.1 For any .p > 0, we have .Mult(Np ) = H ∞ . Moreover, for any .u ∈ H ∞ , Mu Np →Np = u∞ .

.

Proof For .u ∈ H ∞ and .f ∈ Np , the function uf belongs to .Hol(B), and, from the definition of the norm in .Np , it follows that uf p ≤ u∞ f p .

.

This shows that .H ∞ ⊂ Mult(Np ) and .Mu  ≤ u∞ . Now suppose that u is an element in .Mult(Np ). For any integer .m ≥ 1, we have um p = Mum 1p ≤ Mu m 1p .

.

Combining with Theorem 7.5.1, we obtain a positive constant .C > 0 independent of .u, m, and z such that, for all .z ∈ B, |um (z)| ≤ C(1 − |z|2 )−(n+1)/2 um p ≤ C(1 − |z|2 )−(n+1)/2 1p Mu m .

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_9

153

9 Structure of .Np -Spaces in the Unit Ball .B

154

Consequently,  1/m |u(z)| ≤ C(1 − |z|2 )−(n+1)/2 1p Mu .

.

Letting .m → ∞, we conclude that .|u(z)| ≤ Mu  for all .z ∈ B. Therefore, u belongs to H ∞ and .u∞ ≤ Mu . This completes the proof of the theorem.



.

A space .X of functions defined on .B is said to be Möbius-invariant, or simply .Minvariant, if .f ◦  ∈ X for every .f ∈ X and every . ∈ Aut(B). Recall that .kw is the test function introduced in Lemma 8.1.1. Theorem 9.1.2 The space .Np is .M-invariant. Moreover, for any . ∈ Aut(B), we have  C  = M1/ka  =

.

1 + |a| 1 − |a|

 n+1 2

,

where .a = −1 (0). Proof Note that for any automorphism . on .B, we have C = M1/ka ◦ W .

.

Recall that .W is defined by (7.6). According to Theorem 7.3.1, the weighted composition operator .W is a surjective isometry on .Np . Since .1/ka is a bounded function, it is a multiplier of .Np , and since .W is a surjective isometry, it follows that .C  = M1/ka . But, by Theorem 9.1.1, we know that M1/ka  = 1/ka ∞ =

.

 1 + |a|  n+1 2

1 − |a|

.



9.2

An Upper Estimate for  · p

It is immediate from the definition of the norm in .Np that f p ≥ f A2p ,

.

(p > 0).

We now provide an upper estimate for .f p , whenever .p ≤ n.

9.2 An Upper Estimate for  · p

155

Theorem 9.2.1 Let .0 < p ≤ n. Then there exists a positive constant .C = C(n, p) such that, for any .f ∈ Np ,   f p ≤ C

1/2  2 p sup |f (w)| (1 − |z| ) dV (z) . 2

.

B

|w|=|z|

Proof For .a ∈ B, integration in polar coordinates gives  .

B

|f (z)|2 (1 − |a (z)|2 )p dV (z) 

(1 − |z|2 )p (1 − |a|2 )p dV (z) |1 − z, a |2p B  1   (1 − |a|2 )p = 2n dσ (ζ ) dr r 2n−1 (1 − |r|2 )p |f (rζ )|2 |1 − ζ, ra |2p S 0  1  

 (1 − |a|2 )p 2n−1 2 p 2 ≤ 2n dσ (ζ ) dr. r (1 − |r| ) sup |f (rζ )| 2p S |1 − ζ, ra | 0 ζ ∈S |f (z)|2

=

Now, recall that with .a ∈ B and .0 < r < 1, we have  .

S

dσ (ζ ) = |1 − rζ, a |2p =



 S

dσ (ζ ) |1 − ζ, ra |2p

S

dσ (ζ ) |1 − ζ, ra |n+(2p−n)



⎧ ⎪ ⎪ ⎨bounded in B

for 0 < p < n2 ,

1 log 1−r12 |a|2 ≤ log 1−|a| 2 ⎪ ⎪ ⎩ (1 − r 2 |a|2 )n−2p ≤ (1 − |a|2 )n−2p

for p = n2 , for

n 2

< p ≤ n.

Thus, for .0 < p ≤ n, there exists a positive constant C independent of a and r, such that  .

S

(1 − |a|2 )p dσ (ζ ) ≤ C. |1 − rζ, a |2p

It then follows that 

  |f (z)| (1 − |a (z)| ) dV (z) ≤ C 2n 2

.

B

1

2 p

0

 =C

r 2n−1 (1 − |r|2 )p sup |f (w)|2 dr

B



|w|=r

sup |f (w)|2 (1 − |z|2 )p dV (z).

|w|=|z|

Taking supremum over .a ∈ B gives the required inequality.



9 Structure of .Np -Spaces in the Unit Ball .B

156

9.3

The Space Np0

The little .Np0 space is defined as 0 .Np

=

Np0 (B)

   2 2 p = f ∈ Np : lim |f (z)| (1 − |a (z)| ) dV (z) = 0 . |a|→1− B

In this section, we show that polynomials are dense in .Np0 and that this latter space is closed in .Np . Theorem 9.3.1 .Np0 is a closed subspace of .Np , and hence, it is a Banach space by itself. Proof It is elementary that .Np0 is a subspace of .Np , and hence, it remains to show that 0 0 .Np is closed. Consider a sequence .(fn ) ⊂ Np that converges to some .f ∈ Np . We need 0 to show that .f ∈ Np . By the convergence assumption, for any .ε > 0, there exists an .N ≥ 1 such that  f − fn p
N be fixed. Since .fn0 ∈ Np0 , there exists a .δ ∈ (0, 1) such that  sup

.

δ 0, such that

.

 .

(1 − |z|2 )s dμ(w) = 0. |1 − z, w |p+s

lim

|z|→1− B

Then for any .ε > 0, there exists .δ > 0, such that when .δ < |z| < 1,  .

B

(1 − |z|2 )s dμ(w) < ε. |1 − z, w |p+s

(9.13)

We first show that .μ is a p-Carleson measure. Indeed, for .|z| ≤ δ, we have  .

B

(1 − |z|2 )s dμ(w) ≤ |1 − z, w |p+s

 B

dμ(w) μ(B) ≤ < ∞. p+s |1 − z, w | (1 − δ)p+s

This fact and (9.13) show that .μ is a p-Carleson measure. Next, we prove that .μ is a vanishing p-Carleson measure. Let .ξ be in .S. For .r ∈ (0, 1 − δ), put .z = (1 − r)ξ . Then .δ < |z| < 1, and for any .w ∈ Qr (ξ ),  

  |1 − z, w | = (1 − r) 1 − ξ, w + r  ≤ (1 − r)r + r < 2r.

.

Consequently,

.

(1 − |z|2 )s rs 2−(p+s) (1 − |z|)s ≥ = . ≥ p+s p+s p+s |1 − z, w | (2r) rp |1 − z, w |

Using (9.13), we obtain μ(Qr (ξ )) 1 . = p p r r



 dμ(w) ≤ 2

p+s

Qr (ξ )



≤ 2p+s

Qr (ξ )

(1 − |z|2 )s dμ(w) B

|1 − z, w |p+s

which implies .(i).

(1 − |z|2 )s dμ(w) |1 − z, w |p+s

< 2p+s ε,



The following result describes a relationship between functions in .Np and .Np0 and Carleson measures. Theorem 9.5.2 Let .p > 0 and .f ∈ Hol(B). Define

9 Structure of .Np -Spaces in the Unit Ball .B

164

dμf,p (z) = |f (z)|2 (1 − |z|2 )p dV (z).

.

Then the following assertions hold. (i) .f ∈ Np if and only if .μf,p is a p-Carleson measure. (ii) .f ∈ Np0 if and only if .μf,p is a vanishing p-Carleson measure. Moreover, 2 .f p

μf,p (Qr (ξ )) 1  sup = sup p p r r r∈(0,1),ξ ∈S r∈(0,1),ξ ∈S

 |f (z)|2 (1 − |z|2 )p dV (z). Qr (ξ )

Proof For any for .f ∈ Hol(B), we can write 

2 .f p

(1 − |a|2 )p (1 − |z|2 )p dV (z) |1 − a, z |2p a∈B B p   1 − |a|2 = sup dμf,p (z). 2 a∈B B |1 − a, z |

= sup

|f (z)|2

From this, statement .(i) and the asymptotic formula for the norm follow. The statement (ii) is a consequence of Lemma 9.5.1.



.

9.6

Np Spaces Are Not Homeomorphic

In this section, we show that, for .0 < p ≤ n, all .Np -spaces are different topological vector spaces. This result together with Theorem 7.5.1 gives a complete relationship between .Np spaces for all .p > 0. We prove this fact by a construction. To do so, we need some additional tools. Let us denote the point .(1, 0, . . . , 0) ∈ S by .1. On .S, we have the natural rotation-invariant probability measure .σ . Denote by .Wk (S) the space of all k-homogeneous polynomials on .Cn restricted to the unit sphere .S. On .Wk (S), we can consider a chain of norms: for .f ∈ Wk (S), put  f p =

.

S

1/p |f (ξ )|p dσ (ξ )

if .1 ≤ p < ∞ and f ∞ = sup |f (ξ )|.

.

ξ ∈S

9.6 Np Spaces Are Not Homeomorphic

165



p The Banach space . Wk (S),  · p is denoted by .Wk (S). Write  .α(n, p) = | ξ, 1 |p dσ (ξ ), Sn

where .Sn is the unit sphere in .Cn . In particular, it can be computed that    p 1+p (n) n + . .α(n, p) =  2 2 

For two finite-dimensional Banach spaces X and Y , the projection constant .λ(X) of X is a number defined by λ(X) = inf{T  · S, T : X → L∞ , S : L∞ → X, ST = IdX }

.

and the so-called Banach–Mazur distance .d(X, Y ) between X and Y is defined by d(X, Y ) = inf{T  · T −1 , T : X → Y is surjective}.

.

It is well-known that λ(X) ≤ d(X, Y )λ(Y ).

(9.14)

.

Also, λ( 2k ) =

.

1√ α(k, 1) ≥ kπ . α(k, 2) 2

(9.15)

Now we give a brief description of the construction of some important polynomials. To do so, we consider the operator (Tk f )(ξ ) =

.

1 α(n, 2k)

 S

f (η) ξ, η k dσ (η)

(9.16)

is an orthogonal projection from .L2 (S) onto .Wk2 (S). The norm of this projection, considered as an operator from .L∞ (S) onto .Wk∞ (S), is smaller than .2n−1 . We also have .

Moreover,

inf{p2 : p ∈ Wk (S) and p∞ = 1} =



α(n, 2k).

(9.17)

9 Structure of .Np -Spaces in the Unit Ball .B

166

.



dim Wk (S) =

1 . α(n, 2k)

(9.18)

Lemma 9.6.1 For every k, there exists .Pk ∈ Wk (S) with .Pk ∞ = 1 and .Pk 2 ≥ √ π /2n . Proof We apply (9.14) for .X = Wk2 (S) and .Y = Wk∞ (S). By (9.15), (9.16), and (9.18), we obtain   λ(Wk2 (S)) d Wk2 (S), Wk∞ (S) ≥ λ(Wk∞ (S)) √  π 1 . ≥ 21−n 2 α(n, 2k)

.

(9.19)

Let I denote the identity map from .Wk∞ (S) to .Wk2 (S). By the definition of the Banach– Mazur distance, we have   I  · I −1  ≥ d Wk2 (S), Wk∞ (S) .

.

 1 . So finally, by (9.19), we conclude that α(n, 2k)

Then (9.17) gives .I −1  =

√ I  ≥ 2

1−n

.

π . 2



The proof is completed.

Now we return to the main problem. By Lemma 9.6.1, there exists a sequence of homogeneous polynomials .(Pk )k≥1 satisfying .deg(Pk ) = k, Pk ∞ = sup |Pk (ξ )| = 1,

(9.20)

1/2 √ π |Pk (ξ )| dσ (ξ ) ≥ n . 2 S

(9.21)

.

ξ ∈S

and  .

2

Note that the homogeneity of .Pk implies that .|Pk (z)| ≤ |z|k for all .z ∈ B. Let .(mk )∞ k=0 be a sequence of positive integers such that .mk+1 /mk ≥ c > 1 for all .k ≥ 0. Let

9.6 Np Spaces Are Not Homeomorphic

f (z) =

167

∞ 

(z ∈ B).

bk Pmk (z),

.

(9.22)

k=0

Such a function is said to belong to the Hadamard gap class. In the following result, we obtain an estimate for the .Np -norm and .A−q -norm of f . Theorem 9.6.2 Let f be defined as in (9.22), and let .p > 0. Then the following statements hold. (i) For .0 < p ≤ n, we have f 2p 

.

∞  |bk |2 p+1

.

mk

k=0

(ii) For any .q > 0, we have |f |q  sup

.

k

|bk | q . mk

Here, .f p and .|f |q denote the norm of f in the spaces .Np and .A−q , respectively. Proof (i): Since .|Pmk (w)| ≤ |w|mk for all .k ≥ 0 and .w ∈ B, we have

.

sup |f (w)| ≤

|w|=|z|

∞ 

|bk ||z|mk

k=0

for all .z ∈ B. Then we do integration in polar coordinates. Hence, by Theorem 9.2.1,  2 .f p

 0

∞ 1 

|bk |r mk

2

(1 − r 2 )p dr.

k=0

On the other hand, we have 

∞ 1

.

0

k=0

|bk |r mk

2

(1 − r 2 )p dr 

∞  k=0

2−k(p+1)



 2k ≤mj 0,

.

where .J ϕ is the complex Jacobian of .ϕ. Then .Cϕ : Np −→ Np , .p > 0, is a bounded operator with .Cϕ  ≤ δ −1 . Proof Let .f ∈ Np . For .a, z ∈ B, the Schwarz–Pick lemma gives .|ϕ(a) (ϕ(z))| ≤ |a (z)|. Since .ϕ is univalent, it is biholomorphic from .B onto .ϕ(B). This makes the change of variables .w = ϕ(z) possible in the following estimates. More explicitly, we have  δ2

.

 ≤

B

B

|f (ϕ(z))|2 (1 − |a (z)|2 )p dV (z)

|f (ϕ(z))|2 (1 − |ϕ(a) (ϕ(z))|2 )p |J ϕ(z)|2 dV (z)

 =

|f (w)|2 (1 − |ϕ(a) (w)|2 )p dV (w) ϕ(B)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_10

173

10 Composition Operators Between .Np and .Nq

174

 ≤

B

|f (w)|2 (1 − |ϕ(a) (w)|2 )p dV (w) ≤ f 2p .

Then taking supremum over .a ∈ B gives δ 2 Cϕ f 2p ≤ f 2p ,

.

which immediately implies .Cϕ f p ≤ δ −1 f p . Since f was an arbitrary element in .Np , we conclude that .Cϕ is a bounded operator on .Np with .Cϕ  ≤ δ −1 as desired.   Corollary 10.1.2 Suppose .A : Cn −→ Cn is an invertible linear operator and b is a vector in .Cn such that .ϕ(z) = Az + b is a self-mapping of .B. Then .Cϕ : Np −→ Np is bounded for all .p > 0. Proof Since .|J ϕ(z)| = |det(A)| > 0, Theorem 10.1.1 provides the desired conclusion.  

10.2

Boundedness of Cϕ : Np −→ Nq , the General Symbol

For a general self-mapping .ϕ of the unit ball, we give a necessary condition and then a sufficient condition for .Cϕ to be bounded from .Np into .Nq . We make use of a sequence of homogeneous polynomials .{Pm } (see (9.21) in Chap. 9), which satisfy .deg(Pm ) = m, and  Pm ∞ = sup |Pm (ζ )| = 1, and

.

ζ ∈S

S

|Pm (ζ )|2 dσ (ζ ) ≥

π . 4n

(10.1)

We also need the inequality 1+

∞ 

.

2kγ x 2  (1 − x)−γ k

for 0 ≤ x < 1,

(10.2)

k=0

where .γ is a positive number. Indeed, this is already proved in (5.9) Chap. 5 for .x > 1/2. On the other hand, it is clear that .2γ ≥ (1 − x)−γ for .0 ≤ x ≤ 1/2. Therefore, (10.2) holds. The following estimation result will be needed in obtaining the necessary condition for the boundedness of .Cϕ . Lemma 10.2.1 Let p and q be positive numbers, and let .ϕ be a holomorphic self-map of B. Let .BNp denote the closed unit ball of .Np . Suppose .μ is a positive number and E is a measurable subset of .B such that

.

10.2 Boundedness of Cϕ : Np −→ Nq , the General Symbol

175

 |f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) ≤ μ.

sup

.

a∈B,f ∈BNp

E

Then, for any .0 < ε ≤ p + 1, we have  .

sup a∈B E

(1 − |a (z)|2 )q dV (z)  μ. (1 − |ϕ(z)|2 )p+1−ε

Proof The hypothesis implies that for any .a ∈ B and any .f ∈ Np , we have  .

E

|f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) ≤ μf 2p .

(10.3)

Put F (z) = 1 +

∞ 

.

2k(p+1−ε)/2 P2k (z),

(z ∈ B).

k=0

Then, by Theorem 9.6.2, 2 .F p



∞ k(p+1−ε)  2 k=0

2k(p+1)

=

∞ 

2−kε < ∞,

k=0

which shows that F belongs to .Np . Let .Un denote the group of all unitary operators on the Hilbert space .Cn . For each .U ∈ Un , the unitary invariant property of the volume measure shows that .F ◦ U also belongs to .Np and .F ◦ U p = F p . Setting .f = F ◦ U in (10.3) gives  .

E

|F (U ϕ(z))|2 (1 − |a (z)|2 )q dV (z) ≤ μF ◦ U 2p = μF 2p

for all .a ∈ B and .U ∈ Un . Now, fix .a ∈ B. Integration with respect to the Haar measure dU on .Un and using Fubini’s theorem yield   .

E

Un

 |F (U ϕ(z))|2 dU (1 − |a (z)|2 )q dV (z) ≤ μF 2p .

Then, for any .z ∈ B, 

 |F (U ϕ(z))| dU = 2

.

Un

S

   F |ϕ(z)|ζ 2 dσ (ζ )

10 Composition Operators Between .Np and .Nq

176

=

  ∞ 2    + 2k(p+1−ε)/2 P2k (|ϕ(z)|ζ ) dσ (ζ ) 1 S

= 1+

k=0 ∞ 

2k(p+1−ε) (|ϕ(z)|2 )2

k=0

k

 S

|P2k (ζ )|2 dσ (ζ )

  π k 2k(p+1−ε) (|ϕ(z)|2 )2 1+ (by (10.1)) n 4 ∞

≥

k=0

−(p+1−ε)   1 − |ϕ(z)|2

(by (10.2)).

Here, we note that the first equality is a basic identity for unitary operators, while for the third equality, the orthogonality and the homogeneity of .{P2k } are used. Consequently, 

.

(1 − |a (z)|2 )q dV (z) 2 p+1−ε E (1 − |ϕ(z)| )    |F (U ϕ(z))|2 dU (1 − |a (z)|2 )q dV (z) ≤ μF 2p ,  E

U

 

as desired.

We are now ready to prove a necessary condition and a sufficient condition for the boundedness of .Cϕ : Np −→ Nq . Theorem 10.2.2 Let p and q be two positive numbers, and let .ϕ be a holomorphic selfmap of .B. If  .

sup

a∈B B

(1 − |a (z)|2 )q dV (z) < ∞, (1 − |ϕ(z)|2 )n+1

(10.4)

then .Cϕ : Np −→ Nq is bounded. Conversely, if .Cϕ : Np −→ Nq is bounded, then for any .0 < ε ≤ p + 1,  .

sup

a∈B B

(1 − |a (z)|2 )q dV (z) < ∞. (1 − |ϕ(z)|2 )p+1−ε

(10.5)

Proof Suppose (10.4) holds. By Theorem 7.5.1, there is a constant .C > 0 such that for each .f ∈ Np , we have |f (z)|(1 − |z|2 )

.

Hence,

n+1 2

≤ Cf p ,

(z ∈ B).

10.3 Characterizations of Compactness of Cϕ : Np −→ Nq

177

 .

sup

a∈B B

 |f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) ≤ (Cf p )2 sup

a∈B B

(1 − |a (z)|2 )q dV (z), (1 − |ϕ(z)|2 )n+1

which shows that .Cϕ is bounded from .Np into .Nq . Conversely, suppose .Cϕ : Np −→ Nq is bounded. Then  .

sup

a∈B,f ∈BNp

B

|f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) = sup Cϕ f 2q ≤ Cϕ 2 . f ∈BNp

The desired inequality now follows from Lemma 10.2.1.

 

An application of Theorem 10.2.2 immediately gives the following result. In fact, the operator .Cϕ in the corollary is compact, as we see in the next section. Corollary 10.2.3 Let .ϕ be a holomorphic self-mapping of .B such that .ϕ∞ < 1. Then Cϕ : Np −→ Nq is bounded for all .p, q > 0.

.

10.3

Characterizations of Compactness of Cϕ : Np −→ Nq

In this section, we study the compactness of composition operators between .Np -spaces. We use the following result which is a reformulation of the well-known criterion for compactness, i.e., weak convergent sequences are transformed to strong convergent sequences: “A bounded composition operator .Cϕ : Np → Nq is compact if and only if for any bounded sequence .(fm ) ⊂ Np converging to zero uniformly on compact subsets of .B, the sequence .(fm ◦ ϕq ) converges to zero as .m → ∞.” We start with the following result, which provides a necessary condition for .Cϕ to be a compact operator. Lemma 10.3.1 Suppose .Cϕ : Np −→ Nq is a compact composition operator. Let .BNp = {f ∈ Np : f p ≤ 1}. Then the following statements are true. (i) .Cϕ (Np ) ⊂ Nq0 and  .

lim

sup

|a|→1− f ∈BN

p

B

|f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) = 0.

(10.6)

(ii) For any decreasing sequence .{Ak }k≥1 of measurable subsets of the unit ball whose intersection is empty, we have .

lim

k→∞ f ∈BN

 sup

sup

p

a∈B Ak

|f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) = 0.

(10.7)

10 Composition Operators Between .Np and .Nq

178

Proof We first prove that .Cϕ (Np ) is a subset of .Nq0 . Let f be in .Np . For any integer ∞ , and the sequence .m ≥ 1, put .fm (z) = f (mz/(m + 1)). Then each .fm belongs to .H .(fm ) is bounded on .Np and converges to f uniformly on compact subsets of .B. By the compactness criteria mentioned above, the sequence .(Cϕ fm ) converges to .Cϕ f in .Nq as ∞ ⊂ N 0 and .N 0 is a closed subspace .m → ∞. Since each function .Cϕ fm belongs to .H q q of .Np , we conclude that .Cϕ f is an element in .Nq0 . Since f was arbitrary, it follows that the image of .Np under .Cϕ is contained in .Nq0 . Let .ε > 0 be given. Since .Cϕ is compact and its range is contained in .Nq0 , the image 0 .Cϕ (BNp ) is pre-compact in .Nq . Therefore, .Cϕ (BNp ) can be covered by finitely many √ . ε/2-balls. That is, there exists a finite set .{f1 , . . . , fM } ⊂ BNp such that for any .f ∈ BNp , there is a number .j ∈ {1, 2, . . . , M} for which Cϕ (f ) − Cϕ (fj )2q
r,  .

B

|fj (ϕ(z))|2 (1 − |a (z)|2 )q dV (z)
r and .f ∈ Np with .f p < 1, choose .1 ≤ j ≤ M such that (10.8) holds. Combining with (10.9), we have  .

B

|f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z)    |f (ϕ(z)) − fj (ϕ(z))|2 + |fj (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) ≤2 B



= 2Cϕ (f ) − Cϕ (fj )2q + 2 ε ε + = ε. 0 be given. By (10.14), there exists .0 < δ < 1 such that 

 sup

sup

.

δ m0 , by (10.15), (10.16), and (10.18), we have

ε . 3

(10.18)

10 Composition Operators Between .Np and .Nq

182

 2 .Cϕ (fm )q

= sup

a∈B B

|fm (ϕ(z))|2 (1 − |a (z)|2 )q dV (z)  |fm (ϕ(z))|2 (1 − |a (z)|2 )q dV (z)

≤ sup

δtk

 |f (z)|2 (1 − |a (z)|2 )q dV (z) = 0.

10.3 Characterizations of Compactness of Cϕ : Np −→ Nq

183

For each integer .k ≥ 1, define .Ek = {z : |z| ≤ tk } in the case of statement (ii) and Ek = {z : |ϕ(z)| ≤ tk } in the case of statement (iii). Since .{tk }k≥1 is increasing to 1, we see that .{Ek }k≥1 is an increasing sequence of measurable sets and .∪∞ k=1 Ek = B. Furthermore, it is clear that the set .ϕ(Ek ) is compact for each k. The equivalence of (i) and (ii) and the   equivalence of (i) and (iii) now follow from Theorem 10.3.2.

.

Corollary 10.3.5 Let .ϕ be a holomorphic self-map of .B such that .ϕ∞ < 1. Then .Cϕ : Np −→ Nq is compact for all .p, q > 0. Proof By Corollary 10.2.3, the operator .Cϕ is bounded from .Np into .Nq . In addition, condition (iii) in Corollary 10.3.4 is clearly satisfied since the set .{|ϕ(z)| > t} is empty for   all .ϕ∞ < t < 1. Consequently, .Cϕ is compact. By changing the role of .Np to .Np0 in the proofs of Theorems 10.3.2 and 10.3.3, we immediately obtain the following result describing the compactness of composition operators acting between .Np0 and .Nq . Theorem 10.3.6 Let .p, q be positive numbers and .ϕ a holomorphic self-map of .B such that the composition operator .Cϕ : Np0 −→ Nq is bounded. Let .E1 ⊂ E2 ⊂ · · · ⊂ B be an increasing sequence of measurable sets such that .∪k≥1 Ek = B and for each k, the closure .ϕ(Ek ) is compact in .B. Then the following statements are equivalent. (i) .Cϕ is compact from .Np0 into .Nq . (ii) .Cϕ is compact from .Np0 into .Nq0 .

  (iii) . lim supf ∈B 0 supa∈B B\Ek |f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) = 0. Np

k→∞

(iv) The following two conditions are satisfied:



 lim

.

k→∞

sup

 |f (ϕ(z))|2 (1 − |z|2 )q dV (z) = 0;

(10.19)

|f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) = 0.

(10.20)

B\Ek

f ∈BN 0 p

and  .

lim

sup

|a|→1− f ∈B

Np0

B

Here, .BNp0 = {f ∈ Np0 : f p ≤ 1} is the unit ball of .Np0 .

10 Composition Operators Between .Np and .Nq

184

10.4

A Sufficient Compactness Condition

Although Theorems 10.3.2, 10.3.3, and 10.3.6 offer several characterizations of the compactness of composition operators .Cϕ : Np −→ Nq , the conditions are rather abstract and difficult to verify. We shall provide in this section a necessary condition and a sufficient condition for the compactness of .Cϕ directly in terms of .ϕ. These conditions are more useful in applications. Theorem 10.4.1 Let .p, q ∈ (0, n] be two positive numbers and .ϕ a holomorphic self-map of .B such that .Cϕ : Np −→ Nq is bounded. If  .

lim sup

t→1− a∈B |ϕ(z)|>t

(1 − |a (z)|2 )q dV (z) = 0, (1 − |ϕ(z)|2 )n+1

(10.21)

then .Cϕ is compact. Conversely, if .Cϕ : Np → Nq is compact, then for .0 < ε ≤ p + 1,  .

lim sup

t→1− a∈B |ϕ(z)|>t

(1 − |a (z)|2 )q dV (z) = 0. (1 − |ϕ(z)|2 )p+1−ε

(10.22)

Proof Suppose (10.21) holds. As in the proof of Theorem 10.2.2, there is a constant .C > 0 such that for any .f ∈ Np with .f p ≤ 1, |f (z)| ≤ Cf p (1 − |z|2 )−(n+1)/2 ≤ C(1 − |z|2 )−(n+1)/2 .

.

It implies that for any .0 < t < 1 and any .a ∈ B,  .

 |f (ϕ(z))| (1 − |a (z)| ) dV (z) ≤ C 2

2 q

|ϕ(z)|>t

2

(1 − |a (z)|2 )q dV (z). 2 n+1 |ϕ(z)|>t (1 − |z| )

Now (10.21) shows that statement (iii) in Corollary 10.3.4 is satisfied. As a result, .Cϕ is a compact operator from .Np into .Nq . Now suppose .Cϕ : Np −→ Nq is compact. Let .μ > 0 be given. By Corollary 10.3.4, there is a number .tμ ∈ (0, 1) such that for all .tμ < t < 1,  .

sup

a∈B,f ∈BNp

|ϕ(z)|>t

|f (ϕ(z))|2 (1 − |a (z)|2 )q dV (z) ≤ μ.

An application of Lemma 10.2.1 with .E = {z ∈ B : |ϕ(z)| > t} yields  .

sup

a∈B |ϕ(z)|>t

(1 − |a (z)|2 )q dV (z)  μ, (1 − |z|2 )p+1−ε

for all .tμ < t < 1. Since .μ is arbitrary, (10.22) follows.

 

10.4 A Sufficient Compactness Condition

185

As an application of Theorems 10.2.2 and 10.4.1, we mention the following result. Corollary 10.4.2 Suppose .k > 0, p, q, r ∈ (0, n), r ≥ q, ε ∈ (0, q + 1) and .ϕ is a holomorphic self-map of .B. The following statements hold: (i) If .Cϕ : A−k(q+1−ε) −→ A−k(n+1) is a bounded operator, then .Cϕ : Np −→ Nr is a bounded operator. (ii) If .Cϕ : A−k(q+1−ε) −→ A−k(n+1) is a compact operator, then .Cϕ : Np −→ Nr is a compact operator. Proof The boundedness of .Cϕ : A−k(q+1−ε) −→ A−k(n+1) is equivalent to M = sup

.

z∈B

(1 − |z|2 )q+1−ε < ∞. (1 − |ϕ(z)|2 )n+1

(10.23)

By Theorem 10.2.2, it suffices to show that  .

sup

a∈B B

(1 − |a (z)|2 )r dV (z) < ∞. (1 − |ϕ(z)|2 )n+1

Indeed, we have  (1 − |a (z)|2 )r (1 − |a (z)|2 )r dV (z) ≤ M sup dV (z) 2 n+1 2 q+1−ε B (1 − |ϕ(z)| ) a∈B B (1 − |z| )  (1 − |z|2 )r−q−1+ε = M sup(1 − |a|2 )r dV (z) |1 − z, a|2r B a∈B  (1 − |z|2 )r−q−1+ε = M sup(1 − |a|2 )r dV (z). n+1+(r−q−1+ε)+(r+q−n−ε) B |1 − z, a| a∈B

 .

sup a∈B

Therefore,  .

sup(1 − |a|2 )r a∈B

B

(1 − |z|2 )r−q−1+ε dV (z) < ∞, |1 − z, a|n+1+(r−q−1+ε)+(r+q−n−ε)

which implies the desired result. The proof of statement (ii) is similar to that of (i) by using the fact that .Cϕ : A−k(q+1−ε) −→ A−k(n+1) is compact if and only if .

lim

sup

t→1− |ϕ(z)|>t

(1 − |z|2 )q+1−ε = 0. (1 − |ϕ(z)|2 )n+1  

186

10 Composition Operators Between .Np and .Nq

Notes on Chapter 10 The results of this chapter are mainly from [44]. In fact, the one-dimensional case of these results was considered in [73]. A sufficient condition and a necessary condition for .Cϕ to be bounded on .Np of the unit disc were given there. These conditions involve the generalized Nevanlinna counting function introduced by Shapiro. The existence of a sequence (10.1) is proved in [81]. The estimate (10.2) is borrowed from [95].

Np -Type Functions with Hadamard Gaps

.

in the Unit Ball B .

11

A function .f ∈ Hol(B) written in the form f (z) =

∞ 

.

Pnk (z),

k=0

where .Pnk is a homogeneous polynomial of degree .nk , is said to have Hadamard gaps (see, e.g., [89]) if, for some .c > 1 , .

nk+1 ≥c nk

for all .k ≥ 0. Hadamard gap series on spaces of holomorphic functions in .D or in .B has been extensively studied. In this chapter, we study some aspects of Hadamard gap series in .Np -spaces.

11.1

Some Estimates for Volume and Surface Integrals

In the sequel, we need some relationships between the volume measure dv on the unit ball B, normalized so that .v(B) = 1, and the surface measure ds on the unit sphere .S, similarly normalized so that .s(S) = 1. The normalizing constants, namely, the actual volume of .B and the actual surface area of .S, are in general not important. The following result is referred to as integration in polar coordinates. .

Lemma 11.1.1 The measures v and s are related by

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_11

187

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

188





.

B

1

f (z) dv(z) = 2n

 r 2n−1 dr

0

S

f (rζ ) ds(ζ ).

Proof Let .dV = dx1 dy1 · · · dxn dyn be the actual Lebesgue measure in .Cn before normalization, where each .zk is identified with .xk + iyk (.k = 1, 2, . . . , n). Similarly, let dS be the actual surface measure on .S before normalization. For .r > 0, let .V (r) be the actual volume of the ball |z1 |2 + · · · + |zn |2 < r 2

.

and .S(r) be the actual surface area of the sphere |z1 |2 + · · · + |zn |2 = r 2 .

.

Then the Euclidean volume of the solid determined by dS on .S, by .r > 0, and .r + dr, is given by dV =

.

dS [V (r + dr) − V (r)]. S(1)

Hence, by the change of variables .zk = rwk (.1 ≤ k ≤ n), we have  V (r) =

.

|z1 |2 +···+|zn |2 −1. Then  .

S

(n − 1)!m! (n − 1 + |m|)!

(11.1)

m! (n + α + 1) . (n + |m| + α + 1)

(11.2)

|ζ m |2 ds(ζ ) =

and  .

B

|zm |2 dvα (z) =

Proof Identifying .Cn with .R2n and denoting by dV the usual Lebesgue measure on .Cn , we have .c dv = dV , where c is the Euclidean volume of .B. We evaluate the integral  I=

|zm |2 e−|z| dV (z) 2

.

Cn

in two different ways. First, we use Fubini’s theorem to obtain I =

n  

.

k=1

=π

n

R2

(x 2 + y 2 )mk e−(x

n  

∞

2 +y 2 )

dx dy

r mk e−r dr = π n m!.

k=1 0

Next, we use Lemma 11.1.1 about integration in polar coordinates to get  I = 2nc

∞

r

.

2|m|+2n−1 −r 2

e

0



= nc(|m| + n − 1)!

S

 dr

S

|ζ m |2 ds(ζ )

|ζ m |2 ds(ζ ).

Comparing the two results yields  .

S

|ζ m |2 ds(ζ ) =

π n m! . nc(|m| + n − 1)!

In particular, for .m = (0, . . . , 0), we have c=

.

πn . n!

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

190

Consequently,  .

S

|ζ m |2 ds(ζ ) =

(n − 1)!m! . (|m| + n − 1)!

Furthermore, again by Lemma 11.1.1, we have 

 |z | dvα (z) = 2ncα m 2

.

B



1

r

2|m|+2n−1

(1 − r ) dr 2 α

0



1

= ncα

r |m|+n−1 (1 − r 2 )α dr ·

0

S

|ζ m |2 ds(ζ )

(n − 1)!m! . (|m| + n − 1)!

Note that 

1

.

r |m|+n−1 (1 − r 2 )α dr =

0

(n + |m|)(α + 1) . (n + |m| + α + 1)  

Thus, (11.2) follows from (11.1). We also need the following estimates (see (7.13) in Chap. 7).



(1 − |z|2 )p−2k |1 − z, a |2p

.

B

where .0 < k ≤

p < n + 1 − 2k, p = n + 1 − 2k,

(11.3)

p > n + 1 − 2k,

n+1 . In any case, there exists a positive constant C such that 2 

(1 − |a|2 )p

.

11.2

⎧ ⎪ ⎪ ⎨bounded in B, dV (z) log 1 2 , 1−|a| ⎪ ⎪ ⎩ (1 − |a|2 )n+1−p−2k ,

B

(1 − |z|2 )p−2k dV (z) ≤ C, |1 − z, a |2p

(a ∈ B).

(11.4)

Hadamard Gaps in Np -Spaces (p ≤ n)

In this section and the next one, we characterize the holomorphic functions with Hadamard gaps in .Np -space. Here, our focus is on the case .p ∈ (0, n]. Let us denote Mk = sup |Pnk (ξ )|

.

ξ ∈S

and

11.2 Hadamard Gaps in Np -Spaces (p ≤ n)

191

 Ik =

1/2

.

ξ ∈S

|Pnk (ξ )|2 dσ (ξ )

,

where .dσ is the normalized surface measure on .S, that is, .σ (S) = 1. Clearly, for each k ≥ 0, .Mk and .Ik are well-defined.

.

Theorem 11.2.1 Let .p ∈ (0, n], and let f (z) =

∞ 

.

Pnk (z)

k=0

be a series with Hadamard gaps. Consider the following statements. (i) ∞  .

⎛ ⎝

k=0



1 2k(1+p)

⎞ |Mj |2 ⎠ < ∞.

2k ≤n 1 such that nj +1 ≥ cnj for all .j ≥ 0. Then

.

⎛ ⎞ ∞

1 ⎝  ⎠  1 m c

. = n . ≤ j c

c − 1 k

nj ≤k

(11.10)

m=0

Combining (11.9) and (11.10) yields ⎛ ⎞ 1 ⎝  ⎠ k

Mc

, nj ≤

. − −1 ·

k c −1 (−1)k k

nj ≤k

from which it follows that  .

n j ≤ (−1)k

nj ≤k



− − 1 Mc

. k c − 1

Consequently, for any .z ∈ B, by (11.11), we have

∞ 1 Mc  |f (z)| Mc

t − − 1 |z|t =

· · 

(−1) , . t c −1 c − 1 (1 − |z|) +1 1 − |z| t=0

which gives |f (z)|(1 − |z|2 ) 

.

Mc

. c − 1

(11.11)

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

200

This shows that .f ∈ A− . −

.(ii)-Necessity Suppose that .f ∈ A 0 . This means that for any .ε > 0, there exists .δ ∈ (0, 1), such that whenever .δ < |z| < 1, we have |f (z)|(1 − |z|2 ) < ε.

.

Take .N0 ∈ N with .δ < 1 − n1k < 1 whenever .k > N0 . Then for all .k > N0 and .r = 1 − n1k , applying the proof in part (i), we get Mk ≤

.

n k (1 −

· sup |f (z)|(1 − |z|2 )
N0 ).

For each .m ∈ N, we put fm (z) =

∞ 

.

Pnk (z).

k=0

Note that |fm (z)|(1 − |z|2 ) ≤

 m 

.



k=0

 |Pnk (z)| (1 − |z|2 )

 m     z n k Pn ( )|z|  (1 − |z|2 )

=  k |z|  k=0

∞ 11.4 A Characterization of Hμ∞ and Hμ,0

201

≤ K(1 − |z|2 )

m 

|z|nk ≤ Km(1 − |z|2 ) .

k=0

Here, .K = max{M0 , M1 , . . . , Mm }. Thus, .

lim |fm (z)|(1 − |z|2 ) = 0,

|z|→1−1

which means that .fm ∈ A− for each .m ∈ N. Then it suffices to show that .|fm − f | → 0 as .m → ∞. Indeed, for .m > N0 , we have  ∞  ∞ ∞        .|fm (z) − f (z)| =  Pnk (z) ≤ |Mk |z|nk ≤ ε |nk | |z|nk .   k=m+1

k=m+1

k=m+1

Doing as in the proof of part (i), we obtain |fm (z) − f (z)| ≤ε . 1 − |z| =ε



∞ 

nk

 ∞ 

k=m+1

∞ 

=0



⎝

⎛ ⎝

 |z|

s

s=0

⎛

∞ 

=nm+1

≤ε

|nk | |z|

⎞

n j ⎠ |z|

nm+1 ≤nj <



⎞

n j ⎠ |z| ≤ M 

nj ≤

ε . 1 − |z|) +1

Here, .M  is a positive constant which is independent of m. Hence, whenever .m > N0 , we   have .|fm − f | ≤ M  ε, which implies that .f ∈ A−

0 .

11.4

∞ A Characterization of Hμ∞ and Hμ,0

We recall from Sect. 6.4 of Chap. 6 that a positive continuous function .μ on .[0, 1) is called normal if there exist positive .α and .β with .α < β, and .δ ∈ (0, 1), such that ⎧ μ(r) μ(r) ⎪ ⎪ ⎨ (1−r)α is decreasing on [δ, 1), limr→1 (1−r)α = 0, . and ⎪ ⎪ ⎩ μ(r) μ(r) is increasing on [δ, 1), limr→1 (1−r) β = ∞. (1−r)β

(11.12)

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

202

A normal function .μ : [0, 1) → [0, ∞) is decreasing in a neighborhood of 1 and satisfies limr→1− μ(r) = 0. The corresponding weighted-type space in the ball is defined as

.





Hμ∞ = f ∈ O(B) : f = sup |f (z)|μ(|z|) < ∞ ,

.

z∈B

where .μ is normal on .[0, 1). It is well-known that .Hμ∞ is a Banach space with the norm ∞ ∞ . · . The little weighted-type space, denoted by .H μ,0 , is the closed subspace of .Hμ that consists of .f ∈ Hμ∞ satisfying .

lim |f (z)|μ(|z|) = 0.

|z|→1−

∞ are, respectively, In the case .μ(|z|) = (1−|z|2 )α , α > 0, the induced spaces .Hμ∞ and .Hμ,0 ∞ ∞ denoted by .Hα and .Hα,0 . ∞  Pk (z) be a holomorphic function in .B, where .Pk (z) is a homogeneous Let .f (z) = k=0

polynomial of degree k. We put Mk = sup |Pk (ξ )|,

.

ξ ∈S

k ≥ 0.

∞ , The following results are about estimates for .Mk of a function f belonging .Hμ∞ and .Hμ,0 respectively.

Lemma 11.4.1 Let .μ be a normal function on .[0, 1), and let f (z) =

∞ 

.

Pk (z) ∈ Hol(B).

k=0

Then the following assertions hold. (i) If .f ∈ Hμ∞ , then .

1 sup Mk μ(1 − ) < ∞. k k≥0

∞ , then (ii) If .f ∈ Hμ,0

.

1 lim Mk μ(1 − ) = 0. k

k→∞

∞ 11.4 A Characterization of Hμ∞ and Hμ,0

203

Proof (i): Suppose that .f ∈ Hμ∞ . Fix .ξ ∈ S, and denote fξ (w) =

∞ 

.

Pk (ξ )w k =

k=0

∞ 

Pk (ξ w),

(w ∈ D).

k=0

Since .f ∈ Hμ∞ , for a fixed .ξ ∈ S, the function .fξ ∈ Hμ∞ . In this case, for every .r ∈ (0, 1), we have Mk = sup |Pk (ξ )|

.

ξ ∈S

   1 = sup  2π i

  fξ (w) dw  k+1 |w|=r w ξ ∈S     1 f (ξ w)  = sup  dw  k+1 2π ξ ∈S |w|=r w  1 |f (ξ w)| ≤ sup |dw| 2π ξ ∈S |w|=r r k+1 ≤

f . r k μ(r)

In particular, for .r = 1 − k1 , k ≥ 2, we have Mk ≤

.

f (1 −

1 k 1 k ) μ(1 − k )

,

which gives

f 1 Mk μ(1 − ) ≤ ≤ 4 f , k (1 − k1 )k

.

k ≥ 2.

Hence, .

1 sup Mk μ(1 − ) ≤ max{M1 μ(0), 4 f } < ∞. k k≥1

∞ . For any . > 0, there exists .δ ∈ (0, 1) such that (ii): Suppose that .f ∈ Hμ,0 1 .μ(|z|)|f (z)| <  whenever .δ < |z| < 1. Fix .N0 ∈ N satisfying .δ < 1 − k < 1 1 for all .k > N0 . Then, as in the proof of part (i), for .k > N0 and .r = 1 − k , we have

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

204

Mk ≤

.

1 (1 −

1 1 k k ) μ(1 − k )

· sup μ(|z||f (z)| < δ 0, there exists .N0 ∈ N such that

11.5 The Growth Rate in Hμ∞

205

Mnm μ(1 −

.

1 ) N0 .

For each .m ∈ N, put Sm (z) =

m 

.

∞ Pnk (z) ∈ Hμ,0 .

k=0 ∞ , it suffices to prove that . S − f → 0 as .m → ∞. We have To show .f ∈ Hμ,0 m

 ∞  ∞ ∞      |z|nk   , .|Sm (z) − f (z)| =  Pnk (z) ≤ Mnk |z|nk ≤ ε   μ(1 − n1k ) k=m+1 k=m+1 k=m+1  

from which the desired result follows.

11.5

The Growth Rate in Hμ∞

As an application of Theorem 11.4.2, we provide an estimate of the growth rate of functions in .Hμ∞ . To achieve this goal, we need some auxiliary results. For .ξ, ζ ∈ S, we denote d(ξ, ζ ) = (1 − |ξ, ζ |2 )1/2 .

.

Here, d satisfies the triangle inequality. We also write Eδ (ζ ) = {ξ ∈ S : d(ξ, ζ ) < δ},

.

(0 < δ < 1).

We say that a set . ⊂ S is d-separated by .δ if the sets .Eδ (ζ ), where .ζ ∈ , are pairwise disjoint. Lemma 11.5.1 For each .a > 0, there exists a positive integer .N = Nn (a) with the following property: if .δ > 0 and if . ⊂ S is d-separated by .aδ, then . can be decomposed as =

N 

.

j

j =1

in such a way that each .j is d-separated by .δ. Lemma 11.5.2 Suppose . ⊂ S is d-separated by .δ and k is a positive integer. If

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

206

P (z) =



.

z, ζ k ,

(z ∈ B),

ζ ∈

then |P (z)| ≤ 1 +

∞ 

(m + 2)2n−2 e−m

2 δ 2 k/2

.

.

m=1

Theorem 11.5.3 Let .μ be a normal function on .[0, 1). Then there exists a positive integer N = N(n), and there are functions .fj ∈ Hμ∞ (j = 1, . . . , N) such that

.

N  .

|fj (z)| 

j =1

1 , μ(|z|)

(z ∈ B).

Proof First we note that the integer N in the statement of the theorem must be at least 2. Indeed, assume in contrary that .N = 1, i.e., there exists .f ∈ Hμ∞ such that |f (z)| 

.

1 , μ(|z|)

(z ∈ B).

This shows that f has no zeros in .B, and hence there exists .g ∈ Hol(B) such that .f = eg . Then the last inequality means eReg(z) 

.

1 μ(|z|)

⇐⇒

Reg(z)  log

1 , μ(|z|)

z ∈ B.

For each .r ∈ (0, 1), integrating both sides of the inequality above on .rS = {z ∈ B : |z| = r}, we have 

 Reg(z) dσ 

.

rS

log( rS

1 1 ) dσ = log( ) · σ (rS). μ(r) μ(r)

Hence, by the mean value property, Reg(0)  log(

.

1 ), μ(r)

for all r ∈ (0, 1),

which is impossible. Now we prove the theorem by constructing the functions .fj ∈ Hμ∞ satisfying the given property near the boundary only; then by adding a proper constant, we can extend it to the whole unit ball. By normality of .μ on .[0, 1), there exist positive constants .α < β and .δ ∈ (0, 1) that satisfy (11.12). Let .0 < A < 1 be a sufficiently small number for which

11.5 The Growth Rate in Hμ∞

207 ∞ 

.

(m + 2)2n−2 e−m

2 /2A2

≤

m=1

1 . 27

(11.13)

Let .N = Nn ( A2 ) be a positive constant from Lemma 11.5.1 (with .a = integer p large enough satisfying the following conditions:

Take a positive

1 ≥ δ, p

1−

.

A 2 ).

1 1 1 ≤ (1 − )p ≤ , p 2 3 1 pαN

−1

1 , 200

≤

pβN · 2−p 1 − pβN

N− 21

N− 1 −(p2N − 12 −p 2 )

≤

·2

1 . 200

(11.14)

For every positive number .j ≤ N, we put .δj,0 with 2 A2 pj δj,0 =1

.

and inductively define .δj,ν by 2 2 pN δj,ν = δj,ν−1 ,

.

ν ≥ 1.

In this case, we have 2 A2 pνN +j δj,ν = 1.

(11.15)

.

For each fixed .j, ν, denote by . j,ν the maximal subset of .S subject to the condition that is d-separated by .Aδj,ν /2. Then by Lemma 11.5.1, write

j,ν .

 j,ν =

N 

.

(11.16)

j,νN +

=1

in such a way that each .j,νN+ is d-separated by .δj, ν. For each .i, j = 1, 2, . . . , N and .ν ≥ 0, set Pi,νN +j (z) =



.

ξ ∈j,νN+τ i (j )

z, ξ p

νN+j

,

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

208

where .τ i is the i-th iteration of the permutation .τ on .{1, 2, . . . , N} defined by ⎧ ⎨j + 1, .τ (j ) = ⎩1,

j < N, j = N.

By Lemma 11.5.2, (11.13), and (11.15), we have |Pi,νN+j (z)| ≤ 1 +

∞ 

.

(m + 2)2n−2 e

−m2 δj,ν p νN+j 2

m=1

≤ 1+

∞ 

(m + 2)2n−2 e

−

m2 2A2

≤ 2,

(z ∈ B),

m=1

for all .i, j = 1, 2, . . . , N and .ν ≥ 0. Now define gi,j (z) =

.

∞  Pi,νN+j (z) ν=0

μ(1 −

1 pνN+j

)

,

(z ∈ B).

By Theorem 11.4.2, .gi,j ∈ Hμ∞ for each .i, j = 1, 2, . . . , N. We show that for every .ν ≥ 0, .1 ≤ j ≤ N, and .z ∈ B with 1−

.

1 1 ≤ |z| ≤ 1 − 1 − , νN+j + 12 pνN+j p

(11.17)

C there exists .i ∈ {1, 2, . . . , N} such that .|gi,j (z)| ≥ μ(|z|) , where C is some positive constant independent of the choice of .i, j , and z. Fix .ν, j , and .z = η|z| (η ∈ S) for which (11.17) holds. Since d-balls centered at points of . j,ν of radius .Aδj,ν cover .S, by maximality, there exists some .ξ ∈  j,ν such that .η ∈ EAδj,ν (ζ ). Note that by (11.16), .ζ ∈ j,νN + for some .1 ≤ ≤ N . We now estimate .|gi,j (z), as follows.

  ∞   Pi,kN+j (z)    .|gi,j (z)| =   1 k=0 μ(1 − pkN+j )           Pi,νN+j (z)   Pi,kN+j (z)      ≥ −  1 1  μ(1 − pνN+j )  k=ν μ(1 − pkN+j )      νN+j p kN+j   Pi,νN+j (η)  |z| Pi,kN+j (η)  |z| − =   1 1 μ(1 − pνN+j ) k=ν μ(1 − pkN+j ) 

11.5 The Growth Rate in Hμ∞

≥

|z|p

νN+j

209

Pi,νN +j (η)

μ(1 −

1 pνN+j

)

−2

ν−1 

|z|p

k=0

μ(1 −

∞ 

kN+j

1 pkN+j

)

−2

k=ν+1

|z|p μ(1 −

= J1 − J2 − J3 . We estimate the above J ’s separately. The estimation of .J1 : By (11.14) and (11.17), we have |z|p

νN+j

≥ (1 −

.

1 pνN +j

)p

νN+j

≥

1 , 3

which implies J1 =

.

≥

≥ ≥

|z|p

νN+j

Pi,νN +j (η)

μ(1 −

1 pνN+j

)

Pi,νN +j (η) 3μ(1 −

1 pνN+j

|η, ζ |p

νN+j

)

−



ξ ∈j,νN+τ i (j ),ξ =ζ

3μ(1 − 2 27μ(1 −

1 pνN+j

)

1 pνN+j

|η, ζ |p

νN+j

)

.

The estimation of .J2 : By normality of .μ, with every .s ∈ N, we have (1 − .

(1 −

1 )α psN+j 1 )α

≤

p(s+1)N+j

μ(1 − μ(1 −

1 psN+j 1

)

p(s+1)N+j

)

≤

(1 − (1 −

1 )β psN+j . 1 )β

p(s+1)N+j

This gives 1 < pN α ≤

.

Combining this and (11.14) yields J2 = 2

ν−1 

|z|p

k=0

μ(1 −

.

kN+j

1 pkN+j

)

μ(1 − μ(1 −

1 psN+j 1

)

) p(s+1)N+j

≤ pNβ .

kN+j

1 pkN+j

)

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

210

≤2

ν−1 

1 μ(1 −

k=0

=

≤ ≤ ≤

1 pkN+j ν−1 

2 μ(1 −

1 pνN+j

)

⎣

k=0 ν−1 

2 μ(1 −

) ⎡

1 pνN+j

)

μ(1 − μ(1 −

1 pνN+j

)

1 p(ν−1)N+j

μ(1 −

· ) μ(1 −

1 p(ν−1)N+j 1 p(ν−2)N+j

) )

···

μ(1 −

1 p(k+1)N+j

μ(1 −

1 pkN+j

1 pαN (ν−k)

k=0

2 μ(1 −

1 · αN p −1 ) pνN+j 1

1 100μ(1 −

1 pνN+j

)

.

The estimation of .J3 : Since ∞ 

J3 = 2

.

k=ν+1

=

|z|p

kN+j

μ(1 −

) ⎡

∞ 

(ν+1)N+j 2|z|p

μ(1 −

1 pkN+j

⎣

1

) pνN+j k=ν+1

μ(1 − μ(1 −

1 pνN+j

)

1

) pkN+j

⎤ |z|p

kN+j −p (ν+1)N+j

⎦,

using (11.14) and (11.17) as for .J1 , we have ∞  .

k=ν+1

=

∞  k=ν+1

≤

∞ 

⎡ ⎣ ⎡ ⎣

μ(1 − μ(1 −

) pνN+j 1 pkN+j

μ(1 − μ(1 −

⎤

1

)

1 pνN+j

|z|p )

1 p(ν+1)N+j

pβN (k−ν) |z|p

)

kN+j −p (ν+1)N+j

⎦

μ(1 −

1

···

p(k−1)N+j

μ(1 −

kN+j −p (ν+1)N+j

k=ν+1

=

∞ 

pβN pβN (k−ν−1) |z|p

j p kN −p (ν+1)N

k=ν+1

=

∞  k=0

pβN pβN s |z|p

j +(ν+1)N (p sN−1 )

.

1 pkN+j

)

)

⎤ |z|

pkN+j −p(ν+1)N+j

⎦

)

)

⎤ ⎦

11.5 The Growth Rate in Hμ∞

211

Here, .s = k − ν − 1, and we used .psN −1 ≥ s(pN − 1). Thus, ∞ 

J3 ≤

.

pβN pβN s |z|p

j +(ν+1)N (p N−1 )s

k=0 ∞ 

=

" # j +(ν+2)N −p j +(ν+1)N s pβN pβN |z|p .

k=0

Hence, J3 ≤

.

=

≤ ≤

2|z|p

∞ 

(ν+1)N+j

μ(1 −

1 pνN+j

2 μ(1 −

1 pνN+j

k=0

pβN (|z|p

·

νN+j

1 pνN+j

N

2N −pN )

N− 21

· 1 1 ) 1 − pβN · 2−(p2N− 2 −pN− 2 )

1 100μ(1 −

)p

) 1 − pβN |z|pνN+j (p pβN · 2−p

2 μ(1 −

)

" # j +(ν+2)N −p j +(ν+1)N s pβN pβN |z|p

1 pνN+j

)

.

Now combining all these estimates for .J1 , J2 , and .J3 , we obtain |gi,j (z)| ≥ J1 − J2 − J3

1 1 1 2 − − ≥ 27 100 100 μ(1 −

.

1 pνN+j

>

=

≥

≥

)

1 20μ(1 −

1 pνN+j

) μ(1 −

1 20μ(1 −

1 p

νN+j + 12

)

·

μ(1 −

1 β 2

20p μ(1 − 1 β 2

20p μ(|z|)

1 p

,

νN+j + 12

1 p

)

νN+j + 12

1 pνN+j

)

)

11 .Np -Type Functions with Hadamard Gaps in the Unit Ball .B

212

for all .z ∈ B with .1 −

≤ |z| ≤ 1 −

1 pk

1 p

k+ 12

, k ≥ 1. Then take a sequence .(qk ) of positive

1

integers satisfying .0 ≤ qk − pk+ 2 < 1, and for each .j = 1, 2, . . . , N, take a sequence 2 2 .(εj,ν ) satisfying .A qνN+j ε j,ν = 1. Choose a sequence of subsets .j,ν ⊂ S with the following property: for each nonnegative integer .ν, the set .∪N

=1 j,νN + is a maximal subset of .S which is d-separated by .Aεj,ν /2 and each .j,νN + . For each .i, j = 1, 2, . . . , N and .ν ≥ 0, we set 

Qi,νN+j (z) =

z, ξ qνN+j

.

ξ ∈j,νN+τ i (j )

and define hi,j (z) =

.

∞  Qi,νN+j (z) ν=0

μ(1 −

1 qνN+j

)

.

It is clear that .hi,j is in the Hadamard gap series, since for each .ν ≥ 0, we have

.

qνN+j q(ν−1)N+j

1

≥

p νN+ 2 p

(ν−1)N+ 12

+1

≥

pN > 1. 2

Furthermore, the homogeneous polynomials .Qi,νN+j are uniformly bounded by 2 as above. Then each .hi,j ∈ Hμ∞ by Theorem 11.4.2. Moreover, by some modifications of the previous arguments, it shows that for each .ν ≥ 0, .1 ≤ j ≤ N , and .z ∈ B with 1−

1

.

p

νN+j + 12

≤ |z| ≤ 1 −

1 pνN+j +1

,

C

p there exists .i ∈ {1, 2, . . . , N} such that .|hi,j (z)| ≥ μ(|z|) , where .Cp is some positive constant independent of the choice of .i, j , and z. Consequently,

N N   .

i=1 j =1

for all .1 −

1 p

k+ 12

≤ |z| ≤ 1 −

1 ,k pk+1

N  N  .

i=1 j =1

|hi,j (z)| ≥

Cp , μ(|z|)

≥ 1. Finally, we arrive to the following:

|gi,j (z)| + |hi,j (z)| ≥

C , μ(|z|)

for all .z ∈ B close enough to the boundary and for some positive constant C.

 

11.5 The Growth Rate in Hμ∞

213

Corollary 11.5.4 There exist some positive integer N and functions .fj ∈ Hα∞ (j = 1, 2, . . . , N) such that N  .

|fj (z)| 

j =1

1 , (1 − |z|2 )α

(z ∈ B).

Notes on Chapter 11 Hadamard gap series on different spaces of holomorphic functions in higher dimensions, namely, in the unit ball .B, have been extensively studied, e.g., see [10, 59, 89, 107] and references therein. Lemma 11.1.1 is given in [108]. Corollary 11.2.3 is an extension of the results of [74]. The result in [74] is a particular case of the assertion (i) in Theorem 11.3.1. The main results in Sect. 11.4 are from papers [47, 48]. Lemmas 11.5.1 and 11.5.2 are, respectively, Lemmas 2.2 and 2.3 in [12]. Recently, in [55], some simplifications to the formulas in Theorem 11.2.1 have been discovered. Namely, condition (i) can be replaced by

.

∞  Mk2 p+1

k=0

nk

< ∞,

and condition (iv) by

.

∞  L2k p+1

k=0

nk

< ∞.

N (p, q, s)-Type Spaces in the Unit Ball of Cn

.

.

12

In this chapter, a new class, which is a generalization of .Np and .Np -type spaces, is studied. Besides basic properties, several other topics are covered, including the distance between Bergman-type spaces and .N (p, q, s)-type spaces. The results and their proofs are mainly developed from the corresponding ones in the .Np spaces. Nevertheless, they have independent interests.

12.1

N (p, q, s) as a Functional Banach Space

Let dλ(z) =

.

dV (z) . (1 − |z|2 )n+1

This is an important Möbius-invariant measure, i.e.,  .

B

 f (z) dλ(z) =

B

f ◦ (z) dλ(z),

for each .f ∈ L1 (λ) and an automorphism . of .B. Let .p, q, s > 0. The .N (p, q, s)-space consists of holomorphic functions f on .B, for which  f  = sup

1/p



.

a∈B B

|f (z)| (1 − |z| ) (1 − |a (z)| ) p

2 q

2 ns

dλ(z)

< ∞.

The corresponding little space .N 0 (p, q, s), which is a subspace of .N (p, q, s), includes those .f ∈ N (p, q, s) with © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. H. Khoi, J. Mashreghi, Theory of Np Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-39704-2_12

215

12 .N (p, q, s)-Type Spaces in the Unit Ball of .Cn

216

 .

lim

|a|→1 B

|f (z)|p (1 − |z|2 )q (1 − |a (z)|2 )ns dλ(z) = 0.

When .p = 2, q = n + 1, .N (2, n + 1, s) coincides with the .Nns -space. In particular, by Theorem 7.6.1 in Chap. 7, when .p = 2, q = n + 1, and .s > 1, we have N (2, n + 1, s) = A−

.

n+1 2

(B).

Similar results hold for the little space. Recall that for .a ∈ B, .a denotes the automorphism of .B. For .0 < R < 1, we put BR = {z : |z| < R}

.

and D(a, R) = a ({z ∈ B : |z| < R}) = {z ∈ B : |a (z)| < R} .

.

First, we need the following result. Lemma 12.1.1 Let .p ≥ 1, and let .q, s > 0. Then the point evaluation .Kz : f → f (z) is −q a continuous linear functional on .N (p, q, s). Moreover, .N (p, q, s) ⊆ A p (B). Proof For each .f ∈ N (p, q, s) and .a0 ∈ B, we have  f  = sup

.

p

a∈B B

|f (z)|p (1 − |z|2 )q (1 − |a (z)|2 )ns dλ(z)



≥ D(a0 ,1/2)

|f (z)|p (1 − |z|2 )q (1 − |a0 (z)|2 )ns dλ(z)



|f (z)|p (1 − |z|2 )q dλ(z)

≥ Cn,s D(a0 ,1/2)

 = Cn,s

B1/2

|f (a0 (w))|p (1 − |a0 (w)|2 )q dλ(w)

(change variable z = a0 (w))  (1 − |a0 |2 )q (1 − |w|2 )q−n−1 = Cn,s |f (a0 (w))|p dV (w) |1 − a0 , w|2q B1/2  2 q ≥ Cq,s,n (1 − |a0 | ) |f (a0 (w))|p dV (w) B1/2

(1 − |a0 |2 )q |f (a0 )|p ≥ Cq,s,n

(because |f ◦ a0 (·)|p is subharmonic),

12.1 N (p, q, s) as a Functional Banach Space

217

where .Cn,s is a constant depending on n and s and .Cq,s,n and .Cq,s,n are some constants depending on .q, s, and n. This shows that for any .z ∈ B,

|f (z)| 

.

f  q

(1 − |z|2 ) p

,

which implies that the point evaluation is a continuous linear functional, as well as −q .N (p, q, s) ⊆ A p (B).

 Now we show that .N (p, q, s) is a functional Banach space when .p ≥ 1 and .q, s > 0. Theorem 12.1.2 Let .p ≥ 1 and .q, s > 0. The .N (p, q, s)-space is a functional Banach space. Proof Since .N (p, q, s) is a normed space with respect to the norm .·, it suffices to show the completeness of .N (p, q, s). Let .(fm ) be a Cauchy sequence in .N (p, q, s). From this, by Lemma 12.1.1, it follows that .(fm ) is a Cauchy sequence in the space .Hol(B), and hence it converges to some .f ∈ Hol(B). We show that .f ∈ N (p, q, s). Indeed, there exists a .0 ∈ N such that for all .m,  ≥ 0 , .fm − f  ≤ 1. Take and fix an arbitrary .a ∈ B; by Fatou’s lemma, we have  .

B

|f (z) − f0 (z)|p (1 − |z|2 )q (1 − |a (z)|2 )ns dλ(z) 

≤ lim

→∞ B

|f (z) − f0 (z)|p (1 − |z|2 )q (1 − |a (z)|2 )ns dλ(z)

≤ lim f − f0 p ≤ 1, →∞

which implies that  f − f0  = sup

.

a∈B B

|f (z) − f0 (z)|p (1 − |z|2 )q (1 − |a (z)|2 )ns dλ(z) ≤ 1,

and hence .f  ≤ 1 + f0  < ∞. By Lemma 12.1.1, the point evaluations on .N (p, q, s) are continuous linear functionals. Therefore, .N (p, q, s) is a functional Banach space.  − pq

Next, we show that when .p ≥ 1, .q > 0 and .s > 1, .N (p, q, s) = A precisely, we have the following result. Theorem 12.1.3 Let .p ≥ 1, .q, s > 0. If .s > 1 −

q−kp n , − pq

N (p, q, s). In particular, when .s > 1, .N (p, q, s) = A

(B). More

k ∈ (0, q/p], then .A−k (B) ⊆

.

(B).

12 .N (p, q, s)-Type Spaces in the Unit Ball of .Cn

218

Proof Suppose .p ≥ 1, .q > 0, and .s > 1 − q−kp n for some .k ∈ (0, q/p]. Then .q + ns − n − 1 − kp > −1, and hence, by formula (7.13) in Chap. 7, for each .a ∈ B, we have ⎧ ⎪ ⎪ ⎨bounded in B,

 .

B

(1 − |z|2 )q+ns−n−1−pk dV (z)  log 1 2 , 1−|a| ⎪ |1 − a, z|2ns ⎪ ⎩ (1 − |a|2 )q−ns−pk ,

if ns + pk < q, if ns + pk = q, if ns + pk > q,

which implies that there exists a positive constant C such that  .

sup(1 − |a|2 )ns a∈B

B

(1 − |z|2 )q+ns−n−1−pk dV (z) ≤ C. |1 − a, z|2ns

(12.1)

Let .f ∈ A−k (B). By (12.1), we have  f  = sup p

.

a∈B B

|f (z)|p (1 − |z|2 )q (1 − |a (z)|2 |)ns dλ(z)



= sup

a∈B B

≤ ≤

p |f |k

|f (z)|p (1 − |z|2 )pk (1 − |z|2 )q−pk (1 − |a (z)|2 )ns dλ(z) 

sup(1 − |a| )

2 ns

a∈B

B

(1 − |z|2 )q+ns−n−1−pk dV (z) |1 − a, z|2ns

p C|f |k ,

which shows that .A−k (B) ⊆ N (p, q, s). Furthermore, if .s > 1, then taking .k = − pq

have .A result.

12.2

q p,

we

(B) ⊆ N (p, q, s). Combining this fact with Theorem 12.1.1, we get the desired



The Closure of Polynomials in N (p, q, s)-Type Spaces

We start by showing that .N 0 (p, q, s) by itself is a Banach space. Lemma 12.2.1 Let .p ≥ 1 and .q, s > 0. The little space .N 0 (p, q, s) is a closed subspace of .N (p, q, s), and hence, it is a Banach space. Proof Since .N 0 (p, q, s) is a subspace of .N (p, q, s), we show .N 0 (p, q, s) is complete. If .(fn ) is a Cauchy sequence in .N 0 (p, q, s), then by Theorem 12.1.2, there exists a limit .f ∈ N (p, q, s) of .(fn ). For any .ε > 0, we can find an .N ∈ N, such that f − fn 
N . Since .fn0 ∈ N 0 (p, q, s), there exists a .δ ∈ (0, 1) such that whenever .δ < |a| < 1,  sup

.

δ 0. The space .N (p, q, s) contains the set of polynomials if and only if .ns + q > n. Proof Necessity Assume in contrary that .ns + q ≤ n. Then .τ = n − ns − q ≥ 0. We show that the constant function .f (z) ≡ 1 does not belong to .N (p, q, s). Indeed, we have  f p = sup

.

a∈B B

(1 − |z|2 )q (1 − |a (z)|2 )ns dλ(z)

= sup(1 − |a|2 )ns a∈B



 B

(1 − |z|2 )q+ns−n−1 dV (z) |1 − a, z|2ns

1 dV (z) (taking a = 0) 2 τ +1 B (1 − |z| )  1  1 r 2n−1 1 dr  dr = ∞,  2 1+τ 1+τ 0 (1 − r ) 1/2 (1 − r) ≥

which is a contradiction. Sufficiency Combining Lemma 12.2.2 and Theorem 12.2.3 yields the desired result.

12.3

The Space N∗ (p, q, s)

The invariant Green’s function is defined as .G(z, a) = g(a (z)), where



12.3 The Space N∗ (p, q, s)

223

g(z) =

.

n+1 2n



1

|z|

(1 − t 2 )n−1 t −2n+1 dt.

The following property of g is important in the sequel. Lemma 12.3.1 Let .n ≥ 2 be an integer. Then there are positive constants .C1 and .C2 such that for all .z ∈ B\{0}, C1 (1 − |z|2 )n |z|−2(n−1) ≤ g(z) ≤ C2 (1 − |z|2 )n |z|−2(n−1) .

.

Proof We have .

g(z)

lim

|z|→0 (1 − |z|2 )n |z|−2(n−1)

=

n+1 4n(n − 1)

and .

lim

|z|→1

g(z) n+1 . = 4n2 (1 − |z|2 )n |z|−2(n−1)

From the above two identities and the continuity of .g(z), the desired result follows.



For .p ≥ 1 and .q, s > 0 and .n ≥ 2, define the .N∗ (p, q, s)-type space and a corresponding little subspace by N∗ (p, q, s) =

  p f ∈ H (B) : f ∗ = sup |f (z)|p (1 − |z|2 )q Gs (z, a) dλ(z) < ∞

.

a∈B B

and .

N∗0 (p, q, s) =

  f ∈ N∗ (p, q, s) : lim |f (z)|p (1 − |z|2 )q Gs (z, a) dλ(z) = 0 . |a|→1 B

We note, by Lemma 12.3.1, that .·  ·∗ , which means that .N∗ (p, q, s) ⊆ N (p, q, s). Combining this fact with the proof of Lemma 12.1.1, Theorem 12.1.2, and Lemma 12.2.1, we have the following result. Theorem 12.3.2 For .p ≥ 1, and .q, s > 0, and .n ≥ 2, the following assertions are true. (i) .N∗ (p, q, s) is a functional Banach space.

12 .N (p, q, s)-Type Spaces in the Unit Ball of .Cn

224

(ii) .N∗0 (p, q, s) is a closed subspace of .N∗ (p, q, s). − pq

(iii) .N∗ (p, q, s) ⊆ A

(B).

As in the .N (p, q, s) space, in general, not every .N∗ (p, q, s) contains all the polynomials. For instance, by Corollary 12.2.4 and the fact that . ·    · ∗ , we see that in case .ns + q ≤ n, the function .f ≡ 1 does not belong to .N∗ (p, q, s). However, we have the following result. Lemma 12.3.3 Let .p ≥ 1, let .q, s > 0, and let .n ≥ 2. Then the space .N∗ (p, q, s) n contains the set of all polynomials if and only if .ns + q > n and .s < n−1 . Proof Necessity Consider the constant function .f (z) ≡ 1. By Lemma 12.3.1, we have 

p

f ∗ = sup

.

a∈B B

(1 − |z|2 )q Gs (z, a) dλ(z)

 ≥  

B

B

(1 − |z|2 )q Gs (z) dλ(z) (1 − |z|2 )q

 =

B

(1 − |z|2 )ns dλ(z) |z|2(n−1)s

(1 − |z|2 )q+ns−n−1 dV (z) |z|2(n−1)s

= I1 + I2 , where  I1 =

.

B1/2

(1 − |z|2 )q+ns−n−1 dV (z) |z|2(n−1)s

and  I2 =

.

B\B1/2

(1 − |z|2 )q+ns−n−1 dV (z). |z|2(n−1)s

There are now two cases to consider. ns + q ≤ n. In this case,

Case I:

.

 I2 



.

B\B1/2

(1 − |z|2 )q+ns−n−1 dV (z) 

which is a contradiction.

1

1/2

1 dr = ∞, (1 − r)n+1−q−ns

12.3 The Space N∗ (p, q, s)

Case II:

s≥

n n−1 .

.

225

In this case, 

I1 



1

.

|z|2(n−1)s

B1/2

dV (z) 

1/2

0

1 dr = ∞, r 2(n−1)s−2n+1

which contradicts the assumption that .f ∈ N∗ (p, q, s). n . Let P be a polynomial defined on .B. Then Sufficiency Suppose .ns + q > n and .s < n−1 we have  p .P ∗ = sup |P (z)|p (1 − |z|2 )q Gs (z, a) dλ(z) a∈B B



 sup

a∈B B

(1 − |z|2 )q Gs (z, a) dλ(z)



 sup

a∈B B

(1 − |a (z)|2 )ns dλ(z) |a (z)|2(n−1)s

(1 − |z|2 )p



= sup

a∈B B

(1 − |a (w)|2 )q 

= sup(1 − |a|2 )q

(1 − |w|2 )ns dλ(w) (change variable w = a (z)) |w|2(n−1)s

(1 − |w|2 )q+ns−n−1 dV (w). |w|2(n−1)s |1 − a, w|2q

B

a∈B

(P is bounded on B)

For each .a ∈ B, consider  .

B

(1 − |w|2 )q+ns−n−1 dV (w) = I1,a + I2,a , |w|2(n−1)s |1 − a, w|2q

where  I1,a =

(1 − |w|2 )q+ns−n−1 dV (w) |w|2(n−1)s |1 − a, w|2q

.

B1/2

and  I2,a =

.

B\B1/2

(1 − |w|2 )q+ns−n−1 dV (w). |w|2(n−1)s |1 − a, w|2q

For .I1,a , by the proof of necessary part, we have  I1,a 

.

B1/2

1 dV (w) < ∞ |w|2(n−1)s

12 .N (p, q, s)-Type Spaces in the Unit Ball of .Cn

226

independent of the choice of a, which implies that .sup(1 − |a|2 )q I1,a < ∞. a∈B

For .I2,a , by formula (7.13) in Chap. 7, we have  I2,a 

.

B\B1/2

 



B

(1 − |w|2 )q+ns−n−1 dV (w) |w|2(n−1)s |1 − a, w|2q

(1 − |w|2 )q+ns−n−1 dV (w) |1 − a, w|2q

⎧ ⎪ ⎪ ⎨bounded in B,

1 log 1−|a| 2, ⎪ ⎪ ⎩ 2 (1 − |a| )ns−q ,

if ns > q if ns = q if ns < q,

which shows that .sup(1 − |a|2 )q I2,a < ∞. Thus, .P ∗ < ∞, and the proof is complete. a∈B



Lemma 12.3.3 leads us to the following description of .N (p, q, s)-spaces by the invariant Green’s function. Theorem 12.3.4 Let .p ≥ 1, let .q, s > 0, and let .n ≥ 2. If .s < n N∗ (p, q, s). In particular, if .1 < s < n−1 , then − pq

N (p, q, s) = N∗ (p, q, s) = A

.

n n−1 ,

then .N (p, q, s) =

(B).

Proof It is clear .N∗ (p, q, s) ⊆ N (p, q, s). We show that .N (p, q, s) ⊆ N∗ (p, q, s), n whenever .s < n−1 . Take .f ∈ N∗ (p, q, s). For each .a ∈ B, we have  .

   =

B

B

B

|f (z)|p (1 − |z|2 )q Gs (z, a) dλ(z) |f (z)|p (1 − |z|2 )q

(1 − |a (z)|2 )ns dλ(z) |a (z)|2(n−1)s

|f (a (w))|p (1 − |a (w)|2 )q

(change variable w = a (z)) = J1,a + J2,a , where

(1 − |w|2 )ns dλ(w) |w|2(n−1)s

12.3 The Space N∗ (p, q, s)

227

 J1,a =

.

B1/2

|f (a (w))|p (1 − |a (w)|2 )q

(1 − |w|2 )ns dλ(w) |w|2(n−1)s

and  J2,a =

.

B\B1/2

|f (a (w))|p (1 − |a (w)|2 )q

(1 − |w|2 )ns dλ(w). |w|2(n−1)s

For .J2,a , we have  J2,a 

.

 ≤  =

B\B1/2

B

B

|f (a (w))|p (1 − |a (w)|2 )q (1 − |w|2 )ns dλ(w)

|f (a (w))|p (1 − |a (w)|2 )q (1 − |w|2 )ns dλ(w) |f (z)|p (1 − |z|2 )q (1 − |a (z)|2 )ns dλ(z) ≤ f p .

For .J1,a , taking into account .s < Lemma 12.3.3, we have p



J1,a ≤ |f |q/p

.

  f p

B1/2

n n−1 ,

(1 − |w|2 )ns−n−1 dV (w) |w|2(n−1)s 1

B1/2

by Lemma 12.1.1, and the proof of

|w|2(n−1)s

dV (w) ≤ Mf p ,

for some .M > 0, which is independent of a choice of a. Combining the two estimates yields  .

B

|f (z)|p (1 − |z|2 )q Gs (z, a) dλ(z)  f p ,

which gives .f ∗  f . In particular, when .1 < s
0, |f (a (w))| ≥

.

δ , 2

(|w| < r).

Hence, 

p

f ∗ = sup

.

a∈B B

|f (z)|p (1 − |z|2 )q Gs (z, a) dλ(z)



≥  

B

B

|f (z)|p (1 − |z|2 )q Gs (z, a0 ) dλ(z) |f (z)|p (1 − |z|2 )q

 =

B

(1 − |a0 (z)|2 )ns dλ(z) |a0 (z)|2(n−1)s

|f (a0 (w)|p (1 − |a0 (w)|2 )q

(1 − |w|2 )ns−n−1 dV (w) |w|2(n−1)s

(change variable z = a (w))  (1 − |w|2 )ns−n−1 ≥ |f (a0 (w)|p (1 − |a0 (w)|2 )q dV (w) |w|2(n−1)s |w| 1− qn , holds. Moreover, −q

if .s > 1, then by Theorem 12.1.3, .N (p, q, s) = A p (B), which was already studied in the previous chapter. Hence, it remains to study the case .s ≤ 1. We keep the notations Mk = sup |Pnk (ξ )|

.

ξ ∈S

and  Lk,p =

.

S

1/p |Pnk (ξ )|p dσ (ξ ) ,

(p ≥ 1),

where .dσ is the normalized surface measure on .S, that is, .σ (S) = 1. These quantities clearly are well-defined for each .k ≥ 0 and .p ≥ 1.   Theorem 12.4.1 Let .p ≥ 1, let .q > 0, and let .max 0, 1 − qn < s ≤ 1. Suppose that f (z) =

∞ 

.

Pnk (z)

k=0

is a series with Hadamard gaps. Consider the following statements:

(i)

∞  .

k=0

⎛ 1 2k(ns+q−n)

⎝



⎞ Mj ⎠ < ∞. p

2k ≤nj