Geometric Qp Functions (Frontiers in Mathematics)

Frontiers in Mathematics Advisory Editorial Board Luigi Ambrosio (Scuola Normale Superiore, Pisa) Leonid Bunimovich (...

Author: Jie Xiao

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Frontiers in Mathematics

Advisory Editorial Board

Luigi Ambrosio (Scuola Normale Superiore, Pisa) Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Ecole Normale Supérieure, Paris) Gennady Samorodnitsky (Cornell University, Rhodes Hall) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg)

Jie Xiao

Geometric QP Functions

Birkhäuser Verlag Basel . Boston . Berlin

Author: Jie Xiao Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL A1C 5S7 Canada e-mail: [email protected]

Narayanaswami Vanaja Department of Mathematics University of Mumbai Vidyanagari Marg Mumbay 400098 India e-mail: [email protected]

Christian Lomp Departamento de Matemática Pura Faculdade de Ciências Universidade do Porto Rua Campo Alegre 687 4169-007 Porto Portugal e-mail: [email protected]

Robert Wisbauer Institute of Mathematics +HLQULFK+HLQH8QLYHUVLW\'VVHOGRUI Universitätsstr. 1 40225 Düsseldorf Germany e-mail: [email protected]

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A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek 'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD¿H detailed bibliographic data is available in the Internet at .

,6%1%LUNKlXVHU9HUODJ%DVHO±%RVWRQ±%HUOLQ This work is subject to copyright. All rights are reserved, whether the whole or part RIWKHPDWHULDOLVFRQFHUQHGVSHFL¿FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVH RILOOXVWUDWLRQVUHFLWDWLRQEURDGFDVWLQJUHSURGXFWLRQRQPLFUR¿OPVRULQRWKHUZD\V and storage in data banks. For any kind of use permission of the copyright owner must be obtained. %LUNKlXVHU9HUODJ32%R[&+%DVHO6ZLW]HUODQG Part of Springer Science+Business Media &RYHUGHVLJQ%LUJLW%ORKPDQQ=ULFK6ZLW]HUODQG Printed on acid-free paper produced from chlorine-free pulp. TCF f Printed in Germany ,6%1 H,6%1 ,6%1 ZZZELUNKDXVHUFK

To Xianli and Sa

Contents Preface 1 Preliminaries 1.1 Background . . . . . . . . . . . . . . . . . . 1.2 Logarithmic Conformal Mappings . . . . . . 1.3 Conformal Domains and Superpositions . . 1.4 Descriptions via Harmonic Majorants . . . 1.5 Regularity for the Euler–Lagrange Equation 1.6 Notes . . . . . . . . . . . . . . . . . . . . .

ix

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1 1 7 12 16 19 22

2 Poisson versus Berezin with Generalizations 2.1 Boundary Value and Brownian Motion . . . . . . . 2.2 Derivative-free Module via Poisson Extension . . . 2.3 Derivative-free Module via Berezin Transformation 2.4 Mixture of Derivative and Quotient . . . . . . . . . 2.5 Dirichlet Double Integral without Derivative . . . . 2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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25 25 29 32 35 40 45

3 Isomorphism, Decomposition and Discreteness 3.1 Carleson Measures under an Integral Operator 3.2 Isomorphism to a Holomorphic Morrey Space . 3.3 Decomposition via Bergman Style Kernels . . . 3.4 Discreteness by Derivatives . . . . . . . . . . . 3.5 Characterization in Terms of a Conjugate Pair 3.6 Notes . . . . . . . . . . . . . . . . . . . . . . .

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47 47 52 57 64 67 71

4 Invariant Preduality through Hausdorﬀ Capacity 4.1 Nonlinear Integrals and Maximal Operators 4.2 Adams Type Dualities . . . . . . . . . . . . 4.3 Quadratic Tent Spaces . . . . . . . . . . . . 4.4 Preduals under Invariant Pairing . . . . . . 4.5 Invariant Duals of Vanishing Classes . . . .

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73 73 81 84 89 96

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viii

Contents 4.6

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Cauchy Pairing with Expressions and Extremities 5.1 Background on Cauchy Pairing . . . . . . . . 5.2 Cauchy Duality by Dot Product . . . . . . . 5.3 Atom-like Representations . . . . . . . . . . . 5.4 Extreme Points of Unit Balls . . . . . . . . . 5.5 Notes . . . . . . . . . . . . . . . . . . . . . .

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107 107 113 118 123 133

6 As Symbols of Hankel and Volterra Operators 6.1 Hankel and Volterra from Small to Large Spaces . 6.2 Carleson Embeddings for Dirichlet Spaces . . . . . 6.3 More on Carleson Embeddings for Dirichlet Spaces 6.4 Hankel and Volterra on Dirichlet Spaces . . . . . . 6.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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135 135 138 144 150 159

7 Estimates for Growth and Decay 7.1 Convexity Inequalities . . . . . . . . 7.2 Exponential Integrabilities . . . . . . 7.3 Hadamard Convolutions . . . . . . . 7.4 Characteristic Bounds of Derivatives 7.5 Notes . . . . . . . . . . . . . . . . .

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163 163 169 177 182 188

8 Holomorphic Q-Classes on Hyperbolic Riemann Surfaces 8.1 Basics about Riemann Surfaces . . . . . . . . . . . . 8.2 Area and Seminorm Inequalities . . . . . . . . . . . 8.3 Intermediate Setting – BMOA Class . . . . . . . . . 8.4 Sharpness . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Limiting Case – Bloch Classes . . . . . . . . . . . . . 8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

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191 191 195 201 207 215 223

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Bibliography

227

Index

239

Preface The aim of the book Geometric Qp Functions is to document the rich structure of the holomorphic Q functions which are geometric in the sense that they transform naturally under conformal mappings, with particular emphasis on the last few years’ development based on interaction between geometrical function and measure theory and other branches of mathematical analysis, including complex variables, harmonic analysis, potential theory, functional analysis, and operator theory. The book comprises eight chapters in which some results appear for the ﬁrst time. The ﬁrst chapter begins with a motive and a very brief review of the mostly standard characterizations of holomorphic Q functions presented in the author’s monograph — Springer’s LNM 1767: Holomorphic Q Classes — followed by some further preliminaries on logarithmic conformal maps, conformal domains and superpositions and harmonic majorants with an application to Euler–Lagrange equations. The second chapter gives function-theoretic characterizations by means of Poisson extension and Berezin transform with two more generalized variants. The third chapter takes a careful look at isomorphism, decomposition, and discreteness of spaces via equivalent forms of the generalized Carleson measures. The fourth chapter discusses invariant preduality through Hausdorﬀ capacity, which is a useful tool to classify negligible sets for various ﬁne properties of functions. The ﬁfth chapter develops some essential properties of the Cauchy dualities via both weak factorizations and extreme points of the target function spaces. The sixth chapter shows particularly that each holomorphic Q function can be treated as a symbol of the holomorphic Hankel and Volterra operators acting between two Dirichlet spaces. The seventh chapter deals with various size estimates involving functions and their exponentials and derivatives. Finally, the eighth chapter handles how much of the basic theory of holomorphic Q functions can be carried over the hyperbolic Riemann surfaces by sharpening the area and isoperimetric inequalities and settling the limit spaces. Although this book may be more or less regarded as a worthy sequel to the previously-mentioned monograph, it is essentially self-contained. And so, without reading that monograph, readers can understand the contents of this successor, once they are familiar with some basic facts on geometric function-measure theory and complex harmonic-functional analysis. For further background, each chapter ends with brief notes on the history and current state of the subject. Readers may

x

Preface

consult those notes and go further to study the references cited by this book for more information. As is often the case, the completion of a book is strongly inﬂuenced by some organizations and individuals. This book has been no exception. Therefore, the author would like to deliver a word of thanks to: Natural Sciences and Engineering Research Council of Canada as well as Faculty of Science, Memorial University of Newfoundland, Canada, that have made this book project possible; next, a number of people including (in alphabetical order): D. R. Adams (University of Kentucky), A. Aleman (Lund University), R. Aulaskari (University of Joensuu), H. Chen (Nanjing Normal University), K. M. Dyakonov (University of Barcelona), P. Fenton (University of Otago), T. Hempﬂing (Birkh¨auser Verlag AG), M. Milman (Florida Atlantic University), M. Pavlovic (University of Belgrade), J. Shapiro (Michigan State University), A. Siskakis (University of Thessaloniki), K. J. Wirths (Technical University of Braunschweig), Z. Wu (University of Alabama), G. Y. Zhang (Polytechnic University of New York), R. Zhao (State University of New York at Brockport) and K. Zhu (State University of New York at Albany), who have directly or indirectly assisted in the preparation of this book. Last but not least, the author’s family, who the author owes a great debt of gratitude for their understanding and moral support during the course of writing.

St. John’s Fall 2005 – Summer 2006

J. Xiao [email protected]

Chapter 1

Preliminaries Our major goal in this chapter is to deal with some of the necessary preliminary results motivating a further study of the holomorphic Q classes via the following ﬁve sections: • • • • •

Background; Logarithmic Conformal Mappings; Conformal Domains and Superpositions; Descriptions via Harmonic Majorants; Regularity for the Euler–Lagrange Equation.

1.1 Background In order to make this book self-contained and accessible, we start with some notation and terminology, provide with the readers a motive from geometric function theory, and look over several principal properties presented in the monograph: Holomorphic Q Classes, LNM 1767, Springer-Verlag, 2001. First of all, let us have a look at the concept of conformal radius which plays an important role in geometric function theory. For every simply connected domain Ω on the ﬁnite complex plane C whose boundary ∂Ω contains at least two points, denote by F (Ω) the class of all univalent functions (conformal mappings) deﬁned on Ω whose images are subsets of the open unit disk D = {z ∈ C : |z| < 1} with boundary T = {z ∈ C : |z| = 1} such that f (z0 ) = 0 for a ﬁxed point z0 ∈ Ω. Then there is an f0 ∈ F(Ω) such that sup |f (z0 )| = |f0 (z0 )|.

f ∈F (Ω)

This extremal function f0 is called a Riemann mapping. Since it is unique up to a rotation, we can deﬁne −1 rΩ (z0 ) = |f0 (z0 )|

2

Chapter 1. Preliminaries

and call it the conformal radius of Ω at the point z0 . By the superposition f0 with a standard M¨ obius transform of D: w−z σw (z) = , z, w ∈ D, 1 − wz ¯ yields that f (z, w) = σf0 (z) (f0 (w)) =

f0 (z) − f0 (w) 1 − f0 (z)f0 (w)

,

z, w ∈ Ω,

is a Riemann mapping with f (z, z) = 0. Consequently, the conformal radius at an arbitrary point z ∈ Ω can be evaluated by rΩ (z) =

1 − |f0 (z)|2 . |f0 (z)|

Accordingly, if f −1 : D → Ω stands for the inverse of f , then f −1 (0) = z0 and rΩ f −1 (z) = (1 − |z|2 )|(f −1 ) (z)|, z ∈ D. Even more interestingly, the conformal radius also appears in the study of geometric elliptic equations. To see this, suppose Ω ⊂ C is a hyperbolic domain, a planar domain whose boundary ∂Ω contains at least two points. Consider the Liouville equation ∂2 u(z) = 4e2u(z) , ∆u(z) = 4 ∂z∂ z¯ where ∂ ∂ ∂ ∂ ∂ ∂ = 2−1 −i = 2−1 +i and for z = x + iy. ∂z ∂x ∂y ∂ z¯ ∂x ∂y If

U (z) = sup{u(z) : ∆u(z) = 4e2u(z) },

z ∈ Ω,

stands for the maximal solution of the Liouville equation, then RΩ (z) = e−U(z) is called the hyperbolic radius of Ω at z. When Ω is simply connected and f : Ω → D is a Riemann mapping, the maximal solution U exists and is given by the following formula: 1 − |f (z)|2 = − log rΩ (z), z ∈ Ω. U (z) = − log |f (z)| It is worth pointing out that RΩ (z) provides a complete, conformally ﬂat metric, called hyperbolic or Poincar´e metric, on Ω of constant negative curvature whose line element is |dz| = eU(z) |dz| ds = RΩ (z)

1.1. Background

3

and whose scalar curvature is determined by ∆ log eU(z) = −e−2U(z) ∆U (z) = −4. e2U(z) Moreover, if g : D → Ω is a universal covering map, then RΩ g(z) = (1 − |z|2 )|g (z)|, z ∈ D K =−

and hence the Bloch semi-norm: sup (1 − |z|2 )|g (z)| z∈D

is independent of the covering but dependent only on the domain. On the other hand, we should note that any hyperbolic domain possesses Green’s function. More precisely, for z, w ∈ Ω, let − log |z − w| be the fundamental singularity of the Laplace operator ∆. Then gΩ (z, w), the Green function of Ω, is determined by two conditions: (i) gΩ (z, w)+log |z −w| is real-valued harmonic on Ω and continuous on Ω∪∂Ω; (ii) gΩ (z, w) = 0 for z ∈ ∂Ω. Here and henceforth, log denotes an abbreviation of the logarithmic function with base e. Consequently, sΩ (z) = exp lim gΩ (z, w) + log |z − w| w→z

is called the harmonic radius of Ω at z. Note that if Ω is simply connected, then the Green function gΩ (z, w) = − log |f (z)| where f : Ω → D is a Riemann mapping with f (w) = 0, and hence sΩ (z) = rΩ (z). In this case, by the maximum principle and the Koebe 1/4 Theorem, we can dominate rΩ (z) via the distance of z to ∂Ω from above and below: 4−1 rΩ (z) ≤ d(z, ∂Ω) = inf |w − z| ≤ rΩ (z). w∈∂Ω

A further estimate of rΩ (z) can be worked out by the following a priori estimate for gΩ (·, ·). Example 1.1.1. Let p ∈ [0, ∞), Γ(·) be the Gamma function, and dm stand for the two-dimensional Lebesgue measure element on C. Suppose Ω ⊂ C is a simply connected hyperbolic domain. Then 2 p 2p gΩ (z, w) dm(w) ≤ m Ω , π rΩ (z) ≤ Γ(p + 1) Ω where equality occurs when Ω is a disk centered at z. Equivalently, 2 p 2p |f |2 − log |σζ | dm ≤ |f |2 dm π (1 − |ζ|2 )|f (ζ)| ≤ Γ(p + 1) D D holds for any surjectively conformal mapping f : D → Ω with f (ζ) = z, where equality holds when f is the identity.

4

Chapter 1. Preliminaries

Proof. Note that if f : D → Ω is conformal and onto, then gΩ (z, w) = gΩ f (ζ), f (η) = − log |σζ (η)|,

z = f (ζ), w = f (η).

So it suﬃces to check the second part of Example 1.1.1. To this end, let ∞

f (η) =

η ∈ D,

aj η j ,

j=0

be the Taylor expansion of f at 0. Then 2 Ep (f, 0) = |f (ζ)|2 (− log |η|)p dm(η) D

=

2π

∞

j 2 |aj |2

j=1

1

r2(j−1) (− log r)p rdr.

0

This formula easily implies 2 Ep (f, 0) ≥ 2π|a1 |2

1

r(− log r)p dr = 2−p πΓ(p + 1)|f (0)|2 .

0

Meanwhile, integrating by parts, we derive 2 Ep (f, 0)

= ≤

π

∞

2

j|aj |

j=1 ∞

πp

j|aj |2

1

=

0 ∞

2−p πΓ(p + 1) 2−p Γ(p + 1)

1

r(− log r)p−1 dr

j=1

=

(− log r)p dr2j

0

j|aj |2

j=1

D

|f (η)|2 dm(η).

Accordingly, 2−p Γ(p + 1)π|f (0)|2

≤ ≤

|f (η)|2 (− log |η|)p dm(η) −p 2 Γ(p + 1) |f (η)|2 dm(η). Ω

D

Substituting f ◦ σζ (η) for f in the last estimate, we see the desired assertion right away. Obviously, the equalities over there are valid for f (ζ) = ζ and Ω = D.

1.1. Background

5

These observations motivate the main issue of this book. From now on, besides the previously-introduced notation, we will use H as the class of all holomorphic mappings from D to C. By Ep (f, w) =

D

2

|f (z)|

12 p − log |σw (z)| dm(z)

we mean the weighted energy integral associated with p ∈ [0, ∞), f ∈ H and w ∈ D. Using this term, we can symbolically write out a holomorphic Q space, as follows. Deﬁnition 1.1.2. For p ∈ [0, ∞) let Qp be the class of all f ∈ H such that

f Qp = sup Ep (f, w) < ∞. w∈D

Each Qp enjoys a large number of useful properties and has many applications; in particular, it is a dual Banach space under the norm Ep (f ) + |f (0)|, and Aut(D), the group of all conformal mappings from D to itself, acts on Qp via the isometry: Ep (f ◦ φ) = Ep (f ) for any φ ∈ Aut(D). Perhaps, more important is that the spaces Qp , p ∈ [0, ∞), form a nested scale of conformally invariant function spaces, but also bear natural connections to other such spaces as stated below: First, if p = 0, Qp = D, the Dirichlet space of all f ∈ H obeying

f D =

D

12 |f (z)| dm(z) < ∞.

2

Second, if p = 1, then Qp = BMOA, the John–Nirenberg space of all analytic (holomorphic) functions of bounded mean oscillation; that is, f ∈ BMOA if and only if f ∈ H and its nontangential limit, identiﬁed with f , exists a.e. on the unit circle T = {z ∈ C : |z| = 1} and satisﬁes |dζ| < ∞,

f BMOA = sup |I|−1 |f (ζ) − fI | 2π I⊆T I where the supremum is taken over all open subarcs I ⊆ T with |dζ| |dζ| and fI = |I|−1 f (ζ) . |I| = 2π 2π I I Third, if p > 1, then Qp = B, the Bloch space of those f ∈ H satisfying

f B = sup (1 − |z|2 )|f (z)| < ∞. z∈D

Obviously, the cases p ∈ (0, 1) are of independent interest, and worth being further studied. The forthcoming essential characterizations of Qp , whose proofs can be found in [Xi3], will help us with our later development of the Qp theory.

6

Chapter 1. Preliminaries

Theorem 1.1.3. Let p ∈ (0, 1) and f ∈ H. Then the following statements are equivalent: (i) f ∈ Qp . (ii) f Qp ,1 = supw∈D Fp (f, w) < ∞, where Fp (f, w) =

12 p |f (z)|2 |σw (z)|−2 − 1 dm(z) < ∞.

(iii) sup D

w∈D

(iv) sup p

1

0

w∈D

D

12 p |f (z)|2 1 − |σw (z)|2 dm(z) .

|σw (z)|
|f (z)|2 dm(z) (1 − r)p−1 dr

12 < ∞.

(v) For any natural number n ≥ 2, i.e., n − 1 ∈ N, the n-th derivative f (n) of f satisﬁes sup w∈D

(vi) For f (z) = sup w∈D

D

|f

∞ k=0

(n)

12 2 p 2 2n−2 (z)| 1 − |σw (z)| (1 − |z| ) dm(z) < ∞. 2

ak z k with ak = f (k) (0)/k!,

∞ n (1 − |w|2 )p (m + 1)am+1 Γ(n − m + p) n−m 2 w ¯ (n + 1)p+1 m=0 (n − m)! n=0

12 < ∞.

(vii) dµf,p (z) = |f (z)|2 (1 − |z|2 )p dm(z) is a p-Carleson measure on D; that is,

f Qp ,2 = sup

−p 1 |I| µf,p S(I) 2 < ∞,

S(I)⊆D

where the supremum ranges over all Carleson boxes S(I) = {rξ ∈ D : 1−|I| ≤ r < 1, ξ ∈ I} based on open subarcs I of T. 12 p (viii) sup E1 (f, u)dm(u) 1 − |σw (z)|2 (1 − |z|2 )−2 dm(z) < ∞. w∈D

D

D

(ix) There are a complex number c, an inner function I and an outer function O such that f = cIO with O ∈ Qp and sup w∈D

D

12 p |O(z)|2 1 − |I(z)|2 1 − |σw (z)|2 (1 − |z|2 )−2 dm(z) < ∞.

1.2. Logarithmic Conformal Mappings

7

(x) The nontangential limit f (ξ) exists for a.e. ξ ∈ T and satisﬁes 12 |f (ζ) − f (η)|2 |dζ||dη| < ∞,

f Qp ,3 = sup |I|−p |ζ − η|2−p I⊆T I I where the supremum is taken over all open subarcs I ⊆ T. (xi) There are functions f1 , f2 ∈ H such that f = f1 + f2 and f1 , f2 are essentially bounded on T and obey the ﬁniteness condition of (x). Remark 1.1.4. The equivalence amongst (i), (ii), (iv), (v), (vi) and (vii) is also valid for p ∈ [1, ∞). In addition, for f ∈ Qp , p ∈ (0, 1), we have

f Qp ≈ f Qp ,1 ≈ f Qp,2 ≈ f Qp ,3 . Here and thereafter, X ≈ Y means both X Y and Y X; that is, there are positive constants c1 and c2 such that both X ≤ c1 Y and Y ≤ c2 X hold. There is no general procedure known concerning the investigation of all classes Qp . Such a procedure seems unlikely to exist, given the rich variety of analytic, geometric, and even probabilistic phenomena which can be modeled by these spaces. However, we will try to get a better understanding of Qp through

· Qp , · Qp ,k , k = 1, 2, 3 and their variations.

1.2 Logarithmic Conformal Mappings The preceding Theorem 1.1.3 yields that log(1 − z) is a typical member of all Qp , p ∈ (0, ∞), and accordingly that log f lies in p∈(0,∞) Qp for any univalent function f : D → C \ {0}. In what follows, we will obtain more about this fact via searching for the best estimate based on symmetrization. Deﬁnition 1.2.1. Given a domain Ω ⊆ C. We call it circularly symmetric provided that for any r ∈ (0, ∞) one of the following three conditions is valid: (i) Ω ∩ {z ∈ C : |z| = r} = ∅; (ii) Ω ∩ {z ∈ C : |z| = r} = {z ∈ C : |z| = r}; (iii) Ω ∩ {z ∈ C : |z| = r} is a single arc on {z ∈ C : |z| = r} which contains {r} and is symmetric about the real line R. Moreover, the circular symmetrization of Ω is the domain Ω∗ determined in the following way: (iv) If {θ ∈ [−π, π] : reiθ ∈ Ω} = ∅, then Ω∗ ∩ {z ∈ C : |z| = r} = ∅; (v) If {θ ∈ [−π, π] : reiθ ∈ Ω} = [−π, π], then Ω∗ ∩ {z ∈ C : |z| = r} = {z ∈ C : |z| = r};

8

Chapter 1. Preliminaries

(vi) If {θ ∈ [−π, π] : reiθ ∈ Ω} is neither ∅ nor [−π, π], and if its one-dimensional Lebesgue measure is α, then Ω∗ ∩ {z ∈ C : |z| = r} equals {reiθ : 2|θ| < α}; (vii) If 0 ∈ Ω, then 0 ∈ Ω∗ and vice versa. Clearly, Ω∗ is a circularly symmetric domain, and Ω is circularly symmetric if and only if Ω = Ω∗ . At the same time, Ω∗ is simply connected whenever Ω is simply connected; see also [Hay, p. 115]. Lemma 1.2.2. Let Ω ⊂ C be a simply connected hyperbolic domain and ﬁx z0 ∈ Ω. Let gΩ (·, z0 ) and gΩ∗ (·, |z0 |) be the Green functions of Ω and Ω∗ with poles at z0 and |z0 | respectively. Set gΩ (z, z0 ) = 0 for z ∈ C \ Ω and gΩ∗ (z, |z0 |) = 0 for z ∈ C \ Ω∗ . Then: (i) For any r ∈ (0, ∞), sup gΩ (rξ, z0 ) ≤ sup gΩ∗ (rξ, |z0 |).

ξ∈T

ξ∈T

(ii) For any r ∈ (0, ∞) and any p ∈ [1, ∞), p p gΩ (rξ, z0 ) |dξ| ≤ gΩ∗ (rξ, |z0 |) |dξ|. T

T

(iii) Ω = Ω∗ provided z0 > 0 and p p gΩ (rξ, z0 ) |dξ| = gΩ∗ (rξ, |z0 |) |dξ|, T

T

for some r ∈ (0, ∞) and some p ∈ (1, ∞). Proof. (i) and (ii) are consequences of [Bae1, Theorem 5], and (iii) is a corollary of [EsSh, Theorem 1]. Theorem 1.2.3. Let f : D → C \ {0} be univalent with f (0) = 1. Then: (i) For p ∈ (0, ∞),

log f Qp

≤ 8π

1

0

1 + r p −1 log r dr 1−r

12 .

(ii) For p ∈ [1, ∞),

log f Qp

≤ 16

0

1

T

1 p |dξ| log rdr |1 − r2 ξ 2 |2 r

with equality if and only if log f B = 4.

12 ,

1.2. Logarithmic Conformal Mappings

9

−1 f ◦ σw , Dw = fw (D), and gDw (·, 1) be Proof. For w ∈ D, let fw = f (w) the Green function of Dw and be deﬁned to equal 0 outside Dw . Then a simple calculation with the conformal transform σw plus the conformal invariance of the Green function yields

log f 2Qp

1 p fw (z) 2 = sup dm(z) log |z| w∈D D fw (z) p gDw (z, 1) |z|−2 dm(z) = sup w∈D Dw ∞ p gDw (rξ, 1) |dξ| r−1 dr. = sup 0

w∈D

T

2 . This map is conformal with Consider the Koebe-like map: k(z) = 1+z 1−z Ω = k(D) = C \ (−∞, 0], and consequently, the Green function gΩ (·, 1) of Ω with pole at 1 obeys 1 + √z √ , z ∈ Ω, gΩ (z, 1) = log 1− z which is also extended to be 0 on (−∞, 0]. If D∗w stands for the circularly symmetric domain of Dw and gD∗w (·, 1) is the Green function of Dw with pole at 1 — of course, the value of gD∗w (·, 1) is set to be 0 outside D∗w , then D∗w ⊆ Ω and hence gD∗w (z, 1) ≤ gΩ (z, 1),

z ∈ C.

(i) In case of p ∈ (0, ∞), Lemma 1.2.2 (i) is applied to imply

log f 2Qp

= ≤

sup w∈D 2π

0

≤ as desired.

T

p sup gDw (rξ, 1) r−1 dr p sup gD∗w (rξ, 1) r−1 dr

ξ∈T ∞

2π 0

p sup gΩ (rξ, 1) r−1 dr

ξ∈T

1 + √r p √ r−1 dr log 1 − r 0 1 1 + r p −1 log 8π r dr, 1−r 0

=

p gDw (rξ, 1) |dξ| r−1 dr

ξ∈T ∞

2π

≤

0 ∞

0

≤

∞

2π

∞

10

Chapter 1. Preliminaries (ii) When p ∈ [1, ∞), we use Lemma 1.2.2 (ii) to deduce p gDw (rξ, 1) |dξ| r−1 dr sup w∈D 0 T ∞ p gD∗w (rξ, 1) |dξ| r−1 dr sup w∈D 0 T ∞ p gΩ (rξ, 1) |dξ| r−1 dr T 0 p gΩ (z, 1) |z|−2 dm(z) Ω 2 k (z) (− log |z|)p dm(z) D k(z) 1 |dξ| 16 (− log r)p rdr, 2 2 2 0 T |1 − r ξ |

log f 2Qp

= ≤ ≤ = = =

∞

as required. Moreover, the above inequality becomes an equality for f = k. Next we handle the situation of equality. If log f B = 4, then log f Qp ≥

log k Qp . In fact, since {fwn } is a normal family, we may assume that it converges locally uniformally to a univalent function F on D with F (0) = 1. Accordingly, 2 f (wn ) = lim |f (0)| = |F (0)| = 4 lim (1 − |wn | ) n→∞ f (wn ) n→∞ wn which implies F (z) = k(ηz) for some η ∈ T thanks to the uniqueness of the Koebe function. Furthermore,

log f 2Qp

log fwn 2Qp fwn (z) 2 ≥ (− log |z|)p dm(z) f (z) w D n 2 F (z) → (− log |z|)p dm(z) as n → ∞. F (z) D =

This gives log f Qp ≥ log k Qp . Note that log f Qp ≤ log k Qp as proved already. So we derive

log f Qp = log k Qp . On the other hand, suppose log f Qp = log k Qp . Then there is a sequence {wn } in D such that fwn (z) 2 lim (− log |z|)p dm(z) = log k 2Qp . n→∞ D fwn (z)

1.2. Logarithmic Conformal Mappings

11

Observe that {fwn } forms a normal family. Thus we may assume that it converges to a univalent function F ∈ H uniformly on compacta of D. Accordingly, F (0) = 1 and 2 F (z)

log k 2Qp = (− log |z|)p dm(z). D F (z) Suppose that gF (D) (·, 1) and gF (D)∗ (·, 1) are the Green functions of F (D) and its circularly symmetric domain F (D)∗ with pole at 1 respectively, and suppose that their values are 0 on C \ F (D) and C \ F (D)∗ respectively. Then using Lemma 1.2.2 (ii) again as well as F (D)∗ ⊆ Ω, we further get F (z)

log k 2Qp = (− log |z|)p dm(z) D F (z) ∞ p gF (D) (rξ, 1) |dξ|r−1 dr = 0 ∞ T p ≤ gF (D)∗ (rξ, 1) |dξ|r−1 dr 0 ∞ T p gΩ (rξ, 1) |dξ|r−1 dr ≤ 0 T 2 k (z) = (− log |z|)p dm(z) k(z) D = which yields ∞ 0

T

log k 2Qp ,

p gF (D) (rξ, 1) |dξ|r−1 dr

∞

= 0 ∞ T = 0

T

p gF (D)∗ (rξ, 1) |dξ|r−1 dr p gΩ (rξ, 1) |dξ|r−1 dr.

If p ∈ (1, ∞), then an application of Lemma 1.2.2 (iii) produces F (D) = F (D)∗ , and so F (D) ⊆ Ω which yields gF (D) (rξ, 1) ≤ gΩ (rξ, 1). Consequently, gF (D) (rξ, 1) = gΩ (rξ, 1) and hence F (D) = Ω. This means that there is an η ∈ T such that F (z) = k(ηz). Because {fwn } is convergent to F uniformly on compacta of D, it follows that 2 f (wn ) = lim |f (0)| = |F (0)| = |k (0)| = 4. lim (1 − |wn | ) n→∞ f (wn ) n→∞ wn Obviously, this limit ensures that log f B ≥ 4. However, it is well known that

log f B ≤ 4. Therefore log f B = 4. If p = 1, then log f Q1 = log k Q1 implies that there exists a sequence {wn } in D such that 2 fwn (z) 2 k (z) lim (− log |z|)dm(z) = (− log |z|)dm(z). n→∞ D fwn (z) D k(z)

12

Chapter 1. Preliminaries

As before, we may assume that {fwn } converges to a nonzero univalent function F with F (0) = 1 uniformly on compacta of D. With this, we get 2 2 F (z) k (z) (− log |z|)dm(z) = (− log |z|)dm(z). D F (z) D k(z) If gF (D) (·, 0) and gk(D) (·, 0) stand for the Green functions of F (D) and k(D) with pole at 0, and being 0 outside of F (D) and k(D), respectively, then

∞

−∞

∞

−∞

gF (D) (x + iy, 0) − gk(D) (x + iy, 0) dydx = 0.

Writing

s

H(x, s) = −s

gF (D) (x + iy, 0) − gk(D) (x + iy, 0) dy,

s ∈ [0, ∞),

we ﬁnd H(x, s) ≥ 0; see the proofs of Theorem 2 and Proposition 6 in [Gir1] (see (6) in [Gir2]). Accordingly, lims→∞ H(x, s) ≥ 0 for x ∈ (−∞, ∞). So 0≤

∞

lim H(x, s)dx = 0

−∞ s→∞

which gives lims→∞ H(x, s) = 0 for a.e. x ∈ (−∞, ∞). Note that since H(x, s) is increasing and continuous with s ≥ 0, we conclude that H(x, s) = 0 for s ∈ [0, ∞) and that gF (D) (x + iy, 0) − gk(D) (x + iy, 0) = 0. Accordingly, F (D) = k(D) and hence f (z) = k(ηz) for some η ∈ T which naturally deduces log f B = 4.

1.3 Conformal Domains and Superpositions The purpose of this section is two-fold. One is to give a criterion for a simply connected domain Ω ⊆ C to be a conformal Qp domain. The other is to characterize the membership of a conformal mapping in Qp by means of the geometric structure of the image domain Ω and to give an application of this geometric criterion to the superposition acting between two holomorphic Q classes. Clearly, when f is a conformal mapping from D into Ω, a proper subdomain of C, we see that f ∈ D if and only if the area of the image of f (D): f (D) dm is ﬁnite. This simple observation leads to a description of the conformal Qp domains as follows.

1.3. Conformal Domains and Superpositions

13

Theorem 1.3.1. Let p ∈ [0, ∞) and Ω ⊂ C be a simply connected hyperbolic domain. Then the following statements are equivalent: (i) Ω is a conformal Qp domain; that is, every conformal mapping f : D → Ω lies in Qp . (ii) supw∈Ω d(w, ∂Ω) < ∞. Proof. Suppose (ii) is valid. If f : D → f (D) ⊆ Ω is conformal, then d(w, ∂f (D)) ≤ d(w, ∂Ω),

w ∈ f (D).

Note that this conformal mapping f obeys the following well-known estimate: d f (z), ∂f (D) ≤ (1 − |z|2 )|f (z)| ≤ 4d f (z), ∂f (D) , z ∈ D. So we read oﬀ f ∈ B and so f ∈ Qp . This proves (i). Conversely, assuming (i) holds, we verify (ii). It suﬃces to demonstrate that every simply connected domain Ω ⊂ C with supw∈Ω d(w, ∂Ω) = ∞ must contain a simply connected domain Ω∗ with the same property. To this end, choose a sequence {wj } in Ω such that |wj+1 | − |wj | > 4j

and D(wj , 2j) = {w ∈ C : |w − wj | < 2j} ⊆ Ω,

j ∈ N.

It is clear that the closures of the disks D(wj , j) = {w ∈ C : |w − wj | < j} are mutually disjoint. Inductively, we can take distinct points: ξ1 ∈ {w ∈ C : |w−w1 | = 1} and ζj , ξj+1 ∈ {w ∈ C : |w−wj | = j} for

j ≥ 2,

and obtain a sequence of mutually disjoint, piecewise linear, simple arcs Aj , j ∈ N, made up of segments parallel to either the real or the imaginary axis in such a way in Ω, joins ξj and ζj+1 , and contains no other points that each Aj is contained ∞ of the closure of j=1 D(wj , j). From this construction it turns out that for each j ∈ N there is a positive j such that Uj = {w ∈ C : inf ξ∈Aj |ξ − w| < j /2} are also mutually disjoint. Now setting Ω1 = D(w1 , 1) and Ωj =

j k=1

j−1 D(wk , k) ∪ Uj ,

j − 1 ∈ N,

k=1

we derive that each Ωj is not only a bounded domain whose boundary is a simple closed curve but also enjoys the following inclusion chain: D(wj , j) ⊆ Ωj ⊆ Ωj+1 ⊆ Ω, j ∈ N, ∞ hence producing that Ω∗ = j=1 Ωj ⊆ Ω is a simply connected domain which contains arbitrarily large disks. Due to this reason, we see that any conformal mapping from D onto Ω∗ is not in B and hence not in Qp , and so that Ω is not a conformal Qp domain, contradicting the assumption (i). This completes the argument.

14

Chapter 1. Preliminaries

In order to give a geometric criterion for a conformal mapping from D into a simply connected hyperbolic domain Ω ⊂ C to belong to Qp , we say that a function f ∈ H is of the conformally invariant Besov class Bq , q ∈ (1, ∞), provided

f Bq =

D

2 q−2

|f (z)| (1 − |z| ) q

q1 dm(z) < ∞.

Here the conformal invariance of Bq simply means:

f ◦ σw Bq = f Bq ,

w ∈ D.

Theorem 1.3.2. Let p ∈ [0, ∞) and Ω ⊂ C be a simply connected hyperbolic domain. If f : D → Ω is a surjectively conformal mapping, then: (i) f ∈ Qp when and only when supw∈Ω d(w, ∂Ω) < ∞. p (ii) f ∈ Bp+2 when and only when Ω d(w, ∂Ω) dm(w) < ∞. Proof. Note again that f (D) = Ω and d f (z), ∂f (D) ≤ (1 − |z|2 )|f (z)| ≤ 4d f (z), ∂f (D) ,

z ∈ D.

So the assertion (i) follows from the equivalence f ∈ Qp ⇔ f ∈ B and the assertion (ii) follows from the conformal change of variable w = f (z). Corollary 1.3.3. Let q ∈ (1, ∞). Suppose {wj } is a sequence of points in C. Assume {rj } and {hj } are sequences of positive numbers with the following property: for each j ∈ N, 0 ≤ arg wj ≤ π/4, |wj | ≤ |wj+1 |/2, rj < |wj |/4 and |jj | < min{rj , rj+1 }/3. ∞ Then Ω = j=1 (Dj ∪ Rj ) is a simply connected hyperbolic domain of C, where Dj = {w ∈ C : |w − wj | < rj } and Rj is the rectangle whose longer symmetry axis is the segment [wj , wj+1 ] and whose shorter side has length 2hj . Furthermore, a conformal mapping from D onto Ω belongs to Bq if and only if ∞ j=1

rjq +

∞

hq−1 |wj+1 − wj | < ∞. j

j=1

Proof. First of all, note that the distance to the origin increases as one moves along the segment [wj , wj+1 ] from wj to wj+1 . So nj=1 (Dj ∪ Rj ) is simply connected for each n ∈ N, and consequently, Ω is simply connected. Next, suppose f : D → Ω is conformal and surjective. Then from Theorem 1.3.2 (ii) and the above-required property it is not hard to ﬁgure out that f ∈ Bq

1.3. Conformal Domains and Superpositions

15

if and only if ∞ > ≈ ≈

∞

q−2 d(w, ∂Ω) dm(w) +

Dj j=1 ∞ j=1 ∞

Dj

rjq +

j=1

q−2 d(w, ∂Ω) dm(w)

Rj

q−2 d(w, ∂Dj ) dm(w) +

∞

Rj

q−2 d(w, ∂Rj ) dm(w)

hq−1 |wj+1 − wj |, j

j=1

as desired.

Using the previous construction of unbounded conformal mappings in Bq , q > 1 we can characterize the superposition associated with Qp . Theorem 1.3.4. Let p ∈ [0, ∞). For f ∈ H and an entire function φ on C set Sφ (f ) = φ(f ). Then the following statements are equivalent: (i) Sφ sends Qp to B. (ii) Sφ sends Bp+2 to B. (iii) φ is linear; that is, φ(w) = aw + b, where a, w, b ∈ C. Proof. (i)⇒(ii) This implication is trivial thanks to Bp+2 ⊂ Qp . (ii)⇒(iii) Suppose (ii) holds. If (iii) fails, then φ is not a constant mapping, and hence by Liouville’s Theorem on the entire functions there is a sequence of points {wj } in C such that |wj+1 | ≥ |wj | and |φ (wj )| ≥ 22j for j ∈ N. Note that at least one of the eight octants: arg−1 [(k−1)π/4, kπ/4) , k = 1, 2, ..., 8, contains inﬁnitely many elements of {wj }. Using a rotation if necessary, we may assume that 0 ≤ arg wj < π/4, and then use Corollary 1.3.3 to derive that if |w1 | > 2,

rj = 2−j

1

and hj = 2−j−2 |wj − wj+1 |− p+1 ,

and if Ω is the simply connected domain given in Corollary 1.3.3 and f : D → Ω is a surjectively conformal mapping, then it follows that f ∈ Bp+2 owing to Corollary 1.3.3. Choosing zj ∈ D such that f (zj ) = wj for j ∈ N, we ﬁnd limj→∞ |zj | = 1, and consequently, |φ (wj )||f (zj )|(1 − |zj |2 ) ≈ |φ (wj )|dΩ (wj , ∂Ω) rj−1 → ∞ as j → ∞. This yields Sφ (f ) ∈ B, a contradiction. (iii)⇒(i) If φ(w) = aw + b, where a, w, b ∈ C, then φ is a constant a and hence Sφ (f ) (z) = af (z) which implies (i) thanks to Qp ⊆ B.

16

Chapter 1. Preliminaries

The following consequence of Theorem 1.3.4 tells us that the action of superposition between any two conformally invariant spaces from the Q families and the Besov families is trivial. The reason behind this phenomenon is that any conformal mapping from C onto C is linear, namely, of the form: aw + b. Corollary 1.3.5. Let φ be an entire function on C. Suppose X and Y are any two members in {Qp }p∈(0,∞) and {Bp+2 }p∈(0,∞) . Then Sφ sends X to Y if and only if φ is linear whenever X ⊆ Y or φ is constant whenever X ⊆ Y. Proof. This follows from Theorem 1.3.4 right away.

1.4 Descriptions via Harmonic Majorants In this section, we investigate Qp through a solvable function determined by the integral inducing · Qp ,1 . To begin, we need the following notion. Deﬁnition 1.4.1. Given a connected open proper subset Ω of the extended complex plane Ce with boundary ∂Ω. Let u be a function from ∂Ω to the extended real number system Re = R ∪ {±∞}. Then (i) LP(u, Ω) = f : f is subharmonic and bounded above on Ω with lim sup f (z) ≤ f (ξ), ξ ∈ ∂Ω z→ξ

and UP(u, Ω) = f : f

is supharmonic and bounded above on Ω with lim inf f (z) ≥ f (ξ), ξ ∈ ∂Ω z→ξ

are respectively called the lower and upper Perron families associated with u and Ω. (ii) If sup{f (z) : f ∈ UP(u, Ω)} = inf{f (z) : f ∈ LP(u, Ω)}, z ∈ Ω is a ﬁnite function, then u is called solvable on ∂Ω, and hence, it is the solution of the Dirichlet problem with boundary value u. (iii) If u = −∞, then we say that it has the least harmonic majorant U : D → R whenever U is harmonic on D, u ≤ U on D, and U ≤ V on D for any harmonic function V : D → R with u ≤ V on D. Deﬁnition 1.4.1, and the following proposition whose (ii) is Harnack’s Principle, are classical and can be found in [Con].

1.4. Descriptions via Harmonic Majorants

17

Proposition 1.4.2. Let Ω ⊂ C be a domain with boundary ∂Ω. (i) If {un } is an increasing sequence of solvable functions on ∂Ω and u(z) = limn→∞ un (z) for z ∈ Ω, then u is solvable on ∂Ω and sup{f (z) : f ∈ UP(un , Ω)} converges to sup{f (z) : f ∈ UP(u, Ω)} uniformly on compacta of Ω, or else sup{f (z) : f ∈ UP(u, Ω)} = inf{f (z) : f ∈ LP(u, Ω)} = ∞, z ∈ Ω and sup{f (z) : f ∈ UP(un , Ω)} converges to ∞ uniformally on compacta of Ω. (ii) If {un } is an increasing sequence of harmonic functions from Ω to R, and if {un (z0 )} is bounded for some z0 ∈ Ω, then {un } converges uniformly on compacta of Ω to a harmonic function from Ω to R. Lemma 1.4.3. Given a function f : D → R. For (r, ξ) ∈ [0, 1) × T let f (r, ξ) = sups∈[0,r] f (sξ). Then f (·, ξ) is increasing on (0, 1) for each ξ ∈ T. Furthermore: (i) If f is continuous on D, then f (r, ·) is continuous on T for each r ∈ [0, 1). (ii) If f is continuous and bounded on D, then 2 r − |z|2 |dξ| f (r, ξ) F (z) = sup 2 2π r∈[0,1) T |rξ − z| is a harmonic function on D, and lim f (r, ·) = f (1, ·) = sup f (s·)

r→1−

s∈[0,1)

is a solvable function on T with F being the solution of its Dirichlet’s problem. Proof. The fact that 0 ≤ r1 < r2 < 1 ⇒ f (r1 , ξ) ≤ f (r2 , ξ),

ξ∈T

is a straightforward consequence of the deﬁnition of f (r, ξ). (i) In order to prove the continuity, we may assume that for a ﬁxed r ∈ [0, 1), f (r, ·) is discontinuous at some point ξ0 ∈ T. Then there exists an 0 > 0 such that for any n ∈ N one can ﬁnd such a ξn ∈ T that |ξn − ξ0 | <

1 n

and |f (r, ξn ) − f (r, ξ0 )| ≥ 0 .

Since f is continuous on D, we can conclude that f (r, ·) is attained. Accordingly, there are r0 , rn ∈ [0, r] such that f (r, ξ0 ) = f (r0 ξ0 ) and f (r, ξn ) = f (rn ξn ).

18

Chapter 1. Preliminaries

Clearly, there is a subsequence {rnk } which converges to s0 ∈ [0, r]. This plus the continuity of f implies f (rnk ξnk ) → f (s0 ξ0 ) and thus f (r0 ξ0 ) − f (s0 ξ0 ) = |f (s0 ξ0 ) − f (r0 ξ0 )| ≥ 0 . Nevertheless, there exists a natural number k0 such that k > k0 ⇒ |f (r0 ξnk ) − f (r0 ξ0 )| <

0 . 2

Note that f (r0 ξnk ) ≤ f (r, ξnk ) = f (rnk ξnk ). So, if k > k0 , then f (r0 ξ0 ) <

0

0 + f (rnk ξnk ) → + f (s0 ξ0 ) 2 2

and hence 0 < 0, a contradiction. (ii) For r ∈ [0, 1) and z ∈ D let 2 r − |z|2 |dξ| F (r, z) = f (r, ξ) . 2 |rξ − z| 2π T Then F (r, z) is harmonic in z ∈ D and hence f (r, ·) is a solvable function on T. Also, by the boundedness of f on D, we obtain 0 ≤ r1 < r2 < 1 ⇒ F (r1 , z) ≤ F (r2 , z) ≤ sup f (w) < ∞,

z ∈ D.

w∈D

Consequently, if {rn } ⊂ (0, 1) is increasing and convergent to 1, then it follows from Harnack’s Principle (Proposition 1.4.2 (ii)) that F (z) = limn→∞ F (rn , z) is harmonic in z ∈ D. Using the boundedness of f on D, we ﬁnd that f (1, ·) = limr→1− f (r, ·) is ﬁnite on T, and so that it is a solvable function on T as the boundary value of F , due to Proposition 1.4.2 (i). Returning to our principal business, we have the following result. Theorem 1.4.4. Let p ∈ (0, ∞) and f ∈ H. Then f ∈ Qp if and only if f (1, ·) = limr→1− f (r, ·) is bounded on T, where p |f (z)|2 1 − |σsξ (z)|2 dm(z), r ∈ (0, 1). f (r, ξ) = sup s∈[0,r]

D

In this case, f (1, ·) is solvable on T and the solution to its corresponding Dirichlet’s problem F : D → C is the least harmonic majorant of the family of harmonic functions {F (r, ·)}r∈(0,1) , where 2 r − |z|2 |dξ| f (r, ξ) , z ∈ D. F (r, z) = 2 |rξ − z| 2π T

1.5. Regularity for the Euler–Lagrange Equation

19

Proof. The equivalence is derived immediately from (i)⇔(ii) in Theorem 1.1.3 and its extension to p ∈ [1, ∞). Concerning the remaining part, we ﬁrst prove that if f ∈ Qp , then p 2 |f (z)|2 1 − |σw (z)|2 dm(z) Fp (f, w) = D

is a continuous and bounded function on D. Of course, the boundedness is trivial. To see the continuity, let f be nonconstant, ﬁx w0 ∈ D, and assume that δ > 0 ensures {z ∈ D : |σw (z)| ≤ δ} ⊂ D. Then the mapping q(z, a) =

1 − |a|2 p , |1 − a ¯z|2

is uniformly continuous on {z ∈ C : |z| ≤ 1} × {z ∈ D : |σw (z)| ≤ δ}. Accordingly, for any > 0 there exists s > 0 such that |z1 − z2 | < s and |a1 − a2 | < s imply |q(z1 , a1 ) − q(z2 , a2 )| <

D

−1 |f (z)| (1 − |z| ) dm(z) .

2

2 p

Accordingly, |w − w0 | < s yields Fp (f, w) 2 − Fp (f, w0 ) 2

≤

D

|f (z)|2 (1 − |z|2 )p |q(z, w) − q(z, w0 )|dm(z)

< .

2 Now for (r, ξ) ∈ (0, 1) × T let f (r, ξ) = sups∈[0,r] Fp (f, sξ) . Then an ap2 plication of Lemma 1.4.3 with Fp (f, ·) implies the desired assertion.

1.5 Regularity for the Euler–Lagrange Equation In this section, we will see that the concept of harmonic majorants can be used to settle the regularity of a solution to the Euler–Lagrange equation. Given a constant λ ≥ 0. Consider the problem of solving the Euler–Lagrange equation ∆u(z) = λu(z), z ∈ D subject to the Dirichlet-type energy condition 2 |∇u(z)|2 + λ u(z) dm(z) < ∞, D

where ∇ = (2∂/∂z, 2∂/∂ z¯) stands for the gradient vector. As well-known, this problem is solvable (cf. [BergSc, p. 258]), and each solution u is C ∞ on D. If

20

Chapter 1. Preliminaries

λ = 0, then any solution u is harmonic on D, and hence there is a holomorphic function f such that u = f . Moreover, if this harmonic solution u satisﬁes Ep (u) = sup |∇u(z)|2 (1 − |σw (z)|2 )p dm(z) < ∞, p ∈ [0, 1], w∈D

D

then f ∈ Qp and thus f ∈ 0
f Hq =

q1

sup r∈(0,1)

|f (rξ)| |dξ| q

T

< ∞,

and H∞ is the set of all f ∈ H with

f H∞ = sup |f (z)| < ∞. z∈D

Thus, |f |q admits a harmonic majorant. Since |u| ≤ |f |, we get that |u|q has a harmonic majorant for any q > 0. Even more is true. Theorem 1.5.1. Let p ∈ (0, 1) and λ ≥ 0 be a constant. If u satisﬁes ∆u = λu on D and Ep (u) < ∞, then |u|q admits a harmonic majorant on D for all q > 0. Proof. The argument is split into two cases. Case 1: q ∈ (0, 2/p]. In this case, the following estimate for u is elementary: |u|q ≤ (|u| + 1)q ≤ (|u| + 1)2/p (|u|2/p + 1). So, if u is such that ∆u = λu on D and Ep (u) < ∞, then |∇u(z)|2 (1 − |z|2 )p dm(z) ≤ Ep (u) < ∞, D

and hence |u|2/p admits a harmonic majorant owing to Theorem 1 in [Ya]. Therefore, |u|q admits a harmonic majorant. Case 2: q ∈ (2/p, ∞). Note that p ∈ (0, 1). So q > 2/p implies q > 2. Without loss of generality, we may assume that u is nonconstant. A simple calculation with q > 2 gives ∆(|u|q ) = q(q − 1)|u|q−2 |∇u|2 + qλ|u|q ≥ 0 which indicates that |u|q is subharmonic on D. Suppose r ∈ (1/4, 1). Applying the Green formula to |u|q and log r/|z| on the annulus {z ∈ D : 1/4 < |z| < r}, and taking ∂/∂n as the radial derivative, we have r I(r) = π log ∆ |u(z)|q dm(z) = I1 (r) + I2 (r) + I3 (r), |z| 1/4<|z|
1.5. Regularity for the Euler–Lagrange Equation

21

∂ r log |dζ|, ∂n |ζ| |ζ|=r r ∂ log |u(ζ)|q |dζ|, I2 (r) = − |ζ| ∂n |ζ|=1/4

where

I1 (r) = −

|u(ζ)|q

and I3 (r) = −4

|ζ|=1/4

|u(ζ)|q |dζ|.

Clearly, I1 (r) is equal to 2πVr (0), where Vr is the Poisson integral of |u|q on {z ∈ D : |z| = r}. Observe that Vr is the least harmonic majorant of |u|q over {z ∈ D : |z| < r} and is nondecreasing with r. So, if we can prove sup1/4
I(r) < ∞,

sup 1/4
I2 (r) < ∞ and

sup 1/4
I3 (r) < ∞.

Since I3 (r) is actually a constant (independent of r) and limr→1 I2 (r) is ﬁnite due to the smoothness of |u|q in case of q > 2, it remains to prove sup1/4
thanks to

− log |z| ≤ 4(1 − |z|2 ),

1/4 < |z| < 1.

If z = x + iy, then ∂ 2 u 2 ∂ 2 u 2 ∂ 2 u 2 2−1 ∆ |∇u|2 = + 2 + + λ|∇u|2 ≥ 0, ∂x2 ∂x∂y ∂y 2 and hence |∇u|2 is subharmonic on D, and so is |∇u ◦ σw |2 for any w ∈ D. A simple calculation with this subharmonicity produces 2 (1 − |w|2 )|∇u(w)|

|∇u ◦ σw (0)|2 |∇u ◦ σw (z)|2 (1 − |z|2 )p dm(z)

Ep (u),

=

|z|≤ 12

w ∈ D.

As a result, we get |u(z)| |u(0)| +

Ep (u) log

1 , 1 − |z|

z ∈ D.

22

Chapter 1. Preliminaries

Note that both p ∈ (0, 1) and q ∈ (2, ∞) ensure (1 − |z|)1−p log

1 1 − |z|

q−2 1,

z ∈ D.

So II(r)

1/4<|z|<1

|u(z)|q−2 |∇u(z)|2 (1 − |z|2 )dm(z)

+ 1/4<|z|<1

1/4<|z|<1

|u(z)|q (1 − |z|2 )dm(z)

|u(0)| + Ep (u) log

+ 1/4<|z|<1

|u(0)| + Ep (u) log

|u(0)|

q−2 1/4<|z|<1

q−2 + Ep (u) 2 q + |u(0)|

q−2

1 1 − |z|

|∇u(z)|2 (1 − |z|2 )dm(z)

q

(1 − |z|2 )dm(z)

|∇u(z)|2 (1 − |z|2 )dm(z)

1/4<|z|<1

1/4<|z|<1

|∇u(z)|2 (1 − |z|2 )p dm(z)

(1 − |z|2 )dm(z)

q 1 (1 − |z|2 )dm(z) 1 − |z| 1/4<|z|<1 q |u(0)|q−2 Ep (u) + |u(0)|q + Ep (u) 2 < ∞.

q + Ep (u) 2

1 1 − |z|

The proof is complete.

log

1.6 Notes Note 1.6.1. Section 1.1 summarizes the ﬁrst four sections of [BanFlu] and the main function-theoretic characterizations of Qp presented in [Xi3]. Example 1.1.1 is a special case of [Ban, (2.21)] which is veriﬁed via the isoperimetric inequality. A broad exploration of the Qp spaces naturally leads to a treatment of some other Qp type function spaces and classes — see e.g., [AuMW], [EsWuXi], [Lis], [Lix], [PerR], [Rat], [Wul] and [Zhu3]. From 1999 to 2006, the scale of Qp spaces was generalized in diﬀerent ways to higher dimensions — see also [CnDe], [GuCDS], [GuMa], [CeKaSo], [ReKa], [Hu], [LiO], [ElGRT], [BernGRT] and [Sha]. Recently, the basic theory of the Qp -spaces was extended to bounded symmetric domains — see also [Eng].

1.6. Notes

23

Note 1.6.2. Section 1.2 is basically an adaptation of the sixth section of [DoGiVu]. Related to this topic, we can say something about the Schwarz derivative. If f is univalent on D, then SD(f )(z) =

f

− 2−1

f

f 2 f

is called the Schwarz derivative of f . It is well known that log f ∈ B

with

log f B ≤ 6

and log f belongs to BMOA if and only if |SD(f )(z)|2 (1 − |z|2 )3 dm(z) is a 1-Carleson measure on D; see [BisJo]. Using Theorem 1.1.3 (v), we can readily ﬁnd: if p ∈ (0, 1) then log f ∈ Qp must imply that |SD(f )(z)|2 (1 − |z|2 )p+2 dm(z) is a p-Carleson measure on D. Although being unable to prove whether or not the converse holds too, we believe that it should be true since the Schwarz derivative can be regarded as a type of second derivative, measuring the rate of change of the best approximating M¨ obius mappings rather than linear transforms. Note 1.6.3. Section 1.3 is a combination of [DoGiVu, Theorem 4.2] and [BuFeVu, Section 2]. Two further remarks are in order. The ﬁrst one is about Sφ on the limiting Besov space B1 which consists of all functions f ∈ H with

f B1 = |f (z)|dm(z) < ∞. D

Since B1 ⊂ H∞ , the chain rule gives that if φ is an entire function on C, then Sφ (f ) = f Sφ (f ) + (f )2 Sφ (f ), f ∈ B1 and so that Sφ (f ) ∈ B1 . In other words, the nonlinear operator Sφ always maps B1 to itself; see also [BuFeVu, Proposition 5] and [AlvVMV] for the action of Sφ between B and the Bergman spaces. The second one is about Sφ from the Blochtype spaces B α into Qp and vice versa. Recall that f ∈ B α , α ∈ (0, ∞) means: f ∈H

and f Bα = sup (1 − |z|2 )α |f (z)| < ∞. z∈D

Using Theorem 1.3.4 and the existing result [Xi3, Theorem 2.1.1] that there are two f1 , f2 ∈ B α such that (1 − |z|2 )α |f1 (z)| + |f2 (z)| ≈ 1, z ∈ D,

24

Chapter 1. Preliminaries

one can readily get Sφ : B α → Qp , p ∈ [0, ∞), if and only if φ satisﬁes φ = 0 whenever α > 1 or φ(w) = aw + b whenever α = 1 or

sup w∈D

D

(1 − |z|2 )−2α (1 − |σw (z)|2 )p dm(z) < ∞ whenever α ∈ (0, 1).

The case α ∈ (0, 1) can be found in [Xio, Theorem 2]. In addition, Sφ : Qp → B α holds for p ∈ [0, ∞) and α ∈ (0, 1) (or α = 1) if and only if φ is constant (or φ(w) = aw + b); see also [Xio, Theorem 1] for α ∈ (0, 1). However, if α ∈ (1, ∞), then f ∈ B α is equivalent to f ∈H

with

sup (1 − |z|2 )α−1 |f (z)| < ∞, z∈D

and hence the inclusion Qp ⊆ B with growth |f (z)|

log

2

f Qp , 1 − |z|

f ∈ Qp ,

the proof of [AlvVMV, Theorem 9] and its immediate remark yield that Sφ : Qp → B α when and only when φ is an entire function either of order ρ = lim supr→∞ (log r)−1 log log max{|φ(z)| : |z| = r} less than 1 or of order ρ = 1 and type τ = lim supr→∞ r−ρ log max{|φ(z)| : |z| = r} = 0. A full description of the boundedness of Sφ acting between two Bloch-type spaces is given in [Xu]. Note 1.6.4. Section 1.4 is a direct outcome of reading [AuReTo] and its followup paper [RaReTo]. The key fact used in proving Theorem 1.4.4 is Fp (f, ·) is continuous. This fact can be veriﬁed by the following basic estimate: 1 − |σw1 (z)|2 (1 − |w1 |)(1 − |w2 |) 4 ≤ , z, w1 , w2 ∈ D. ≤ 4 1 − |σw2 (z)|2 (1 − |w1 |)(1 − |w2 |) For details, see also [WiXi2, Lemma 3.2]. Note 1.6.5. Section 1.5 is essentially [Xi4, Section 3.3]. However, the proof of Theorem 1.5.1 is motivated by the argument for [Ya, Theorem 1].

Chapter 2

Poisson versus Berezin with Generalizations We are given z ∈ D. Recall that (2π)−1 f ◦ σz (ζ)|dζ| T

and π −1

D

f ◦ σz (ζ)dm(ζ)

are the Poisson extensions of f ∈ L1 (T) (the Lebesgue 1-space on T) and the Berezin transformation of f ∈ L1 (D) (the Lebesgue 1-space on D), respectively. These two operators play an important role in the function theory of Hardy spaces and Bergman spaces. In this chapter, we use both to deduce directly or indirectly certain comparable derivative-free properties of each Qp space via the following ﬁve aspects: • • • • •

Boundary Value and Brownian Motion; Derivative-free Module via Poisson Extension; Derivative-free Module via Berezin Transformation; Mixture of Derivative and Quotient; Dirichlet Double Integral without Derivative.

2.1 Boundary Value and Brownian Motion The non-holomorphic Q classes on T, which we denote Qp (T), −∞ < p < ∞, are the classes of all functions f ∈ L1 (T) that satisfy 12 |f (ζ) − f (η)|2

f Qp,3 = sup |I|−p |dζ||dη| < ∞. |ζ − η|2−p I⊆T I I

26

Chapter 2. Poisson versus Berezin with Generalizations

The range of interest is p ∈ (0, 1). The forthcoming characterization involving Brownian motions is prepared for the characterization of the holomorphic Q classes in terms of the Poisson integrals of some absolute value functions. Following the known BMO-case; see also [Pet], we may assume that (Ω, B, P ) is a probability space. For t ≥ 0 denote by γz,t (ω) the two-dimensional Brownian motion in D starting at the point z ∈ D at time t = 0. Given z ∈ D, put τz (ω) = inf{t ≥ 0 : |γz,t (ω)| = 1}, which is the ﬁrst hitting time of γz,t on T. Write γ˜z,t (ω) = γz,min{t,τz (ω)} (ω) for the Brownian motion starting at z ∈ D which is absorbed at its ﬁrst point of impact on T, at time τz (ω). For a sub-σ-algebra: A ⊆ B, and a function f ∈ L1 (Ω, B, P ), it is known that ν(A) = A f dP for A ∈ A is absolutely continuous with respect to P |A, and so by the Radon–Nikodym Theorem there exists an A-measurable L1 function g such that ν(A) = A gdP for all A ∈ A. Because this equation determines g uniquely a.e., g = E(f |A) is called the conditional expectation of f with respect to A. If one knows for each set of A whether or not it contains ω, then E(f |A)(ω) is the expected value of f . In particular, E(f ) = Ω f dP . The Poisson extension fˆ of a function f ∈ L1 (T) from T to D is just the following operator acting on f : 1 − |z|2 |dζ| ˆ , z ∈ D. f (z) = f (ζ) |ζ − z|2 2π T Lemma 2.1.1. Let p ∈ (0, 1) and f ∈ L2 (T). Then the following statements are equivalent: (i) f ∈ Qp (T). (ii) |∇fˆ(z)|2 (1 − |z|2 )p dm(z) is a p-Carleson measure on D. (iii) sup |∇fˆ(z)|2 (1 − |σw (z)|2 )p dm(z) < ∞. w∈D

(iv) (v) (vi) (vii)

D

1 − u dm(z) ¯z 2 ˆ |∇f (u)| log < ∞. sup dm(u) (1 − |σw (z)|2 )p z − u (1 − |z|2 )2 w∈D D D 2 dm(z) < ∞. |f |2 (z) − fˆ(z) (1 − |σw (z)|2 )p sup (1 − |z|2 )2 w∈D D dm(z) 2 2 ˆ sup |f (γz,τz (ω) )| dP (ω) − |f (z)| (1 − |σw (z)|2 )p < ∞. (1 − |z|2 )2 w∈D D Ω dm(z) sup < ∞. E |f (γz,τz )|2 − |fˆ(z)|2 (1 − |σw (z)|2 )p (1 − |z|2 )2 w∈D D

2.1. Boundary Value and Brownian Motion D

(ix) sup w∈D

E D

0

Ω

0

τ

2 ∇fˆ(˜ γz,t ) dt dP (ω) (1 − |σw (z)|2 )p

τ (ω)

(viii) sup w∈D

27

2 ∇fˆ(˜ γz,t ) dt (1 − |σw (z)|2 )p

dm(z) < ∞. (1 − |z|2 )2

dm(z) < ∞. (1 − |z|2 )2

Proof. Step 1: (i)⇔(ii)⇔(iii) This follows from [Xi3, Theorem 7.1.1 and Lemma 4.1.1] — see also Theorem 1.1.3 (viii). Step 2: (iii)⇔(iv)⇔(v) The well-known Green formula is applied to infer that under z ∈ D and p ∈ (0, 1), 2 ∂ 2 (1 − |u|2 )p log |σz (u)|dm(u) (1 − |z|2 )p = π D ∂u∂ u ¯ 1 − u¯z dm(u) = 2p (1 − p|u|2 )(1 − |u|2 )p log u − z (1 − |u|2 )2 D 1 − u ¯z dm(u) ≈ (1 − |u|2 )p log . u − z (1 − |u|2 )2 D This estimate, together with Fubini’s Theorem, Hardy–Littlewood’s identity (cf. [Ga, p. 238]) and the conformal transform σw (z), implies |∇fˆ(z)|2 (1 − |σw (z)|2 )p dm(z) D = |∇fˆ σw (z) |2 (1 − |z|2 )p dm(z) D 1 − u¯z dm(z) (1 − |u|2 )p dm(u) ≈ |∇fˆ σw (z) |2 log u−z (1 − |u|2 )2 D D 1 − u ¯z dm(u) dm(z) (1 − |σw (u)|2 )p ≈ |∇fˆ(z)|2 log u−z (1 − |u|2 )2 D D 2 dm(z) ≈ . |f |2 (z) − fˆ(z) (1 − |σw (z)|2 )p (1 − |z|2 )2 D So, the desired equivalence follows. Step 3: (i)⇔(vi)⇔(vii) Given a continuous function g on T. Fix z ∈ D and let νz (A) = P ω : γz,τz (ω) ∈ A for every Lebesgue measurable subset A of T. Changing variable yields g γz,τz (ω) (ω) dP (ω) = g(ζ)dνz (ζ). Ω

T

Using Kakutani’s Theorem in [Kak], we ﬁnd that the right-hand-side function is harmonic on D with g being the boundary value function on T, and hence it is

28

Chapter 2. Poisson versus Berezin with Generalizations

just gˆ(z). Note that this is true for all continuous functions on T. Accordingly, we ﬁnd 1 − |z|2 |dζ| , ζ ∈ T, dνz (ζ) = |ζ − z|2 2π which is a conformally invariant probability measure on T in the sense of dνz (ζ) = dνσw (z) (σw (ζ)),

w, z ∈ D, ζ ∈ T.

In other words, if S ⊆ T is Lebesgue measurable, then the hitting probability of γz,t in S is the harmonic measure of S at z: P

1 − |z|2 1 ω : γz,τz (ω) (ω) ∈ S = νz (S) = |dζ|. 2π S |ζ − z|2

Accordingly, this equation, together with the previous Steps 1-2, implies the required equivalence. Step 4: (i)⇔(viii)⇔(ix) For z ∈ D, t ≥ 0, and a Lebesgue measurable set S ⊆ D, let µz,t (S) = P {ω : γ˜z,t (ω) ∈ S} . From the density formula associated with the two-dimensional Brownian motions (cf. [Pet, p. 20]), it follows that 1 µz,t (S) ≤ P {ω : γz,t (ω) ∈ S} = 2πt

|z|2 dm(z), exp − 2t S

and so that µz,t is absolutely continuous with respect to dm. Accordingly, there is a density function δ(z, t, w) such that δ(z, t, w) ≤

|z − w|2 1 exp − 2t 2t

and µz,t (S) = π −1

δ(z, t, w)dm(w). S

Using Hunt’s main result in [Hun], we obtain that if δ(z, t, w) is determined via P {ω : γ˜z,t (ω) ∈ S} = π −1 δ(z, t, w)dm(w) S

for each Lebesgue measurable set S ⊆ D, then 0

∞

1 − wz ¯ δ(z, t, w)dt = log z−w

a.e. w, z ∈ D.

2.2. Derivative-free Module via Poisson Extension Hence

1 − wz ¯ dm(z) 2 ˆ |∇f (z)| log w−z π D

29

∞ 2 ˆ |∇f (z)| δ(w, t, z)dt dm(z) 0 D ∞ 2 ˆ |∇f (z)| δ(w, t, z)dm(z) dt 0 D ∞ 2 ˆ |∇f (z)| dµw,t (z) dt 0 D

∞ 2 |∇fˆ(˜ γw,t )| dP (ω) dt

≈ ≈ ≈ ≈

0

t<τ (ω)

≈

Ω

≈

E 0

τ (ω)

0 τ

2 ˆ |∇f γ˜w,t (ω) | dt dP (ω)

2 ˆ |∇f (˜ γw,t )| dt ,

establishing the desired equivalence due to Steps 1–2 above.

2.2 Derivative-free Module via Poisson Extension As with Qp , we shall apply Lemma 2.1.1 to discuss the eﬀect of absolute values on the behavior of f ∈ Qp and at the same time, the value distribution of f ∈ Qp . Theorem 2.2.1. Let p ∈ (0, 1) and f ∈ H2 . Then the following statements are equivalent: (i) f ∈ Qp . ζ − z 2 |f (ζ) − f (η)|2 dm(z) log |dζ||dη| < ∞. (ii) sup |ζ − η|2 η−z (1 − |σw (z)|2 )−p w∈D T T D dm(z) (iii) sup < ∞. |f − f (z)|2 (z)(1 − |σw (z)|2 )p (1 − |z|2 )2 w∈D D (iv) |f | ∈ Qp (T) and 2 |f |(z) − |f (z)|2 (1 − |σw (z)|2 )p sup w∈D

D

dm(z) < ∞. (1 − |z|2 )2

(v) |f | ∈ Qp (T) and there is a nonnegative harmonic function h such that |f | ≤ h on D and 2 dm(z) h(z) − |f (z)|2 (1 − |σw (z)|2 )p < ∞. sup (1 − |z|2 )2 w∈D D

30

Chapter 2. Poisson versus Berezin with Generalizations

(vi) If

Nf (w, z) =

then

sup w∈D

D

C

−

f (zj )=w

log |σzj (z)| 0

, ,

w ∈ f (D), w ∈ C \ f (D),

Nf (v, z)dm(v) (1 − |σw (z)|2 )p

dm(z) < ∞. (1 − |z|2 )2

Proof. Step 1: (i)⇔(ii)⇔(iii) Note that fˆ = f for f ∈ H2 . So, from Lemma 2.1.1 with f ∈ H2 it follows that f ∈ Qp if and only if 2 dm(z) < ∞. sup |f |2 (z) − f (z) (1 − |σw (z)|2 )p (1 − |z|2 )2 w∈D D This equivalence, along with 2(|f − f (z)|2 )(z) = 2 |f |2 (z) − |f (z)|2 2 1 − |z|2 |f (ζ) − f (η)|2 = ¯ − z η¯| |dζ||dη| |1 − z ζ||1 T T 1 2 f (ζ) − f (η) 2 1 = − (1 − |z|2 )2 |dζ||dη| ζ − η ζ − z η − z T T ζ − z 2 f (ζ) − f (η) 2 = log (1 − |z|2 )2 |dζ||dη|, ζ − η η − z T T implies the desired equivalence right away. Step 2: (i)⇒(iv) If f ∈ Qp , then f ∈ Qp (T) and hence |f | ∈ Qp (T) thanks to |f (ζ)| − |f (η)| ≤ |f (ζ) − f (η)|, ζ, η ∈ T. Clearly,

2 |f |(z) ≤ |f |2 (z),

Thus,

z ∈ D.

2 dm(z) |f |(z) − |f (z)|2 (1 − |σw (z)|2 )p (1 − |z|2 )2 w∈D D dm(z) sup , |f |2 (z) − |f (z)|2 (1 − |σw (z)|2 )p (1 − |z|2 )2 w∈D D sup

≤

and consequently, (iv) follows from Lemma 2.1.1. | = h on D. Step 3: (iv)⇒(v) This follows from |f | ≤ |f Step 4: (v)⇒(i) Let |f | ∈ Qp (T) and let there be a nonnegative harmonic function h such that |f | ≤ h on D and 2 dm(z) < ∞. h(z) − |f (z)|2 (1 − |σw (z)|2 )p sup (1 − |z|2 )2 w∈D D

2.2. Derivative-free Module via Poisson Extension

31

An application of Lemma 2.1.1 with |f | ∈ Qp (T) yields 2 dm(z) |(z) (1 − |σw (z)|2 )p < ∞. sup |f |2 (z) − |f (1 − |z|2 )2 w∈D D | is the least nonnegative harmonic function greater than or equal to Note that |f | ≤ h on D and |f | on D (see e.g., [Du, p. 28]). So |f 2 2 |f |2 (z) − |f (z)|2 ≤ |f |2 (z) − |f |(z) + h(z) − |f (z)|2 ,

z ∈ D.

These inequalities imply that Lemma 2.1.1 (v) holds for f , and thus f ∈ Qp . Step 5: (i)⇔(vi) Note that if f ∈ H2 , then 1 − u¯z dm(u) = |f (u)|2 log Nf (v, z)dm(v), z ∈ D. u−z D C This, together with Lemma 2.1.1 (iv), implies the desired equivalence.

Corollary 2.2.2. Let p ∈ (0, 1). Then: (i) For a function ψ obeying ψ > 0 a.e. on T, log ψ ∈ L1 (T) and ψ ∈ L2 (T), there exists a function f ∈ Qp such that |f | = ψ a.e. on T if and only if ψ ∈ Qp (T) and 2 dm(z) ψ(z) − exp log ψ 2 (z) (1 − |σw (z)|2 )p < ∞. sup (1 − |z|2 )2 w∈D D (ii) A universal covering map f of a domain Ω ⊆ C is in Qp if and only if p dm(z) sup gΩ v, f (z) dm(v) 1 − |σw (z)|2 < ∞. (1 − |z|2 )2 w∈D D Ω Proof. (i) If ψ ∈ Qp (T) satisﬁes the required supremum condition, then we deﬁne the outer function ζ + z |dζ| Oψ (z) = exp log ψ(ζ) , z ∈ D. 2π D ζ −z ψ(z) for any z ∈ D, Since |Oψ (ζ)| = ψ(ζ) for a.e. ζ ∈ T and |Oψ (z)| = exp log we conclude from Theorem 2.2.1 that Oψ ∈ Qp which is the desired function. On the other hand, if there exists a function f ∈ Qp such that |f | = ψ a.e. on T, then f H2 < ∞ and hence f can be written as IO — product of the inner function I and the outer function O. Since |I| ≤ 1 on D and |I| = 1 a.e. on T as well as ζ + z |dζ| log ψ(ζ) , z ∈ D, O(z) = exp 2π D ζ −z

32

Chapter 2. Poisson versus Berezin with Generalizations

we have |f | ≤ |O| on D and |f | = ψ = |O| a.e. on T. Accordingly, 2 dm(z) ψ 2 (z) (1 − |σw (z)|2 )p sup − exp log ψ(z) (1 − |z|2 )2 w∈D D 2 |(z) − |f (z)|2 (1 − |σw (z)|2 )p dm(z) < ∞. |f ≤ sup (1 − |z|2 )2 w∈D D (ii) If f is a universal covering map of Ω, it can be seen from [Ne, p. 209] that Nf (v, z) = gΩ (v, f (z)),

v ∈ Ω, z ∈ D,

and hence the required equivalence follows from Theorem 2.2.1 (vi).

2.3 Derivative-free Module via Berezin Transformation The hyperbolic distance between two points in D is given by 1 + |σ (z )| 12 z1 2 , z1 , z2 ∈ D. dD (z1 , z2 ) = log 1 − |σz1 (z2 )| This distance determines the hyperbolic or Bergman metric on D, is invariant under Aut(D), and produces the hyperbolic open ball B(z, r) = {w ∈ D : dD (z, w) < r},

z ∈ D, r > 0.

Meanwhile, for p ∈ (0, ∞) and β ∈ (−1, ∞), the weighted Bergman space Aq,β is the collection of all f ∈ H satisfying 1q q 2 β

f Aq,β = |f (z)| (1 − |z| ) dm(z) < ∞. D

It is known that (see e.g. [Zhu2, Chapter 2]) q1

f Aq,β ≈ |f (z)|q (1 − |z|2 )q+β dm(z) . D

The right-hand-side integral is of the Dirichlet type. The next result indicates that in some cases these integrals have equivalent bidisk forms without derivatives. Lemma 2.3.1. Let p ∈ (0, 2) and f ∈ H. Then |f (z)|2 (1 − |z|2 )p dm(z) < ∞ I1 (f ) = D

if and only if

|f (z) − f (w)|2 (1 − |z|2 )p dm(z)dm(w) < ∞. I2 (f ) = |1 − z w| ¯4 D D

In this case, I1 (f ) ≈ I2 (f ).

2.3. Derivative-free Module via Berezin Transformation

33

Proof. First of all, note that |F (w) − F (0)|2 dm(w) ≈ |F (w)|2 (1 − |w|2 )2 dm(w) D

D

holds for any F ∈ H; see e.g. [Zhu1, Theorem 4.27]. So, this comparability result with F (w) = f ◦ σz (w) implies I2 (f ) = (1 − |z|2 )p−2 |f ◦ σz (w) − f (z)|2 dm(w) dm(z) D D ≈ (1 − |z|2 )p−2 |(f ◦ σz ) (w)|2 (1 − |w|2 )2 dm(w) dm(z) D D 1 − |w|2 2 ≈ (1 − |z|2 )p |f (w)|2 dm(w) dm(z). |1 − z w| ¯2 D D If I1 (f ) < ∞, then by Fubini’s Theorem and p ∈ (0, 2), we have (1 − |w|2 )2 (1 − |z|2 )p I2 (f ) ≈ |f (w)|2 dm(z) dm(w) I1 (f ). |1 − z w| ¯4 D D Conversely, if I2 (f ) < ∞, then for r ∈ (0, 1] and z ∈ D, we apply the following estimates (cf. [Zhu1, §4.3]): 1 |f (w)|2 dm(w) |f (z)|2 m B(z, r) B(z,r) and

(1 − |w|2 )2 1 1 , ≈ ≈ |1 − z w| ¯4 (1 − |z|2 )2 m B(z, r)

w ∈ B(z, r)

to get I2 (f )

D

2 p

(1 − |z| )

1 m B(z, r)

2

|f (w)| dm(w) dm(z) I1 (f ). B(z,r)

We are done. Now we bring the Berezin transformation into play. In so doing, put −1 ˜ f (z) = π f (ζ)|σz (ζ)|2 dm(ζ), z ∈ D, D

and Ar (f )(z) =

1 m B(z, r)

f (ζ)dm(ζ), B(z,r)

z ∈ D, r > 0.

34

Chapter 2. Poisson versus Berezin with Generalizations

Theorem 2.3.2. Let p ∈ (0, 2), r ∈ (0, ∞) and f ∈ A2,0 . Then the following statements are equivalent: (i) f ∈ Qp . p |f (z) − f (ζ)|2 1 − |σw (z)|2 dm(z)dm(ζ) < ∞. 4 ¯ |1 − z ζ| w∈D D D

(ii) sup

(iii) sup w∈D

D

(iv) sup w∈D

D

(v) sup w∈D

D

p |f |2 (z) − |f (z)|2 1 − |σw (z)|2

dm(z) < ∞. (1 − |z|2 )2

p Ar (|f |2 )(z) − |Ar (f )(z)|2 1 − |σw (z)|2 2 p Ar f − Ar (f )(z) (z) 1 − |σw (z)|2

dm(z) < ∞. (1 − |z|2 )2

dm(z) < ∞. (1 − |z|2 )2

Proof. Step 1: (i)⇔(ii) We have that f ∈ Qp if and only if supw∈D I1 (f ◦ σw ) < ∞ which is by Lemma 2.3.1 equivalent to supw∈D I2 (f ◦ σw ) < ∞. So the desired equivalence follows from a simple calculation with the conformal map σw . Step 2: (ii)⇔(iii) A routine computation with f ∈ A2,0 gives |f |2 (z) − |f (z)|2

= π

−1

D

= π

−1

D

|f ◦ φz (ζ) − f (z)|2 dm(ζ) |f (ζ) − f (z)|2

(1 − |z|2 )2 ¯ 4 dm(ζ). |1 − z ζ|

Accordingly,

= =

I2 (f ◦ σw ) p dm(z) (1 − |z|2 )2 1 − |σw (z)|2 |f (ζ) − f (z)|2 dm(ζ) ¯4 (1 − |z|2 )2 |1 − z ζ| D D p dm(z) |f |2 (z) − |f (z)|2 1 − |σw (z)|2 π . (1 − |z|2 )2 D

This implies the required equivalence. Step 3: (iii)⇒(iv)⇒(v)⇒(i) From Step 1 and (1 − |w|2 )2 1 ≈ , |1 − z w| ¯4 (1 − |z|2 )2

w∈D

2.4. Mixture of Derivative and Quotient it turns out that |f |2 (z) − |f (z)|2

35

|f (ζ) − f (η)|2 (1 − |z|2 )4 dm(ζ)dm(η) ¯ 4 |1 − z η¯|4 |1 − z ζ| D D |f (ζ) − f (η)|2 (1 − |z|2 )4 dm(ζ)dm(η) ¯ 4 |1 − z η¯|4 |1 − z ζ| B(z,2r) B(z,2r) |f (ζ) − f (η)| 2 dm(ζ)dm(η) m B(z, 2r) B(z,2r) B(z,2r) Ar |f − Ar (f )(z)|2 (z). ≈

Namely, (iii) implies (iv). In the meantime, the elementary estimates 2 |f (ζ) − c|2 dm(ζ), |f (0)|

c∈C

B(0,r)

and

(1 − |z|2 )2 1 ¯ 4 ≈ m B(z, r) , |1 − z ζ|

yield 2 2

2

(1 − |z| ) |f (z)|

ζ ∈ B(z, r)

|f ◦ σz (ζ) − Ar (f )(z)|2 dm(ζ)

B(0,r)

≈

|f (ζ) − Ar (f )(z)|2

B(z,r)

Consequently, I1 (f ◦ σw )

= D

1 − |z|2 2 ¯ 2 dm(ζ) |1 − z ζ|

≈

Ar |f − Ar (f )(z)|2 (z)

≈

Ar (|f |2 )(z) − |Ar (f )(z)|2 .

p |f (z)|2 1 − |σw (z)|2 dm(z) p Ar |f − Ar (f )(z)|2 (z) 1 − |σw (z)|2

dm(z) (1 − |z|2 )2 D p dm(z) ≈ . Ar (|f |2 )(z) − |Ar (f )(z)|2 1 − |σw (z)|2 (1 − |z|2 )2 D

In other words, (iv) implies (v) which implies (i).

2.4 Mixture of Derivative and Quotient In this section, we extend the characterization presented in the last section to the mixed situation involving derivative and quotient.

36

Chapter 2. Poisson versus Berezin with Generalizations

Lemma 2.4.1. Let p ∈ (0, ∞) and q ∈ [0, ∞). (i) Suppose F ∈ Hp and F (0) = 0. If q ∈ (0, min{2, p}], then p |F (ζ)| |dζ| |F (z)|p−q |F (z)|q (1 − |z|2 )q−1 dm(z). T

D

Moreover, if q ∈ [2, p + 2), then |F (z)|p−q |F (z)|q (1 − |z|2 )q−1 dm(z) |F (ζ)|p |dζ|. D

T

(ii) Suppose F ∈ Ap,0 and F (0) = 0. If q ∈ [0, p + 2), then |F (z)|p dm(z) ≈ |F (z)|p−q |F (z)|q (1 − |z|2 )q dm(z). D

D

Proof. For any F ∈ Ap,0 , let I(F ; p, q) = |F (z)|p−q |F (z)|q (1 − |z|2 )q dm(z). D

First of all, we recall two basic facts for f ∈ H, fr (z) = f (rz), r ∈ (0, 1) and p ∈ (0, ∞). The ﬁrst one is the Hardy–Stein identity which reads as: |fr (z)|p−2 |fr (z)|2 (− log |z|)dm(z).

fr pHp = 2π|f (0)|p + p2 D

The second one is the following Littlewood–Paley inequalities:

fr pHp |f (0)|p + I(fr ; p, p), p ∈ (0, 2] and |f (0)|p + I(fr ; p, p) fr pHp , p ∈ [2, ∞). Next, we check (i) and (ii). (i) For the ﬁrst inequality, let q ∈ (0, min{2, p}]. Clearly, the inequality is valid for p = q = 2. So we just consider following two cases. Case 1: 0 < q < 2 < p. Using the Hardy–Stein identity, H¨ older’s inequality and the second Littlewood–Paley inequality above for Fr (z) = F (rz), we derive

Fr pHp |Fr (z)|p−2 |Fr (z)|2 (− log |z|)dm(z) D

p−2 2−q I(Fr ; p, q) p−q I(Fr ; p, p) p−q p−2 p(2−q) I(F ; p, q) p−q Fr Hp p−q ,

2.4. Mixture of Derivative and Quotient

37

thereupon producing F pHp I(F ; p, q). Case 2: 0 < q < p < 2. By the ﬁrst Littlewood–Paley inequality above, H¨ older’s inequality and the Hardy–Stein identity for Fr , we ﬁnd

Fr pHp

I(Fr ; p, p) 2−p p−q I(Fr ; p, q) 2−q I(Fr ; p, 2) 2−q p(p−q) 2−p I(F ; p, q) 2−q Fr H2−q , p

thereupon implying F pHp I(F ; p, q). As with the second estimate, suppose q ∈ [2, p + 2). Since F ∈ Hp , this function can be written as F = BG where G has no zeros with G Hp = F Hp and B is a Blaschke product. Accordingly, |F |p−q |F |q ≤ 2q−1 (|G|p |B|p−q |B |q + |B|p |G|p−q |G |q ). p

Since G = 0, letting h = G q yields h ∈ Hq , |h |q = pq −1 |G|p−q |G |q and |B(z)|p |G(z)|p−q |G (z)|q (1 − |z|2 )q−1 dm(z) D |h (z)|q (1 − |z|2 )q−1 dm(z)

D

h qHq ≈ F pHp .

For the other estimate we use the Carleson embedding for Hp — see [Ga, pp. 238-239] to get |G(z)|p |B(z)|p−q |B (z)|q (1 − |z|2 )q−1 dm(z) sup C(a) G pHp , D

a∈D

where C(a) =

D

1 − |a|2 |B(z)|p−q |B (z)|q (1 − |z|2 )q−1 dm(z). |1 − a ¯z|2

The desired inequality will be established by proving supa∈D C(a) < ∞. In so doing, we are making use of z → σa (z), (1 − |z|2 )|B (z)| ≤ 1 − |B(z)|2 and the Hardy–Stein identity to derive |B ◦ σa (z)|p−q |(B ◦ σa ) (z)|q (1 − |z|2 )q−1 dm(z) C(a) = D |B ◦ σa (z)|q−2 |(B ◦ σa ) (z)|2 (1 − |z|2 )dm(z)

D

B ◦ σa qHq 1.

(ii) Proving that estimate amounts to verifying I(F ; p, 0) ≈ I(F ; p, q). To do so, let us consider two cases as follows.

38

Chapter 2. Poisson versus Berezin with Generalizations Case 1: q ∈ (0, 2]. If p ∈ (0, q), then by the H¨ older inequality, 1− pq p , I(F ; p, 0) I(F ; p, p) I(F ; p, q) q I(F ; p, 0)

and hence I(F ; p, 0) I(F ; p, q). At the same time, if p ∈ [q, ∞), then by the ﬁrst inequality in (i) we have that for r ∈ (0, 1), |F (rζ)|p |dζ| |F (rz)|p−q |rF (rz)|q (− log |z|)q−1 dm(z) T

D

and consequently, integrating both sides with respect to rdr and changing variable twice (z → rz and t = |z|/r), we get

1 r q−1 p−q q q−2 log I(F ; p, 0) |F (z)| |F (z)| r dm(z) rdr |z| 0 |z|
1 r q−1 p−q q p−1 log |F (z)| |F (z)| r dr dm(z) |z| |z| D |F (z)|p−q |F (z)|q (− log |z|)q dm(z)

D

I(F ; p, q).

In the last estimate we have used the fact that |F (z)|p−q |F (z)|q is subharmonic and the inequality −|z| log |z| ≤ 1 − |z|2 . To prove I(F ; p, q) I(F ; p, 0), applying the Hardy–Stein identity to Fr , integrating the induced identity with respect to rdr and using the inequality

1

r log |z|

r dr ≥ (1 − |z|2 )/4, |z|

we further get

r dr dm(z) |z| |z| D

1 r p−2 2 |F (z)| |F (z)| log dm(z) rdr ≈ |z| 0 |z|
I(F ; p, 2)

|F (z)|p−2 |F (z)|2

1

r log

≈ I(F ; p, 0). Accordingly, we can use the H¨ older inequality with q ∈ (0, 2] to obtain q 1− q2 I(F ; q, q) ≤ I(F ; p, 2) 2 I(F ; p, 0) I(F ; p, 0), as desired.

2.4. Mixture of Derivative and Quotient

39

Case 2: q ∈ (2, ∞). Note that if p − q + 2 ∈ (0, ∞) and r ∈ (0, 1), then the second inequality in (i) holds, namely, p−q q 2 q−1 |F (rz)| |rF (rz)| (1 − |z| ) dm(z) |F (rζ)|p |dζ|. D

T

Thus, by integration from r to 1 with respect to rdr, we obtain I(F ; p, q) I(F ; p, 0). To establish the opposite inequality, we observe 1 r 1 dr ≈ (1 − |z|2 ) 1 + log 2 . r log |z| |z| |z| This comparability result, along with the H¨ older inequality and the estimate 2p 1 |z|p− q 1 + log 2 1, z ∈ D, |z| produces I(F ; p, 0)

D

|F (z)|p−2 |F (z)|2

1

r log |z|

r dr dm(z) |z|

1− 2q 2 F (z) p dm(z) I(F ; p, q) q z D 2 1− 2q I(F ; p, q) q , I(F ; p, 0)

thereupon reaching I(F ; p, 0) I(F ; p, q).

The following characterization extends Theorem 2.3.2 from q = 0 to q > 0. Theorem 2.4.2. Given p ∈ (0, 2) and q ∈ [0, 4). For f ∈ H let Ξ(f ; ζ, z) =

(1 − |ζ|2 )|f (ζ)| , ζ, z ∈ D. |f (ζ) − f (z)|

Then f ∈ Qp if and only if. q p |f (z) − f (ζ)|2 sup Ξ(f ; ζ, z) 1 − |σw (z)|2 dm(z)dm(ζ) < ∞. 4 ¯ |1 − z ζ| w∈D D D Proof. Suppose the last condition is valid. We then employ Lemma 2.4.1 (ii) with F = f ◦ σw − f (w) to obtain 2 (1 − |w|2 )|f (w)| |f ◦ σw (z) − f (w)|2 dm(z) D |f ◦ σw (z) − f (w)|2−q |(f ◦ σw ) (z)|q (1 − |z|2 )q dm(z). D

40

Chapter 2. Poisson versus Berezin with Generalizations

Integrating the last estimate against (1 − |σa (w)|2 )p dm(w), a ∈ D, and changing the variable z → σw (z), we ﬁnd q p 2 |f (z) − f (ζ)|2 Ξ(f ; ζ, z) 1 − |σa (z)|2 dm(z)dm(ζ), Fp (f, a) 4 ¯ |1 − z ζ| D D thereupon deriving f ∈ Qp . Conversely, suppose f ∈ Qp . Note that by the change of variable λ = σa (w), a ∈ D, (1 − |σa (w)|2 )p (1 − |λ|2 )p 2 2 dm(w) = (1 − |a| ) dm(λ) |1 − wu| ¯ 4 ¯λ − a¯ u + λ¯ u|4 D D |1 − a (1 − |a|2 )2 (1 − |λ|2 )p = ¯ a (u)|4 dm(λ) |1 − a¯ u|4 D |1 − λσ

(1 − |a|2 )p (1 − |u|2 )p−2 , u ∈ D. 1−a ¯u|2p

So, using the change of variable: λ = σw (z) and Lemma 2.4.1 (ii) again, we ﬁnd q |f (z) − f (ζ)|2 2 p (z)| dm(z)dm(ζ) Ξ(f ; ζ, z) 1 − |σ a ¯4 |1 − z ζ| D I(f ◦ σw − f (w); 2, q)(1 − |w|2 )−2 (1 − |σa (w)|2 )p dm(w) = D |f ◦ σw (z) − f (w)|2 dm(z) (1 − |w|2 )−2 (1 − |σa (w)|2 )p dm(w) D D (1 − |σa (w)|2 )p |(f ◦ σw ) (z)|2 (1 − |z|2 )2 dm(z) dm(w) (1 − |w|2 )2 D D (1 − |σa (w)|2 )p |f (u)|2 (1 − |u|2 )2 dm(w) dm(u) |1 − wu| ¯ 4 D D 2 Fp (f, a) f 2Qp ,1 , whence implying the desired assertion.

2.5 Dirichlet Double Integral without Derivative In this section, we use the derivative-free description of the weighted Dirichlet integral over the bidisk to characterize Qp functions. Lemma 2.5.1. Let f ∈ H. If α ∈ (−∞, 1/2], β, γ ∈ (−1, ∞) and min{β, γ} + 2α > −1, then |f (z)|2 (1 − |z|2 )1−2α dm(z) D |f (z) − f (w)|2 (1 − |z|2 )β (1 − |w|2 )γ dm(z)dm(w). ≈ ¯w|3+2α+β+γ D D |1 − z

2.5. Dirichlet Double Integral without Derivative

41

Proof. It suﬃces to consider the case β = γ since β ≥ γ implies 2γ−β (1 − |z|2 )β (1 − |w|2 )β |1 − z¯w|3+2α+2β

≤

(1 − |z|2 )β (1 − |w|2 )γ |1 − z¯w|3+2α+β+γ

≤

2β−γ (1 − |z|2 )γ (1 − |w|2 )γ . |1 − z¯w|3+2α+2γ

If α = 1/2, then a direct computation yields |f (z) − f (0)|2 (1 − |z|2 )β dm(z) |f (z)|2 (1 − |z|2 )β+2 dm(z). D

D

Applying this inequality to f ◦ σw , we derive I(f ; w) = |f (z) − f (w)|2 |1 − z¯w|−4−2β (1 − |z|2 )β dm(z) D = (1 − |w|2 )−(2+β) |f ◦ σw (z) − f ◦ σw (0)|2 (1 − |z|2 )β dm(z) D 2 −(2+β) (1 − |w| ) |(f ◦ σw ) (z)|2 (1 − |z|2 )β+2 dm(z) D |f (z)|2 |1 − z¯w|−2β−4 (1 − |z|2 )β+2 dm(z). D

Integrating I(f ; w) over D with respect to (1 − |w|2 )β dm(w) and changing the order of integration, we further get (1 − |w|2 )β

f 2D |f (z)|2 dm(w) (1 − |z|2 )β+2 dm(z) 4+2β |1 − wz| ¯ D D I(f ; w)(1 − |w|2 )β dm(w) D

thanks to β > −1. As for the opposite inequality, we make the following estimates: (1 − |z|2 )2 |f (z)|2

= ≈

|(f ◦ σz ) (0)|2 |f ◦ σz (u) − f ◦ σz (0)|2 dm(u) |u|<1/2 |f ◦ σz (u) − f ◦ σz (0)|2 (1 − |u|2 )β dm(u) D |f (z) − f (w)|2 (1 − |w|2 )β dm(w), (1 − |z|2 )β+2 |1 − wz| ¯ 4+2β D

and then integrate both sides of the last inequalities against (1 − |z|2 )−2 dm(z) to produce 2 I(f ; w)(1 − |w|2 )β dm(w),

f D D

42

Chapter 2. Poisson versus Berezin with Generalizations

as desired. ∞ Now suppose α < 1/2. Setting δ = 3/2 + α + β, f (z) = k=0 ak z k , and for any integer m, 1 , m ≥ 0, 1m = 0 , m < 0, plus making substitutions: z = reiθ , w = seiφ , t = φ − θ and ζ = seit , we have

= ≈ ≈ ≈ ≈

I(f ; β, δ) |f (z) − f (w)|2 (1 − |z|2 )β (1 − |w|2 )β dm(z)dm(w) 2δ |1 − z ¯ w| D D 1 π 1 π |f (reiθ ) − f (sei(θ+t) )|2 (1 − r2 )β (1 − s2 )β rsdθdrdtds |1 − rseit |2δ 0 −π 0 −π 1 π 1 ∞ |rk − sk eikt |2 |ak |2 (1 − r2 )β (1 − s2 )β rsdrdtds it |2δ |1 − rse 0 −π 0 k=1 1 ∞ |rk − ζ k |2 2 |ak | (1 − r2 )β (1 − s2 )β rdm(ζ)dr 2δ 0 D |1 − rζ| k=1 ∞

|ak |2 I(k),

k=1

where

1

I(k) = 0

D

|qr (ζ)|2 (1 − r2 )β (1 − s2 )β rdm(ζ)dr with qr (ζ) =

rk − ζ k . (1 − rζ)δ

If we can prove I(k) ≈ k 2α for k ∈ N, we will then have ∞

I(f ; δ, β) ≈

2 2α

|ak | k

k=1

≈

D

|f (z)|2 (1 − |z|2 )1−2α dm(z),

thereby establishing the desired comparability. 1 In so doing, let B(x, y) = 0 tx−1 (1 − t)y−1 dt. Then for x, y ∈ (0, ∞) and j, k ∈ N,

=

B(j + 1 + x, y) − B(j + k + 1 + x, y) 1 k−1 tj+x tn (1 − t)y dt 0

≈

k−1

n=0

(n + j + 1)−y−1 ≈ (j + 1)−y − (k + j + 1)−y .

n=0

2.5. Dirichlet Double Integral without Derivative

43

Since qr (ζ) is a member of A2,β , it can be represented by the reproducing formula for this space. Consequently, I(k) 1 = 0

2 1 + β 1 − |ζ|2 β qr (η)(1 − |η|2 )β dm(η) π (1 − r2 )−1 dm(ζ)rdr (1 − η¯ζ)2+β 0 D D 2 1 j j 2 β ∞ D qr (η)¯ η ζ (1 − |η| ) dm(η) 1 − |ζ|2 β (1 − r2 )−1 dm(ζ)rdr B(j + 1, β + 1) 0 D j=0 ⎛ 2 ⎞ j 2 2 β 1 ∞ (1 − |η| q (η)¯ η )(1 − r ) dm(η) ⎟ r D ⎜ ⎠ rdr. ⎝ B(j + 1, β + 1) 0

= ≈

≈

D

|qr (ζ)|2 (1 − |ζ|2 )β dm(ζ)(1 − r2 )β rdr

1

j=0

Note also that qr (η) = (rk − η k )

∞ Γ(n + δ)rn η n . Γ(δ)Γ(n + 1) n=0

Thus, we can conclude qr (η)¯ η j (1 − |η|2 )β dm(η) D

≈

1j−k Γ(j − k + δ)B(j + 1, β + 1)rj−k Γ(j + δ)B(j + 1, β + 1)rk+j − , Γ(j + 1) Γ(j − k + 1)

so that I(k) ≈ lim

m→∞

where

∞ j=m−k+1

I1 (j, k) + 2

m−k

I1 (j, k) − I2 (j, k) ,

j=0

2 B(j + 1, β + 1)B(j + k + 1, β + 1) Γ(j + δ) I1 (j, k) = 2 Γ(j + 1)

and I2 (j, k) =

2 B(j + k + 1, β + 1) Γ(j + k + δ)Γ(j + δ) . Γ(j + k + 1)Γ(j + 1)

By the above estimate on B(·, ·), it follows that I1 (j, k) ≈ j −1−β (k + j)−1−β j 2δ−2 j 2α−1 , j ∈ N,

44

Chapter 2. Poisson versus Berezin with Generalizations

and so that limm→∞ estimates

m

j=m−k+1 I1 (j, k)

= 0 due to α ∈ (−∞, 1/2). Using the

Γ(j + δ − 1) ≈ (j + 1)α+δ−3/2 and B(k, δ − 1) ≈ k 1−δ , Γ(j + 1/2 − α) we further get that I1 (j, k) − I2 (j, k) is equal to the product of both B(k + j + 1, β + 1)Γ(j + δ)Γ(β + 1) Γ(j + 1)Γ(1/2 − α) and B(j + δ, 1/2 − α) − B(j + k + δ, 1/2 − α), whence producing I1 (j, k) − I2 (j, k) ≈ (j + 1)δ−1 (K + j + 1)−β−1 (j + 1)α−1/2 − (j + k + 1)α−1/2 . Combining the previous estimates we ﬁnally obtain I(k) ≈

lim

m→∞

≈ ≈

m−k

(j + 1)δ−1 (K + j + 1)−β−1 (j + 1)α−1/2 − (j + k + 1)α−1/2

j=0

∞

xδ−1 (k + x +−β−1 xα−1/2 − (k + x)α−1/2 dx 0 ∞ k 2α t2α+β (1 + t)α−β−3/2 (1 + t)1/2−α − t1/2−α dt ≈ k 2α , 0

as required.

Clearly, Lemma 2.5.1 extends Lemma 2.3.1 from one parameter to three. As a direct application of Lemma 2.5.1 we derive the following result which is a natural generalization of Theorem 2.3.2. Theorem 2.5.2. Given p ∈ (0, ∞), β, γ ∈ (−1, ∞) and min{β, γ} − p > −2. Let f ∈ H. Then f ∈ Qp if and only if |f (z) − f (w)|2 (1 − |a|2 )p (1 − |z|2 )β (1 − |w|2 )γ dm(z)dm(w) sup ¯w|4−p+β+γ |1 − a ¯z|p+β−γ |1 − a ¯w|p+γ−β a∈D D D |1 − z is ﬁnite. Proof. Applying Lemma 2.5.1 to f ◦ σa with p = 1 − 2α, and making use of the identities 1 − |σa (z)|2 =

(1 − |a|2 )(1 − |z|2 ) 1 − |a|2 , σ ◦ σ (z) = z, and |σ (z)| = , a a a |1 − a ¯z|2 |1 − a ¯z|2

we reach the assertion.

2.6. Notes

45

2.6 Notes Note 2.6.1. Section 2.1 is taken from in [Xi4, 3.1]. A good source for more on the Brownian motions associated with BMO is [Pet]. Note 2.6.2. Section 2.2 comes from [Xi4, 3.3]. For the extremal case p = 1 of Theorem 2.2.1 (iii), see the main result in [Kob1] of which a special form is: f ∈ BMOA if and only if |f − f (w)|2 has the least harmonic majorant for any w ∈ D. And in this case, Theorem 2.2.1 (iii) is replaced by f (z)|2 )(z) = sup (|f − z∈D

≈

|2 (z) − |f (z)|2 sup |f z∈D |(z)2 − |f (z)|2 < ∞. sup |f z∈D

See [Leu, Section 3]. As with the limiting case p → 0, we have however noted that f ∈ D if and only if f ∈ H2 and |f (ζ) − f (η)|2 |dζ||dη| < ∞. |ζ − η|2 T T See e.g. [Pee]. This means that Theorem 2.2.1 (ii) is natural. Moreover, regarding Corollary 2.2.2 (i), we may ask an interesting question: is there a function ψ > 0 / Qp ? This question has a a.e. on T with ψ ∈ Qp (T), log ψ ∈ L1 (T) but Oψ ∈ positive answer for p = 1 — see also [Dy], although the answer is unknown for p ∈ (0, 1). Note 2.6.3. Section 2.3 is produced by considering a special case of the weight functions considered in the main results of [WulZh2]. It is well known that f ∈ B = Q2 if and only if f ∈ A2,0 and f (z)|2 )(z) = sup (|f −

|2 (z) − |f (z)|2 sup |f

z∈D

z∈D

≈

sup (Ar (|f |2 )(z) − |Ar (f )(z)|2 ) z∈D

≈

sup Ar (|f − Ar (f )(z)|2 )(z) < ∞. z∈D

See also [Zhu1, Theorem 7.1.9]. In other words, when extending Theorem 2.3.2 to the end point case p = 2, we should obtain the corresponding characterization of the Bloch space. Meanwhile, if considering the other endpoint p = 0, then we ﬁnd that f ∈ D is described by f ∈ H obeying |f (z) − f (w)|2 dm(z)dm(w) < ∞. |1 − z w| ¯4 D D See e.g., [Zhu1, Theorem 5.3.4].

46

Chapter 2. Poisson versus Berezin with Generalizations

Note 2.6.4. Section 2.4 is taken from [Xi5] whose Berezin-type characterization is a by-product of [Kw] that describes the holomorphic diagonal Besov space Bp = {f ∈ H : f ∈ Ap,p−2 }, p ∈ (1, ∞) respectively the Bloch space B by means of α |f (z) − f (w)|p Ξ(f ; z, w) dm(z)dm(w) < ∞, |1 − wz| ¯ 4 D D where α ∈ [0, 2 + p), respectively α |f (z) − f (w)|p Ξ(f ; z, w) sup (1 − |σa (w)|2 )2 dm(z)dm(w) < ∞, |1 − wz| ¯ 4 a∈D D D where p ∈ (0, ∞), α ∈ [0, 2 + p). In addition, Lemma 2.4.1 (ii) is from [Kw], and Lemma 2.4.1 (i) due to [KimKw] and [Lue]. Note 2.6.5. Section 2.5 comprises [RocWu, Theorem 1] and its proof and conformally invariant version (cf. [Xi5]). The argument for the case α = 1/2 of Lemma 2.5.1 is motivated by the proofs of [ArFiJaPe, Lemmas 3.2–3.3–3.4] — on the other hand, this special case can be found in [ArFiPe2, Proposition 3.6].

Chapter 3

Isomorphism, Decomposition and Discreteness When studying a space of holomorphic functions, it is very useful to be able to establish an isomorphism of the space to a well-understood space, or to express every member in the space as a linear combination of some appropriate elementary functions, or to obtain a discrete presentation of the space. In this chapter, we will see that each Qp function has such representations through a consideration of the following ﬁve aspects: • • • • •

Carleson Measures under an Integral Operator; Isomorphism to a Holomorphic Morrey Space; Decomposition via Bergman Style Kernels; Discreteness by Derivatives; Characterization in Terms of a Conjugate Pair.

3.1 Carleson Measures under an Integral Operator For p > 0, we write CMp for the class of all p-Carleson measures — nonnegative Radon measures µ on D with µ S(I)

µ CMp = sup < ∞. |I|p I⊆T Lemma 3.1.1. Let p, q ∈ (0, ∞) and µ be nonnegative Radon measures µ on D. Then µ ∈ CMp if and only if

µ CMp ,q = sup

w∈D

D

(1 − |w|2 )q dµ(z) < ∞. |1 − wz| ¯ p+q

48

Chapter 3. Isomorphism, Decomposition and Discreteness

Proof. Suppose µ CMp ,q < ∞. Given a Carleson box S(I) = {z ∈ D : 1 − h ≤ |z| < 1, |θ − arg z| ≤ h} with h = |I|. Letting w = (1 − h)ei(θ+h/2) , we achieve (1 − |w|2 )q µ(S(I)) , ¯ p+q |I|p z∈S(I) |1 − wz|

µ CMp ,q ≥ µ(S(I)) inf

which implies µ CMp µ CMp ,q < ∞. On the other hand, if µ ∈ CMp , then µ CMp < ∞. When |w| < 3/4, we have (1 − |w|2 )q dµ(z) µ(D) µ CMp . ¯ p+q D |1 − wz| When 3/4 ≤ |w| < 1, we put En = {z ∈ D : |z − w/|w|| < 2n (1 − |w|)} and then obtain µ(En ) µ CMp 2np (1 − |w|2 )p for n ∈ N. Note that 1 (1 − |w|2 )q , z ∈ E1 , p+q |1 − wz| ¯ (1 − |w|2 )p and for n ≥ 1 and E0 = ∅, (1 − |w|2 )q 1 2(p+q)n , z ∈ En \ En−1 . p+q |1 − wz| ¯ 2 (1 − |w|2 )p Thus

µ CMp ,q

≤

sup

∞

w∈D n=1

sup

w∈D n=1

En \En−1

∞

µ CMp

(1 − |w|2 )q dµ(z) |1 − wz| ¯ p+q

µ(En ) − |w|2 )p

22n(p+q) (1 ∞

2−nq ,

n=1

as desired.

For a, b > 0 we deﬁne formally a Ta,b operator; that is, the following integral operator: (1 − |w|2 )b−1 Ta,b f (z) = f (w)dm(w), z ∈ D. ¯ a+b D |1 − wz| In the next result we show that CMp is invariant under Ta,b . 1+p Lemma 3.1.2. Let p ∈ (0, 2), a > 2−p 2 , b > 2 and f be Lebesgue measurable on 2 2 p D. If |f (z)| (1−|z| ) dm(z) belongs to CMp then |Ta,b f (z)|2 (1−|z|2 )p+2a−2 dm(z) also belongs to CMp .

3.1. Carleson Measures under an Integral Operator Proof. Assuming and

49

dµf,p (z) = |f (z)|2 (1 − |z|2 )p dm(z)

dµTa,b f,p (z) = |Ta,b f (z)|2 (1 − |z|2 )p+2a−2 dm(z),

we verify that µf,p CMp < ∞ yields µTa,b f,p CMp < ∞. To that end, we use 2k I as the subarc of T with the same center as I and the length 2k |I| for k ∈ N with k ≤ − log2 |I|. Given any Carleson box S(I). Under µf,p CMp < ∞ we make the following estimates: µTa,b f,p S(I) = |Ta,b f (z)|2 (1 − |z|2 )p+2a−2 dm(z) S(I)

≤

+

S(I)

S(2I)

S(I)

S(2I)

S(I)

|1 − wz| ¯ a+b

2

D\S(2I)

2 dm(w)

dm(z) (1 − |z|2 )2−p−2a

|f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b

+ =

|f (w)|(1 − |w|2 )b−1

D\S(2I)

|f (w)|(1 − |w|2 )b−1 |1 − wz| ¯ a+b

dm(z) (1 − |z|2 )2−p−2a

2 dm(z) dm(w) (1 − |z|2 )2−p−2a

Int1 + Int2 .

For Int1 , we shall apply the classical Schur’s Lemma for bounded operators on L2 (D). Indeed, we consider k(z, w) =

(1 − |z|2 )b−1+p/2 (1 − |w|2 )b−1−p/2 |1 − wz| ¯ a+b

and its induced integral operator f (w)k(z, w) dm(w).

(Tf )(z) = D

Using [Zhu1, Lemma 4.2.2] (or [Xi3, Lemma 1.4.1]), we derive that if p p p p max − a − , − b < γ < min a − 1 + , b − 1 − , 2 2 2 2 then

D

and

D

k(z, w)(1 − |w|2 )γ dm(w) (1 − |z|2 )γ k(z, w)(1 − |z|2 )γ dm(z) (1 − |w|2 )γ .

50

Chapter 3. Isomorphism, Decomposition and Discreteness

Accordingly, the operator T is bounded from L2 (D) to L2 (D). Once the function f in Tf is replaced by g(w) = (1 − |w|2 )p/2 |f (w)|1S(2I) (w), where 1E stands for the characteristic function of E, we ﬁnd

Int1

D D

2 g(w)k(z, w) dm(w) dm(z)

D

|g(z)|2 dm(z) |I|p µf,p CMp .

To handle Int2 , we note that both ¯ 2n |I| |1 − wz|,

and

S(2n I)

z ∈ S(I), w ∈ S(2n+1 I) \ S(2n I),

(1 − |z|2 )c−2 dm(z) (2n |I|)c ,

c > 1,

hold for all n ∈ N ∪ {0}. When writing D \ S(2I) =

∞

S(2n+1 I) \ S(2n I),

n=1

and using a > (2 − p)/2 as well as p ∈ (0, 2), we then get by the H¨ older inequality

Int2 ∞

2 |f (w)|(1 − |w|2 )b−1 dm(z) dm(w) a+b 2 )2−p−2a |1 − wz| ¯ (1 − |z| n+1 n S(I) n=1 S(2 I)\S(2 I)

2 ∞ |f (w)| dm(z) n −(a+b) (2 |I|) dm(w) 2 )1−b 2 )2−p−2a (1 − |w| (1 − |z| n+1 S(I) n=1 S(2 I)

2 ∞ |f (w)| 2a+p n −(a+b) |I| (2 |I|) dm(w) 2 1−b S(2n+1 I) (1 − |w| ) n=1 ∞

2 12 2a+p n −a− p n+1 2 |I| (2 |I|) I) µf,p S(2 |I|p

∞

n=1

2−na

2

µf,p CMp .

n=1

The foregoing estimates on Int1 and Int2 give µTa,b f,p ∈ CMp .

As an immediate application of Lemma 3.1.2, we can estimate the distance of a Bloch function to the Qp space.

3.1. Carleson Measures under an Integral Operator

51

Theorem 3.1.3. For p ∈ (0, 2) and f ∈ B let distB (f, Qp ) = inf{ f − g B : g ∈ Qp }. Then distB (f, Qp ) ≈ inf > 0 : 1Ω (f ) (z)(1 − |z|2 )p−2 dm(z) ∈ CMp , where Ω (f ) = {z ∈ D : (1 − |z|2)|f (z)| ≥ } and 1E stands for the characteristic function of a set E. Proof. Because of f ∈ B, this function has the following integral representation: (1 − |w|2 )f (w) 1 dm(w) = f1 (z) + f2 (z), f (z) = f (0) + π D w(1 ¯ − wz) ¯ 2 where 1 π

f1 (z) = f (0) + and f2 (z) =

1 π

Ω (f )

Note that |f1 (z)| ≤ f B

D\Ω (f )

D

(1 − |w|2 )f (w) dm(w) w(1 ¯ − wz) ¯ 2

(1 − |w|2 )f (w) dm(w). w(1 ¯ − wz) ¯ 2

1 − |w|2 1Ω (f ) (w) dm(w). |1 − z w| ¯ 3 1 − |w|2

2 p−2

So, if 1Ω (f ) (z)(1 − |z| ) dm(z) ∈ CMp then from Lemma 3.1.2 it follows that (1 − |z|2 )p |f1 (z)|2 dm(z) ∈ CMp and consequently f1 ∈ Qp . At the same time, we have |f2 (z)| ≤

|1 − z w| ¯ −3 dm(w) (1 − |z|2 )−1 , z ∈ D, D

hence getting distB (f, Qp ) ≤ f − f1 B . This gives distB (f, Qp ) inf > 0 : 1Ω (f ) (z)(1 − |z|2 )p−2 dm(z) ∈ CMp . In order to verify the reversed version of the above-established inequality, we suppose distB (f, Qp ) < inf > 0 : 1Ω (f ) (z)(1 − |z|2 )p−2 dm(z) ∈ CMp = 0 . Of course, we only consider the case 0 > 0 — otherwise there is nothing to argue. Then, there exists an 1 such that 0 < 1 < 0 and distB (f, Qp ) ≤ 1 . Hence, by deﬁnition, we can ﬁnd a function g ∈ Qp such that f − g B < 1 . Now for any

∈ ( 1 , 0 ) we have 1Ω (f ) (z)(1 − |z|2 )p−2 dm(z) ∈ / CMp .

52

Chapter 3. Isomorphism, Decomposition and Discreteness

But, f − g B < 1 yields (1 − |z|2 )|g (z)| > (1 − |z|2 )|f (z)| − 1 ,

z ∈ D,

and so 1Ω (f ) (z) ≤ 1Ω−1 (g) (z),

z ∈ D.

The last inequality implies 1Ω−1 (g) (z)(1 − |z|2 )p−2 dm(z) ∈ / CMp . But 1Ω−1 (g) (z)(1 − |z|2 )p−2 dm(z) ≤ ( − 1 )−2 |g (z)|2 (1 − |z|2 )p 1Ω−1 (g) (z)dm(z) gives

1Ω−1 (g) (z)(1 − |z|2 )p−2 dm(z) ∈ CMp ,

a contradiction. Therefore, we must have 0 ≤ distB (f, Qp ), as required.

3.2 Isomorphism to a Holomorphic Morrey Space For p ∈ (−∞, ∞), denote by L2,p (T) the Morrey space of all Lebesgue measurable functions f on T that satisfy

f L2,p (T)

12 −p 2 = sup |I| |f (ζ) − fI | |dζ| < ∞, I⊆T

I

where the supremum as before ranges over all subarcs I of T. Clearly, L2,0 (T) and L2,1 (T) coincide with L2 (T) and BMO(T) respectively. Moreover, BMO(T) ⊆ L2,p1 (T) ⊆ L2,p2 (T) ⊆ L2 (T),

0 < p2 < p1 < 1.

Even more surprisingly, there is a closed relationship between L2,p (T) and Qp (T). To see this, we need a characterization of the Morrey space. Theorem 3.2.1. Let p ∈ (0, 2) and f be Lebesgue measurable on T. Then the following statements are equivalent: (i) f ∈ L2,p (T). (ii) If dµf (z) = |∇fˆ(z)|2 (1 − |z|2 )dm(z) then µf ∈ CMp . |∇fˆ(z)|2 1 − |σw (z)|2 dm(z) < ∞. (iii) sup (1 − |w|2 )1−p w∈D

(iv) sup (1 − |w|2 )1−p w∈D

D

D

|∇fˆ(z)|2 − log |σw (z) dm(z) < ∞.

3.2. Isomorphism to a Holomorphic Morrey Space

53

12 fˆ(z)|2 (z) < ∞. (v) f L2,p (T),1 = sup (1 − |z|2 )1−p |f − z∈D

12 |2 (z) − |fˆ(z)|2 (vi) f L2,p (T),2 = sup (1 − |z|2 )1−p |f < ∞. z∈D

Proof. By Lemma 3.1.1, Hardy–Littlewood’s identity 1 |∇fˆ(z)|2 (1 − |z|2 )dm(z) ≈ |∇fˆ(z)|2 log dm(z) |z| D D ≈ |f (ζ) − fˆ(0)|2 |dζ|, T

and some calculations with σw , we can ﬁgure out that (ii)⇔(iii)⇔(iv)⇔(v)⇔(vi). Thus, it suﬃces to prove (i)⇔(v). Suppose (i) holds. For each nonzero z ∈ D let Iz be the subarc of T with center z/|z| and length 1 − |z|, and for z = 0 let Iz = T. Moreover, for n = 0, 1, . . . , N − 1 let Jn = 2n Iz where N is the smallest natural number such that 2N |Iz | ≥ 1. Finally, let JN = T. With these, we make the following estimates for a given point z ∈ D:

|f − fˆ(z)|2 (z) 1 − |z|2 |f (ζ) − fIz |2 |dζ| |1 − z¯ζ|2 T N −1 1 − |z|2 |f (ζ) − fJ0 |2 + |dζ| |1 − z¯ζ|2 J0 n=0 Jn+1 \Jn N −1 2 −1 −2n (1 − |z| ) |f (ζ) − fJ0 |2 |dζ| + 2 |J0 |−1

J0

Jn+1 \Jn

n=0

|f (ζ) − fJ0 |2 |dζ| +

J0

N −1 n=0

2−n |Jn+1 |

Jn+1 \Jn

|f (ζ) − fJ0 |2 |dζ|.

Note that the Cauchy–Schwarz inequality and f ∈ L2,p (T) imply |fJn+1 − fJn | ≤

2

|f (ζ) − fJn+1 ||dζ| |Jn+1 | Jn+1

12 2 |f (ζ) − fJn+1 |2 |dζ| |Jn+1 | Jn+1

2|Jn+1 | 2

p−1 2

f L2,p (T)

1+(p−1)(n+1)/2

(1 − |z|)(p−1)/2 f L2,p(T) .

54

Chapter 3. Isomorphism, Decomposition and Discreteness

So |fJn+1 − fJ0 | ≤ |fJn+1 − fJn | + · · · + |fJ1 − fJ0 | max{1, 2n(p−1)/2 }n(1 − |z|)

p−1 2

f L2,p (T) .

Accordingly, 1

≤

|Jn+1 | 1

|f (ζ) − fJ0 |2 |dζ|

Jn+1

2 12 |f (ζ) − fJn+1 | |dζ| + |fJn+1 − fJ0 |

|Jn+1 |

2

Jn+1

2

n max{1, 2n(p−1)}(1 − |z|)p−1 f 2L2,p (T) . Finally, we obtain ∞ n2 max{1, 2n(p−1) } |f − fˆ(z)|2 (z) (1 − |z|2 )p−1 f 2L2,p (T) . 2n n=0

In other words, (v) is valid because of p ∈ (0, 2). Conversely, if (v) holds, then for any subarc I ⊂ T, we take such a z ∈ D that z/|z| is the center of I and |z| = 1 − |I|, to get ¯ 1 − |z|2 , |1 − ζz|

ζ ∈ I,

and then −p

|I|

2

|f (ζ) − fI | |dζ|

≤

I

−p

|f (ζ) − fˆ(z)|2 |dζ| 1 − |z|2 2 1−p (1 − |z| ) |f (ζ) − fˆ(z)|2 ¯ 2 |dζ| |1 − ζz| I fˆ(z)|2 )(z) (1 − |z|2 )1−p (|f −

4|I|

I

f 2L2,p(T),1 ,

which gives (i) right away.

Corollary 3.2.2. For p ∈ (0, 2) let H2,p = L2,p ∩ H2 be the holomorphic Morrey space on D. If f ∈ H2 , then the following statements are equivalent: (i) f ∈ H2,p (T). (ii) |f (z)|2 (1 − |z|2 )dm(z) ∈ CMp . 2 1−p (iii) sup (1 − |w| ) |f (z)|2 1 − |σw (z)| dm(z) < ∞. w∈D

D

3.2. Isomorphism to a Holomorphic Morrey Space (iv) sup (1 − |w|2 )1−p

D

w∈D

55

|f (z)|2 − log |σw (z)| dm(z) < ∞.

12 (v) sup (1 − |z|2 )1−p |f − f (z)|2 (z) < ∞. z∈D

12 (vi) sup (1 − |z|2 )1−p |f < ∞. |2 (z) − |f (z)|2 z∈D

Proof. Note that fˆ = f for f ∈ H2 . So the desired result follows from Theorem 3.2.1. Now, it is natural to compare Qp with H2,p . In so doing, we consider the following formal α-order derivative of f ∈ H at z ∈ D: f

(α)

Γ(b + α) (z) = πΓ(b)

D

w ¯α−1 (1 − |w|2 )b−1 f (w) dm(w), (1 − wz) ¯ b+α

b + α > 0, b > 1.

Here, x stands for the smallest integer greater than or equal to x ∈ R, and Γ(·) is still the Gamma function. A straightforward calculation gives ! 0 , n < α − 1 + 1, n (α) (z ) = Γ(b+n+α−1−α−1 )Γ(n+1) n−1−α−1 z , n ≥ α − 1 + 1. Γ(b+n)Γ(n−α−1 ) and thus illustrates that if α = n ∈ N, then f (α) is just the usual n-th derivative of f . The next result tells us that each holomorphic Q class is isomorphic to a holomorphic Morrey space. Theorem 3.2.3. Let p ∈ (0, 2), α ∈ ( 2−p 2 , ∞) and f ∈ H. Then the following statements are equivalent: (i) f ∈ Qp . (ii) |f (α) (z)|2 (1 − |z|2 )2α−2+p dm(z) ∈ CMp . (iii) f (

1−p 2 )

∈ H2,p .

Consequently, f ∈ H2,p if and only if f (

p−1 2 )

∈ Qp .

Proof. Step 1: (i)⇔(ii) If f ∈ Qp , then by Theorem 1.1.3 we see that |f (z)|2 (1 − |z|2 )p dm(z) is a p-Carleson measure, and so is |f (α) (z)|2 (1 − |z|2 )2α−2+p dm(z) thanks to an application of Lemma 3.1.2 to |f

(α)

Γ(b + α) (z)| ≤ πΓ(b)

D

(1 − |w|2 )b−1 |f (w)| dm(w), |1 − wz| ¯ b+α

z ∈ D.

56

Chapter 3. Isomorphism, Decomposition and Discreteness

Conversely, suppose |f (α) (z)|2 (1 − |z|2 )2α−2+p dm(z) ∈ CMp . Then we consider the following function g(z) generated by f (α) : Γ(b + 1)z α−1 (1 − |ζ|2 )b+α−2 (α) g(z) = f (ζ)dm(ζ). ¯ b+1 πΓ(b + α − 1) D (1 − ζz) From Lemma 3.1.2 it follows that |g(z)|2 (1 − |z|2 )p dm(z) ∈ CMp . To establish the relationship between g and f , we express f and f (α) as the Taylor series below: f (z) =

∞

aj z j

and f (α) (z) =

j=0

∞

aj,α z j ,

z ∈ D.

j=0

If m = α − 1, then m ≥ 0 (if otherwise m ≤ −1, then α − 1 ≤ −1 and hence α ≤ 0 but α > (2−p)/2 > 0, a contradiction). Consequently, the formula regarding (z j )(α) implies Γ(j + b + α)Γ(j + m + 2) aj,α = aj+m+1 , j ∈ N ∪ {0}. Γ(j + 1)Γ(j + m + 1 + b) This, along with a calculation with the integral determining g, implies ∞ Γ(j + b + 1) aj,α z j+m g(z) = Γ(j + b + α) j=0

=

∞ Γ(j + b + 1)Γ(j + m + 2) aj+m+1 z j+m . Γ(j + m + b + 1)Γ(j + 1) j=0

Case 1: m = 0. This forces α = 1 due to α > (2 − p)/2 > 0 and thus g = f . Of course, f ∈ Qp . Case 2: m > 0. In this case, we use β(x, y) = Γ(x)Γ(y) Γ(x+y) as the Beta function, and then obtain ∞ β(j + b + 1, m) (j + m + 1)aj+m+1 z j+m . g(z) = β(j + 1, m) j=0 Writing sm (z) =

m

jaj z

j−1

j=1

and h(z) =

∞ j=0

1−

β(j + b + 1, m) aj+m+1 z j+m+1 , β(j + 1, m)

we ﬁnd f (z) = g(z) + sm (z) + h (z). Note that sm is a polynomial and |g(z)|2 (1 − |z|2 )p is a p-Carleson measure. Thus, in order to prove f ∈ Qp it is enough to verify h ∈ D. To this end, we observe that the Beta function enjoys 0 < 1−

β(j + b + 1, m) (b + 1)m ≤ , β(j + 1, m) j+m+1

j ∈ N ∪ {0}.

3.3. Decomposition via Bergman Style Kernels

57

Accordingly,

h 2D

= π

∞

1−

j=0

β(j + b + 1, m) 2 (j + m + 1)|aj+m+1 |2 β(j + 1, m)

∞ |aj+m+1 |2 j=0

j+m+1

.

Notice also that the assumption that |f (α) (z)|2 (1 − |z|2 )2α−2+p dm(z) ∈ CMp gives ∞ |aj,α |2 ≈ |f (α) (z)|2 (1 − |z|2 )2α−2+p dm(z) < ∞. 2α−1+p (j + 1) D j=0 So

∞

|aj,α | ≈

|aj+m+1 |2 |aj,α | ⇒ < ∞. (j + 1)α (j + m + 1)p−1 j=0

Finally, we reach

h 2D

∞ j=0

|aj+m+1 |2 < ∞, (j + m + 1)p−1

as required. Step 2: (i)⇔(iii) Since the ﬁrst step reveals that f ∈ Qp if and only if (z)|2 (1 − |z|2 )dm(z) is an element of CMp , we conclude from Corollary |f 1−p 3.2.2 that f ∈ Qp is equivalent to f ( 2 ) ∈ H2,p . p−1 Taking (i)⇔(ii) into account, we realize that g = f ( 2 ) ∈ Qp is equivalent 3−p to |g ( 2 ) |2 (1 − |z|2 )dm(z) ∈ CMp ; that is, |f (z)|2 (1 − |z|2 )dm(z) ∈ CMp , i.e., f ∈ H2,p thanks to Corollary 3.2.2. We are done. ( 3−p 2 )

3.3 Decomposition via Bergman Style Kernels As a special case of the foregoing α-derivatives, we ﬁnd that if f ∈ H satisﬁes Fp (f, 0) < ∞, then by the following integral reproducing formula: (1 − |z|2 )b−1 b p+1 . f (z) dm(z), b > f (w) = π D (1 − z¯w)b+1 2 When approximating the above integral by a Riemann sum, we are naturally led to sketching a decomposition result for functions in Qp . As with this aspect, we will still use B(z, r) = {w ∈ D : dD (z, w) < r} as the hyperbolic 0 < r-ball centered at z ∈ D. Moreover, for τ > 0 we say that a sequence of points {zj } in D is τ -separated, respectively τ -dense, provided inf dD (zj , zk ) ≥ τ, respectively D =

j =k

∞ j=1

B(zj , τ ).

58

Chapter 3. Isomorphism, Decomposition and Discreteness We need two basic lemmas as follows.

Lemma 3.3.1. For b > 0, z, w ∈ D let kw (z) =

(1−|z|2 )b−1 . (1−¯ z w)b+1

|kw (z) − kw (z0 )| dD (z, z0 )|kw (z0 )|,

If dD (z, z0 ) ≤ 1, then

w ∈ D.

Proof. Noting the estimate 1 − |z|2 ≈ 1 − |z0 |2 ≈ |1 − z0 z|,

as

dD (z, z0 ) ≤ 1,

and using the mean-value theorem for derivatives, we obtain that if dD (z, z0 ) ≤ 1/2, then 1 − z w b+1 1 − |z|2 b−1 1 − |z|2 b−1 kw (z) 0 + − 1 1 − ≤ 1 − kw (z0 ) 1 − |z0 |2 1 − |z0 |2 1 − z¯w |z − z0 | |z − z0 | + 2 1 − |z0 | |1 − z¯w| |σz0 (z)| dD (z, z0 ),

as required.

Lemma 3.3.2. For any τ ∈ (0, 1], there is a τ /2-separated and τ -dense sequence {zj } in D such that any z ∈ D lies in at most ﬁnite many of {B(zj , 2τ )}. Proof. Given τ > 0, we can choose a sequence of points {zj } in D that is τ /2separated and τ -dense. Let B(z0 , 2τ ) be ﬁxed and covered by {B(zj , τ /4)}nj=1 with n inf dD (zj , zk ) ≥ τ /8 and B(zj , τ /16) ⊆ B(z0 , 2τ ). j =k

j=1

Clearly, {B(zj , τ /16)}nj=1 are disjoint, and so n

m B(zj , τ /16) ≤ m B(z0 , 2τ ) .

j=1

By a direct calculation, there is a constant κ > 0 independent of τ and {zj }nj=0 such that m B(z0 , 2τ ) ≤ κ min m B(zj , τ /16) . 1≤j≤n

Thus n ≤ κ and therefore, there exists a natural number N such that any hyperbolic 2τ -ball can be covered by N hyperbolic τ /4-balls. +1 Suppose now that there is a point z ∈ D that belongs to ∩N k=1 B(zjk , 2τ ). Then zjk ∈ B(z, 2τ ) for k = 1, 2, . . . , N + 1. According to the existence of N , we may assume that B(z, 2τ ) is covered by {B(wn , τ /4)}N n=1 . Thus there is some +1 . From this we in turn derive B(wn , τ /4) that contains two points of {zjk }N k=1 that the hyperbolic distance between these two points is less than τ /2, which contradicts the fact that {zj } is τ /2-separated.

3.3. Decomposition via Bergman Style Kernels

59

For simplicity, we use δa as the Dirac measure with mass 1 at a. Below is the expected decomposition theorem for Qp , p ∈ (0, 2), in terms of the normalized b Bergman-type kernels (1 − |zj |2 )/(1 − zj z) , b > (1 + p)/2. Theorem 3.3.3. Let p ∈ (0, 2) and b > such that:

1+p 2 .

Then there is a sequence {zj } in D

∞ 2 2 p (i) For any complex-valued sequence {λ}∞ j=1 satisfying j=1 |λj | (1−|zj | ) δzj ∈ CMp , the function ∞ 1 − |z |2 b j f (z) = λj 1 − z z j j=1 belongs to Qp with ∞ " " " " |λj |2 (1 − |zj |2 )p δzj "

f Qp "

CMp

j=1

(ii) If f ∈ Qp , then f (z) =

∞ j=1

λj

.

1 − |z |2 b j 1 − zj z

for some sequence of complex numbers {λ}∞ j=1 obeying ∞ " " " " |λj |2 (1 − |zj |2 )p δzj " "

CMp

j=1

f Qp .

Proof. For τ > 0, suppose that {zj } is a sequence constructed in Lemma 3.3.2. ∞ (i) Suppose dνp = j=1 |λj |2 (1 − |zj |2 )p δzj is in CMp . Then ∞

|λj |2 (1 − |zj |2 )p < ∞.

j=1

Using 2b > 1 + p and the Cauchy–Schwarz inequality, we have that for z in any compacta of D, ⎞ 12 ⎛ ⎞ 12 ⎛ ∞ ∞ 1 − |z |2 b 2 2b−p (1 − |z | ) j j ⎠ < ∞. |λj | |λj |2 (1 − |zj |2 )p ⎠ ⎝ ≤⎝ 1 − z z (1 − |z|)2b j j=1 j=1 j=1

∞

Accordingly, f deﬁned in (ii) belongs to H, but also obeys

f (w) = b

∞ j=1

λj zj

(1 − |z |2 )b j , (1 − zj w)b+1

w ∈ D.

60

Chapter 3. Isomorphism, Decomposition and Discreteness

A direct calculation with z = σzj (ζ) gives

m B(zj , τ /4) (1 − |zj |2 )b−1 (1 − |z|2 )b−1 dm(z) = , ¯w)b+1 Cj (1 − zj w)b+1 B(zj ,τ /4) (1 − z where 2bπ(e5τ − 1)2 (e5τ + 1)−2 Cj = 1 − (e5τ − 1)2 (e5τ + 1)−2 |zj |2 1 − (4e5τ )b (1 + e5τ )−2b is bounded. Consequently, we obtain f (w)

=

=

Cj (1 − |zj |2 ) (1 − |z|2 )b−1 1B(zj ,τ /4) (z) dm(z) (1 − z¯w)b+1 m (B(zj , τ /4) D j=1 ⎛ ⎞ ∞ (1 − |z|2 )b−1 ⎝ Cj (1 − |zj |2 ) 1B(zj ,τ /4) (z)⎠ dm(z). b λj zj ¯w)b+1 j=1 m B(zj , τ /4) D (1 − z

b

∞

λj zj

To prove f ∈ Qp , let ∞ λj Cj zj (1 − |zj |2 )1B(zj ,τ /4) (z) . F (z) = m (B(zj , τ /4) j=1

According to Lemma 3.1.2, it suﬃces to verify that the measure dµF,p (z) = |F (z)|2 (1 − |z|2 )p dm(z) belongs to CMp . To this end, we employ the boundedness of Cj and the disjointness of {B(zj , τ /4)} to establish the following estimates for any w ∈ D: 1 − |w|2 p dµF,p (z) |1 − wz| ¯ 2 D (1 − |σw (z)|2 )p |F (z)|2 dm(z) = D

∞

1 − |zj |2 1B(zj ,τ /4) (z)(1 − |σw (z)|2 )p dm(z) m B(zj , τ /4) D j=1

2 ∞ |λj |(1 − |zj |2 ) (1 − |σw (z)|2 )p dm(z) m B(z , τ /4) j ∪B(z ,τ /4) j j=1 2 |λj | (1 − |σw (zj )|2 )p

|λj |2

D

νp CMp , which in turn implies µF,p ∈ CMp thanks to Lemma 3.1.1.

3.3. Decomposition via Bergman Style Kernels

61

(ii) To handle this part, for those points {zj } ﬁxed at the beginning of the argument, pick a sequence of Lebesgue measurable sets {Dj } (cf. [CoiRoc, p.18]) such that ∞ τ Dj . B(zj , ) ⊆ Dj ⊆ B(zj , τ ); D = 4 j=1 Without loss of generality, we may also assume that |zj | > 0 for j ∈ N. If f ∈ Qp , then f ∈ A2,p and hence by Lemma 3.3.2 (i),

f (w)

(1 − |z|2 )b−1 f (z) dm(z) (1 − z¯w)b+1 D ∞ (1 − |z|2 )b−1 b f (z) dm(z). π j=1 Dj (1 − z¯w)b+1 b π

= =

Putting S(f )(w) =

∞ (1 − |zj |2 )b−1 1 f (zj )m(Dj ) , π j=1 zj (1 − zj w)b

we get by the triangle inequality and the deﬁnition of kw (z), |f (w) − S(f ) (w)| ∞ b (1 − |z|2 )b−1 f (zj )(1 − |zj |2 )b−1 = − f (z) dm(z) π j=1 Dj (1 − z¯w)b+1 (1 − zj w)b+1 + , ≤ 1

2

where 1

∞ b = |f (z)||kw (z) − kw (zj )|dm(z) π j=1 Dj

and 2

=

∞ b |f (z) − f (zj ) |kw (zj )|dm(z). π j=1 Dj

Using Lemma 3.3.1 we have 1

τ

D

|f (z)||kw (z)|dm(z).

62

Chapter 3. Isomorphism, Decomposition and Discreteness

2

we change variables and make simple supremum estimates to get ∞ ≤ |f (z) − f (zj )||kw (zj )|dm(z)

To control

2,

j=1

B(zj ,τ )

2 1 − |zj |2 = |f (σzj (z) − f (σzj (0))| |kw (zj )|dm(z) |1 − zj z|2 j=1 B(0,τ ) 2 ∞ 1 − |zj |2 s(1 − |zj |2 ) ≤ sup |f (ζ)| |kw (zj )|dm(z) (1 − s0 )2 ζ∈B(zj ,τ ) |1 − zj z|2 j=1 B(0,τ ) ∞

≤

∞ s(1 − |zj |2 ) j=1

(1 − s0

)2

sup ζ∈B(zj ,τ )

|f (ζ)|m B(zj , τ ) |kw (zj )|,

where

eτ − e−τ e − e−1 and s0 = . τ −τ e +e e + e−1 To continue our argument, we note four basic facts as follows: First, there is constant c1 > 0 independent of τ ∈ (0, 1] and z ∈ D such that c1 |f (ζ)| ≤ |f (ξ)|dm(ξ); m B(ζ, τ ) B(ζ,τ ) s=

Second, there is a constant c2 > 0 independent of τ ∈ (0, 1], ζ ∈ D and zj such that ζ ∈ B(zj , r) ⇒ B(ζ, τ ) ⊆ B(zj , 2τ ) and m B(zj , τ ) ≤ c2 m B(ζ, τ ; Third, there is a constant c3 > 0 independent of τ ∈ (0, 1], z ∈ D and zj such that 1 − |zj |2 ≤ c3 (1 − |z|2 ),

z ∈ B(zj , 2τ );

Fourth, Lemma 3.3.1 and its symmetric form imply dD (z, z0 ) ≤ 1 ⇒ kw (z) ≈ kw (z0 ),

w ∈ D.

With the help of the above estimates and Lemma 3.3.2 we then have ∞ s |f (z)|(1 − |z|2 )|kw (z)|dm(z) 2

j=1

Putting the estimates on |f (w) − S(f ) (w)|

s

D

Dj

|f (z)|(1 − |z|2 )|kw (z)|dm(z).

and 2 together, we obtain τ |f (z)||kw (z)|dm(z) D +s |f (z)|(1 − |z|2 )|kw (z)|dm(z), 1

D

w ∈ D.

3.3. Decomposition via Bergman Style Kernels Since f ∈ Qp , we conclude that |f (z)|2 (1 − |z|2 )p dm(z) and

63

2 |f (z)|(1 − |z|2 ) (1 − |z|2 )p dm(z)

are in CMp (cf. Theorem 1.1.3 or 3.2.3). Applying this to Lemma 3.1.2, we ﬁnd

f − S(f ) Qp ,1 s f Qp ,1 . If τ is small enough then so is s, and hence the operator S(f ) is invertible with the bounded inverse being S−1 = (Id − (Id − S))−1 =

∞

(Id − S)n ,

n=0

where Id stands for the identity operator acting on Qp . With this, we obtain ∞ (1 − |z |2 )b−1 1 −1 j f (z) = SS−1 (f )(z) = S (f ) (zj ) m(Dj ) π j=1 zj (1 − zj z)b =

∞ j=1

where λj =

λj

1 − |z |2 b j , 1 − zj z

S−1 (f ) (zj )m(Dj ) . πzj (1 − |zj |2 )

Of course, we have to verify dµ{zj },p =

∞

|λj |2 (1 − |zj |2 )p δzj ∈ CMp .

j=1

To do so, we observe that the mean-value-inequality 1 |F (z)|2 (1 − |z|2 )p dm(z) |F (zj )|2 (1 − |zj |2 )p m B(zj , τ /20) B(zj ,τ /20) holds for any F ∈ H. Thus, for any w ∈ D we apply the disjointness of {Dj } to obtain 1 − |w|2 p dµ{zj },p (z) ¯ 2 D |1 − wz| ∞ 1 − |w|2 p −1 |S (f ) (zj )|2 m(Dj )(1 − |zj |2 )p−2 ≤ |1 − wz| ¯ 2 j=1 2 ∞ m(Dj )(1 − |zj |2 )−1 |S−1 (f ) (z)|2 (1 − |σw (z)|2 )p dm(z) m B(zj , τ /20) B(zj ,τ /20) j=1 |S−1 (f ) (z)|2 (1 − |σw (z)|2 )p dm(z) D

S−1 (f ) 2Qp ,1 ,

64

Chapter 3. Isomorphism, Decomposition and Discreteness

which implies by Lemma 3.1.1 the desired result.

3.4 Discreteness by Derivatives Given a sequence of points S = {zj } in D. We are interested in the relation between the geometry of S and the values which derivatives of functions in Qp can take on S. Once done, a discrete form of Qp will be established. Using the fact that for p ∈ (0, ∞), Qp is a subspace of B we ﬁnd that sup (1 − |zj |2 )|f (zj )| ≤ f B f Qp ,

f ∈ Qp .

j∈N

∞ This means that the sequence Tf = {(1 − |zj |2 )f (zj )}∞ j=1 is in (S). On the basis of Theorem 3.3.3 we then ask for the relationship between {Tf : f ∈ Qp } and the ∞sequential space SCMp (S) consisting of all complex numbers {cj } such that j=1 (1−|zj |2 )p |cj |2 δzj belongs to CMp . In so doing, we introduce a variation of Schur’s Lemma for the sequences.

Lemma 3.4.1. Given p ∈ [0, ∞) and a sequence {zj } in D. Let A = (an,k ) be an inﬁnite matrix with nonnegative entries. If there are a sequence {hj } of positive numbers and two positive constants c1 and c2 independent of w ∈ D such that an,k hn ≤ c1 hk , k ∈ N n =k

and

an,k hk (1 − |σw (zk )|2 )p ≤ c2 hn (1 − |σw (zn )|2 )p ,

n ∈ N,

k =n

then

2 an,k sn (1 − |σw (zk )|2 )p ≤ c1 c2 |sn |2 (1 − |σw (zn )|2 )p . k

n =k

n

Proof. Using the hypothesis, Cauchy–Schwarz’s inequality and Fubini’s Theorem, we have 2 p an,k sn 1 − |σw (zk )|2 n =k

k

≤

k

≤

c1

n =k

c1 c2

p 1 − |σw (zk )|2 an,k |sn |2 h−1 n

n =k

|sn |2 h−1 n

n

≤

an,k hn

p hk an,k 1 − |σw (zk )|2

k =n

p |sn | 1 − |σw (zn )|2 , 2

n

as required.

3.4. Discreteness by Derivatives

65

This technique will be used in the argument for the following result. Theorem 3.4.2. Given p ∈ (0, 2). Let {zj } be a sequence of points in D with τ = inf j =k dD (zj , zk ). (i) If τ > 0, then T : Qp → SCMp is injective. (ii) If T : Qp → SCMp is surjective, then τ > 0. Conversely, there is a τ0 > 0 such that if τ > τ0 , then T : Qp → SCMp is surjective. Proof. (i) Suppose τ > 0. Then {B(zj , τ /2)} are disjoint. If f ∈ Qp , then dµf,p (z) = (1 − |z|2 )p |f (z)|2 dm(z) ∈ CMp , and hence for any w ∈ D we use 1 − |z|2 ≈ 1 − |zj |2 and |1 − zj w| ≈ |1 − z¯w| as dD (z, zj ) <

τ 2

and the mean-value-inequality for f to get ∞ p 2 (1 − |zj |2 )|f (zj )| 1 − |σzj (w)|2 j=1

p ∞ (1 − |zj |2 )2 1 − |σzj (w)|2 |f (z)|2 dm(z) m B(z , τ /2) j B(zj ,τ /2) j=1 ∞ p |f (z)|2 1 − |σzj (w)|2 dm(z) j=1

B(zj ,τ /2)

2 Fp (f, w) f 2Qp ,1 .

That is to say, Tf ∈ SCMp , as desired. (ii) Again, noting Qp ⊆ B, we have that if f ∈ Qp , then the Bergman reproducing formula for f gives 2 1 − |z|2 f (z) dm(z), w ∈ D. f (w) = π D (1 − w¯ z )3 By the proof of Lemma 3.3.1 with b = 2 we ﬁnd that d (w, z ) 1 − |z |2 3 j D j f (zj ) . f (w) − 1 − wzj 1 − |w|2 Consequently, |(1 − |zj |2 )f (zj ) − (1 − |zk |2 )f (zk )| f B dD (zj , zk ) f Qp dD (zj , zk ).

66

Chapter 3. Isomorphism, Decomposition and Discreteness

Of course, if T maps Qp onto SCMp , then there are functions fj ∈ Qp that have

fj Qp ,1 uniformly bounded and that satisfy 1 , j = k, (1 − |zk |2 )f (zj ) = 0 , j = k. Hence τ > 0. Conversely, let τ be suﬃciently large. To prove that T is onto, it is enough to ﬁnd the right inverse of T. As a good approximation of the inverse, we are motivated by the decomposition theorem for Qp to consider the following R operator acting on {sj } ∈ SCMp : R({sj })(z) =

∞ sn 1 − |zn |2 b , bzn 1 − zn z n=1

b>

1+p . 2

The proof of Theorem 3.3.3 tells us that R({sj }) ∈ Qp whenever {sj } ∈ SCMp . Now a simple calculation reveals that for the identity operator Id on SCMp , (Id − TR)({sj }) =

−

n =k

sn

(1 − |zk |2 )(1 − |zn |2 )b . (1 − zn zk )b+1

So to prove that the operator norm of Id − TR on SCMp can be small enough, let an,k =

(1 − |zk |2 )(1 − |zn |2 )b . (1 − zn zk )b+1

According to Lemma 3.1.1 we are going to show that there is a constant c > 0 independent of w ∈ D such that ∞ ∞ 2 sn an,k (1 − |σw (zk )|2 )p ≤ c |sn |2 (1 − |σw (zn )|2 )p . k=1 n =k

n=1

By Lemma 3.4.1, it suﬃces to demonstrate that there exists suﬃciently small constant c > 0 independent of w ∈ D such that an,k (1 − |zn |2 )γ ≤ c(1 − |zk |2 )γ Jk = n =k

and Kn =

p p an,k (1 − |zk |2 )γ 1 − |σw (zk )|2 ≤ c(1 − |zn |2 )γ 1 − |σw (zk )|2 ,

k =n

where γ is determined by max{p, 1 − b} < γ < min{b + 2, 1 + p + 2b}.

3.5. Characterization in Terms of a Conjugate Pair

67

When γ ∈ (1 − b, b + 2), we can easily work out the following estimate: (1 − |z|2 )γ+b−2 dm(z) Jk (1 − |zk |2 ) b+1 dD (z,zk )>τ /2 |1 − zk z| (1 − |z|2 )γ+b−2 2 γ (1 − |zk | ) dm(z). |1 − z|2γ−2 dD (z,0)>τ /2 Similarly, when γ ∈ (p, 1 + p + 2b) we can achieve Kn

=

(1 − |zk |2 )1+p+γ |1 − zn zk |b+1 |1 − wz ¯ k |2p k =n (1 − |z|2 )p+γ−1 2 p 2 b (1 − |w| ) (1 − |zn | ) dm(z) |1 − zn z|b+1 |1 − wz| ¯ 2p k =n B(zk ,τ /2) (1 − |z|2 )p+γ−1 2 p 2 b dm(z) (1 − |w| ) (1 − |zn | ) b+1 |1 − wz| ¯ 2p dD (z,zn )>τ /2 |1 − zn z| (1 − |w|2 )p (1 − |z|2 )p+γ−1 |1 − zn z|2b−2γ dm(z) 2 −(p+γ) |1 − wz ¯ n + (w − zn )z|2p (1 − |zn | ) dD (z,0)>τ /2 (1 − |z|2 )γ−1−p 2 γ 2 p (1 − |zn | ) 1 − |σw (zn )| dm(z). |1 − z|2γ−2b dD (z,0)>τ /2 (1 − |w|2 )p (1 − |zn |2 )b

Owing to (1 − |z|2 )γ+b−2 dm(z) < ∞ and |1 − z|2γ−2 D

D

(1 − |z|2 )γ−1−p dm(z) < ∞, |1 − z|2γ−2b

for any γ between max{p, 1 − b} and min{b + 2, 1 + p + 2b}, we can insure (1 − |z|2 )γ+b−2 dm(z) = 0 lim τ →∞ d (z,0)>τ /2 |1 − z|2γ−2 D

and lim

τ →∞

dD (z,0)>τ /2

(1 − |z|2 )γ−1−p dm(z) = 0. |1 − z|2γ−2b

In other words, the operator norm of Id − TR, thanks to Lemma 3.4.1, will be as small as we wish as τ → ∞. It follows that TR = Id − (Id − TR) is invertible. Consequently, R(TR)−1 is the operator that maps SCMp onto Qp , and the proof is ﬁnished.

3.5 Characterization in Terms of a Conjugate Pair We now utilize the covering lemma — Lemma 3.3.2 to study the representation of Qp functions based on a conjugate pair. To this end, we need the following result.

68

Chapter 3. Isomorphism, Decomposition and Discreteness

Lemma 3.5.1. Let p ∈ (0, ∞) and f ∈ H. Then: 2 (i) Fp (f, 0) = p f (z)zf (z)(1 − |z|2 )p−1 dm(z). D 2 (ii) Fp (f, 0) |f (z)f (z)|(1 − |z|2 )p−1 dm(z). D

The reversed inequality holds for f ∈ D2,p if and only if p ∈ (1, ∞). ∞ Proof. (i) Assuming f (z) = j=0 aj z j , and integrating by parts, we get 2 Fp (f, 0) ⎞ ⎛ 1 ∞ ∞ ⎝ = jaj rj−1 ζ j−1 jaj rj−1 ζ¯j−1 |dζ|⎠ (1 − r2 )p rdr 0

= π

T

∞

j=1

j 2 |aj |2

j=1 ∞

= pπ

j|aj |2

j=1

j=1 1

rj−1 (1 − r)p dr

0

1

rj (1 − r)p−1 dr.

0

Meanwhile, we also have f (z)zf (z)(1 − |z|2 )p−1 dm(z) D ⎞ ⎛ 1 ∞ ∞ ⎝ = aj r j ζ j jaj rj ζ¯j |dζ|⎠ r(1 − r2 )p−1 dr 0

= π

T

∞

j=0

j|aj |2

j=1

j=0

1

0

rj (1 − r)p−1 dr,

hence reaching the required formula. (ii) The inequality follows from (i) right away. Now if p ∈ (1, ∞) then by the Cauchy–Schwarz inequality, |f (z)zf (z)|(1 − |z|2 )p−1 dm(z) D

12 |f (z)|2 (1 − |z|2 )p−2 dm(z) ≤ Fp (f, 0) D Fp (f, 0) |f (0)| + Fp (f, 0) . Next we prove that if p ∈ (0, 1), then the last inequality cannot be true in general. To do so, set 12 2 p−1 Gp (f ) = |f (z)f (z)|(1 − |z| ) dm(z) . D

3.5. Characterization in Terms of a Conjugate Pair

69

Suppose otherwise Gp (f ) Fp (f, 0) for any f ∈ D2,p . Then for any f1 , f2 ∈ D2,p with f2 ≡ 0 we have Gp (f1 ± f2 , 0) Fp (f1 ± f2 , 0) Fp (f1 , 0) + Fp (f2 , 0). Using the identity f1 f2 + f1 f2 =

(f1 + f2 )(f1 + f2 ) − (f1 − f2 )(f1 − f2 ) 2

and the triangle inequality, we further get Hp (f1 , f2 )

|f1 (z)f2 (z)|(1 − |z|2 )p−1 dm(z) |f1 (z)|≤|f2 (z)| 2 2 |f1 (z)f2 (z)|(1 − |z|2 )p−1 dm(z) Fp (f1 , 0) + Fp (f2 , 0) + =

2 2 Fp (f1 , 0) + Fp (f2 , 0) . Consequently,

|f1 (z)|≤|f2 (z)|

=

|f1 (z)f2 (z)|(1 − |z|2 )p−1 dm(z) D Hp (f1 , f2 ) + |f1 (z)f2 (z)|(1 − |z|2 )p−1 dm(z)

2 2 Fp (f1 , 0) + Fp (f2 , 0) .

Ip (f1 , f2 ) =

|f1 (z)|≥|f2 (z)|

Nevertheless, the last estimate would yield f2 ≡ 0, a contradiction. As a matter of fact, the techniques used in the last two sections allow us to ﬁnd a positive number τ0 close to 1 with the following property: If τ ∈ (τ0 , 1) and {zj } satisﬁes inf m =n |σzm (zn )| ≥ τ , then T{zj },p (f ) = {(1 − |zj |2 )1+p/2 f (zj )} is a surjective operator from D2,p to 2 . Keeping this in mind, we now take a sequence {zj } on D such that inf m =n |σzm (zn )| ≥ τ and D is covered by the pseudo-hyperbolic disks ∆(zj , r) = {z ∈ D : |σzj (z) < r} where r ∈ (3−1/2 , 1). Then, for each j ∈ N there is a point wj in the closure ∆(zj , r) of ∆(zj , r) satisfying |f2 (wj )| = max{|f2 (z)| : z ∈ ∆(zj , r)}. From the covering of D it follows that ∞ |f2 (z)|2 (1 − |z|2 )p−2 dm(z) (1 − |wj |2 )p |f2 (wj )|2 . D

j=1

Note also that |σwm (zn )| ≤ r, s = 2r(1 + r2 )−1 and m = n imply |σwm (wn )| ≥

|σzm (zn )| − s . 1 − s|σzm (zn )|

70

Chapter 3. Isomorphism, Decomposition and Discreteness

Hence, r ∈ (3−1/2 , 1) yields |σwm (wn )| ≥ δ = (τ − s)(1 − sτ )−1 > τ0

as

τ > (τ0 + s)(1 + sτ0 )−1 ,

and consequently T{wj },p : D2,p → 2 is onto. Furthermore, from the Open Mapping Theorem it turns out that for any {cj } ∈ 2 there exists an f1 ∈ D2,p such that

f1 D2,p {cj } 2 and cj = (1 − |wj |2 )1+p/2 f1 (wj ). With the preceding constructions, we derive ∞

(1 − |wj |2 )p |f2 (wj )|2

12

j=1

∞

sup

{cj } 2 ≤1 j=1

|f2 (wj )|(1 − |wj |2 )p/2 |cj |

∞

|f2 (wj )f1 (wj )|(1 − |wj |2 )1+p

f D2,p ≤1 j=1 ∞ sup |f1 (z)f2 (z)|(1 − |z|2 )p−1 dm(z)

f1 D2,p ≤1 j=1 ∆(wj ,δ/2)

sup

sup

f1 D2,p ≤1

Ip (f1 , f2 )

2 1 + Fp (f2 , 0) , hence producing D

|f2 (z)|2 (1 − |z|2 )p−2 dm(z) < ∞,

p ∈ (0, 1).

This ﬁniteness is not valid unless f2 ≡ 0. We are done.

The above lemma leads to the following description involving the conjugate pair. Theorem 3.5.2. Let p ∈ (0, ∞) and f ∈ H. Then: p−1 2 f (z)f (z)σw (z)σw (z) 1 − |σw (z)|2 dm(z). (i) f Qp ,1 = p sup w∈D

(ii) f 2Qp ,1 sup

w∈D

D

D

p−1 |f (z)f (z)||σw (z)σw (z)| 1 − |σw (z)|2 dm(z).

The reversed inequality holds for all f ∈ Qp if and only if p ∈ (1, ∞).

3.6. Notes

71

Proof. (i) Applying Lemma 3.5.1 (i) to f ◦ σw and doing some simple calculations with the conformal mapping σw , we can verify the assertion. (ii) The forward inequality trivially follows from (i). Regarding the opposite estimate, we choose f (z) = log(1 − z) to ﬁnd f 2Qp < ∞ but p−1 |f (z)f (z)||σw (z)σw (z)| 1 − |σw (z)|2 dm(z) sup w∈D D ≥ | log(1 − z)||z||1 − z|−1 (1 − |z|2 )p−1 dm(z) D ≥ − log(1 − |z|2 ) |z|(1 − |z|2 )p−2 dm(z)

|1−z|<1−|z|2 1 2

t

0

= ∞

− log(1 − t2 ) (1 − t2 )p−2 dt

for p ∈ (0, 1).

3.6 Notes Note 3.6.1. Section 3.1 is more or less a by-product of [WuXie1], [WuXie2] and [Zha2]. Lemma 3.1.1 has a higher dimensional version — see also [KotLR]. The operator in Lemma 3.1.2 is a generalized combination of Lemmas 4.1.1 and 7.2.2 in [Xi1]. Theorem 3.1.3 for p ∈ (0, 1] is a special case of [Zha2, Theorem 2] — in particuar, the case p = 1 goes back to Jones’ result (cf. [An] and [GhZ]). Furthermore, using the second-order derivatives of f1 and f2 , Fubini’s Theorem and the limiting case p = 0 of Lemma 3.1.1 (cf. Lemma 6.4.1 (i) of this book), we can readily extend Theorem 3.1.3 to p = 0 — more precisely, if f ∈ B and B1 ⊆ X ⊆ D, then 1Ω (f ) (z) distB (f, X ) = inf{ f − g B : g ∈ X } ≈ inf > 0 : dm(z) < ∞ . 2 2 D (1 − |z| ) Here, we should also observe distB (f, D) ≤ distB (f, X ) ≤ distB (f, B1 ). But, it is still an open problem to give an equivalent estimate (similar to the above description) for distB (f, H∞ ) – see [An] and [AnClPo]. Since B1 ⊂ H∞ ⊂ BMOA, we conjecture that if f ∈ B, then 1Ω (f ) (z) ∞ distB (f, H ) ≈ inf > 0 : sup dm(z) < ∞ . ¯2 w∈D D |1 − z w| Of course, the key issue is to prove that the left-hand-side distance is greater than or equal to a constant multiple of the right-hand-side inﬁmum.

72

Chapter 3. Isomorphism, Decomposition and Discreteness

Note 3.6.2. Section 3.2 essentially consists of [WuXie1, Corollary 4] and the disc analogue of [WuXie2]. Note that the results have been extended to p ∈ (0, 2). So 1−p Theorem 3.2.3 (iii) reveals that in case of p ∈ (1, 2), f ( 2 ) has radial limiting values on T even though f ∈ Qp = B may not have. Of course, Theorem 3.2.3 (ii) generalizes Theorem 1.1.3 (v) that has been also reproved in Wulan–Zhu [WulZh1] using the Schur Lemma, but also the corresponding result in [RocWu] for p = 1 and α > 0. Note 3.6.3. Section 3.3 comes basically from [WuXie1]. However, the argument presented in this section is motivated by [Zhu1, §4.3–4.4] and so oﬀers a slightly diﬀerent approach. In the cases of BMOA (BMOA = Q1 ) and Bloch (B = Qp , p ∈ (1, 2)), see [Roc2, Theorem 2.10], [Zhu2, Theorems 5.29 and 3.23] and [Xi1, Theorem 3.2]. Up to this point, we should note that if p ∈ (1, 2), then ∞ " " " " sup |λj | < ∞ ⇔ " |λj |2 (1 − |zj |2 )p δzj "

j∈N

CMp

j=1

< ∞.

Note 3.6.4. Section 3.4 is taken from an unpublished manuscript [Xi5], and has showed that the idea on interpolation by functions in Bergman spaces (cf. [Roc1] and Roc2) can be carried over to Qp , p ∈ (0, 2) including BMOA and B; see also [Roc1], [Roc2], and [Sun], [BoNi], [Sei], [Sch] and [Xi1] for more information on the interpolations by values of functions in BMOA or B. Note 3.6.5. Section 3.5 is taken from [WiXi1]. Here it is perhaps appropriate to mention that the Cauchy–Schwarz inequality yields: for p > 0 and f ∈ H, |f (z)f (z)|(1 − |z|)p−1 dm(z) ≤ |f (z)|2 (1 − |z|)p−1 dm(z), D

D

where equality occurs at f (z) ≡ z. This inequality may be viewed as a holomorphic version of either the Opial inequality discussed in [AgPa] or the logarithmic Sobolev-type inequality presented in [Bec].

Chapter 4

Invariant Preduality through Hausdorﬀ Capacity As a characteristic of every holomorphic Q class, · Qp is conformal invariant. So, it is natural and important to look into the structure of predual space of each Qp under the invariant pairing D f g¯ dm. In this chapter, we do this job through the following aspects: • • • • •

Nonlinear Integrals and Maximal Operators; Adams Type Dualities; Quadratic Tent Spaces; Preduals under Invariant Pairing; Invariant Duals of Vanishing Classes.

4.1 Nonlinear Integrals and Maximal Operators This section shows some applications of the notion of the nonlinear Choquet integral of a function with respect to the nonadditive Hausdorﬀ capacity in maximal operators of the Hardy–Littlewood and nontangential types. Given p ∈ (0, 1], for a set E ⊆ T let ⎧ ⎫ ∞ ∞ ⎨ ⎬ p (E) = inf |Ij |p : E ⊆ Ij , H∞ ⎩ ⎭ j=1

j=1

be the p-dimensional Hausdorﬀ capacity of E, where the inﬁmum is taken over countably many open arcs Ij ⊆ T satisfying E ⊆ j Ij . If one had required the length of Ij to satisfy |Ij | ≤ for some positive number , then one would have

74

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

written Hp (E). The classical Hausdorﬀ measure of E of dimension p is H0p (E) = lim→0 Hp (E). It is well known that H0p (·) is a outer measure, but not ﬁnite in p p general, whereas H∞ (·) is. Moreover, H0p (·) and H∞ (·) have the same null sets. On p the other hand, if the inﬁmum in the deﬁnition of H∞ (E) is taken over coverings p,d (E), is of E by dyadic subarcs of T, then the corresponding capacity, denoted H∞ called the dyadic p-dimensional Hausdorﬀ capacity of E. Since each subarc I has two adjacent dyadic subarcs I and I satisfying I ⊂ I ∪ I and |I| = |I | = |I |, we conclude p,d p p,d 2−1 H∞ (E) ≤ H∞ (E) ≤ H∞ (E). In the above and the below, a dyadic subarc of T is an arc I ⊆ T of the form iθ e : j21−k π ≤ θ < (j + 1)21−k π , k ∈ {0} ∪ N, j = 0, 1, 2, . . . , 2k − 1. Properties of the dyadic Hausdorﬀ capacity needed include: p,d p,d lim H∞ (Ej ) = H∞ (E)

j→∞

whenever Ej increases to E or Ej is compact and decreasing to E. And, p,d p,d p,d p,d (E1 ∪ E2 ) + H∞ (E1 ∩ E2 ) ≤ H∞ (E1 ) + H∞ (E2 ). H∞ p At the same time, H∞ is a monotone and countably subadditive set function on p is the class of all subsets of T which vanishes on the empty set. Furthermore, H∞ an outer capacity in the sense of p p H∞ (E) = inf{H∞ (O) : open O ⊃ E}. p Associated with the Hausdorﬀ capacity H∞ (·) is the Choquet integral of a function f on T:

T

p |f |dH∞ =

∞ 0

p {ζ ∈ T : |f (ζ)| > t} dt. H∞

Here the right-hand-side integral is the usual Lebesgue integral. As a matter of p {ζ ∈ T : f (ζ) > t} is nonincreasing with t ≥ 0 and hence Lebesgue fact, H∞ measurable. In particular, p p 1E dH∞ = H∞ (E) T

for E ⊆ T, where 1E is the characteristic function on E. Below are four basic and useful properties of the Hausdorﬀ capacity and its Choquet integral.

4.1. Nonlinear Integrals and Maximal Operators

75

Lemma 4.1.1. Given p ∈ (0, 1], let f and {fj }∞ j=1 be nonnegative functions on T. Then p p = κ T f dH∞ for any constant κ ≥ 0; (i) T κf dH∞ p p -a.e. on T; that is, f = 0 on T except on E (ii) T f dH∞ = 0 ⇐⇒ f = 0 H∞ p where H∞ (E) = 0; p1 1 p f dH ; (iii) T f (ζ) p |dζ| ≤ ∞ T ∞ ∞ p p (iv) T j=1 fj dH∞ ≤ 2 j=1 T fj dH∞ . Proof. (i) and (ii) are straightforward. For (iii), we use the simple inequality p 1 1 (E) ≤ H∞ (E) p , H∞ to derive 1 f (ζ) p |dζ| T

= p−1

0

∞

E⊆T

1−p 1 {ζ ∈ T : f (ζ) > t} t p dt H∞

p1 t p d 1 H∞ {ζ ∈ T : f (ζ) > s} ds dt ≤ dt 0 0 ∞ p1 p 1 = H∞ {ζ ∈ T : f (ζ) > s} ds

∞

≤

0

∞ 0

=

T

p H∞

{ζ ∈ T : f (ζ) > s} ds

p f dH∞

p1

p1

.

p,d Finally, (iv) follows from the sublinearity of H∞ .

Note that is not additive for p ∈ (0, 1). This can be veriﬁed by dividing T into 2m subarcs with equal length and letting m → ∞. Of course, the inequality p H∞ (·)

k j=1

Ij

p f dH∞

S j

Ij

p f dH∞ ,

f ≥0

does not hold for any non-overlapping dyadic subarcs {Ij }kj=1 . However, we can establish a weak form — the circular version of Melnikov’s Covering Lemma. Lemma 4.1.2. Let {Ij } be a collection of non-overlapping dyadic subarcs of T. Then there is a subcollection {Ijk } such that p p (i) Ijk ⊆I |Ijk | ≤ 2|I| for any dyadic subarc I of T; p p (ii) H∞ j Ij ≤ 2 k |Ijk | ;

76 (iii)

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity k

Ijk

p f dH∞ ≤2

S k

Ijk

p f dH∞ for any nonnegative function f on T.

Proof. Let j1 = 1. If j1 , . . . , jk have been determined to ensure (i), then we deﬁne jk+1 as the ﬁrst index such that {Ij1 , . . . , Ijk+1 } satisﬁes (i). Then {Ijk } forms a maximal subcollection of Ij obeying (i). We further show that this subcollection ensures (ii). To that end, choose j such that jk < j < jk+1 for some k. Then according to the choice of jk , we choose a dyadic subarc Jj ⊇ Ij such that |Ijn |p + |Ij |p > 2|Jj |p . Ijn ⊆Jj ,n≤k

Consequently, we have

|Ijn |p > |Jj |p .

Ijn ⊆Jj ,n≤k

Clearly, we may assume that k |Ijk |p is convergent — otherwise (ii) is trivial. Under this assumption, we see that the sequence {|Jj |p }j is bounded and accordingly, we may consider the collection {Kn } of the maximal arcs in the collection {Jn }, and then obtain Ij ⊆ Ijm ∪ Kjm . m

j

This gives p H∞

Ij

m

p ≤ H∞

j

≤

p Ijm + H∞ Kn

m

|Ijm |p +

m

≤ 2

n

|Kjm |p

m

|Ijm |p ,

m

as required in (ii). To establish (iii), let {Dn } be a sequence of dyadic subarcs of T such that ζ:ζ∈ Ijk & f (ζ) > t ⊆ Dn , t > 0. n

k

Then using (i), we get 2 |Dn |p ≥ |Ijk |p n Ijk ⊆Dn

n

=

n Ijk ⊆Dn

≥

n Ijk ⊆Dn

≥

k

p H∞ (Ijk )

p {ζ : ζ ∈ Ijk & f (ζ) > t} H∞

p {ζ : ζ ∈ Ijk & f (ζ) > t} . H∞

4.1. Nonlinear Integrals and Maximal Operators

77

Consequently, we have by deﬁnition p p Ijk & f (ζ) > t} ≥ H∞ {ζ : ζ ∈ Ijk & f (ζ) > t} , {ζ : ζ ∈ 2H∞ k

k

which, along with an integration over (0, ∞) with respect to dt, gives the desired inequality. For a function f ∈ L1 (T), we write Mf (ζ) = sup |I|−1

|f (η)||dη|

ζ∈I

I

for the Hardy–Littlewood maximal function of f , where the supremum ranges over all arcs I ⊆ T containing ζ. As above, if the supremum is taken over all dyadic subarcs of T, then we get a dyadic maximal function, denoted Md f . It is clear that Md f (ζ) ≤ Mf (ζ) but not the reverse. Even though, we have the inequality p p {ζ ∈ T : Mf (ζ) > t} ≤ 3p H∞ {ζ ∈ T : Md f (ζ) > 4−1 t} , H∞ and hence Md f controls Mf in the following sense: p p Mf (ζ)dH∞ (ζ) ≤ 22 3p Md f (ζ)dH∞ (ζ). T

T

p Theorem 4.1.3. Let q > 0, p ∈ (0, min{1, q}) and T |f |q dH∞ < ∞. Then: q p q p (i) T (Mf ) dH∞ T |f | dH∞ . p p (ii) H∞ {ζ ∈ T : Mf (ζ) > t} t−p T |f |p dH∞ . Proof. Without loss of generality, we may assume that f is nonnegative on T. (n) (i) For each integer n, let {Ij }j be a family of non-overlapping dyadic subarcs of T such that (n) {ζ ∈ T : 2n < f (ζ) ≤ 2n+1 } ⊆ Ij j

and

(n) p {ζ ∈ T : 2n < f (ζ) ≤ 2n+1 } . |Ij |p ≤ 2H∞

j

If Jn =

(n)

Ij

and g =

2(n+1)q 1Jn ,

n

j

then f q ≤ g. We consider two cases as follows. Case 1: q ∈ [1, ∞). For this, we have 2(n+1)q M 1I (n) . (Mf )q ≤ M(f q ) ≤ Mg ≤ n

j

j

78

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Note that if ζJ is the center of a given subarc J of T, then M(1J )(ζ) min{1, |ζ − ζJ |−1 |J|},

ζ ∈ T.

So, this, along with Lemma 4.1.1 (iv), implies p p (Mf )q dH∞ 2(n+1)q M 1I (n) dH∞ T

n

2(n+1)q

n

j

T

j

(n)

|Ij |p

j

2

(n+1)q

p {ζ ∈ T : 2n < f (ζ) ≤ 2n+1 } H∞

n

T

p {ζ ∈ T : f (ζ)q > t} dt H∞

2(n−1)q

n

2nq

p f q dH∞ .

Case 2: p < q < 1. As for this case, we employ f ≤ n 2n+1 1Jn and again Lemma 4.1.1 (iv) to achieve q p q p dH∞ M(1I (n) ) (Mf ) dH∞ ≤ 2(n+1)q T

T

n

2(n+1)q

n

j

2(n+1)q

n

T

T

q p M(1I (n) ) dH∞ j

(n)

|Ij |p

j

j

j

p f q dH∞ .

(ii) dyadic subarcs of T with For t > 0, let {Ij } be the collection of maximal |Ij |−1 Ij f > t. Then {ζ ∈ T : Md f (ζ) > t} = j Ij . Using Lemma 4.1.1 (iii) we get p p −1 −p p |Ij | ≤ t f (ζ)|dζ| t f p dH∞ . Ij

Ij

Furthermore, applying Lemma 4.1.2 to {Ij }, we can ﬁnd a subcollection {Ijk } of {Ij } such that p p p {ζ ∈ T : Md f (ζ) > t} t−p H∞ f p dH∞ t−p S f p dH∞ , k

as desired.

Ijk

k

Ijk

4.1. Nonlinear Integrals and Maximal Operators

79

For ζ ∈ T let Λ(ζ) be the convex hull of {z ∈ D : |z| < 2−1 } ∪ {ζ} at ζ ∈ T. Associated with this region is the nontangential maximal function N(f ) of a function f on D. It is deﬁned by N(f )(ζ) = sup |f (z)|. z∈Λ(ζ)

¯ = D ∪ T, and µ be a nonnegative Radon measure Theorem 4.1.4. Let p ∈ (0, 1], D on D. Then the following statements are equivalent: (i) µ ∈ CMp . (ii) There is a constant C > 0 such that p |f |dµ ≤ C N(f )dH∞ D

T

holds for every complex-valued function f on D with the right-hand-side integral being ﬁnite. (iii) For every q > p, there is a constant C > 0 such that q p |f | d|µ| ≤ C |f |q dH∞ D

T

¯ with f being complex-valued harmonic on D holds for every function f on D and the right-hand-side integral being ﬁnite. (iv) For each t > 0 there is a constant C > 0 such that p |f |p dH∞ tp µ {z ∈ D : |f (z)| > t} ≤ C T

¯ with f being complex-valued harmonic on D holds for every function f on D and the right-hand-side integral being ﬁnite. Proof. We demonstrate the result by showing (i)⇒(ii)⇒(iii)⇒(i)⇒ and (iv)⇔(i). p < ∞, then (i)⇒(ii) Assume that µ is a p-Carleson measure. If T N(f )dH∞ for t ≥ 0, let {Ij } be a sequence of open subarcs on T such that {ζ ∈ T : N(f )(ζ) > t} ⊆

Ij .

j

Clearly, there is a sequence of disjoint open subarcs {Jk } on T such that J . From this it follows that k k Jk =

Ij ⊆Jk

Ij

and |Jk |p ≤

Ij ⊆Jk

|Ij |p

j Ij

=

80

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

hold for each k. Consequently, |Jk |p ≤ |Ij |p = |Ij |p . k Ij ⊆Jk

k

j

If |f (z)| > t, there exists some Jk such that |Jk | > 1 − |z| and N(f )(ζ) > t when ζ ∈ Jk , and thus z ∈ S(Jk ). Namely, S(Jk ). {z ∈ D : |f (z)| > t} ⊆ k

Accordingly, µ {z ∈ D : |f (z)| > t} ≤ µ S(Jk ) ≤ µ CMp |Jk |p ≤ µ CMp |Ij |p . k

j

k

Taking the inﬁmum over all such coverings and integrating over (0, ∞) with respect to dt, we obtain ∞ |f |dµ = µ {z ∈ D : |f (z)| > t} dt 0 D ∞ p ≤ µ CMp H∞ {ζ ∈ T : N(f )(ζ) > t} dt 0 p = µ CMp N(f )dH∞ . T

(ii)⇒(iii) Suppose (ii) is valid. According to Theorem 4.1.4 (i), we read oﬀ q p p (M(f )) dH∞ |f |q dH∞ , q > p. T

T

Since the function f in (ii) is complex-valued harmonic on D and its boundary value f exists on T, we have the following well-known estimate: N(f )(ζ) = sup |f (z)| M(f )(ζ),

ζ ∈ T.

z∈Λ(ζ)

By applying (ii) to |f |q , q > p, we get a constant C > 0 such that p p p |f |q dµ ≤ C N(|f |q )dH∞ C M(|f |q )dH∞ C |f |q dH∞ . D

T

T

T

So, (iii) follows. (iii)⇒(i) Let I ⊆ T be an open arc. Consider the complex-valued harmonic function fI generated by the characteristic function 1I of I: 1I (ζ)(1 − |z|2 ) fI (z) = |dζ|, z ∈ D. |ζ − z|2 T

4.2. Adams Type Dualities

81

It is easy to see that if z ∈ S(I), then 1 |fI (z)|. Because of (iii) implies p |fI |q dµ C 1I dH∞ C|I|p µ S(I) D

T

p 1I dH∞ ≈ |I|p ,

T

for a constant C > 0. Thus (i) follows. (iv)⇔(i) If (iv) holds, then for the above-deﬁned fI we have p µ S(I) µ {z ∈ T : 1 |fI (z)|} ≤ C 1I dH∞ C|I|p . T

In other words, µ ∈ CMp . Conversely, if (i) holds, then (ii) is true, and hence (iv) follows from p {ζ ∈ T : N(f )(ζ) > t} tp µ CMp H∞ p {ζ ∈ T : M(f )(ζ) > t} tp µ CMp H∞ p

µ CMp |f |p dH∞ .

tp µ {z ∈ D : |f (z)| > t} ≤

T

4.2 Adams Type Dualities ¯ we denote by C(X) and M(X) the space of continuous complexFor X = T or D, valued functions and ﬁnite complex-valued Radon measures on X, respectively. When µ ∈ M(X), we use |µ| as the total variation of µ, which is a nonnegative measure determined by the property that if dµ = f dν where ν is a nonnegative measure, then d|µ| = |f |dν. The Riesz Theorem states that the dual of C(X) is M(X). Using this well-known dual result, we can establish the following Adams Duality Theorem. Theorem 4.2.1. Given p ∈ (0, 1]. Let L1,p (T) be the Morrey space of elements µ from M(T) for which

µ L1,p = sup |I|−p |µ|(I) < ∞, I

p ) stands for the where the supremum is taken over all subarcs I of T. If L1 (H∞ p 1,p completion of C(T) with respect to T |f |dH∞ , then L (T) is isomorphic to the p dual of L1 (H∞ ) under the pairing T f dµ. Proof. Because of µ ∈ L1,p (T), the linear functional L(f ) = T f dµ is well deﬁned p on L1 (H∞ ) since for any µ-measurable E we have p |µ|(E) ≤ µ L1,p H∞ (E)

82

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

p (·). This in turn gives by using the deﬁnition of H∞ p |L(f )| ≤ |f |d|µ| ≤ µ L1,p |f |dH∞ , T

T

p f ∈ L1 (H∞ ).

As a result, we ﬁnd that the norm L of L is not greater than µ L1,p . p Conversely, if L is a continuous linear functional on L1 (H∞ ), then there is a µ ∈ M(T) such that L is expressed as L(f ) = T f dµ for f ∈ C(T), thanks to p C(T) ⊆ L1 (H∞ ). But then for any g ∈ C(T), gd|µ| ≤ |g|d|µ| T T ≤ sup f dµ : f ∈ C(T) & |f | ≤ |g| p |f |dH∞ : f ∈ C(T) & |f | ≤ |g| ≤ L sup T p ≤ L

|g|dH∞ . T

Now, given a subarc I of T and arbitrary small number > 0, let I be the subarc of T centered at the center of I and with length |I| + , and choose a function g ∈ C(T) with g = 1 on I and g = 0 on T \ I . Then p |µ|(I) ≤ L H∞ (I ) which implies µ L1,p ≤ L .

Using Theorem 4.2.1, we develop a technique that is often easier to dominate the Choquet integrals against Hausdorﬀ capacity. Corollary 4.2.2. For p ∈ (0, 1] let L1,p + (T) be the class of all nonnegative elements in L1,p (T). If f is nonnegative and lower semicontinuous on T, then p 1,p ≤ 1 . f dH∞ ≈ sup f dµ : µ ∈ L1,p (T) &

µ

L + T

T

p,d p,d ) be the completion of C(T) under the norm T |f |dH∞ . Then Proof. Let L1 (H∞ 1 p,d it is a Banach space, and hence the canonical mapping from L (H∞ ) into its second dual has norm p,d p,d |g|dH∞ = sup gdµ : µ ∈ L1,p g ∈ L1 (H∞ ). + (T) & µ L1,p ≤ 1 , T

T

Now since f , nonnegative and lower semicontinuous on T, can be approximated from below by a nonnegative and increasing sequence {fj }∞ j=1 in C(T), we have p,d 1,p fj dH∞ = sup fj dµ : µ ∈ L1,p (T) &

µ

≤ 1 L + T T ≤ sup f dµ : µ ∈ L1,p + (T) & µ L1,p ≤ 1 . T

4.2. Adams Type Dualities

83

Using the limit p,d p,d lim H∞ {ζ ∈ T : fj (ζ) ≥ t} = H∞ {ζ ∈ T : f (ζ) > t} ,

t > 0,

j→∞

we obtain p,d p,d 1,p f dH∞ ≤ sup f dµ : µ ∈ L1,p (T) &

µ

≤ 1 ≤ 2 f dH∞ , L + T

T

T

as desired. Returning to D, we have the following analogue of Theorem 4.2.1.

¯ whose Theorem 4.2.3. Given p ∈ (0, 1]. Let CCMp be the space of all µ ∈ M(D) 1 p total variation |µ| is in CMp . If LN (H∞ ) is the completion of all functions ¯ with respect to f ∈ C(D) p p

f LN 1 (H∞ = N(f )dH∞ , ) T

p ) under the pairing then CCMp is isomorphic to the dual space of LN 1 (H∞

¯ D

f dµ.

Proof. Using the implication from (i) to (ii) of Theorem 4.1.4 we easily ﬁnd that p ). Conversely, every µ ∈ CCMp induces a bounded linear functional on LN 1 (H∞ 1 p ¯ ⊆ suppose that L is a bounded linear functional on LN (H∞ ). Now since C(D) p LN 1 (H∞ ), we employ the above-mentioned Riesz Theorem to produce a µ ∈ ¯ such that M(D) ¯ p L(f ) = f dµ and |L(f )| ≤ L

f LN 1 (H∞ ) for f ∈ C(D). ¯ D

¯ we have But then for every g ∈ C(D), ) ¯ gd|µ| ≤ sup p . f dµ : f ∈ C(D) & |f | ≤ |g| ≤ L

g LN 1 (H∞ ) ¯ D

D

To verify that |µ| is actually a p-Carleson measure on D, we consider an open subarc I of T and assume that I ⊆ T, for > 0 small enough, is an open arc with the same center as I’s and with length |I| + , and take such a nonnegative ¯ that g(z) = 1 for z ∈ S(I) and g(z) = 0 for D\ ¯ S(I ). This choice implies g ∈ C(D) N(g) 1I and p p |µ|(S(I)) ≤ |µ|(S(I )) = gd|µ| ≤ L

g LN 1 (H∞ ) L |I | . ¯ D

Thus |µ| ∈ CMp .

84

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

4.3 Quadratic Tent Spaces We call f ∈ Tp∞ provided f is Lebesgue measurable on D and satisﬁes −p

f Tp∞ = sup

12

|I|

2

T (I)

I⊆T

2 p

|f (z)| (1 − |z| ) dm(z)

< ∞,

where the supremum runs over all open subarcs I of T, and T (E) = {reiθ ∈ D : dist(eiθ , T \ E) > 1 − r} is the tent over the set E ⊂ T. Moreover, if

f Tp1 = inf ω

D

2

−1

|f (z)| (ω(z))

2 −p

(1 − |z| )

12 dm(z) < ∞,

then we say f ∈ Tp1 , where the inﬁmum is taken over all nonnegative functions ω p on D with ω LN 1 (H∞ ) ≤ 1. Note that the last condition implies that if z ∈ D\{0} and Iz is the subarc of T centered at z/|z| with length 1 − |z|, then |ω(z)| ≤ inf N(ω)(ζ). ζ∈Iz

Furthermore, the last inequality is multiplied by 1Iz and integrated in ζ with p respect to dH∞ to produce p (1 − |z|)p |ω(z)| ω LN 1 (H∞ ) 1.

Thus, f Tp1 = 0 must yield f = 0 a.e. on D. ∞ ∞ Lemma 4.3.1. Given p ∈ (0, 1]. If j=1 fj Tp1 < ∞, then f = j=1 fj ∈ Tp1 with ∞

f Tp1 ≤ 2 j=1 fj Tp1 . Proof. Without loss of generality, we may assume fj Tp1 > 0 for all j = 1, 2, . . .. p For any ∈ (0, 1) we take ωj ≥ 0 such that ωj LN 1 (H∞ ) ≤ 1 and −1 |fj (z)|2 ωj (z) (1 − |z|2 )−p dm(z) ≤ (1 + ) fj 2Tp1 . D

Using the Cauchy–Schwarz inequality we derive 2

|f | ≤

∞

fj

j=1

Let

Tp1

ωj

∞

ωj fj Tp1

−1

|fj |2 .

j=1

∞ ∞ −1

fj Tp1 ωj fj Tp1 . ω= 2 j=1

j=1

4.3. Quadratic Tent Spaces Then

85

p

ω LN 1 (H∞ )

≤

T

p,d N(ω)dH∞

∞ ∞ −1 p,d = 2

fj Tp1

fj Tp1 N(ωj ) dH∞ T

j=1

j=1

≤ 1. At the same time, by Fatou’s Lemma we have −1 |f (z)|2 ω(z) (1 − |z|2 )−p dm(z) D

≤ 2

∞

∞ −1 −1 2 2 −p

fj Tp1 |fj (z)| ωj (z) (1 − |z| ) dm(z)

fj T 1

j=1

p

j=1 ∞

= 2(1 + )

fj Tp1

D

2 ,

j=1

thereupon deriving the desired estimate.

It is an immediate consequence of Lemma 4.3.1 that Tp1 is complete under the quasi-norm · Tp1 . To better understand the structure of Tp1 , we introduce the concept of a Tp1 -atom. Given p ∈ (0, 1]. A function a on D is called a Tp1 -atom provided there exists a subarc I of T such that a is supported in the tent T (I) and satisﬁes |a(z)|2 (1 − |z|)−p dm(z) ≤ |I|−p . T (I)

Theorem 4.3.2. Let p ∈ (0, 1]. Then: there is a sequence of Tp1 -atoms {aj } and an 1 -sequence (i) f ∈ Tp1 if and only if {λj } such that f = ∞ j=1 λj aj . Moreover,

f Tp1 ≈ |||f |||Tp1 = inf

∞ j=1

|λj | : f =

∞

λj aj ,

j=1

where the inﬁmum is taken over all possible atomic decompositions of f ∈ Tp1 . Consequently, Tp1 is a Banach space under the norm ||| · |||Tp1 . (ii) Tp∞ is isomorphic to the dual of Tp1 under the pairing f, g = D f g¯dm. Proof. (i) Given a Tp1 -atom a. By deﬁnition there is an arc I ⊂ T centered at ζI such that the support of a is contained in T (I) and |a(z)|2 (1 − |z|)−p dm(z) ≤ |I|−p . T (I)

86

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Now for > 0, put −p

ω(z) = |I|

! min 1,

|I| 2|z − ζI |

p+ * .

Note that ω(z) = |I|−p as |z − ζI | ≤ |I|/2 and ω(z) falls oﬀ as z ∈ D \ S(I). Thus, for each ζ ∈ T we derive ! p+ * |I| −p N(ω)(ζ) |I| min 1, . 2|ζ − ζI | As a result, we have p N(ω)dH∞

≤

T

|ζ−ζI |≤|I|/2

1 + |I|

1.

p N(ω)dH∞

|I|−(p+)

+ |ζ−ζI |>|I|/2

p N(ω)dH∞

t−p/(p+) dt

0

Since ω ≈ |I|−p on T (I), we conclude −1 2 −p p |a(z)| ω(z) (1 − |z|) dm(z) ≈ |I| T (I)

T (I)

|a(z)|2 (1 − |z|)−p dm(z) 1.

Accordingly, a ∈ Tp1 with a Tp1 1. This, plus Lemma 4.3.1, further yields that if < ∞, then ∞ {aj } is a sequence of Tp1 -atoms and ∞ j=1 |λj | j=1 λj aj is convergent in · Tp1 to a function f ∈ Tp1 with f Tp1 ∞ |λ |. j j=1 Conversely, suppose f ∈ Tp1 . If f Tp1 = 0, then f = 0 a.e. on D and hence there is nothing more to argue. If f Tp1 = 0, then there exists an ω ≥ 0 on D such that −1 |f (z)|2 ω(z) (1 − |z|)−p dm(z) ≤ 2 f 2Tp1 . D

For each integer k let Ek = {ζ ∈ T : N ω(ζ) > 2k } and choose a sequence of open arcs {Ij,k } on T such that p Ek ⊆ Ij,k and |Ij,k |p ≤ 2H∞ (Ek ). j

j

Consequently, there exists a sequence of disjoint open subarcs {Jj,k } of T such that j Jj,k = j Ij,k . This implies Jj,k = Il,k ⊆Jj,k Ij,k and so by p ∈ (0, 1], j

|Jj,k |p ≤

j

Il,k ⊆Jj,k

|Il,k |p =

j

|Ij,k |p .

4.3. Quadratic Tent Spaces Note that Ek ⊆ Now, putting

j

87

Jj,k and {Jj,k } are disjoint for each k. So T (Ek ) ⊆ Tj,k = S(Jj,k ) \

j

S(Jj,k ).

S(Jn,m ),

m>k n

we derive that {Tj,k } are disjoint for all diﬀerent choices of j, k but also that for any l ∈ N the following inclusion holds: l

Tj,k ⊇ T (El ) \

S(Jn,m ).

m>l n

k=−l j

This inclusion implies

Tj,k ⊇

j,k

T (Ek ) \

S(Jn,m ).

l m>l n

k

From the deﬁnition of Ek and

+

T

p N(ω)dH∞ ≤ 1 it follows that

T (Ek ) = {z ∈ D : ω(z) > 0}

k

and S m>l

S n

S(Jn,m )

dm

|Jn,m |p

m>l n

p H∞ (Em ) → 0 as l → ∞.

m>l

Noticing also that ω is allowed to be 0 only when f = 0, we get f = a.e. on D, where 1Tj,k is the characteristic function of Tj,k . Now, if aj,k and

p = f 1Tj,k |Jj,k |

j,k

f 1Tj,k

− 12 |f (z)|2 (1 − |z|)−p dm(z) Tj,k

12 |f (z)|2 (1 − |z|)−p dm(z) ,

λj,k = |Jj,k |

p Tj,k

then f=

λj,k f aj,k .

j,k

It is easy to see that aj,k is a Tp1 -atom, but also {λj,k } is 1 -summable due to the following estimate established by z ∈ Tj,k ⇒ ω(z) ≤ 2k+1

88

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

and the Cauchy–Schwarz inequality: 12 k+1 −1 p |λj,k | ≤ 2 2 |Jj,k | 2 |f (z)|2 ω(z) (1 − |z|)−p dm(z) j,k

Tj,k

j,k

≤

2k+1 |Jj,k |p

j,k

f Tp1

k

f Tp1

12 j,k

2k

|Jj,k |p

f Tp1

T

Tj,k

12

j

12 p 2 k H∞ (Ek )

k

12 −1 |f (z)|2 ω(z) (1 − |z|)−p dm(z)

p N(ω)dH∞

12

.

By Lemma 4.3.1 it follows that j,k λj,k f aj,k converges to f in · Tp1 . Since |||·|||Tp1 is a norm, Lemma 4.3.1 is applied to yield that Tp1 is a Banach space under this norm. (ii) Let f ∈ Tp1 , g ∈ Tp∞ and dµg,p (z) = |g(z)|2 (1 − |z|2 )p dm(z). Then µg,p ∈ CMp and hence it follows from Theorem 4.1.4 and its proof that 2 p |g(z)|2 ω(z)(1 − |z|2 )p dm(z) ≤ g 2Tp∞ ω LN 1 (H∞ ) ≤ g Tp∞ D

holds for all ω in the deﬁnition of f ∈ Tp1 . An application of the Cauchy–Schwarz inequality gives 12 2 −1 2 −p |f (z)g(z)|dm(z) ≤ |f (z)| (ω(z)) (1 − |z| ) dm(z)

g Tp∞ . D

D

∈ Tp∞

induces a bounded linear functional on Tp1 via the above Thus every g pairing f, g = D f g¯dm. Conversely, let L be a continuous linear functional on Tp1 and ﬁx an open arc I ⊆ T centered at ζI . If f is supported in T (I) with f ∈ L2 (T (I)), then 2 2 −p −p |f (z)| (1 − |z| ) dm(z) |I| |f |2 dm. T (I)

T (I)

Now for > 0, set −p

ω(z) = |I|

! min 1,

|I| 2|z − ζI |

p+ *

As proved in the ﬁrst part of (i) we once again obtain ! p+ * |I| −p N(ω)(ζ) |I| min 1, , 2|ζ − ζI |

.

ζ ∈ T.

4.4. Preduals under Invariant Pairing This yields p N(ω)dH∞

89

≤

T

|ζ−ζI |≤|I|/2

1 + |I| 1.

p N(ω)dH∞ +

|I|−(p+)

|ζ−ζI |>|I|/2

p N(ω)dH∞

t−p/(p+) dt

0

From the fact that the above-selected ω satisﬁes ω −1 ≈ |I|p on T (I) it follows that |f (z)|2 (ω(z))−1 (1 − |z|2 )−p dm(z) |f |2 dm,

f 2Tp1 ≤ T (I)

T (I)

and so that f ∈ Tp1 . Hence L induces a continuous linear functional on L2 (T (I)), product. Taking an and acts with some function gI ∈ L2 (T (I)) via the L2 -inner increasing family of closed subarcs {Ij } of T such that T = j Ij , we thus get a function g such that L(f ) = f g¯dm for all f ∈ Tp1 supported in T (I). D

To establish g ∈ Tp∞ , let f (z) = (1 − |z|2 )p g(z)1T (I) (z). Then L(f ) =

T (I)

|g(z)|2 (1 − |z|2 )p dm(z)

12

L |I|

2

p

and hence

−p

|I|

T (I)

T (I)

2 p

|g(z)| (1 − |z| ) dm(z)

,

12 2

2 p

|g(z)| (1 − |z| ) dm(z)

In other words, g ∈ Tp∞ with g Tp∞ L .

L .

4.4 Preduals under Invariant Pairing Recall that a holomorphic function f on D is of the Sobolev class H (denoted W11 sometimes, see [ArFi]) if f belongs to the Hardy space H1 , that is,

f H = f H1 = sup |f (rζ)||dζ| < ∞. r∈(0,1)

T

90

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

A famous duality theorem of Feﬀerman-type is that BMOA is isomorphic to the dual of H with respect to the invariant pairing r 2π f, ginv = f g dm = lim f (reiθ )g (reiθ )rdθdr. r→1

D

0

0

Meanwhile, the dual of D under ·, ·inv is isomorphic to D. Here and henceforth, for simplicity we prefer to work modulo constants, and by invariance we mean f ◦ φ, g ◦ φinv = f, ginv , Thanks to

φ ∈ Aut(D).

H ⊂ D ⊂ Qp ⊂ BMOA,

p ∈ (0, 1),

the predual of Qp under ·, ·inv should contain H but be contained in D. This observation leads to the discovery of a new kind of holomorphic space. Given p ∈ (0, 1]. Let Pp be the class of all holomorphic functions f on D with 12 2 −1 2 2−p |(zf (z)) | (ω(z)) (1 − |z| ) dm(z) < ∞,

f Pp = inf ω

D

where the inﬁmum ranges over all nonnegative functions ω on D with p

ω LN 1 (H∞ ) ≤ 1.

Using the implication 2 −p p

ω LN 1 (H∞ , ) ≤ 1 ⇒ ω(z) (1 − |z| )

we derive that if f ∈ Pp , then 2 |(zf (z)) |(1 − |z|2 ) dm(z) f 2Pp . D

This, along with Lemma 4.3.1, implies readily that Pp is a quasi-Banach space. Theorem 4.4.1. Let p ∈ (0, 1]. Then Qp is isomorphic to the dual of Pp under ·, ·inv . Proof. On the one hand, let f ∈ Pp and g ∈ Qp . Note that ·, ·inv has another expression (zf (z)) g (z)(1 − |z|2 )dm(z). f, ginv = D

So, using the Cauchy–Schwarz inequality, we get

|f, ginv | 12 1 2 −1 2 2−p 2 |(zf (z)) | (ω(z)) (1 − |z| ) dm(z)

ω LN 1 (H p ) g Qp ,1 ∞ D

g Qp ,1

2

D

−1

|(zf (z)) | (ω(z))

2 2−p

(1 − |z| )

12 dm(z)

4.4. Preduals under Invariant Pairing

91

for any ω required in Pp . Thus, each member of Qp induces a bounded linear functional on Pp . Conversely, suppose L is a bounded linear functional on Pp . Let D(f )(z) = (zf (z)) (1 − |z|2 ),

z ∈ D.

Then D is an isometric map from Pp into Tp1 . Since Tp1 is a Banach space under ||| · |||Tp1 , it follows from the Hahn–Banach Extension Theorem that we can select a function h ∈ Tp∞ such that L(f ) = (zf (z)) (1 − |z|2 )h(z)dm(z), f ∈ Pp . D

Note again that |f (z)|2 dm(z)

D

D

D

|(zf (z)) |2 (1 − |z|2 )2 dm(z) |(zf (z)) |2 (ω(z))−1 (1 − |z|2 )2−p dm(z)

for any ω used in Pp and consequently Pp is a subspace of D. Thus we get by an application of the reproducing formula for D, (wf (w)) (1 − |w|2 ) −1 (zf (z)) = 2π dm(w), f ∈ Pp . (1 − wz) ¯ 3 D Accordingly, L(f ) = =

2π

−1

(wf (w)) D

D

h(z)(1 − |z|2 ) dm(z) (1 − |w|2 )dm(w) (1 − wz) ¯ 3

f, ginv ,

where g(w) = 2π −1

0

w

D

h(z)(1 − |z|2 ) dm(z) du. (1 − u¯ z )3

The argument will be done if we can prove g ∈ Qp . To do so, it suﬃces to show that (1 − |w|)p |g (w)|2 dm(w) is a p-Carleson measure. But, this follows from an application of Lemma 3.1.2 to the inequality |h(z)|(1 − |z|2 ) |g (w)| ≤ dm(z), |1 − w¯ z |3 D and the fact that |h(z)|2 (1 − |z|2 )p dm(z) is a p-Carleson measure.

To see that for p ∈ (0, 1), Pp lies properly between H and D, we need an estimate for the radial maximal operator of the generalized Poisson integral of an H1 -function on T with respect to the Hausdorﬀ capacity.

92

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Lemma 4.4.2. Given p ∈ (0, 1] let f ∈ H1 (T). If f (η)(1 − |z|2 )2−p |dη|, Ip f (z) = |z − η|2 T then

¯ z ∈ D,

T

p N(Ip f )dH∞

p sup |Ip f (rζ)|dH∞ (ζ) f H1 .

T r∈(0,1)

Proof. We begin with proving the ﬁrst estimate. Note that Ip f (z)(1 − |z|2 )p−1 is harmonic on D. So if z = reiθ and Dz = {seit ∈ D : |s − r| < 1 − r & |t − θ| < 1 − r}, then we have the following sub-mean-value inequality for q ∈ (0, 1): |Ip f (z)|q (1 − |z|)−2 |Ip f (w)|q dm(w). |w−z|<1−r

Consequently, if z ∈ Λ(λ) where |z| ≥ 1/2 and λ = eiu ∈ T, and ζ ∈ T, then |Ip f (z)|q (1 − |z|)−2 |Ip f (seit )|q dm(seit ) (1 − |z|)−2

Dz θ+(1−|z|) 1

2|z|−1

θ−(1−|z|)

−1

|Ip f (seit )|q sdsdt

θ+(1−|z|)

sup |Ip f (seit )|q dt.

(1 − |z|)

θ−(1−|z|) s∈(0,1)

Now that z ∈ Λ(λ) implies |θ − u| ≤ c(1 − |z|) for some constant c > 1, we derive θ − (1 − |z|), θ + (1 − |z|) ⊆ u − (1 + c)(1 − |z|), u + (1 + c)(1 − |z|) and then

N(Ip (f ))(ζ)

M

sup |Ip f (s·)|

q

1q

(ζ)

,

ζ ∈ T.

s∈(0,1)

Again, M is the Hardy–Littlewood maximal operator. Applying Theorem 4.1.3 (i) and q −1 > 1 to the last inequality, we get p p N(Ip (f ))(ζ)dH∞ (ζ) sup |Ip f (sζ)|dH∞ (ζ). T

T s∈(0,1)

For the second estimate, we should recognize that f can be written as j λj aj on T where j |λj | f H1 and aj is an H1 -atom on T. Recall that an H1 -atom is a function a : T → C which is supported on an open subarc I of T, and satisﬁes −1 and a(ζ)|dζ| = 0.

a ∞ = a L∞ (T) < |I| T

4.4. Preduals under Invariant Pairing

93

So, we have by Lemma 4.1.1 (iv), p sup |Ip f (sζ)|dH∞ (ζ) |λj | T s∈(0,1)

j

p sup |Ip aj (sζ)|dH∞ (ζ)

T s∈(0,1)

|λj |

j

f H1 , provided that the following is veriﬁed: p sup |Ip a(sζ)|dH∞ (ζ) 1 T s∈(0,1)

for any H1 -atom a on T.

In so doing, we may assume that the H1 -atom a has the support I centered at ζI . Using a ∞ < |I|−1 , we are now able to derive that for s ∈ (0, 1), |a(η)|(1 − s2 )2−p |dη| −1 |dη| |I| |I|−p , ζ ∈ T, |Ip a(sζ)| ≤ 2 |ζ − sη| |ζ − η|p I I which implies

sup |Ip a(sζ)| |I|−p ,

ζ ∈ T.

s∈(0,1)

Meanwhile, using T a(η)|dη| = 0, a ∞ < |I|−1 and the mean value theorem for derivatives, we further obtain that for s ∈ (0, 1), a(η) a(η) |dη| − |Ip a(sζ)| = (1 − s2 )2−p 2 2 |ζ − sζI | T |ζ − sη| (1 − s2 )2−p (1 − s2 )2−p − ≤ a ∞ |dη| 2 |ζ − sη| |ζ − sζI |2 I |I|2−p |ζ − ζI |−2

as |ζ − ζI | > 2|I|.

Accordingly, sup |Ip a(sζ)| |I|2−p |ζ − ζI |−2

as |ζ − ζI | > 2|I|.

s∈(0,1)

Therefore, if µ ∈ L1,p + , then sup |Ip a(sζ)|dµ(ζ) T s∈(0,1)

=

+ |ζ−ζI |≤2|I|

1 + |I|2−p

1.

0

|ζ−ζI |>2|I|

(2|I|)−2

sup |Ip a(sζ)|dµ(ζ) s∈(0,1)

µ {ζ ∈ T : |ζ − ζI |−2 > t} dt

Therefore, the desired estimate follows from Corollary 4.2.2.

94

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Theorem 4.4.3. Let p ∈ (0, 1] and µ be a nonnegative Radon measure on D. Then: (i) There is a constant C > 0 such that D |Ip (f)|dµ ≤ C f H1 holds for any f ∈ H1 if and only if µ ∈ CMp . (ii) H ⊆ Pp ⊆ D holds with

f D f Pp f H ,

f ∈ H .

In particular, H = P1 and Pp = D. Proof. (i) If µ ∈ CMp , then Lemma 4.4.2 and the argument for the implication (i)⇒(ii) in Theorem 4.1.4 leads to p |Ip (f)|dµ µ CMp N Ip (f ) dH∞ µ CMp f H1 f ∈ H1 . D

T

Conversely, suppose that µ satisﬁes the above-assigned inequality. Given a subarc I ⊂ T with center ζI , let g(z) = (1 − (1 − |I|)ζI z)−2 . Then g H1 ≈ |I|−1 . Using the formula −2 Ip (g)(z) = 2π(1 − |z|2 )1−p 1 − (1 − |I|)ζI z , we derive

µ S(I) |I|−1−p

D

|Ip (g)|dµ ≤ C g H1 C|I|−1 ,

thus implying µ ∈ CMp . (ii) Because Pp ⊆ D has been established above, we here show H ⊆ Pp . Let f ∈ H with f H > 0 and ω(z) = |(zf (z)) |(1 − |z|2 )1−p f −1 H . Then (zf (z)) ∈ H1 and hence it is applied to produce |(zf (z)) |2 (ω(z))−1 (1 − |z|2 )2−p dm(z) D |(zf (z)) |2 ≤ f H (1 − |z|2 )dm(z) D |(zf (z)) | f H zf (z) H

f 2H .

In the meantime, since f ∈ H , g(z) = (zf (z)) can be represented by the Poisson integral of its boundary value function g(ζ) = f (ζ) + ζf (ζ), ζ ∈ T: 1 − |z|2 1 |dζ|. g(ζ) (zf (z)) = 2π T |ζ − z|2

4.4. Preduals under Invariant Pairing

95

This formula implies |Ip g(z)| = 2π(1 − |z|2 )1−p |(zf (z)) |. So, by Lemma 4.4.2 we have −1 p p N(Ip g)dH∞

ω LN 1 (H∞ ) f H T p f −1 sup |Ip g(sζ)|dH∞ (ζ) H T s∈(0,1)

f −1 H g H1 1,

thereupon producing f ∈ Pp . Of course, Pp = D, for if it were not, then it would follow that Qp = D, a contradiction, since the duals of Pp and D are Qp and D respectively under the invariant paring ·, ·inv . In order to see Pp = H as p ∈ (0, 1), it suﬃces to prove H = P1 due to BMOA = Qp and Theorem 4.4.1. To this end, we handle two situations. On the one hand, if f ∈ H , then by taking ω(z) = |(zf (z)) | f −1 H (naturally, f H > 0 is assumed, otherwise, there is nothing to do) we apply the proof above to establish |(zf (z)) |2 (ω(z))−1 (1 − |z|2 )dm(z) f 2H . D

Note that −1

ω LN 1 (H∞ 1 ) ≈ N(ω) L1 (T) (zf (z)) H1 f 1. H

So, f ∈ P1 . Namely, H ⊆ P1 . On the other hand, if f ∈ P1 , then for any nonnegative ω with

N(ω) L1 (T) ≈ ω LN 1 (H∞ 1 ) 1, we can take an account of the well-known square function characterization of H1 , the Cauchy–Schwarz inequality and Fubini’s Theorem to derive

zf (z) H

=

(zf (z)) H1 T

Λ(ζ)

12

|(zf (z)) |2 dm(z)

12

T

Λ(ζ)

D

|dζ|

2

−1

|(zf (z)) | (ω(z)) 2

−1

|(zf (z)) | (ω(z))

dm(z)

1 N(ω)(ζ) 2 |dζ|

12 1 (1 − |z| )dm(z)

N(ω) L2 1 (T) . 2

96

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Hence we get zf (z) H f P1 < ∞. Because of P1 ⊆ D, a further calculation shows f ∈ H and then P1 ⊆ H . Therefore P1 = H .

4.5 Invariant Duals of Vanishing Classes Note that the inclusion D ⊂ Qp , p > 0 can be improved by imbedding the vanishing Q class — Qp,0 — all functions f ∈ Qp with lim|w|→1 Ep (f, w) = 0, between D and Qp . So it seems reasonable and possible to use Qp,0 to give another look at the predual structure of Qp , but also to explore more properties on the dual of Qp.0 . To begin with, we consider the dyadic partition of D by letting E = {En }n∈N be the family of the sets z ∈ D : 2−(k+1) < 1 − |z| ≤ 2−k & 2−(k+1) πj ≤ arg z < 2−(k+1) π(j + 1) , where k ∈ N ∪ {0} and 0 ≤ j < 2k+2 . And, for each n ∈ N we choose an to be the center of En . Lemma 4.5.1. Given p ∈ (0, 1). For n ∈ N let Xn respectively Yn consist of those functions g, respectively f , holomorphic on D for which 12 2 2 p |g (z)| (1 − |σan (z)| ) dm(z) < ∞,

g Xn = D

respectively,

f Yn =

D

12 |(zf (z)) |2 (1 − |σan (z)|2 )−p (1 − |z|2 )2 dm(z) < ∞.

Then the dual of Xn is isomorphic to Yn under ·, ·inv . Moreover,

f Yn ≈ sup{|f, ginv | : g Xn ≤ 1},

f ∈ Yn ,

where the constants involved in the foregoing and following estimates are independent of f and an . Proof. For g ∈ Xn and f ∈ Yn we use the Cauchy–Schwarz inequality to produce |g, f inv | = g (z)(zf (z)) (1 − |z|2 )dm(z) ≤ g Xn f Yn ; D

that is, f induces a continuous linear functional on Xn . On the other hand, suppose that L is a continuous linear functional on Xn . Then L can be extended, with preserving the norm, to a continuous linear func2 2 p D, (1 − |σ (z)| ) dm(z) and consequently, there exists a function tional on L an 2 2 −p 2 2 h ∈ L D, (1 − |σan (z)| ) (1 − |z| ) dm(z) such that −1 g (z)h(z)(1 − |z|2 )dm(z), g ∈ Xn , L(g) = g, h = π D

4.5. Invariant Duals of Vanishing Classes and

L =

D

2

97

2 −p

|h(z)| (1 − |σan (z)| )

12 (1 − |z| ) dm(z) . 2 2

Note that if g ∈ Xn , then g (z) has the reproducing formula g (w)(1 − |w|2 ) −1 dm(w), z ∈ D. g (z) = 2π (1 − wz) ¯ 3 D Accordingly, if g ∈ Xn , then by Fubini’s Theorem, −1 g (w)(wf (w)) (1 − |w|2 )dm(w), L(g) = π D

where f is an element in H determined by −1 h(z)(1 − z¯w)−3 (1 − |z|2 )dm(z), (wf (w)) = 2π D

w ∈ D.

Observe that the function f is holomorphic in D and that |h(z)|2 (1 − |σan (z)|2 )−p (1 − |z|2 )2 dm(z) < ∞. D

So it remains to prove ¯w|2p ¯z|2p 2 |1 − a 2 |1 − a |(wf (w)) | dm(w) |h(z)| dm(z), a ∈ D, (1 − |w|2 )p−2 (1 − |z|2 )p−2 D D where the constant depends only on p. To do so, we introduce the functions F and H by h(w) = H(w)(1 − |w|2 )−β

and (zf (z)) = F (z)(1 − |z|2 )−β ,

where β ∈ (1 + p, 3 − p). Then the required demonstration is equivalent to showing |F (w)|2 |1 − a ¯w|2p |H(z)|2 |1 − a ¯z|2p dm(w) dm(z), a ∈ D, 2 p+2β−2 2 p+2β−2 D (1 − |w| ) D (1 − |z| ) where F (w) = 2π −1 (1 − |w|2 )β For a ∈ D, put

D

H(z)(1 − |z|2 )1−β (1 − z¯w)−3 dm(z).

¯z|2p dm(z) dµa (z) = (1 − |z|2 )2−p−2β |1 − a

and consider the linear operator T : H → F. If we can prove the operator norm estimates:

T Lq (D,dµa )→Lq (D,dµa ) ≤ Cq for q = 1, ∞,

98

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

where C1 and C∞ are constants independent of a ∈ D, then the required estimate follows from the Riesz–Thorin Theorem. The case q = ∞ follows from (1 − |z|2 )1−β |1 − z¯w|−3 dm(z) |F (w)| ≤ 2π −1 (1 − |w|2 )β H L∞ (D,dµa ) D

H L∞ (D,dµa ) .

However, in the case of q = 1, we note that the formula of F implies |F | dµa D |1 − a ¯w|2p ≤ 2π −1 (1 − |z|2 )1−β |H(z)| dm(w) dm(z). 2 p+β−2 |1 − z ¯w|3 D D (1 − |w| ) Hence it is enough to establish |1 − a ¯w|2p K(a) = dm(w) ≤ C(1 − |z|2 )1−p−β |1 − a ¯z|2p , 2 p+β−2 |1 − z ¯w|3 D (1 − |w| ) where C is independent of a and z. To that end, we make a change of variable w = σz (λ) for a given z ∈ D, obtaining 1 − |σz (λ)|2 =

(1 − |λ|2 )(1 − |z|2 ) 1−a ¯z + λ(¯ a − z¯) , , 1−a ¯σz (λ) = |1 − z¯λ|2 1 − z¯λ

and K(a) =

2 1−p−β

(1 − |z| )

D

|1 − a ¯z + λ(¯ a − z¯)|2p dm(λ) |1 − z¯λ|5−2β (1 − |λ|2 )β+p−2

22p |1 − a ¯z|2p |1 − z¯λ|2β−5 (1 − |λ|2 )2−β−p dm(λ) (1 − |z|2 )p+β−1 D |1 − a ¯z|2p , (1 − |z|2 )p+β−1

≤

as desired. In the above estimates we have used the inequalities: 1 + p < β < 3 − p and |1 − a ¯z + λ(¯ a − z¯)| ≤ |1 − a ¯z| + |¯ a − z¯| ≤ 2|1 − a ¯z|.

On the basis of Lemma 4.5.1, we ﬁnd a dual of the vanishing Q space. Theorem 4.5.2. For p ∈(0, 1) let Rp,1 be the class of those holomorphic functions f on D for which f = ∞ n=1 fn , fn ∈ H with San (fn ) =

D

2

2 −p

|(zfn (z)) | (1 − |σan (z)| )

12 (1 − |z| ) dm(z) <∞ 2 2

4.5. Invariant Duals of Vanishing Classes and

f Rp,1 = inf

∞

99

San (fn ),

n=1

where the inﬁmum is taken over all the foregoing representations for which the convergence is uniform on compact subsets of D. Then the dual of Qp,0 is isomorphic to Rp,1 under ·, ·inv . More precisely, every function f ∈ Rp,1 induces a continuous linear functional on Qp,0 by that g, f inv for any g ∈ Qp,0 . Conversely, if L is a continuous linear functional on Qp,0 , then there exists f ∈ Rp,1 such that L(g) = g, f inv for any g ∈ Qp,0 . Moreover,

f Rp,1 ≈ sup |g, f inv | : g ∈ Qp,0 , g Qp ≤ 1 , f ∈ Rp,1 . Proof. If g ∈ Qp,0 , then g ∈ Qp . Thus, f ∈ Rp,1 and Cauchy–Schwarz’s inequality imply ∞ −1 |g, f inv | ≤ π |g (z)(zf (z)) |(1 − |z|2 )dm(z) San (fn ) g Qp ,1 , D

n=1

and consequently, f induces a continuous linear functional L(g) = g, f inv on Qp,0 . To prove that for every continuous linear functional L on Qp,0 there exists an f ∈ Rp,1 such that L(g) = g, f inv for any g ∈ Qp,0 , we use 1 − |σa (z)|2 ≈ 1 − |σan (z)|2 ,

a ∈ En , z ∈ D,

to get that the supremum in the deﬁnition of the · Qp ,1 can be taken over {an }∞ n=1 . Suppose now that each Xn consists of those functions g ∈ H for which 12 2 2 p |g (z)| (1 − |σan (z)| ) dm(z) < ∞.

g Xn = D

Denote by X the direct c0 -sum of {Xn }; that is, the space of holomorphic function sequences {gn }∞ 1 on D such that gn ∈ Xn for every n ∈ N and limn→∞ gn Xn = 0. Equipping X with the norm {gn }∞ 1 X = supn∈N gn Xn , we see that Qp,0 is a normed subspace of X under this new norm. Analogously we deﬁne Y to be the 1 -sum of the spaces Yn , where each Yn is the class of all holomorphic functions f on D with 12 2 2 −p 2 2 |(zf (z)) | (1 − |σan (z)| ) (1 − |z| ) dm(z) < ∞.

f Yn = D

Accordingly, the dual of X is isometrically isomorphic to the 1 -sum of the dual spaces Xn∗ of Xn . Using Lemma 4.5.1, we see that Xn∗ can be replaced by Yn and so the dual of X is isomorphic to Y under the pairing ∞ n=1

gn , fn inv ,

fn ∈ Yn , gn ∈ Xn .

100

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Suppose L is a continuous linear functional on Qp,0 . Using Lemma 4.5.1 with the Hahn–Banach extension of L to X we obtain such an fn ∈ Yn that L(g) =

∞

g, fn inv ,

g ∈ Qp,0 ,

n=1

∞ ∞ and the norm L of L satisﬁes n=1 fn Yn L . Finally, if f = n=1 fn , then this series converges uniformly on compact sets of D. Hence f ∈ H, f Y ≤

∞

fn Yn and L(g) = g, f inv for g ∈ Qp,0 .

n=1

This proves f ∈ Rp,1 with f Rp,1 L , as desired.

Quite naturally, we are led to connect Rp,1 with another space deﬁned in terms of Hausdorﬀ capacity. Theorem 4.5.3. For p ∈ (0, 1) let Rp,2 be the class of all f ∈ H with

f Rp,2 =

inf

ω≥0, ω LN 1 (H p ) ≤1

D

∞

12 −1 2 −p |f (z)| ω(z) (1 − |z| ) dm(z) < ∞.

2

Then the dual of Rp,2 is isomorphic to Qp under ·, ·inv . Consequently,

f Rp,2 sup |f, ginv | : g ∈ Qp , g Qp ≤ 1 , f ∈ Rp,2 . Proof. As proved in Theorem 4.4.1, we have that if f ∈ Rp,2 and g ∈ Qp , then |f, ginv |

D

12 −1 2 −p |f (z)| ω(z) (1 − |z| ) dm(z)

g Qp,1 .

2

As a result, |f, ginv | f Rp,2 g Qp ,1 ; that is, g ∈ Qp induces a continuous linear functional on Rp,2 . Conversely, suppose L is a continuous linear functional on Rp,2 . Then it follows from Theorem 4.3.2 (i) and the Hahn–Banach Extension Theorem that there exists a function g on D such that dµg,p = |g(z)|2 (1 − |z|2 )p dm(z) is a p-Carleson measure on D and L(f ) = f g¯dm, f ∈ Rp,2 . D

Also, it is easy to check that each f ∈ Rp,2 has the reproducing formula −1 f (z) = π f (w)(1 − wz) ¯ −2 dm(w), z ∈ D. D

4.5. Invariant Duals of Vanishing Classes

101

Thus, L(f ) =

−1 −2 f (w) π g(z)(1 − z¯w) dm(z) dm(w) = f, hinv ,

D

D

where h(w) =

0

w

g(z)(1 − z¯ζ)−2 dm(z) dζ, π −1 D

w ∈ D.

Thanks to µg,p ∈ CMp and

|h (w)| ≤ π

−1

D

|g(z)||1 − z¯w|−2 dm(z),

w ∈ D,

we conclude from Lemma 3.1.2 that |h (w)|2 (1 − |w|2 )p dm(w) is a p-Carleson measure on D and then h ∈ Qp which deﬁnes L. Applying a corollary of the Hahn–Banach Extension Theorem (cf. [DeV, p. 48, Corollary 2]) to Rp,2 , we see that if f ∈ Rp,2 is nonconstant, then there exists L, a continuous linear functional on Rp,2 , such that

L = 1

and f Rp,2 = L(f ).

With the help of the foregoing argument, we can ﬁnd a function g ∈ Qp such that

g Qp ≤ 1

and L(f ) = f, ginv ,

f ∈ Rp,2 .

This clearly implies the inequality required in Theorem 4.5.3.

In order to explore the relationship among Pp , Rp,1 and Rp,2 , we need a density result as follows. Lemma 4.5.4. Given p ∈ (0, 1). (i) If [Qp,0 ]∗ stands for the space of those f ∈ H with sup{|f, ginv | : g ∈ Qp,0 , g Qp,1 ≤ 1} < ∞, then Rp,2 is a dense subset of [Qp,0 ]∗ . (ii) If g ∈ Qp and gr (z) = g(rz) for r ∈ (0, 1), then there exists a sequence {rn } in (0, 1) such that lim rn = 1

n→∞

and

lim f, grn inv = f, ginv ,

n→∞

f ∈ Rp,2 .

Proof. (i) Since Qp,0 ⊂ Qp , we conclude by Theorem 4.5.3 that sup |g, f inv | : g ∈ Qp,0 , g Qp ,1 ≤ 1 f Rp,2 . This means Rp,2 ⊆ Rp,1 .

102

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Secondly, we prove that Rp,2p contains all the polynomials. To do so, taking 1. If f is a polynomial, then f H∞ < ∞. ω = 1 on D, we have T N(ω)dH∞ Accordingly,

f Rp,2 ≤ f H∞

D

12 (1 − |z|2 )−p dm(z) f H∞ ,

p ∈ (0, 1).

Thirdly, we prove that the polynomials are dense in Rp,1 . For this, let f ∈ Rp,1 . Then f=

∞

fn

∞

and

n=1

San (fn ) < ∞,

where fn ∈ Yn .

n=1

Since the dual of Xn is isomorphic to Yn , and the hypothesis that g, f inv = 0 for every polynomial g obviously implies f = 0, we derive that the polynomials are dense in each Yn . Consequently, for any > 0, there is a sequence ∞ of polynomials S (f − P ) <

. Choose j ≥ 1 so that Pn such that ∞ a n n n n=1 n=j+1 San (fn ) < , j and deﬁne the polynomial P by P = n=1 Pj . Then

f − P Rp,1 ≤

j

San (fn − Pn ) +

n=1

∞

San (fn ) < 2 .

n=j+1

Combining the above-proved results and Theorem 4.5.3, we get that Rp,2 is dense in Rp,1 , hence proving the desired assertion. (ii) Note that gr Qp ,1 ≤ g Qp ,1 (cf. [WiXi2]). So gr can be treated as a bounded family in the dual space of Rp,2 under ·, ·inv . Furthermore, via the Banach–Alaoglu Theorem, there exists an increasing sequence {rn } in (0, 1) and a function h ∈ Qp such that lim rn = 1

n→∞

and

lim f, grn inv = f, hinv ,

n→∞

f ∈ Rp,2 .

Taking f (z) = z j , j ≥ 0, we get h = g.

Now we are in a position to show that Rp,2 or Pp is another description of either Rp,1 , and conversely, Rp,1 suggests a sort of decomposition theorem for Pp or Rp,2 . Theorem 4.5.5. Given p ∈ (0, 1). Then: (i) Rp,2 is isomorphic to the dual of Qp,0 under ·, ·inv . (ii) Rp,2 = Rp,1 = Pp . (iii) The second dual of Qp,0 is isomorphic to Qp under ·, ·inv .

4.6. Notes

103

Proof. Since (ii) and (iii) follow from (i), Theorems 4.5.3, 4.5.2 and 4.4.1, we just ¯ check (i). As in Lemma 4.5.4 (ii), we deﬁne gw (z) = g(wz) for g ∈ Qp and w ∈ D and then get gw ∈ Qp with gw Qp ,1 ≤ g Qp ,1 , but also gw ∈ Qp,0 for every w ∈ D. For the sake of convenience, denote by BQp and BQp,0 the closed unit balls in Qp and Qp,0 respectively. Then sup |f, ginv | = g∈BQp

sup |f, ginv |,

f ∈ Rp,2 .

g∈BQp,0

Indeed, suppose f ∈ Rp,2 . Then by Lemma 4.5.4 (ii) we have that if g ∈ Qp , then sup |f, gw inv | = sup |f, gζ inv |. ¯ ζ∈D

w∈D

With this, we further get sup |f, ginv | = g∈BQp

≤

sup sup |f, gw inv |

¯ g∈BQp w∈D

sup sup |f, gw inv | g∈BQp w∈D

≤

sup |f, ginv | g∈BQp,0

≤

sup |f, ginv |, g∈BQp

thereupon verifying the above-desired equation. This equation, along with the estimate in Theorem 4.5.3, yields

f Rp,2

sup |f, ginv | f Rp,2 ,

f ∈ Rp,2 ,

g∈BQp,0

which in turn tells us that Rp,2 is isomorphic to a subspace of the invariant dual space of Qp,0 . Using Lemma 4.5.4 (i) we derive that Rp,2 is isomorphic to the dual space of Qp,0 under ·, , ·inv .

4.6 Notes Note 4.6.1. Section 4.1 may be regarded as an expanded version of [Xi4, Subsection 2.1]. Even though, we would like to make a few remarks on this section. The properties (i)–(ii)–(iii) of Lemma 4.1.1 are elementary, and Lemma 4.1.1 (iii) is the unit circle version of [ChaDXi, Theorem (i)] that sharpens [OrVe, Lemma 3]. Lemma 4.1.2 is the circular variation of [Mel, Lemmas 1–2], and its proof is a modiﬁcation of the argument for [OrVe, Lemma 2]. Theorem 4.1.3 is the counterpart of [OrVe, Theorem] which extends [Ad2, Theorem A]. Theorem 4.1.4 (iv) is a new contribution to [Xi4, Section 2.1]. In the proof of Theorem 4.1.4 (i)⇒(ii) we have consulted [AhJe].

104

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

Note 4.6.2. Section 4.2 is an outcome of understanding the circular version of [Ad2, Section 3] and [Xi4, Theorem 2.2]. Motivated by Theorem 4.2.1 we may pose a natural predual problem as follows. For p ∈ (0, 1), let L1,p f un (T) be the Morrey space of complex-valued functions f on T such that sup |I|−p |f (ζ) − fI ||dζ| < ∞. I⊆T

I

What is the predual of under T g(ζ)f (ζ)|dζ| ? Note 4.6.3. Section 4.3 is a by-product of modifying [Xi4, Theorem 2.3] and transferring [DaXi, Section 5] to the unit circle. See also [AlvJMi] and [CoiMeSt] for some related topics. Note 4.6.4. Section 4.4 is basically taken from [Xi4, Section 2.3]. From the formula zf (z) g (z)(1 − |z|2 )dm(z) f, ginv = L1,p f un (T)

D

it follows readily that the extremal holomorphic Besov space B1 = B11,1 comprising all f ∈ H with D

|(zf (z)) |dm(z) < ∞

has the Bloch space B as the invariant dual; see also [ArFi]and [ArFiPe1]. Of ∞ course, B1 ⊂ H — because f ∈ B1 has a representation f = j=1 λj σzj (z) and |σzj (ζ)||dζ| = |dσzj (ζ)| = 2π, T

∞

T

where j=1 |λj | < ∞ and {zj } is a sequence in D. In the proof of Lemma 4.4.2 we have borrowed ideas used in [Ad3, Theorem 2.2], [Ad4, p. 33], [Pa1, Theorem 7.1.8], [Or, Lemma 1.1] and [Kr, Theorems 2.1– 2.2]. Similarly, we can prove that if f (η) Jp f (ζ) = |dη|, p ∈ (0, 1), |ζ − η|p T then

T

f ∈ H1 .

p M(Jp f )dH∞ f H1 ,

In the very ﬁrst part of the proof of Theorem 4.4.3 we have used a trick from [Bae2, p. 27]. Additionally, it is worth mentioning the geometric nature of Theorem 4.4.3. To be more speciﬁc, let f be a conformal map of D onto a domain Ω ⊂ C with the boundary ∂Ω. Then it follows from Theorem 4.4.3 that 12 12 2 |f | dm f Pp |f (ζ)||dζ| = Length(∂Ω), Area(Ω) = D

T

improving the well-known isoperimetric inequality without sharp constant.

4.6. Notes

105

Note 4.6.5. Section 4.5 is established via a modiﬁcation of the main techniques used in [PaXi] and [WiXi2]. Lemma 4.5.1 is crucial and can be also proved via Schur’s Lemma. In fact, it is enough to show the following boundedness: 2 p 2 2−p 2 |1 − a ¯z| (1 − |w| ) |h(w)|K(z, w)dm(w) dm(z) D D 2 2−p |1 − a ¯z|p (1 − |z|2 ) 2 |h(z)| dm(z), D

where the constant does not depend on a, and

p p 2 2−p 1−a 2 (1 − |w|2 ) 2 ¯ z (1 − |z| ) K(z, w) = . 1−a ¯w |1 − wz| ¯ 3 Selecting numbers γ, β such that p < β < 1 and −1 − p2 < γ < −β − p2 , we do some basic integral estimates with the change of variables: w = σz (λ) and z = σw (λ) to deduce K(z, w)(1 − |w|2 )γ |1 − a ¯w|β dm(w) (1 − |z|2 )γ |1 − a ¯z|β , z ∈ D D

and

D

K(z, w)(1 − |z|2 )γ |1 − a ¯z|β dm(w) (1 − |w|2 )γ |1 − a ¯w|β

w ∈ D.

These inequalities justify the required conditions of Schur’s Lemma and then give the desired boundedness. extend to p ∈ [1, 2). Using the Cauchy–Schwarz The deﬁnition of Rp,1 can inequality, we derive that if f = n fn ∈ Rp,1 , p > 1, then f ∈ B1 with 12 ∞ 2 p 2 −2 |(zf (z)) |dm(z) ≤ San (fn ) (1 − |σan (z)| ) (1 − |z| ) dm(z) D

n=1 ∞

D

San (fn ).

n=1

obius-invariant space. So it is our conjecture that Note that B1 is the minimal M¨ Rp,1 = B1 for p ∈ (1, 2) and R1,1 = H with equivalent seminorms. The key issue to settle this conjecture seems to see whether or not Lemma 4.5.1 allows an extension from p ∈ (0, 1) to p ∈ [1, 2). Nevertheless, we can employ the characterization of Qp,0 in terms of the second order derivative (cf. [AuNoZh]) and the methods of proving Lemma 4.5.1 and Theorem 4.5.2 to show that Rp,1 , p ∈ (0, 1) consists of ∞ those f ∈ H for which f = n=1 fn , fn ∈ H is convergent uniformly on compacta of D, and is equipped with the seminorm 12 ∞ 2 2 −p |fn (z)| (1 − |σan (z)| ) dm(z) < ∞, inf n=1

D

106

Chapter 4. Invariant Preduality through Hausdorﬀ Capacity

where the inﬁmum ranges over all above-given representations f . If VMOA and B0 denote the spaces of all f ∈ H with lim|w|→1 E1 (f, w) = 0 and lim|w|→1 (1 − |w|2 )|f (w)| = 0 respectively, then it is known that the invariant duals of these two spaces are isomorphic to H and B1 respectively; see also [ArFiPe1]. Additionally, [Wu4] has claimed that the ﬁrst and second duals of a Qp,0 -type space are found under an appropriate pairing.

Chapter 5

Cauchy Pairing with Expressions and Extremities We have seen from the previous chapter that each holomorphic Q class exists as a dual space with respect to the invariant pairing ·, ·inv , however the dual space may have diﬀerent preduals. In this chapter, we introduce another dual pairing, which is variant under Aut(D), but allows us to handle all the cases of the preduals and duals for the holomorphic Q classes. This pairing is the so-called Cauchy pairing: f, g = lim

r→1

f (rζ)g(rζ)|dζ|. T

One more advantage of the Cauchy pairing is that it enables us to settle some natural questions on the expressions and extremes of the related holomorphic function spaces. The further details distribute the following sections: • • • •

Background on Cauchy Pairing; Cauchy Duality by Dot Product; Atom-like Representations; Extreme Points of Unit Balls.

5.1 Background on Cauchy Pairing Although we intend later to work almost exclusively in the Cauchy pairing attached to Qp , it is most convenient to start abstractly. Suppose X ⊆ H is a Banach space equipped with the norm · X and contains ¯ (comprising all holomorphic functions on a neighborhood of D) ¯ as a dense H(D) ∗ subspace. If X stands for the dual space of X — that is — the space of all complex-valued continuous/bounded linear functionals on X , then the following

108

Chapter 5. Cauchy pairing with Expressions and Extremities

action with respect to the running variable w: Θ(L)(z) = L (1 − z¯w)−1 , L ∈ X ∗ , z ∈ D, deﬁnes a one-to-one mapping from X ∗ into H, and hence this induces an isometric copy: Θ(X ∗ ) of X ∗ , called the Cauchy dual of X , via the norm

Θ(L) Θ(X ∗) = L X ∗ = sup{|L(f )| : f X ≤ 1}. This concept comes from the fact that any L ∈ X ∗ can be written as a Cauchy pairing between f ∈ X and g = Θ(L): −1 −1 L(f ) = (2π) f, g = f (0)g(0) + π lim f (rz)g (rz)(− log |z|2 )dm(z). r→1

D

This is actually a derivative representation of the Cauchy pairing, and reveals that the Cauchy pairing is not invariant under Aut(D); that is, f ◦ σw , g ◦ σw = f, g unless w = 0. According to the Cauchy pairing, we naturally ﬁnd out Θ(D∗ ) ∼ = A2,0 and Θ (A2,0 )∗ ∼ = D. At the same time, Θ(VMOA∗ ) ∼ = BMOA = H1 and Θ (H1 )∗ ∼ are the well-known Feﬀerman–Sarason’s dualities. In addition, for q ∈ [1, ∞) and β ∈ (−1, ∞) let Dq,β = {f ∈ H : f Dq,β = |f (0)| + f Aq,β < ∞} be the Dirichlet type space, then we reach Anderson–Clunie–Pommerenke’s dualities (cf. [AnClPo]): Θ(B0∗ ) ∼ = B. = D1,0 and Θ (D1,0 )∗ ∼ In the above and the below, the symbol “∼ =” means “isomorphic”. The preceding examples lead to the following general result. ¯ ⊆ X ⊆ H. Then Theorem 5.1.1. Let (X , · X ) be a Banach space with H(D) ∗ ¯ and H. Moreover, if H(D) ¯ is dense in Θ(X ∗ ), then Θ(X ) lies between H( D) ∗ comprises f ∈ H for which there is a sequence {fj } in X with Θ Θ(X ∗ ) supj∈N fj X < ∞ and fj (z) → f (z) for each z ∈ D. Consequently,

f Θ

Θ(X ∗ )

∗ = f X ∗∗ = inf{lim sup fj X },

where the inﬁmum ranges over all sequences {fj } described above.

5.1. Background on Cauchy Pairing

109

Proof. For the ﬁrst assertion, we consider Lj ∈ X ∗ which is determined by Lj (f ) = f (j) (0)/j!,

j ∈ N ∪ {0}, f ∈ X ;

we then obtain |Lj (f )| ≤ sup r−j |f (z)| r−j f X , |z|=r

r ∈ (0, 1),

−1

¯ with the Taylor exhence implying lim supj Lj jX ∗ ≤ 1. Assuming g ∈ H(D) ∞ ∞ j pansion g(z) = j=0 aj z , we derive that j=0 aj Lj converges in X ∗ . Note that Θ(Lj ) = z j . So ⎞ ⎛ ∞ ∞ g= aj Θ(Lj ) = Θ ⎝ aj Lj ⎠ ∈ Θ(X ∗ ). j=0

j=0

On the other hand, since supz∈K (1− z¯w)−1 X < ∞ holds for any compact subset K of D, we conclude Θ(X ∗ ) ⊆ H. ∗ ¯ Regarding the second assertion, suppose H(D) is dense in Θ(X ). Since∗∗Θ ∗ ∗ is isometrically isomorphic to X , is injective, we conclude that Θ Θ(X ) the second dual of X , but also the canonical imbedding X → X ∗∗ gives X ⊆ ∗ ∗ ∗ ∗ Θ Θ(X ) . Now for f ∈ Θ Θ(X ) , we can use the hypothesis to ﬁnd a sequence {fj } such that it is convergent to f in the topology of weak-star and

fj X ≤ f Θ

Θ(X ∗ )

∗ .

Consequently, f (z) =

(2π)−1 f, (1 − z¯w)−1

=

(2π)−1 lim fj , (1 − z¯w)−1

=

lim fj (z),

j→∞

j→∞

z ∈ D.

Meanwhile, if f ∈ H and if {fj } is a bounded sequence in X which converges ∗ pointwisely to f on D, then each functional Lfj (Φ) = Φ(fj ) belongs to Θ(X ∗ ) and satisﬁes Lfj (1 − z¯w)−1 = fj (z) → f (z), z ∈ D. ¯ in Θ(X ∗ ), yields that {Lfj } is convergent This, together with the density of H(D) ∗ ∗ weak-star to a functional L ∈ Θ(X ) with L (1 − z¯w)−1 = f (z), z ∈ D and L

Θ(X ∗ )

as desired.

∗ ≤ lim sup fj X , j→∞

110

Chapter 5. Cauchy pairing with Expressions and Extremities

In many situations, we identify a Banach space X isomorphically with an accessible sequence space, say q , q ∈ [1, ∞) and c0 , in order to ﬁgure out the Cauchy dual Θ(X ∗ ) of X . More precisely, let Y be one of these sequence spaces and Φ : X → Y be bounded from above and below. Then there is a sequence {Lj } in X ∗ such that Φ(f ) = {Lj (f )} for any f ∈ X , and hence the adjoint operator Φ∗ : Y ∗ → X ∗ , given by Φ∗ (sj ) = j sj Lj , is bounded and onto. Of course, L ∈ X ∗ can be expressed as L= sj Lj with {sj } ∈ Y ∗ and L X ∗ ≈ inf {sj } Y ∗ j

where the inﬁmum ranges over all sequences {sj } ∈ Y ∗ with the foregoing representation. This amounts to saying that f ∈ Θ(X ∗ ) can be written as f= sj fj with {sj } ∈ Y ∗ , fj = Θ(Lj ) and f Θ(X ∗) ≈ inf {sj } Y ∗ , j

where the inﬁmum ranges over all sequences {sj } ∈ Y ∗ with the last representation. To gain a solid understanding of the previous principle, let us give one more example which will be used later on. Example 5.1.2. Given α ∈ (0, 1), let A−α and A−α be the spaces of f ∈ H with 0

f A−α = sup (1 − |λ|2 )α |f (z)| < ∞ and z∈D

lim (1 − |λ|2 )α |f (z)| = 0,

|z|→1

respectively. Then ∗ ∼ Θ (A−α = D1,−α and Θ (D1,−α )∗ ∼ = A−α . 0 ) If W = {wj } is a discrete subset of D such that inf |z − wj | < κ(1 − |z|), z ∈ D

wj ∈W

holds for a suﬃciently small constant κ ∈ (0, 1), then: (i) f A−α ≈ sup (1 − |wj |)α |f (wj )|, wj ∈W

f ∈ A−α .

(ii) The mapping Φ : f → {(1 − |wj |)α f (wj )}wj ∈W is an isomorphism between A−α and a closed subspace of c0 . 0 (iii) Any element f ∈ D1,−α can be written as cλ (1 − |wj |)α (1 − wj z)−1 with f (z) = wj ∈W

f D1,−α ≈ inf

|cwj |,

wj ∈W

where the inﬁmum ranges over all sequences for the representation of f above.

5.1. Background on Cauchy Pairing

111

Proof. For the Cauchy duality relations, we ﬁrst prove Θ (D1,−α )∗ ∼ = A−α . On −α then the one hand, if g ∈ A sup (1 − |z|)1+α |g (z)| g A−α

z∈D

and hence g produces an element of Θ (D1,−α )∗ thanks to the above-mentioned formula of f, g involving the derivatives of f and g, plus the following estimate f (rz)g (rz)(− log |z|2 )dm(z) D |f (rz)|(− log |z|2 )−α dm(z) ≤ sup (− log |z|2 )1+α |g (z)| D

z∈D

g A−α f D1,−α . Now, suppose L ∈ Θ (D1,−α )∗ . Since f ∈ D1,−α can be approximated by fr in norm, i.e., limr→1 fr − f D1,−α = 0, we can consider ¯ −1 |dζ|. fr (z) = f (rz) = (2π)−1 fr (ζ)(1 − z ζ) T

Clearly, we have −1

lim L(fr ) = lim (2π)

r→1

r→1

T

¯ −1 |dζ|, fr (ζ)L (1 − zrζ)

where L is treated as acting with respect to the running variable z. Since L is bounded, we conclude that for w ∈ D, L (1 − wz) ¯ −1 ≤ L

Θ (D 1,−α )

(1 − wz) ¯ −1 D1,−α ∗

L

Θ (D 1,−α )∗

(1 − |w|)α

.

This yields g(w) = (2π)−1 L (1 − wz) ¯ −1 ∈ A−α with L(f ) = lim L(fr ) = (2π)−1 f, g. r→1

∗ ∼ Next, we give proof for Θ (A−α = D1,−α . If g ∈ D1,−α then it produces 0 ) a−α ∗ ∗ an element of Θ (A0 ) as proved above. Conversely, suppose L ∈ Θ (A−α 0 ) . with fj A−α = 1. For each j ∈ N ∪ {0} let fj (z) = z j , z ∈ D. Then fj ∈ A−α 0 Assuming bj = L(fj ), we derive |bj | ≤ L

∗ Θ (A−α 0 )

fj A−α = L

∗ Θ (A−α 0 )

,

∞ yielding that g(z) = j=0 bj z j has a radius of convergence greater than or equal to 1 and hence is holomorphic on D. Furthermore, we show g ∈ D1,−α with g D1,−α L

∗ Θ (A−α 0 )

.

112

Chapter 5. Cauchy pairing with Expressions and Extremities

To this end, we observe that gr (z) = g(rz) belongs to D1,−α for each r ∈ (0, 1). So, using the previous duality argument we have

gr D1,−α sup

(2π)−1 |g , f | r : f ∈ A−α \ {0} .

f A−α

∗ −α j Since L ∈ Θ (A−α with f (z) = ∞ 0 ) , we conclude that if f ∈ A j=0 aj z , then

fr A−α ≤ f A−α and L(fr ) =

∞

aj bj rj = (2π)−1 f, gr .

j=0

Accordingly, (2π)−1 |gr , f | = (2π)−1 |f, gr | = |L(fr )| ≤ L

∗ Θ (A−α 0 )

f A−α .

This, along with the above norm estimate on gr , gives

gr D1,−α L

∗ Θ (A−α 0 )

.

An application of Fatou’s Lemma to the last inequality yields g ∈ D1,−α with g D1,−α L

∗ Θ (A−α 0 )

.

In the meantime, f ∈ A−α is equivalent to limr→1 f − fr A−α = 0. Therefore, 0 L(f ) = lim L(fr2 ) = lim r→1

r→1

∞

aj bj r2j = (2π)−1 f, g, f ∈ A−α 0 ,

j=0

as desired. The assertions (i) and (ii) can be veriﬁed by the following two implications for a suﬃciently small κ ∈ (0, 1/2) : |z − wj | < κ(1 − |z|) ⇒ 1 − κ < (1 − |wj |)(1 − |z|)−1 < 1 + κ and |σwj (z)| < κ ⇒ |(1 − |z|)α f (z) − (1 − |wj |)α f (wj )| f A−α |σwj (z)|. Of course, (iii) is a straightforward consequence of (i), (ii), and an application of the foregoing general principle to the Cauchy dualities for D1,−α , A−α , and A−α 0 .

5.2. Cauchy Duality by Dot Product

113

5.2 Cauchy Duality by Dot Product However, a careful look at the invariant dual space of Qp,0 , p ∈ (0, 1) (see e.g., Theorem 4.5.2) reveals that it seems more natural to consider certain vector-valued sequence spaces than the Carleson measure type sequence spaces in Theorem 3.4.2. In so doing, recall that given a Banach space Y over C, q (Y) (where q ∈ [1, ∞)) and c0 (Y) are the spaces of sequences { yj Y } that belong to q and c0 respectively. Of course, their norms are deﬁned respectively by

{yj } q (Y) =

yj qY

q−1

and {yj } c0 (Y) = sup yj Y .

j

j

Classically, the dual of c0 (Y) is 1 (Y ∗ ), namely, to every continuous linear functional Λ on c0 (Y) there corresponds a unique sequence {sj } ∈ 1 (Y ∗ ) such that sj (yj ), {yj } ∈ c0 (Y) with Λ (c0(Y))∗ =

sj Y ∗ . Λ({yj }) = j

j

If again X is the given space in Theorem 5.1.1, if Φ is an isomorphism from X onto some closed subspace of c0 (Y) and Φj expresses the projection of Φ on the j-th coordinate, then L ∈ X ∗ can be written as sj Φj (f ) , {sj } ∈ 1 (Y ∗ ) with L X ∗ ≈ inf

sj Y ∗ , L(f ) = j

j

where the inﬁmum ranges over all sequences {sj } obeying the form of L(f ) represented above. Theorem 5.2.1. Given p ∈ (0, ∞), let W = {wj } be a discrete subset of D such that there exists a constant κ ∈ (0, 1) ensuring inf j |z − wj | < κ(1 − |z|), z ∈ D. For q > 0, j ∈ N ∪ {0}, and f ∈ Qp,0 set z q f (w) Φq,0 f (z) = f (0), Φq,j f (z) = (1 − |wj |2 ) 2 j ∈ N. p+q dw, 0 (1 − wj w) 2 Then: (i) Φq (f ) = {Φq,j (f )} is an isomorphism from Qp,0 onto a closed subspace of c0 (D2,p ). ∞ (ii) Each element g ∈ Θ(Q∗p,0 ) can be represented as g = a0 + j=1 aj gj where {aj } ∈ 1 and q hj (w)(− log |w|2 ) z(1 − |wj |2 ) 2 dm(w) with hj D2,2−p ≈ 1. gj (z) = p+q π ¯ 2 (1 − wz) ¯ 2 D (1 − wj w) In this case

g Θ(Q∗p,0 ) ≈ inf {aj } 1 ,

114

Chapter 5. Cauchy pairing with Expressions and Extremities where the inﬁmum is taken over all complex-valued sequences {aj } satisfying the previously-mentioned representation of g.

Proof. (i) This follows from the vanishing version of Lemma 3.1.1, the deﬁnition of Qp,0 and the following implication for z ∈ D: 1 − w z |wj − w| 1 − |w| |wj − w| j <κ⇒ < (1 − κ)−1 and < 1 + κ. ≤1+ 1 − |w| 1 − |wj | 1 − wz ¯ |1 − wz| ¯ (ii) By the above-stated general duality principle we see that any L ∈ Q∗p,0 can be written as ∞ L(f ) = a0 f (0) + aj sj Φq,j (f ) , j=1

where

sj ∈ (D2,p )∗ with sj (D2,p )∗ = 1 and {aj } 1 < ∞.

Note that to each sj ∈ (D2,p )∗ with sj (D2,p )∗ = 1 there corresponds a function hj ∈ D2,2−p such that hj D2,2−p ≈ 1 and g (z)hj (z)(− log |z|2 )dm(z), g ∈ D2,p . sj (g) = g(0)hj (0) + π −1 D

This fact, along with a routine calculation with L (1 − z¯w) , yields the desired result. Obviously, Theorem 5.2.1 is not good enough from a function-theoretic viewpoint. In what follows we characterize the Cauchy dual of Qp,0 in terms of the so-called weak factorization. Given two Banach spaces U, V ⊆ H, let the dot prod∞ uct U V be the class of all functions f ∈ H of the form f (z) = j=1 uj (z)vj (z), z ∈ D, where uj ∈ U, vj ∈ V, and the sum converges on compacta of D. U V becomes a Banach space equipped with the norm

f U V = inf

∞ j=1

uj U vj V : f =

∞

uj vj .

j=1

¯ is dense in both U and V so is it in U V. It is easy to see that if H(D) To identify the Cauchy dual of a vanishing Q class with a dot product, we need a more dedicate treatment for the integrals used in Theorem 5.2.1. Lemma 5.2.2. Given α ∈ [0, ∞), β ∈ (−1, ∞) and −1 < γ < 2(1 + β), let K ∈ H ∞ be such that K(z) = j=0 kj z j with |kj − kj−1 | j α for j ∈ N. Deﬁne ¯ 1 − λz K(z w)(− ¯ log |w|2 )β f (w)dm(w), λ ∈ D, f ∈ D2,γ . Hλ,K f (z) = ¯ D 1 − λw If δ = γ "− 2(β − α) > −1, then Hλ,K is a bounded operator from D2,γ to D2,δ with supλ∈D "Hλ,K D2,γ →D2,δ < ∞.

5.2. Cauchy Duality by Dot Product

115

∞ Proof. Case 1: δ > 1. With this condition, we have that if f (z) = j=0 aj z j ∈ D2,δ , then ∞

f 2D2,δ ≈ |f (z)|2 (1 − |z|2 )δ−2 dm(z) ≈ (1 + j)1−δ |aj |2 . D

j=0 2,γ

Since all polynomials are dense in D , it suﬃces to demonstrate that there is a constant C > 0 independent of λ ∈ D such that Hλ,K f D2,δ ≤ C f D2,γ holds for any polynomial f . To do so, note that Hλ,K f D2,δ is a subharmonic ¯ Thus it is enough to consider function of λ ∈ D and extendable continuously to D.

H1,K f D2,δ thanks to ¯ Hλ,K f (z) = H1,K fλ (λz) where λ ∈ T and fλ (z) = f (λz). To that end, noticing cj,β = 2π

1

0

we get that if f (z) =

r2j (− log r2 )β dr ≈ (j + 1)−(1+β) , j ∈ N ∪ {0},

∞ j=0

aj z j is a polynomial, then

H1,K f (z) = (1 − z)

∞

cn,β an

n=0

where F1 (z) =

∞

(kj − kj−1 )z j

j=1

n

kj z j = F1 (z) + F2 (z) + F3 (z),

j=0 ∞

cn,β an , F2 (z) = k0

n=j

and F3 (z) = −

∞

∞

cj,β aj ,

j=0

kj z j+1 cj,β aj .

j=0

Taking η > γ − 1, ζ > 1 and η + ζ = 2(β − 1), we employ the Cauchy–Schwarz inequality, the estimate of cj,β and the assumption on K to produce ∞ ∞ 2 |kj − kj−1 |2

F1 2D2,δ c a n,β n δ−1 (j + 1) j=1 n=j ⎞ ⎛ ∞ ∞ ∞ |kj − kj−1 |2 ⎝ |an |2 1 ⎠ δ−1 η ζ (j + 1) (n + 1) (n + 1) j=1 n=j n=j ∞ ∞ |kj − kj−1 |2 1−ζ |an |2 j (j + 1)δ−1 (n + 1)η j=1 n=j

∞

n |an |2 (j + 1)2+2α−ζ−δ η (n + 1) n=1 j=1

f 2D2,γ .

116

Chapter 5. Cauchy pairing with Expressions and Extremities

Owing to −1 < γ < 2(1 + β), the Cauchy–Schwarz inequality yields |F2 (z)|2 ≤ |k0 |2

∞

c2j,β (1 + j)δ−1

j=0

∞

(1 + j)1−δ |aj |2 |k0 |2 f 2D2,δ .

j=0

Furthermore, the condition on K gives |kj | (1 + j)1+α for j ∈ {0} ∪ N and thus

F3 2D2,δ

∞

c2j,β (1 + j)1−δ |kj |2 |aj |2 f 2D2,γ .

j=0

The above three estimates imply H1,K f D2,δ f 2D2,γ . Case 2: δ ∈ (−1, 1]. Under this condition, we deﬁne K1 (z) = zK (z) =

∞

kj,1 z j

and K2 (z) = z 2 K (z) =

j=0

∞

kj,2 z j .

j=0

Using the assumption on K once again, we derive |kj,1 − kj−1,1 | j 1+α and |kj,2 − kj−1,2 | j 2+α for j ∈ N, and so that if f ∈ D2,γ , then

Hλ,K f 2D2,δ

≈

|Hλ,K f (0)|2 + (Hλ,K f ) 2D2,2+δ

f 2D2,γ + |(Hλ,K f ) (z)|2 (1 − |z|2 )2+δ dm(z) D 2

f D2,γ + |Hλ,K1 f (z)|2 (1 − |z|2 )δ dm(z) D + |Hλ,K2 f (z)|2 (1 − |z|2 )2+δ dm(z) D

f 2D2,γ + Hλ,K1 f 2D2,2+δ + Hλ,K2 f 2D2,4+δ

f 2D2,γ ,

as desired.

Now we are ready to establish the following factorization theorem for the Cauchy dual of a vanishing Q class. Theorem 5.2.3. Given p ∈ (0, 2), let W = {wj } be a discrete subset of D such that there exists a constant κ ∈ (0, 1) ensuring inf j |z − wj | < κ(1 − |z|) for any z ∈ D. If κ > 0 is small enough, then: D2,2−p . Equivalently, f ∈ D1, 2 D2,2−p if and only (i) Θ(Q∗p,0 ) ∼ = D1,− 2 if (1 − |wj |2 )1− p2 vj , f Θ(Q∗p,0 ) ≈ {cj } 1 < ∞, cj f= 1 − wj z j 2−p

where vj ∈ D2,2−p with vj D2,2−p = 1.

p−2

5.2. Cauchy Duality by Dot Product

117

∗ ∼ (ii) The polynomials are dense in Θ(Q∗p,0 ) and so Θ Θ(Q∗p,0 ) = Qp . 2−p

Proof. (i) Suppose f ∈ Qp,0 and g ∈ D1,− 2 D2,2−p . So to prove that ·, g generates a continuous linear functional on Qp,0 , we just need to check that |f (z)(uv) (z)|(− log |z|2 )dm(z) f Qp ,1 u 1,− 2−p v D2,2−p D

D

2

2−p

for any u ∈ D1,− 2 and v ∈ D2,2−p . Nevertheless, Example 5.1.2 tells us that it is enough to demonstrate that the last inequality holds for p

u(z) = (1 − |wj |)1− 2 (1 − wj z)−1 , as κ > 0 is suﬃciently small. This can be done by handling the following two integrals: p |f (z)|(1 − |wj |)1− 2 |1 − wj z|−2 |v(z)|(− log |z|2 )dm(z) I1 (f ) = D

and

I2 (f ) =

D

p

|f (z)|(1 − |wj |)1− 2 |1 − wj z|−1 |v (z)|(− log |z|2 )dm(z).

In order to control I1 (z), we employ Lemma 3.1.1 with dµf,p (z) = |f (z)|2 (1 − |z|2 )p dm(z) ∈ CMp , the Cauchy–Schwarz inequality and the fact that (1 − |z|2 ) and (− log |z|2 ) give equivalent Bergman spaces to get I1 (f )

12 12 (1 − |wj |2 )2−p 2 2 −p dµf,p (z) |v(w)| (1 − |z| ) dm(z) |1 − wj z|2 D D

f Qp ,1 v D2,2−p .

Similarly, for I2 (f ) we have I2 (f )

12 12 (1 − |wj |2 )2−p 2 2 2−p dµf,p (z) |v (w)| (1 − |z| ) dm(z) |1 − wj z|2 D D

f Qp ,1 v D2,2−p .

Conversely, assuming g ∈ Θ(Q∗p,0 ), we can use Theorem 5.2.1 (ii) to obtain ∞ g = a0 + j=1 aj gj where {aj } 1 g Θ(Q∗p,0 ) and p

gj (z) =

z(1 − |wj |2 )1− 2 π

D

hj (w)(− log |w|2 ) (1 − wj w) ¯

p+q 2

(1 − wz) ¯ 2

dm(w)

118

Chapter 5. Cauchy pairing with Expressions and Extremities

with hj D2,2−p ≈ 1. Clearly, if K(z) = (1 − z)−2 , α = 0, β = 1, γ = 4 − p in Lemma 5.2.2, then p

z(1 − |wj |2 )1− 2 Hwj ,K hj (z) gj (z) = π(1 − wj z) and hence g ∈ D1,−

2−p 2

D2,2−p because of Lemma 5.2.2 with

2−p

D

p−2 (1 − |z|2 )− 2 dm(z) (1 − |wj |) 2 and Hwj ,K hj D2,2−p hj D2,2−p . 2 |1 − wj z| 2−p

(ii) Since the polynomials are dense in D1,− 2 D2,2−p and Qp,0 , we conclude from (i) and Theorem 5.1.1 that the desired assertion is true. Corollary 5.2.4. Given p ∈ (0, 2). Then: (i) Θ(Q∗p1 ,0 ) ⊆ Θ(Q∗p2 ,0 ) for 0 < p2 ≤ p1 < 2. In particular, Θ(Q∗1,0 ) ∼ = H1 and ∗ 1,0 ∼ Θ(Qp,0 ) = D for all p ∈ (1, 2). (ii) f ◦ σw Θ(Q∗p,0 ) ≈ f Θ(Q∗p,0 ) for f ∈ Θ(Q∗p,0 ) and w ∈ D. Proof. (i) and (ii) can be demonstrated by using Theorem 5.2.3 with some simple 2−p calculations with D1,− 2 D2,2−p .

5.3 Atom-like Representations For the dyadic closed intervals [2πk2−n , 2π(k + 1)2−n ] we write Sn,k for the corresponding Carleson boxes. As before, we have that for p ∈ (0, ∞) and f ∈ H, −np sup 2 |f (z)|2 (1 − |z|2 )p dm(z) < ∞ f ∈ Qp ⇔ 0≤k≤2n ,n∈N

Sn,k

and sup 2−np

f ∈ Qp,0 ⇔ lim

n→∞ 0≤k≤2n

|f (z)|2 (1 − |z|2 )p dm(z) = 0. Sn,k

However, if p ∈ (0, 1), then for f ∈ H, f ∈ Qp ⇔

|||f |||2Qp ,3

−p

sup |I|

=

I⊆[0,2π]

and f ∈ Qp,0 ⇔ lim sup |I|−p |I|→0

I

I

I

I

|f (eit ) − f (eis )|2 dtds < ∞ |eit − eis |2−p

|f (eit ) − f (eis )|2 dtds = 0. |eit − eis |2−p

Here |I| = b − a if I is an interval with endpoints a and b.

5.3. Atom-like Representations

119

Lemma 5.3.1. Given p ∈ (0, 1) and f ∈ H. Then there is a sequence {Ij } of intervals contained in [0, 2π] such that limj→∞ |Ij | = 0 and f ∈ Qp ⇔ sup |Ij |−p

j∈N

Ij

and f ∈ Qp,0 ⇔ lim sup |Ij |−p

|f (eit ) − f (eis )|2 dtds < ∞ |eit − eis |2−p

Ij

j→∞

Ij

Ij

|f (eit ) − f (eis )|2 dtds = 0. |eit − eis |2−p

Proof. Without loss of generality, we may assume f (0) = 0. It suﬃces to prove the ﬁrst equivalence. To do so, for f ∈ Qp and 0 ≤ a < b ≤ 2π let −p

b

Ff (a, b) = (b − a)

a

a

b

|f (eit ) − f (eis )|2 dtds. |eit − eis |2−p

We are about to show that the family of functions F = {Ff : f ∈ Qp , |||f |||Qp ,3 = 1} is equicontinuous on any δ-triangle ∆δ = {(a, b) : 0 ≤ a ≤ b − δ < b ≤ 2π}. By so doing, we observe that for (a1 , b), (a2 , b) ∈ ∆δ , a1 < a2 the following estimate holds: b − a p 1 1− |Ff (a1 , b) − Ff (a2 , b)| ≤ Ff (a1 , b) b − a2 2 a2 b |f (eit ) − f (eis )|2 + p dtds δ a1 a1 |eit − eis |2−p = T1 + T2 . Evidently, if a2 is so close to a1 that (a1 , b), (a2 , b) ∈ ∆δ , then T1 tends to 0 uniformly for f ∈ Qp with |||f |||Qp ,3 = 1. To handle T2 , we choose b2 ∈ (a2 , b) and assume

b2

b2

J1 = a1

a1

|f (eit ) − f (eis )|2 dtds and J2 = |eit − eis |2−p

a2

a1

b

b2

|f (eit ) − f (eis )|2 dtds. |eit − eis |2−p

Clearly, J1 ≤ (b2 − a1 )p . To estimate J2 , we note that there is a constant κ > 0 such that |eit − eis | ≥ κ(b2 − a2 ), (t, s) ∈ [b2 , b] × [a1 , a2 ].

120

Chapter 5. Cauchy pairing with Expressions and Extremities

Using q > 1, the H¨ older inequality and the inclusion Qp ⊂ BMOA ⊂ Lq (T), we derive J2

≤

q−1 2π 2π q1 (a2 − a1 )(b − b2 ) q it is 2q |f (e ) − f (e )| dtds 2−p 0 0 κ(b2 − a2 ) q−1 (a2 − a1 )(b − b2 ) q |||f |||2Qp ,3 . 2−p κ(b2 − a2 ) q−1

Accordingly, if a2 tends to a1 with (a2 − a1 ) q (b2 − a2 )2−p → 0 for a suitable b2 , then J1 and J2 approach 0 uniformly on the unit sphere of (Qp , ||| · |||Qp ,3 ). The equicontinuity in the other variable can be veriﬁed similarly. To close the proof, we employ a compactness argument to ﬁnd that for each k ∈ N there are a δk and an Sk ⊂ ∆δk+1 \ ∆δk such that inf

(x,y)∈Sk

|Ff (a, b) − Ff (x, y)| < 3−1 , (a, b) ∈ ∆δk , |||f |||Qp ,3 = 1.

Now, enumerating the collection of intervals with endpoints in ∪Sk , we obtain a sequence of intervals Ij = (aj , bj ) such that |Ij | = bj − aj → 0 and inf |Ff (a, b) − Ff (aj , bj )| < 2−1 |||f |||2Qp ,3 , (a, b) ∈ ∆δk , f ∈ Qp .

j∈N

This proves the desired assertion.

In what follows we give two descriptions of Θ(Q∗p,0 ) using atom-like functions of diﬀerent types. The atom-like function of the ﬁrst type is from D. More precisely, given p ∈ (0, ∞) we consider a function u ∈ H on a Carleson box S(I) based on an arc I ⊆ T satisfying |I|p |u(z)|2 (1 − |z|2 )p dm(z) ≤ 1. S(I)

The desired atom-like function is of the form u(w) a(z) = z (1 − |w|2 )p dm(w). ¯ 2 S(I) (1 − z w) Note that if L ∈ Q∗p,0 is given by L(f ) =

f (w)u(w)(1 − |w|2 )p dm(w), f ∈ Qp,0 ,

S(I)

then

a(z) = L (1 − z¯w)−1

5.3. Atom-like Representations and hence

121

a ∈ Θ(Q∗p,0 ) with a Θ(Q∗p,0 ) = L Q∗p,0 1.

The atom-like function of the second type is based on T. Given p ∈ (0, 1) and I a subinterval of [0, 2π], let v be a Lebesgue measurable function on I such that |v(t) − v(s)|2 p p dtds = |I| |v(t) − v(s)|2 |eit − eis |p−2 dtds ≤ 1 |I| it is 2−p I I |e − e | I I and

b(z) = z I

I

(v(t) − v(s))(e−it − e−is ) it |e − eis |p−2 dtds. (1 − ze−it )(1 − ze−is )

This function b(z) is called the atom-like function of the second type. It is indeed the element of Θ(Q∗p,0 ) generated by the following L, i.e., b(z) = L (1 − z¯eit )−1 , where L is an element of Q∗p,0 determined by L(f ) = v(t) − v(s) f (eit ) − f (eis ) |eit − eis |p−2 dtds, f ∈ Qp,0 . I

Theorem 5.3.2. as

I

(i) If p ∈ (0, 2), then every function f ∈ Θ(Q∗p,0 ) can be written f = f (0) +

∞

cj aj , {cj } ∈ 1 ,

j=1

where {aj } are atom-like functions of the ﬁrst type, and

f Θ(Q∗p,0 ) ≈ |f (0)| + inf

∞

|cj | , f ∈ Θ(Q∗p,0 ),

j=1

for which the inﬁmum is taken over all above-mentioned representations of f. (ii) If p ∈ (0, 1), then every function f ∈ Θ(Q∗p,0 ) can be written as f = f (0) +

∞

cj bj , {cj } ∈ 1 ,

j=1

where {bj } are atom-like functions of the second type, and

f

Θ(Q∗ p,0 )

≈ |f (0)| + inf

∞

|cj | , f ∈ Θ(Q∗p,0 ),

j=1

for which the inﬁmum is taken over all above-mentioned representations of f.

122

Chapter 5. Cauchy pairing with Expressions and Extremities

Proof. (i) We introduce a mapping Ψ from Qp,0 to c0 L2 (D, (1 − |z|2 )p dm(z)) as ∞ follows. Fix an enumeration {Jj }∞ j=1 of the dyadic subarcs of T and let {Sj }j=1 ∞ be the sequence of Carleson boxes based on {Jj }j=1 . For f ∈ Qp,0 set p

Ψ1,0 f = f (0), Ψ1,j f = |Jj |− 2 1Sj f , j ∈ N. Then

2 2 p Ψ1 = {Ψ1,j }∞ j=1 : Qp,0 → c0 L (D, (1 − |z| ) dm(z))

is bounded from above and below. Consequently, Ψ(Qp,0 ) is a closed subspace of c0 L2 (D, (1 − |z|2 )p dm(z)) . This fact and that general duality principle right before Theorem 5.2.1 give that for L ∈ Q∗p,0 and f ∈ Qp,0 , L(f ) can be represented as L(f ) = c0 f (0) + = c0 f (0) +

∞ j=1 ∞

cj

D

gj (z)Ψ1,j f (z)(1 − |z|2 )p dm(z) p

cj |Jj |− 2

j=1

gj (z)f (z)(1 − |z|2 )p dm(z),

Sj

where {cj } ∈ 1 and gj ∈ L2 D, (1 − |z|2 )p dm(z) with gj L2 D,(1−|z|2 )p dm(z) = 1. Moreover, we can select {cj } and {gj } such that {cj } 1 L Q∗p,0 . Note that H ∩ L2 (Sj , (1 − |z|2 )p dm(z)) is a closed subspace of L2 (Sj , (1 − |z|2 )p dm(z)). So an application of the Riesz Representation Theorem produces a function uj ∈ H such that for f ∈ Qp,0 , p

|Jj |− 2

gj (z)f (z)(1 − |z|2 )p dm(z) =

Sj

uj (z)f (z)(1 − |z|2 )p dm(z).

Sj

Letting

aj (z) = z Sj

uj (w) (1 − |w|2 )p dm(w) (1 − z w) ¯ 2

and evaluating L (1 − z¯w)−1 , we derive the desired representation result. (ii) The argument is similar. The key issue is to introduce a map from Qp,0 to c0 L2 ([0, 2π] × [0, 2π], |eit − eis |p−2 dtds) . In so doing, for f ∈ Qp,0 , p ∈ (0, 1) and the sequence of intervals {Ij } determined by Lemma 5.3.1, let Ψ2 f = {Ψ2,j f }, where p Ψ2,0 f = f (0), Ψ2,j f (t, s) = |Ij |− 2 1Ij (t)1Ij (s) f (eit ) − f (eis ) , j ∈ N.

5.4. Extreme Points of Unit Balls

123

Then L ∈ Q∗p,0 can be expressed as L(f ) = =

∞

c0 f (0) +

j=1 ∞

c0 f (0) +

cj

[0,2π] p

cj |Ij |− 2

[0,2π]

Ij

j=1

gj (t, s)Ψ2,j f (t, s)|eit − eis |p−2 dtds gj (t, s) f (eit ) − f (eis ) |eit − eis |p−2 dtds,

Ij

where {cj } ∈ 1 , and gj (·, ·) ∈ L2 ([0, 2π] × [0, 2π], |eit − eis |p−2 dtds) is of the unit norm. As in (i), we can get such {cj } and {gj } that {cj } 1 L Q∗p,0 . Since the class of all functions v(t) − v(s) (where v produces an atom-like function of the second type) is closed in L2 ([0, 2π] × [0, 2π], |eit − eis |p−2 dtds), we conclude from the Riesz Representation Theorem that for each j there is a function vj such that

|vj (t) − vj (s)|2 |eit − eis |p−2 dtds ≤ 1

|Ij |p Ij

Ij

and

gj (t, s) f (eit ) − f (eis ) |eit − eis |p−2 dtds

|Ij |p

Ij

= Ij

Ij

vj (t) − vj (s) f (eit ) − f (eis ) |eit − eis |p−2 dtds.

Ij

Taking bj (z) = z I

I

(vj (t) − vj (s))(e−it − e−is ) it |e − eis |p−2 dtds, (1 − ze−it )(1 − ze−is )

and calculating L (1 − z¯eit )−1 , we get the required assertion.

5.4 Extreme Points of Unit Balls Given a Banach space X with norm · X . The extreme points of the closed unit ball BX = {f ∈ X : f X ≤ 1} are the points which are not a proper convex combination of two diﬀerent points of BX . From the previous discussions we have seen that the second dual space of Qp,0 is isomorphic to Qp under the pairings ·, ·inv and ·, ·. So the well-known Krein–Milman Theorem then implies that the closed unit ball of Qp has extreme points. We show that the closed unit ball of Qp,0 also has extreme points. The main result of this section is two characterizations of the extreme points of the closed unit ball of Qp,0 .

124

Chapter 5. Cauchy pairing with Expressions and Extremities

Theorem 5.4.1. Given p ∈ (0, ∞) and f ∈ Qp . (i) f is an extreme point of BQp,0 with the norm |f (0)| + f Qp ,1 if and only if f is a constant function of module 1 or f (0) = 0 and f Qp ,1 = 1. (ii) If f is an extreme point of BQp with the norm |f (0)| + · Qp ,1 , then |f (0| +

f Qp ,1 = 1. Conversely, if f is a constant function of module 1 or there exists a point w0 ∈ D such that Ep (f, w0 ) = 1, then f is an extreme point of BQp with the norm |f (0)| + f Qp ,1 . Proof. Deﬁning Q0p = {f ∈ Qp : f (0) = 0} and Q0p,0 = {f ∈ Qp,0 : f (0) = 0}, we ﬁnd

Qp = C ⊕ Q0p

and Qp,0 = C ⊕ Q0p,0 .

Thus a routine argument tells us that (|f (0)|, f Qp,1 ) is an extreme point for the closed unit ball of R × R in the Euclidean 1-norm if and only if either f (0) = 0 and f Qp ,1 = 1 or |f (0)| = 1 and f Qp = 1. With this equivalence, we make only the following consideration. (i) Suppose f is an extreme point of BQ0p,0 with respect to the norm f Qp ,1 . If f Qp,1 < 1, then for 0 < < min{1, f −1 Qp ,1 − 1} let f1 = (1 − )f and f2 = (1 + )f. This choice gives f1 , f2 ∈ BQp,0 but f = 2−1 (f1 +f2 ), contradicting the assumption that f is extreme. In order to verify the converse, we may assume f ∈ Q0p,0 with

f Qp,1 = 1. Then we have lim |w|→1 Ep (f, w) = 0 and thus there is an r ∈ (1/2, 1) such that sup|w|>r Ep (f, w) ≤ 1/2. Accordingly, we get 1 = f Qp ,1 = sup Ep (f, w) = max Ep (f, w). w∈D

|w|≤r

Since Ep (f, ·) is continuous on D, it is uniformly continuous on the compact set {w ∈ D : |w| ≤ r}. This yields that Ep (f, w0 ) = 1 is valid for some |w0 | ≤ r. Let now g ∈ Q0p,0 be such that f + g Qp ,1 ≤ 1 and f − g Qp ,1 ≤ 1. Then 2 1 + Ep (g, w0 )

=

2 2 Ep (f, w0 ) + Ep (g, w0 ) 2 2 2−1 Ep (f + g, w0 ) + Ep (f − g, w0 )

≤

1.

=

This forces Ep (g, w0 ) = 0, and so g = 0 on D. In other words, f is extreme. (ii) The argument is essentially included in (i).

5.4. Extreme Points of Unit Balls

125

Here it is worth pointing out that diﬀerent norms produce usually diﬀerent classes of the extreme points. To understand this principle, let us take · Qp ,2 into account. From now on, for p ∈ (0, ∞) and a Lebesgue measurable function f : T → C put |f (ζ) − f (η)|2 Sp (f, I) = (2π)−2 |dζ||dη|. |ζ − η|2−p I I We say that f ∈ Qp (T) provided |||f |||Qp (T) = (2π)−1 f (ζ)|dζ| + sup |I|−p Sp (f, I) < ∞ T

I⊆T

where the supremum is taken over all subarcs I of T. Moreover, by f ∈ Qp,0 we mean lim sup |I|−p Sp (f, I) = 0. →0 I⊂T,|I|≤

Clearly, if p ∈ (0, 1), then the nontangential boundary value function of a function f in Qp or Qp,0 belongs to Qp (T) or Qp,0 (T). Quite interesting is the following assertion. Proposition 5.4.2. (i) Let BMO(T), respectively VMO(T), be the class of all Lebesgue measurable functions f : T → C with 2 f (ζ) − (2π|I|)−1 f (η)|dη| |dζ| = O(|I|), respectively o(|I|), I

I

as subarc I ⊆ T varies. Then Qp1 (T) ⊆ Qp2 (T) and Qp1 ,0 (T) ⊆ Qp2 ,0 (T) for 0 < p1 < p2 < ∞. In particular, Qp (T) = BMO(T) and Qp,0 (T) = VMO(T) for p ∈ (1, ∞). (ii) Let p ∈ (0, ∞) and f ∈ Qp (T) with dQp (T) (f, Qp,0 (T)) = inf{|||f − g|||Qp (T) : Then

dQp (T) (f, Qp,0 (T)) ≈ lim

sup

δ→0 I⊂T,|I|<δ

g ∈ Qp,0 (T)}.

Sp (f, I) |I|p

1/2 .

Proof. (i) The ﬁrst result is trivial. To prove the special case p > 1, it suﬃces to show that each Qp,0 (T) coincides with VMO(T) whenever p > 1. First, we verify VMO(T) ⊆ Qp,0 (T). To do so, we observe that Qp,0 (T) has an integrated

126

Chapter 5. Cauchy pairing with Expressions and Extremities

Lip-characteristic which says that f ∈ Qp,0 (T) if and only if limδ→0 Ip (f, δ) = 0, where for δ ∈ (0, 1), Ip (f, δ) = sup

|I|<δ

|I|

sinp−2

0

πt dt 2

|f (ei(s+t) ) − f (eis )|2 ds. I

Again, we write rI (r > 0) for the arc with length r|I| and the same center as I, and fJ = (2π|J|)−1 J f (ζ)|dζ| for any subarc J of T. Now if f ∈ VMO(T) then for any small > 0 there is a δ ∈ (0, 1/3) such that as |I| < δ, |f (eis ) − f3I |2 ds < 2π |I|, 3I

and hence

|I|

πt dt 2

sinp−2

0

which gives

|I|

sinp−2

0

πt dt 2

|f (eis ) − f3I |2 ds |I|p ,

I

|f (ei(t+s) ) − f3I |2 ds |I|p .

I

Thus, limδ→0 Ip (f, δ) = 0, i.e., f ∈ Qp,0 (T). Next, we show Qp,0 (T) ⊆ VMO(T). As for the case p ∈ (1, 2], the result follows immediately from the deﬁnition. It remains to deal with the case p ∈ (2, ∞). Let f ∈ Qp,0 (T). Then for arbitrarily small > 0 there exists a δ ∈ (1, 1/2) such that |J|−p Sp (f, J) < as |J| < δ. Thus for I ⊆ T with |I| < δ, we have |f (eis ) − f (eit )|2 dsdt I

≤

I

∞ k=1 ∞ k=1 ∞ k=1

1−k 2−k < |s−t| |I| ≤2

|I| 2−p 2k

|I| 2−p 2k

|eis − eit |2−p |f (eis ) − f (eit )|2 dsdt |eis − eit |2−p

|s−t| ≤21−k |I|

22−k I

22−k I

|f (eis ) − f (eit )|2 dsdt |eis − eit |2−p |f (eis ) − f (eit )|2 dsdt |eis − eit |2−p

|I|2 ,

which yields f ∈ VMO(T). (ii) Since dQp (T) (f, Qp,0 (T)) equals 0 when f ∈ Qp,0 (T), it is easy to obtain lim

sup

δ→0 I⊂T,|I|<δ

Sp (f, I) |I|p

1/2 ≤ dQp (T) (f, Qp,0 (T)),

f ∈ Qp (T).

5.4. Extreme Points of Unit Balls

127

To establish the reversed estimate, we recall the Poisson integral: 1 fr (e ) = 2π

is

f (η) T

1 − r2 |dη|, r ∈ (0, 1), f ∈ Qp (T). |η − reis |2

It is evident that fr ∈ Qp,0 (T) and −1

f (ζ) − fr (ζ) = (2π)

1 − r2 ¯ f (ζ) − f (ζ λ) ¯ 2 |dλ|, |1 − rλ| T

ζ = eis ,

λ = ζ η¯.

¯ and using Minkowski’s inequality, we derive that for any Letting Tλ f (ζ) = f (ζ λ) small > 0, |||f − fr |||Qp (T) 1 − r2 |||f − Tλ f |||Qp (T) ¯ 2 |dλ| |1 − rλ| T 1 − r2 1 − r2 |||f − Tλ f |||Qp (T) |dλ| + |||f ||| Q (T) p ¯2 ¯ 2 |dλ| |1 − rλ| |λ|< ≤|λ|≤π |1 − rλ| = Term1 (r) + Term2 (r). Given δ ∈ (0, 1). The Lebesgue Dominated Convergence Theorem yields lim sup |I|−p Sp (f − Tλ f, I) = 0.

λ→0 |I|>δ

In the meantime, the triangle inequality gives sup |I|−p Sp (f − Tλ f, I) sup

|I|≤δ

|I|≤δ

Sp (f, I) . |I|p

Therefore, if → 0 then Term1 (r) sup

|I|<δ

Sp (f, I) |I|p

1/2 .

And, since limr→1 Term2 (r) = 0, we conclude dQp (T) (f, Qp,0 (T)) ≤ lim |||f − fr |||Qp (T) lim r→1

sup

δ→0 I⊂T,|I|<δ

as desired. Let us come back to our topic on the extreme points.

1/2 |I|−p Sp (f, I) ,

128

Chapter 5. Cauchy pairing with Expressions and Extremities

Theorem 5.4.3. For p ∈ (0, ∞), let Q0p (T) = f ∈ Qp (T) : f (ζ)|dζ| = 0 T

and

Q0p,0 (T) = f ∈ Qp,0 (T) : f (ζ)|dζ| = 0 . T

Q0p,0 (T),

then f is an extreme point of BQ0p,0 (T) with the norm (i) If f ∈ |||f |||Qp (T) when and only when Tp (f, ζ) = sup |I|−p Sp (f, I) = 1, ζ ∈ T, ζ∈I

where the supremum is taken over all arcs I ⊆ T containing ζ. (ii) If f ∈ Q0p (T) is an extreme point of BQ0p (T) with the norm ||| · |||Qp (T) , then there are no two distinct points ζ1 , ζ2 in T such that Tp (f, ζ1 ) < 1 and Tp (f, ζ2 ) < 1. Conversely, if f ∈ Q0p (T) is a function such that for all z ∈ T with one possible exception there are Tp (f, ζ) = 1 and |Iζ |−p Sp (f, Iζ ) = 1 on some open subarc Iζ containing ζ, then f is an extreme point of BQ0p (T) with the norm ||| · |||Qp (T) . Proof. (i) Note that if f ∈ Q0p (T), then Tp (f, ·) is lower semicontinuous; that is, {ζ ∈ T : Tp (f, ζ) > t} is an open set for every t > 0. So we have |||f |||Qp (T) =

12 sup Tp (f, ζ) . ζ∈T

Recalling that C(T) is the class of all continuous functions f : T → C, we can obtain Tp (f, ·) ∈ C(T). To see this, we suﬃce to show that Tp (f, ·) is also upper semicontinuous, namely, {ζ ∈ T : Tp (f, ζ) < t} is an open subset of T for every t > 0. In so doing, we ﬁx t > 0 and let ζ ∈ T obey Tp (f, ζ) < t. If Tp (f, ζ) = 0, then by f ∈ Q0p,0 (T), there is a δ > 0 such that 2 Sp (f, I) ≤ Tp (f, ζ) . p |I| I⊆T,|I|<δ sup

Now let J ⊆ T be an open subarc centered at ζ whose arclength |J| = is very small compared to δ. And, for w ∈ J let I ⊂ T contain w. When |I| < δ, we have Sp (f, I)|I|−p < t2 . In the case |I| ≥ δ we take K = J ∪ I. Then K is an open subarc containing ζ and hence 2 Sp (f, I) |K| p Sp (f, K) |K| p Tp (f, ζ) . ≤ ≤ |I|p |I| |K|p |I|

5.4. Extreme Points of Unit Balls

129

Further, putting Tp (f, ζ) = t − τ , τ ∈ (0, t), we derive |I|−p Sp (f, I) < t2 and thus Tp (f, η) < t when η ∈ J

and |J| = < δ t/(t − τ ))2/p − 1 .

On the other hand, if Tp (f, ζ) = 0, then via the analysis on the open subarc K, we can select an open subarc J centered at ζ such that Tp (f, η) = 0 < t as η ∈ J. For the necessity, suppose Tp (f, ·) ≡ 1. Then by Tp (f, ·) ∈ C(T), there is an open subarc J ⊂ T such that supζ∈J Tp (f, ζ) < 1. Choose a function g ∈ Q0p,0 (T) such that |||g|||Qp (T) ≤ 1 − sup Tp (f, ζ), ζ∈J

g = 0 outside J, and g ≡ 0. When I ⊂ T is such that I ∩ J = ∅, we evidently obtain Sp (f + g, I) Sp (f, I) = ≤ 1. p |I| |I|p If I ⊂ T ensures I ∩ J = ∅, then S (f + g, I) 1/2 p |I|p

S (f, I) 1/2 S (g, I) 1/2 p p + |I|p |I|p ≤ sup Tp (f, ζ) + |||g|||Qp (T) ≤

ζ∈J

≤ 1. Thus f +g Qp(T) ≤ 1. Similarly, f −g Qp(T) ≤ 1. Due to g ≡ 0, those inequalities yield that f is not an extreme point of BQ0p,0 (T) , contradicting the assumption. Assuming Tp (f, ·) ≡ 1, we derive that if ζ0 ∈ T, then there exists a sequence of subarcs In containing ζ0 such that Sp (f, In )/|In |p is convergent to 1. Without loss of generality, we may suppose In = (an , bn ), intervals moving counterclockwise. Then, by passing a subsequence, we may also suppose an → a and bn → b. After that, let Iζ0 ⊂ T be the open subarc determined by the interval (a, b), i.e., Iζ0 = (a, b). In this sense, Iζ0 is viewed as the limiting open subarc of In . Accordingly, Sp (f, Iζ0 )/|Iζ0 |p = 1. Although Iζ0 does not necessarily contain ζ0 , it is easy to see that ζ0 belongs to I¯ζ0 — the closure of Iζ0 — a closed subarc [a, b] of T. Observe that since f ∈ Q0p,0 (T), In cannot get small and hence Iζ0 = ∅. Now suppose g ∈ Q0p,0 (T)

with |||f + g|||Qp (T) ≤ 1 and |||f − g|||Qp (T) ≤ 1.

In order to prove that f is an extreme point of BQ0p,0 (T) , we must show that g ≡ 0 on T. Fix ζ0 ∈ T with the open subarc Iζ0 constructed above. Thus, f (z) − f (w) (|z − w|/|Iζ0 |)(2−p)/2 Iζ0 ×Iζ0

130

Chapter 5. Cauchy pairing with Expressions and Extremities

lies in the unit sphere of L2 (Iζ0 × Iζ0 , (2π|Iζ0 |)−2 |dz||dw|). Using |||f + g|||Qp (T) ≤ 1, we also see that f (z) − f (w) + g(z) − g(w) (|z − w|/|Iζ0 |)(2−p)/2 Iζ0 ×Iζ0 is a member of the closed unit ball of L2 (Iζ0 × Iζ0 , (2π|Iζ0 |)−2 |dz||dw|), and similarly when g is replaced by −g. Notice that f (z) − f (w) + g(z) − g(w) f (z) − f (w) = Iζ0 ×Iζ0 (|z − w|/|Iζ0 |)(2−p)/2 Iζ0 ×Iζ0 2(|z − w|/|Iζ0 |)(2−p)/2 f (z) − f (w) − g(z) − g(w) + 2(|z − w|/|Iζ0 |)(2−p)/2 Iζ0 ×Iζ0 and even more importantly, that L2 (Iζ0 ×Iζ0 , (2π|Iζ0 |)−2 |dz||dw|) has the property: every point of the unit sphere is an extreme point of the closed unit ball [Rud, p.84, problem 16]. So, the last equation gives that g(z) − g(w) = 0 on Iζ0 × Iζ0 ; that is, g is a constant on Iζ0 . For each ζ0 ∈ T let Jζ0 denote the largest open subarc covering Iζ0 such that the restriction g|Jζ0 of g on Jζ0 is constant. Obviously, as to ζ0 , η0 ∈ T, Jζ0 ∩ Jη0 = ∅ induces Jζ0 = Jη0 , and so the collection {Jζ0 } can contain at most countably many diﬀerent open subarcs, owing to |T| = 1. Relabel these disjoint ¯ n , where K ¯ n still open subarcs {Kn }. The condition ζ0 ∈ I¯ζ0 implies T = ∪K stands for the closure of Kn . Thus A = T \ ∪Kn consists of all endpoints of all the subarcs Kn . In particular, A is a closed, countable set. If A = ∅, then {Kn } contains only one element, namely T, and thus g is a constant function on T. Note that g|T = 0. Accordingly, g ≡ 0. It remains to consider the case: A = ∅. Using the Baire Category Theorem, we see that the non-empty, countable, closed set A must have an isolated point c ∈ T. ¯ n , we can conclude that there must be two disjoint open subarcs Since T = ∪K Kn and Km such that c is an endpoint for them. Recall that g|Kn and g|Km are constant. If the two constants do not coincide, then g has a jump discontinuity at c, which certainly contradicts the condition g ∈ Q0p,0 (T). If the two constants are the same, then g is constant on the open subarc Kn ∪ {c} ∪ Km , which violates the maximality of the open subarcs {Jζ0 }ζ0 ∈T . Thus the case A = ∅ cannot happen. (ii) Let f be an extreme point of BQ0p (T). Of course, we must have ||f |||Qp (T) = 1. In order to reach our purpose, suppose otherwise that there are two distinct points ζk ∈ T, k = 1, 2 such that Tp (f, ζk ) < 1, k = 1, 2; and use Ik , k = 1, 2 to denote the two open subarcs of T which have {ζ1 , ζ2 } as endpoints. Deﬁne a function g ∈ Q0p (T) by g|I1 = 1 , g|I2 = − 2 , where k > 0, k = 1, 2 are chosen so that g|T = 0 and |||g|||Qp (T) ≤ 1 − max Tp (f, ζk ). k=1,2

Now, let I be a subarc of T. If both ζ1 and ζ2 are not in I, then g is constant on

5.4. Extreme Points of Unit Balls

131

I and consequently, Sp (f + g, I) Sp (f, I) = ≤ 1. p |I| |I|p If one of ζk , for instance, ζ1 is in I, then by the Minkowski inequality, S (f + g, I) 1/2 p ≤ Tp (f, ζ1 ) + |||g|||Qp (T) ≤ 1. |I|p Accordingly, we get |||f + g|||Qp (T) ≤ 1 and similarly |||f − g|||Qp (T) ≤ 1. Because of f = (f + g)/2 + (f − g)/2, the function f is not an extreme point of BQ0p (T) , contradicting the given condition. On the other hand, suppose that f ∈ Q0p (T) is a function such that for all ζ ∈ T with one possible exception η ∈ T there are Tp (f, ζ) = 1 and Sp (f, Iζ ) = |Iζ |p on some open subarc Iζ containing ζ. Now let g ∈ Q0p (T) obey |||f ± g|||Qp (T) ≤ 1. To close the argument, we must show g ≡ 0 on T. Applying the same reasoning as in the proof of the suﬃciency of (i), we declare that g|Iζ is a constant. Since g is locally constant on the connected set T \ {η}, g ≡ 0 by g|T = 0. We close this section by presenting some examples of either extreme points or nonextreme points. Example 5.4.4. Given p ∈ (0, ∞). (i) For integers n = ±1, ±2, . . . let fn (z) = λz n where z ∈ T and |λ| ≡ 1. Then −1 gn = |||fn |||Qp (T) fn are extreme points of BQ0p,0 (T) . (ii) For δ ∈ (0, 1) let ⎧ ⎨

, θ ∈ [0, δ], 1 , θ ∈ [δ, 2π − δ], ⎩ 2π−θ , θ ∈ [2π − δ, 2π). δ

iθ

fδ (e ) =

θ δ

Also put gδ = fδ − 1 + (2π)−1 δ on [0, 2π) and extend it 2π-periodically. Then 0 there exists a δ > 0 such that |||gδ |||−1 Qp (T) gδ is a nonextreme point of BQp,0 (T) . Proof. (i) By Theorem 5.4.3 we have to verify Tp (gn , ζ) = 1 for all ζ ∈ T. As a matter of fact, we use some elementary estimates to obtain Sp (fn , I) = 2

p+1 0

|I|

(|I| − t) sinp−2 (πt) sin2 (nπt)dt

132

Chapter 5. Cauchy pairing with Expressions and Extremities

and so that Tp (fn , ζ) =

2p+1 p |I|∈(0,1] |I|

sup

0

|I|

(|I| − t) sinp−2 (πt) sin2 (nπt)dt.

Thus, |||fn |||Qp (T) = Tp (fn , ζ) for each ζ ∈ T, and then Tp (gn , ·) ≡ 1 follows. (ii) A key observation is that gδ is convergent to 0 as δ → 0. Since fδ is 0 a Lip1-function, we get |||gδ |||−1 Qp (T) gδ ∈ Qp,0 (T). However, we are about to prove that it is not an extreme point of BQ0p,0 (T) . This will be done as long as Tp (fδ , ·) is veriﬁed to be a nonconstant function for some δ. To this end, it suﬃces to verify Tp (fδ , 0) = Tp (fδ , π) for some δ. First, for any subinterval I = (a, b) ⊂ (0, δ) we have Sp (fδ , I) 2p (2π)p−2 b−a (b − a − t)t2 = dt. |I|p (b − a)p δ 2 0 sin2−p 2t Thus Sp (fδ , I) 2p (2π)p−2 ≥ |I|p δ 2+p I⊂(0,δ)

sup

0

δ

(δ − t)t2 dt. sin2−p 2t

Moreover, using lim

1

δ→0 δ p+2

0

δ

(δ − t)t2 22−p 2−p t dt = (p + 1)(p + 2) , sin 2

we can ﬁnd a δ1 ∈ (0, 1) such that

1/2 µ0 2p π p−2 . = δ ∈ (0, δ1 ) ⇒ Tp (fδ , 0) > 2 (p + 1)(p + 2) 2 Second, suppose I ⊆ T is any subarc containing π. We have −1

|I| ≤ (π − δ)/(2π) ⇒ Sp (fδ , I) = 0. And, if 1 ≥ |I| > (π − δ)/(2π), then by the deﬁnition of fδ , |fδ (eis ) − fδ (eit )|2 1 Sp (fδ , I) ≤ Sp (fδ , T) = dsdt, 2 (2π) |eis − eit |2−p Ω 4 where Ω is a domain deﬁned by j=1 Ωj : ⎧ {(s, t) : 0 ≤ s ≤ δ, 0 ≤ t ≤ 2π} , j = 1, ⎪ ⎪ ⎨ {(s, t) : 2π − δ ≤ s ≤ 2π, 0 ≤ t ≤ 2π} , j = 2, Ωj = {(s, t) : 0 ≤ s ≤ 2π, 0 ≤ t ≤ δ} , j = 3, ⎪ ⎪ ⎩ {(s, t) : 0 ≤ s ≤ 2π, 2π − δ ≤ t ≤ 2π} , j = 4. It is not hard to get a δ2 ∈ (0, 1) such that as δ ∈ (0, δ2 ), π − δ p 2πµ 2 |fδ (eis ) − fδ (eit )|2 0 dsdt ≤ , is it 2−p |e − e | 2π 2 Ωj

j = 1, 2, 3, 4.

5.5. Notes

133

Hence Sp (fδ , I) 1 2π p ≤ |I|p (2π)2 π − δ

Ω

µ0 2 |fδ (eis ) − fδ (eit )|2 dsdt ≤ . is it 2−p |e − e | 2

Consequently, Tp (fδ , π) ≤ µ0 /2 whenever δ ∈ (0, δ2 ). Therefore there exists a δ3 ∈ (0, min{δ1 , δ2 }) such that Tp (fδ3 , 0) > µ0 /2 and Tp (fδ3 , π) ≤ µ0 /2. This concludes the proof.

5.5 Notes Note 5.5.1. Section 5.1 is mainly taken from [AlCaSi, Section 3]. For some related information on the Cauchy duality, see also [RosSh] and [DuRoSh]. In the proof of Example 5.1.2 (i) and (ii), we have actually used the concept of a sampling set for A−α ; see also [HedKZ, Chapter 5]. Moreover, the fact that there are sequences in D for which the representation in Example 5.1.2 (iii) holds is well known — see [Zhu2, Theorem 2.30]. Note 5.5.2. Section 5.2 is a slight modiﬁcation of [AlCaSi, Section 4]. Corollary 5.2.4 (i) gives another description of the well-known fact that H1 = H2 H2 . Of course, this corollary also implies the following interesting factorization: D1,0 = D1,−

2−p 2

D2,2−p

for

p ∈ (1, 2),

which has not been obtained until now. A more interesting reference on the dot product or weak factorization is [CohnVe]. Note 5.5.3. Section 5.3 is a sort of combination of Sections 2 and 6 in [AlCaSi]. Although the atom-like representations in Theorem 5.3.2 have no requirements on the compact support and the vanishing moment of either a(z) or b(z), they provide a good realization of Theorem 4.3.2 (i) on the Cauchy dual of Qp,0 . Additionally, Theorem 5.3.2 may be viewed as a sort of dual form of Theorem 3.3.3. Note 5.5.4. Section 5.4 consists of [WiXi2, Section 3] and [WiXi3]. There are several papers on the extreme points in the Bloch spaces — see e.g., [CiWo], [AnRo], [Wi1] [Wi2], [Col], and [CoheCol]. For a discussion on the extreme points in VMO and BMO, see [AxSh]. And the notion of VMO is originally from [Sara].

Chapter 6

As Symbols of Hankel and Volterra Operators The various holomorphic Q classes have good boundedness properties and consequently are often proper settings for the applications of complex analysis to, for instance, operator theory. In this chapter we will see that the Qp spaces induce bounded holomorphic Hankel and Volterra operators acting between two Dirichlet-type spaces. This feature of function-theoretic operator theory will be shown naturally via the following components: • • • •

Hankel and Volterra from Small to Large Spaces; Carleson Embeddings for Dirichlet Spaces; More on Carleson Embeddings for Dirichlet Spaces; Hankel and Volterra on Dirichlet Spaces.

6.1 Hankel and Volterra from Small to Large Spaces ¯ by For f ∈ H we consider two linear operators initially deﬁned on g ∈ H(D) f (rζ)g(rζ) Hf g(z) = lim |dζ|, z ∈ D, r→1 T 1 − zrζ¯

and Vf g(z) =

0

z

g(w)f (w)dw, z ∈ D.

The former is called a holomorphic Hankel operator and the latter is called a holomorphic Volterra operator. The name of Hankel is motivated by the fact that on the canonical basis {z n } one has: m n φ(ζ)f (ζ)g(ζ)|dζ| ⇒ Hφ (z , z ) = 2π φ(ζ)ζ −(m+n) |dζ|, Hφ (f, g) = T

T

136

Chapter 6. As Symbols of Hankel and Volterra Operators

which is associated with a Hankel matrix, while the name of Volterra is motivated z by the Volterra integral 0 f (w)dw. As a matter of fact, the Volterra operators may be regarded as Riemann–Stieltjes operators which generalize the classical Cesaro operators. p Note that if f ∈ D1,−(1− 2 ) , p ∈ (0, 2), then the mean-value-inequality for |f | produces p

p

|f (z)|(1 − |z|)1+ 2 f D1,−(1− p2 ) and so D1,−(1− 2 ) ⊆ D2,p . Furthermore, the inclusion is proper. To see this, we ﬁrst observe f (z) =

∞

aj z j ∈ D2,p ⇔

j=0

∞

|aj |2 (1 + j)1−p < ∞,

j=0

and next use Hardy’s inequality (cf. [Du, p. 48]) ∞

|aj |(j + 1)−1 f H1 , f (z) =

j=0

∞

aj z j ∈ H 1

j=0

to yield that, for f (z) =

∞ j=0

∞ >

D 1

=

0

∞ j=0 ∞

p

aj z j ∈ D1,−(1− 2 ) , p

|f (z)|(1 − |z|2 ) 2 −1 dm(z) p

(1 − r2 ) 2 −1 |aj |

0

1

T

|f (rζ)||dζ| rdr p

rn (1 − r2 ) 2 −1 rdr p

|aj |(1 + j)− 2 .

j=0

Accordingly, it is quite natural to obtain the forthcoming characterization of Qp in terms of the holomorphic Hankel and Volterra operators. Theorem 6.1.1. Let p ∈ (0, 2) and f ∈ H. Then the following statements are equivalent: p

(i) Hf extends to a continuous operator from D1,−(1− 2 ) to D2,p . p

(ii) Vf is a continuous operator from D1,−(1− 2 ) to D2,p . (iii) f ∈ Qp . p

Proof. (i)⇔(iii) If f ∈ Qp , then for g ∈ D1,−(1− 2 ) and h ∈ D2,2−p we use the p density of polynomials in D1,−(1− 2 ) and D2,2−p , plus Theorem 5.2.3, to get |Hf g, h| = |f, gh| (|f (0)| + f Qp ,1 ) g D1,−(1− p2 ) h D2,2−p .

6.1. Hankel and Volterra from Small to Large Spaces

137

Since the Cauchy dual of D2,2−p is isomorphic to D2,p , we conclude from the just-established estimate that Hf g ∈ D2,p with

Hf D1,−(1− p2 ) →D2,p |f (0)| + f Qp ,1 . p

Conversely, if Hf extends to a continuous operator from D1,−(1− 2 ) to D2,p , then the operator norm Hf D1,−(1− p2 ) →D2,p is ﬁnite with |f, gh| = |Hf g, h| Hf D1,−(1− p2 ) →D2,p g D1,−(1− p2 ) h D2,2−p ,

¯ g, h ∈ H(D). p

Of course, this assertion yields that f belongs to the Cauchy dual of D1,−(1− 2 ) D2,2−p , i.e., f ∈ Qp with |f (0)| + f Qp ,1 Hf D1,−(1− p2 ) →D2,p , owing to Theorem 5.2.3. p (ii)⇔(iii) If Vf sends D1,−(1− 2 ) to D2,p continuously, then for a ∈ D and its p associated function ga (z) = (1 − |a|2 )1− 2 (1 − a ¯z)−1 , we have ga D1,−(1− p2 ) 1, and so that 12 (1 − |a|2 )2−p 2 2 p |f (z)| (1 − |z| ) dm(z) |1 − a ¯z|2 D = Vf ga D2,p ≤ Vf D1,−(1− p2 ) →D2,p ga D1,−(1− p2 ) Vf D1,−(1− p2 ) →D2,p ,

implying f ∈ Qp thanks to Lemma 3.1.1. Conversely, if f ∈ Qp then for g ∈ p D1,−(1− 2 ) we use Example 5.1.2 to choose {wj } ⊂ D and {cj } ∈ 1 such that g(z) =

∞ j=1

p

cj (1 − |wj |2 )1− 2 (1 − wj z)−1 with {cj } 1 g D1,−(1− p2 ) .

By Minkowski’s inequality and Lemma 3.1.1, we derive

Vf g D2,p

≤

12 1 − |wj |2 )2−p 2 2 p |f (z)| (1 − |z| ) dm(z) |cj | |1 − wj z|2 D j=1

∞

{cj } 1 f Qp ,1

g D1,−(1− p2 ) f Qp ,1 ,

thereupon establishing the desired boundedness of Vf .

For an application of the previous theorem, we can characterize pointwise p multipliers of D1,−(1− 2 ) into D2,p . To this end, let Mf g = f g for any two functions f and g.

138

Chapter 6. As Symbols of Hankel and Volterra Operators

Corollary 6.1.2. Let p ∈ (0, 2) and f ∈ H. Then Mf is a continuous operator from p D1,−(1− 2 ) to D2,p if and only if f ∈ H ∞ ∩ Qp . Proof. Assuming

Uf g(z) =

0

z

f (w)g (w)dw, f, g ∈ H, p

we have that Uf is a bounded operator from D1,−(1− 2 ) to D2,p if and only if p f ∈ H∞ . In fact, if f ∈ H∞ , then using D1,−(1− 2 ) ⊂ D2,p we derive

Uf g D2,p ≤ f H∞ Conversely, if Uf : D

D

12 |g (z)|2 (1 − |z|2 )p dm(z) f H∞ g D1,−(1− p2 ) .

1,−(1− p 2)

→ D2,p is bounded, then

12 |f (z)| |g (z)| (1 − |z| ) dm(z) D 12 p Uf D1,−(1− p2 ) →D2,p |g(0)| + |g (z)|(1 − |z|2 ) 2 −1 dm(z) .

2

2

2 p

D

If g takes ga as in Theorem 6.1.1, then the mean-value-inequality for |f | over the disk {z ∈ D : |z − a| ≤ 1−|a| 2 }, a ∈ D, implies 2 −2 |f (a)| (1 − |a|) |f (z)|2 dm(z)

D

|z−a|≤ 1−|a| 2

|f (z)|2 |ga (z)|2 (1 − |z|2 )p dm(z),

and hence

f H∞ Uf D1,−(1− p2 ) →D2,p < ∞. Upon noticing

(Mf g) (z) = (Vf g) (z) + (Uf g) (z), z ∈ D,

we can use the previous result on Uf and Theorem 6.1.1 about Vf to get the desired assertion.

6.2 Carleson Embeddings for Dirichlet Spaces Theorem 6.1.1 (ii) leads to the following Carleson embedding theorem. Theorem 6.2.1. Let p ∈ (0, 2) and µ be a nonnegative Radon measure on D. Then p |g|2 dµ g 2 1,−(1− p2 ) , g ∈ D1,−(1− 2 ) D

if and only if µ ∈ CMp .

D

6.2. Carleson Embeddings for Dirichlet Spaces

139

Proof. Suppose that µ is a measure satisfying the above integral inequality for p p ¯z)−1 g ∈ D1,−(1− 2 ) . Fix a ∈ D. Then the function ga (z) = (1 − |a|2 )1− 2 (1 − a 1,−(1− p ) 2 belongs to D with ga D1,−(1− p2 ) 1. For any subarc I ⊂ T centered at ζI , set a = (1 − |I|)ζI . Accordingly, µ S(I) (1 − |a|2 )2−p dµ(z) 1, p |I| |1 − a ¯z|2 S(I) p

namely, µ ∈ CMp . Conversely, let µ ∈ CMp . Then for g ∈ D1,−(1− 2 ) , using Example 5.1.2 we can ﬁnd {λj } 1 g D1,−(1− p2 ) and {aj } ⊂ D such that g(z) = j λj gaj . Furthermore, we use the Minkowski inequality to conclude 12 12 2 |g| dµ ≤ |λj | |gaj (z)|2 dµ(z) {λj } 1 g D1,−(1− p2 ) , D

D

j

as desired. 1,−(1− p 2)

1,−(1− p 2)

⊂ D2,p , a very natural question is: can one replace D Since D by D in Theorem 6.2.1? To answer this, we introduce the Bessel capacity and its corresponding strong type inequality. For α ∈ (0, 1) let Kα (θ) = |θ|α−1 for θ ∈ [−π, π] and be extended to be a 2π-periodical function on R. Next, set π iθ Kα (θ − t)g(eit )dt, g ∈ L2 (T). Kα ∗ g(e ) = 2,p

−π

After that, for a subset E of T, let τ (E) denote the corresponding subset in the interval [−π, π] obtained by the natural identiﬁcation of this interval with R. With these notions, we deﬁne the L2α -Bessel capacity of a set E ⊆ T as Capα (E) = inf g 2L2 (T) : g ∈ L2 (T), g ≥ 0, Kα ∗ g ≥ 1 on τ (E) . It is well known that Capp (·) is a subadditive set function and Capα (I) ≈ |I|1−2α , whenever α ∈ (0, 1/2) holds for any subarc I of T. Therefore p (E), E ⊆ T, p ∈ (0, 1). Cap 1−p (E) H∞ 2

H0p (E)

< ∞ then Cap 1−p (E) = 0. Meanwhile, if 2 Below is the so-called strong type inequality for the Bessel capacity deﬁned on T. Lemma 6.2.2. Let α ∈ (0, 12 ] and L2+ (T) = {f : f ≥ 0, f ∈ L2 (T)}. Then ∞ Capα {ζ ∈ T : Kα ∗ f (ζ) > t} dt2 f 2L2 (T) , f ∈ L2+ (T). 0

140

Chapter 6. As Symbols of Hankel and Volterra Operators

Proof. To verify this assertion, we need the Bessel capacitary notion and its strong type estimate on R. For α > 0 and x ∈ (−∞, ∞), set α ∞ α−1 π|x|2 t dt α − , t 2 exp − Gα (x) = (4π) 2 Γ 2 t 4π t 0 where Γ(·) is still the classical Gamma function. If x is near 0, then ⎧ ⎨ |x|α−1 , α ∈ (0, 1), log |x|−1 , α = 1, Gα (x) ≈ ⎩ 1 , α > 1. The Bessel capacity Capα;R (E) of a set E ⊆ R is deﬁned by Capα;R (E) = inf{ g 2L2 (R) : g ≥ 0, Gα ∗ g ≥ 1 on E}, where the inﬁmum over the empty set is taken usual convolution of Gα and g. In particular, if small length, then |E|1−2α Capα;R (E) ≈ (− log |E|)−1

to be inﬁnity, and Gα ∗ g is the E is an interval with suﬃciently

, α ∈ (0, 12 ), , α = 12 . Now, we make a claim: Capα (E) ≈ Capα;R τ (E) for any E ⊆ T. To verify this claim, suppose f ≥ 0 is in L2 (T) and set f (x) , x ∈ [−2π, 2π], h(x) = 0 , x ∈ R \ [−2π, 2π].

Clearly, h ≥ 0, h ∈ L2 (R) with h L2 (R) f L2 (T) and ∞ π f (x − y)Kα (y)dy h(x − y)Gα (y)dy, x ∈ [−π, π], −π

−∞

thanks to Gα ≈ Kα on [−π, π]. Accordingly, Capα;R τ (E) Capα (E). To get the opposite estimate, it suﬃces to consider any set E for which Capα (E; R) is small. Accordingly, assume Capα (E; R) < for any small number > 0. Then there is a function h obeying h L2 (R) < , h ≥ 0 and Gα ∗ h ≥ 1 on E. Note that Gα decays exponentially at inﬁnity. So the Cauchy–Schwarz inequality ensures that 1 as → 0. h(x − y)Gα (y)dy < 2 |y|≥π But 1 ≤ 2

|y|<π

h(x − y)Gα (y)dy ≈

|y|<π

h(x − y)Kα (y)dy

when x ∈ τ (E).

6.2. Carleson Embeddings for Dirichlet Spaces

141

Now, let f1 and f2 be the periodic extensions of h on [−2π, 0] and [0, 2π] respectively. Then f = c(f1 + f2 ) with an appropriate constant c > 0 will be a test function for Capα (E). Thus, Capα (E) ≤ f 2L2 (T) ≤ c2 h 2L2 (R) , so that

Capα (E) ≤ c2 Capα;R τ (E) .

By the preceding claim, we see that the desired strong type inequality for Capα (·) is a consequence of the strong type inequality for Capα;R (·) below: 0

∞

Capα;R {x ∈ R : Gα ∗ h(x) > t} dt2 h 2L2 (R) , 0 ≤ h ∈ L2 (R).

See also [AdHe, p. 189, Theorem 7.1.1]. Theorem 6.2.3. Let µ be a nonnegative Radon measure on D. (i) If p ∈ [1, 2), then

D

|g|2 dµ g 2D2,p , g ∈ D2,p

is equivalent to µ ∈ CMp . (ii) If p ∈ [0, 1), then

D

|g|2 dµ g 2D2,p , g ∈ D2,p

n n is equivalent to µ j=1 S(Ij ) Cap 1−p ( j=1 Ij ) whenever {Ij }nj=1 are 2 disjoint subarcs of T. Proof. (i) The necessity follows immediately from taking p

¯ −p , gw (z) = (1 − |w|2 )− 2 (1 − wz)

w ∈ D.

As for the suﬃciency, it suﬃces to consider p ∈ (1, 2) since the case p = 1 is the well-known Carleson theorem — see also [Du, p. 157, Theorem 9.3] for a proof. In this case, if µ ∈ CMp then for ﬁxed r ∈ (0, 1), µ B(z, r) µ CMp < ∞, sup p z∈D (1 − |z|) where B(z, r) is the hyperbolic ball with radius r and center z. Using Lemma 3.3.2

142

Chapter 6. As Symbols of Hankel and Volterra Operators

and its notation we get that if g ∈ D2,p , then g ∈ A2,p−2 and hence D

∞

2

|g| dµ ≤

j=1

B(zj ,τ )

|g|2 dµ

∞ µ B(zj , τ )

j=1 ∞

sup z∈B(zj ,τ )

|g(z)|2

µ B(zj , τ ) |g(z)|2 (1 − |z|)p−2 dm(z) p (1 − |z |) j B(z ,2τ ) j j=1

µ CMp |g(z)|2 (1 − |z|)p−2 dm(z),

D

as desired.

n (ii) Assume µ obeys the integral condition for g ∈ D2,p . Let E = j=1 Ij and f be a test function for E — namely — f ≥ 0 and f ∈ L2 (T) with K 1−p ∗ f ≥ 1 2 on τ (E). Thus K 1−p ∗ f ≥ 1Ij for each j = 1, 2, . . . , n. Note that 1 Ij — the Poisson 2

extension of 1Ij — is greater than or equal to n on j=1 S(Ij ). Accordingly,

1 4

on S(Ij ). So we ﬁnd K α∗f ≥

1 4

n 2 2 S(Ij ) ≤ 16 |K µ α ∗ f | dµ f L2 (T) . D

j=1

This clearly produces n n S(Ij ) Cap 1−p Ij . µ 2

j=1

j=1

Now for the converse, suppose that µ satisﬁes that geometric condition for all ﬁnite disjoint collections of subarcs of T. Let f ∈ L2 (T) obey f ≥ 0. Recall iθ iθ 1−p ∗ f (e ) is the nontangential maximal function of K 1−p ∗ f at e , that N K 2 2 and M(g) stands for the Hardy–Littlewood maximal function of g. Then it is well known that N(ˆ g ) M(g) and M(f ) L2 (T) f L2 (T) hold for any g ∈ L2 (T). For t > 0 let K be a compact subset of the level set 1−p ∗ f (z) > t}. Et (f ) = {z ∈ D : K 2

Then there are ﬁnitely many points {wj }nj=1 and subarcs {Iwj } centered at wj /|wj | n such that {S(2Iwj )} covers K. The union j=1 2Iwj can be written as the disjoint

6.2. Carleson Embeddings for Dirichlet Spaces

143

unionof subarcs {Jj }. Clearly, 2Iwj is contained in one of the subarcs Jj and so K ⊆ j S(Jj ). Moreover, our assumption on the nontangential maximal function force iθ 1−p ∗ f )(e ) > t} 2Iwj ⊆ {eiθ : N(K 2

and hence the same thing is valid for each Jj . Consequently, iθ 1−p ∗ f )(e ) > t} µ(K) Cap 1−p {eiθ : N(K 2

2

and the regularity of µ yields iθ 1−p ∗ f )(e µ Et (f ) Cap 1−p {eiθ : N(K ) > t} . 2

2

Using Lemma 6.2.2 and M K 1−p ∗ f ≤ K 1−p ∗ M(f ), 2

we obtain 2 1−p ∗ f | dµ |K D

∞

=

2

0 ∞ 0 ∞ 0 ∞ 0

2

µ Et (f ) dt2 2 iθ 1−p ∗ f )(e ) > t} dt Cap 1−p {eiθ : N(K 2

2

Cap 1−p {eiθ : M(K 1−p ∗ f )(eiθ ) > t} dt2 2

2

Cap 1−p {eiθ : K 1−p ∗ M(f )(eiθ ) > t} dt2 2

2

M(f ) 2L2 (T)

f 2L2 (T) . α

Note that the Fourier coeﬃcients of Kα are of the form cn (1 + n2 )− 2 where an ≈ 1 for all n = 0, ±1, ±2, . . . and the Hardy space H2 is determined by its Taylor coeﬃcients or as the space of harmonic extensions of functions f ∈ L2 (T) whose Fourier coeﬃcients are zero for n = −1, −2, . . .. So if g ∈ D2,p , then there exists a function f ∈ H2 such that 1−p ∗ f (z) and g D 2,p ≈ f H2 ≈ f L2 (T) . g(z) = K 2

Therefore, we ﬁnally derive D

as desired.

|g|2 dµ f 2L2 (T) ≈ g 2D2,p ,

144

Chapter 6. As Symbols of Hankel and Volterra Operators

6.3 More on Carleson Embeddings for Dirichlet Spaces Motivated by Theorem 6.2.3, we give the following deﬁnition of Carleson measures involving the square Dirichlet spaces. Deﬁnition 6.3.1. Let p ∈ (0, ∞). (i) A nonnegative Radon measure µ on D is called a D2,p -Carleson measure if µ S(I) < ∞, p ∈ [1, ∞),

µ CMDp = sup |I|p I⊆T where S(I) still denotes the Carleson box based on the arc I ⊆ T, and n µ S(I ) j j=1 < ∞, p ∈ (0, 1),

µ CMDp = sup n Cap 1−p I j j=1 2

where the supremum is taken over all ﬁnite sequences {Ij }nj=1 of disjoint subarcs on T. (ii) A function f ∈ H is of Wp class provided that |f (z)|2 (1 − |z|2 )p dm(z) is a D2,p -Carleson measure on D. The norm of f ∈ Wp is determined by

f Wp =

sup

g D2,p ≤1

D

12 |g(z)| |f (z)| (1 − |z| ) dm(z) . 2

2

2 p

Referring to the p-Carleson measure characterization of Qp , we read oﬀ Wp ⊆ Qp — but the problem is whether or not its equality can occur. Of course, it turns out from Theorem 6.2.3 that this is the case whenever p ≥ 1. However, we will see that this is no longer true for p ∈ (0, 1). We start with the concept of an interpolating sequence for D2,2−p which can be identiﬁed with A2,−p whenever p ∈ (0, 1). We say that a sequence {wj } in D is an interpolating sequence for D2,2−p provided that the interpolation operator p

I(f ) = {f (wj )(1 − |wj |)1− 2 } : D2,2−p → 2 is surjective. Lemma 6.3.2. Given p ∈ (0, 1). For j ∈ N ∪ {0} and k = 0, 1, . . . , 2n+4 − 1, set Sj,k = {reiθ : 1 − 2−j < r < 1 − 2−j−1 & πk2−j−3 < θ < π(k + 1)2−j−3 } and

ζj,k = (1 − 32−2−n ) exp 2−j−4 (2k + 1)πi .

6.3. More on Carleson Embeddings for Dirichlet Spaces

145

Let Λ = {ζ1 , ζ2 , · · · } be an enumeration of {ζj,k }. Then: (i) Λ is a ﬁnite union of interpolating sequences for D2,2−p . (ii) There is a constant κp > 0 depending only on p such that for any g ∈ D2,p and every nonnegative Radon measure µ, |g|2 dµ ≤ κp g 2D2,p µ CMp + |g(ζ)|2 µ(Sζ ) , D

ζ∈Λ

where Sζ denotes the unique set Sj,k containing ζ. Proof. (i) Fix l − 4 ∈ N and put Λj = ζn,k : k = m2l + j & m, n ∈ N ∪ {0} , j = 0, 1, . . . , 2l − 1. According to [DuSc, p.182], each Λj is a rotation of the so-called Luecking set with parameter γ = 322−l and then [DuSc, p. 183, Corollary] gives that the so-called upper Seip density of {Λj } is equal to 3(log 2)−1 22−l . If 3/2l−3 < (1−p) log 2, then it follows from [HedKZ, p. 158, Theorem 5.22] that each Λj is an interpolating sequence for D2,2−p = A2,−p , as desired. (ii) For w ∈ D and ∈ (0, 1) let ∆(w, r) ⊆ D be the disk with center w, radius (1 − |z|). If is small enough then D = w∈Λ ∆(w, ) and any z ∈ D belongs to at most ﬁnite many ∆(w, ), w ∈ Λ. At the same time, we claim that if 0 < 1 < 2 < 1 then there is a constant κ( 1 , 2 ) > 0 such that for all w ∈ Λ and any g ∈ H one has sup |g(z) − g(w)|2 ≤ κ( 1 , 2 ) |g |2 dm. z∈∆(w,1 )

∆(w,2 )

To see this, we employ the Taylor expansion of g at w: g(z) − g(w) =

∞

cj (z − w)j , |z − w| < 1 − |w|,

j=1

the Cauchy–Schwarz inequality and Parseval’s identity to deduce |g(z) − g(w)|2

≤

∞

∞ 2j 2j j|cj |2 2 (1 − |w|) j −1 ( 1 −1 ) 2

j=1

= π −1

∆(w,2 )

hence establishing the claim.

j=1

|g |2 dm

∞ j=1

2j , j −1 ( 1 −1 2 )

146

Chapter 6. As Symbols of Hankel and Volterra Operators Taking ∈ (0, 1) such that D = D

|g|2 dµ ≤ 2

w∈Λ

∆(w, ), we derive

sup |g(z) − g(w)|2 µ(Sw ) + 2

w∈Λ z∈Sw

w∈Λ

µ(Sw )

∆(w,2 )

|g |2 dm +

−p µ(Sw ) (1 − |w|)

∆(w,2 )

w∈Λ

+

µ(Sw )|g(w)|2

w∈Λ

µ(Sw )|g(w)|2

w∈Λ

|g (z)|2 (1 − |z|)p dm(z)

µ(Sw )|g(w)|2

w∈Λ

g 2D2,p µ CMp +

µ(Sw )|g(w)|2 ,

w∈Λ

hence giving the desired estimate.

Theorem 6.3.3. Let p ∈ (0, 1) and g ∈ D2,p . Then the following statements are equivalent: p

(i) Mg D2,2−p ⊆ D1,−(1− 2 )

D2,2−p .

(ii) Ug Qp ⊆ D2,p . (iii) CMp ⊆ {µ : µ is a nonnegative Radon measure on D with

D

|g|2 dµ < ∞}.

Proof. (iii)⇒(ii) This is evident because of f ∈ Qp ⇔ |f (z)|2 (1 − |z|2 )p dm(z) ∈ CMp . (ii)⇒(i) By the Cauchy–Schwarz inequality and the Closed Graph Theorem, (ii) implies f (z) zg(z)h(z) (1 − |z|2 )dm(z) D = f ghdm D

12 12 2 2 −p |f (z)| |g(z)| (1 − |z| ) dm(z) |h(z)| (1 − |z| ) dm(z) D D |f (0)| + f Qp,1 h D2,2−p ,

2

2

p

hence giving gh ∈ D1,−(1− 2 )

2 p

D2,2−p owing to Theorem 5.2.3.

6.3. More on Carleson Embeddings for Dirichlet Spaces

147

(i)⇒(iii) Suppose (i) holds. For any nonnegative Radon measure µ on D, we take Lemma 6.3.2 into account. More precisely, given an interpolating sequence Γ for D2,2−p . Then, for any ﬁnite set Ω ⊆ Γ there exists a function u0 ∈ D2,2−p and a constant C > 0 depending only upon Γ such that u0 (ζ) =

p (1 − |ζ|) 2 −1 g(ζ) µ(Sζ ) , ζ ∈ Ω, 0 , ζ ∈Γ\Ω

and

u0 2D2,2−p ≤ C

|g(ζ)|2 µ(Sζ ).

ζ∈Ω

Consequently, if u0 = 0 then u = u0 u0 −1 D 2,2−p produces .

|g(ζ)|2 µ(Sζ ) ≤

ζ∈Ω

√ p C g(ζ)u(ζ)(1 − |ζ|)1− 2 µ(Sζ ). ζ∈Ω

p

Now (i) implies gu ∈ D1,−(1− 2 ) Theorem again that

gu D1,−(1− p2 ) D2,2−p ≤

D2,2−p . So it follows from the Closed Graph

sup

v D2,2−p =1

gv D1,−(1− p2 ) D2,2−p = κ(g) < ∞.

This means that gu can be expressed as g(z)u(z) =

∞

p

cj (1 − |λj |2 )1− 2 (1 − λj z)−1 hj (z)

j=1

where λj ∈ D,

hj ∈ D2,2−p ,

hj D2,2−p = 1 and {cj } 1 ≤ 2κ(g).

Meanwhile, since Γ is an interpolating sequence for D2,2−p , we have

|v(ζ)|2 (1 − |ζ|)2−p 1, v D2,2−p ≤ 1.

ζ∈Ω

Putting all the above estimates together and using the Cauchy–Schwarz inequality, the implication 1 − λz ¯ |z − ζ| 1, λ ∈ D, z ∈ Sζ ⇒ ≤1+ ¯ 1 − |ζ| 1 − λζ

148

Chapter 6. As Symbols of Hankel and Volterra Operators

and Lemma 3.1.1, we derive . |g(ζ)|2 µ(Sζ ) ζ∈Γ

.

=

|g(ζ)|2 µ(Sζ )

ζ∈Ω

√ 2 C

≤

(1 − |λ|2 )1− p2 1− p 2 |v(ζ)|(1 − |ζ|) µ(Sζ ) ¯ |1 − λζ| λ∈D, v D2,2−p ζ∈Ω sup

12 (1 − |λ|2 )2−p κ(g) sup µ(S ) ζ ¯ 2 |1 − λζ| λ∈D 12 (1 − |λ|2 )2−p κ(g) dµ(z) ¯ 2 |1 − λz| D

1

2 κ(g) µ CM . p

With the help of this treatment, we can handle the general case. Since the sequence Λ in Lemma 6.3.2 is a union of, say Np , interpolating sequences Γj for D2,2−p , we can use Lemma 6.3.2 to get ⎞ ⎛ Np |g|2 dµ ≤ κp ⎝ g 2D2,p µ CMp + |g(ζ)|2 µ(Sζ )⎠ D

j=1 ζ∈Γj

2 ,

µ CMp g 2D2,p + κ(g)

thereupon establishing (iii). We next verify Wp = Qp for p ∈ (0, 1).

Theorem 6.3.4. Let p ∈ (0, 1). Then Wp is a proper subspace of Qp . Equivalently, p D1,−(1− 2 ) D2,2−p is a proper subspace of D2,p D2,2−p . Proof. Theorem 6.3.3 suggests that we show only that there are f ∈ Qp , g ∈ D2,p satisfying Vf g D2,p = ∞. To that end, it is enough to show that the conditions in Theorem 6.3.3 cannot hold for all g ∈ D2,p . By Theorem 6.2.3, we have µ j S(Ij ) 2 , sup |g| dµ ≈ sup Cap 1−p ( j Ij )

g D2,p ≤1 D 2

where the supremum on the right-hand-side ranges over all ﬁnite disjoint unions of subarcs Ij of T. Suppose that Theorem 6.3.3 is valid for all g ∈ D2,p . Then by Theorem 6.3.3(i) and the Closed Graph Theorem we derive sup

v D2,2−p =1

gv D1,−(1− p2 ) D2,2−p g D2,p , g ∈ D2,p .

6.3. More on Carleson Embeddings for Dirichlet Spaces

149

Accordingly,

µ j S(Ij ) sup µ CMp Cap 1−p ( j Ij ) 2 m holds for all µ ∈ CMp . For every ﬁnite union E = l=1 Il of disjoint subarcs m Il ⊂ T with midpoint eiθl , let µE = l=1 |Il |p δl where δl is the Dirac measure at (1 − |Il |)eiθl . Then µE

m

m S(Il ) = |Il |p Cap 1−p (E) µ CMp .

l=1

l=1

2

Note that µE (S(I)) = 0 whenever |I| < min1≤l≤m |Il |. So, we will construct a Cantor-like set having positive p-dimensional Hausdorﬀ measure. To begin with, let φn = 2−n/p and note that φn −2φn+1 are positive for all n ∈ N. Remove from [0, 1] the open middle interval of length 1 − 2φ1 , denoted by I(2−1 ), obtaining a set K1 which is the union of two closed intervals [0, φ1 ] and [1−φ1 , 1]. Remove from each of these intervals the open middle intervals of length φ1 −2φ2 , denoted by I(2−2 ) and I(3 · 2−2 ), obtaining a set K2 which is the union of four closed intervals of length φ2 . Now proceed by induction. Assume that Kn−1 is constructed. Remove from the diﬀerent intervals making up this set the open middle intervals of length

φn−1 − 2φn , denoted by I(2−n ), . . . , I((2n − 1)2−n ), to obtain Kn . Let K = ∞ n=1 Kn . Then K = {e2πiθ : θ ∈ K ⊂ [0, 1]} is the natural identiﬁcation of K on T. Now En = j Ij,n is the union of subarcs containing K obtained at the n-th stage of the construction of K. More precisely, the 2n subarcs Ij,n of arclength rn are obtained by removing from the middle of each Ik,n−1 a subarc of arclength (1 − 2r)|Ik,n−1 |. In particular, if Ij,n and Ij+1,n are obtained from Ik,n−1 in this way, then by 2 = r−p , |Ij,n |p + |Ij+1,n |p = 2rnp = r(n−1)p = |Ik,n−1 |. To estimate µ CMp , recall ﬁrst that if µEn (S(I)) > 0, then |I| ≥ rn . Denote by κ(I, n) the number of subarcs Ij,n that intersect the arc 2I and notice that if κ(I, n) ≤ 2, then |I|−p µEn (S(I)) ≤ 2, and if κ(I, n) > 2, then |I|−p µEn (S(I)) ≤ 2

m−1

r(n−l)p + |I|−p µEn−m (S(I))

as

κ(I, n − m + 1) > 2.

l=0

The process stops when m is the smallest integer with κ(I, n − m) ≤ 2. In this case we get |I| > rn−m+1 , producing µEn (S(I)) ≤ 2 1 + (1 − rp )−1 . |I|p

150

Chapter 6. As Symbols of Hankel and Volterra Operators

On the other hand, since H0p (K) ≤ µEn

S(Ij,n ) = 1,

j

we conclude 0 = Cap 1−p (K) = lim Cap 1−p (En ), 2 2 n→∞ which however yields limn→∞ µEn j S(Ij,n ) = 0, a contradiction.

6.4 Hankel and Volterra on Dirichlet Spaces In this section, we show that Wp behaves very much like Qp with respect to the Hankel and Volterra operators acting on D2,p . Lemma 6.4.1. Let Tf (z) =

D

f (w)(1 − |w|2 )b−2 dm(w). |1 − z w| ¯b

(i) If β ∈ (−∞, −1) and b > 1 + (1 + β)/2, then T is a continuous operator on L2 D, (1 − |z|2 )β dm(z) . (ii) If α ∈ (−∞, 1/2], β > max{−1, −1 − 2α} and b > max

β + 3 β + 3 , −α , 2 2

then |Tf (z)|2 (1 − |z|2 )β dm(z) is a D2,1−2α -Carleson measure on D whenever |f (z)|2 (1 − |z|2 )β dm(z) is a D2,1−2α -Carleson measure on D. Proof. (i) Rewriting (1 − |z|2 )b−β−2 Tf (z) = f (z)k(z, w)(1 − |z|2 )β dm(z) where k(z, w) = , |1 − z¯w|b D and noticing

D

and

D

k(z, w)(1 − |z|2 )γ+β dm(z) (1 − |w|2 )γ k(z, w)(1 − |w|2 )γ+β dm(z) (1 − |z|2 )γ

provided that γ satisﬁes max{1 − b, −1 − β} < γ < min{0, b − 2 − β} which is valid under the assumption on β and b, we can use the Schur Lemma to derive the required continuity of the operator.

6.4. Hankel and Volterra on Dirichlet Spaces

151

(ii) Thanks to (i), it suﬃces to prove that if dµ(z) = |f (z)|2 (1 − |z|2 )β dm(z) -Carleson measure, thenMTf − T ◦ Mf is a continuous operator from is D 2,1−2α 2 to L D, (1 − |z|2 )β dm(z) . To do so, note that the formula D 2,1−2α

2 2 f (z) g(w) − g(z) MTf g(w) − T ◦ Mf g(w) = dm(z) D |1 − z¯w|b (1 − |z|2 )2−b holds for any g ∈ D2,1−2α . So we consider two cases as follows. Case 1: α = 1/2. Since Lemma 2.5.1 is valid for this α, we apply this case to g, and employ the Cauchy–Schwarz inequality to get |MTf g(w) − T ◦ Mf g(w)|2 (1 − |z|2 )β dm(z) D |g(w) − g(z)|2 |f (z)|2 (1 − |z|2 )β dm(z) dm(z) |1 − z¯w|2b (1 − |z|2 )4−2b+β D D |f (z)|2 (1 − |z|2 )β dm(z) g 2D2,1−2α . D

Case 2: α < 1/2. Since the above-deﬁned measure µ is a D2,1−2α -Carleson measure, from the fact that for > 0 and α < 1/2, the function ¯z)α−1/2−/2 ga (z) = (1 − |a|2 )/2 (1 − a belongs to D2,1−2α we see (1 − |a|2 ) |1 − a ¯z|2α−1− dµ(z) µ CMD1−2α . sup a∈D

D

Now, selecting such an > 0 that those assumptions for Lemma 6.4.1 remain valid with β being replaced by β − , we use Lemma 2.5.1 and the Cauchy–Schwarz inequality again to deduce |MTf g(w) − T ◦ Mf g(w)|2 (1 − |z|2 )β dm(z) D |f (z)|2 (1 − |z|2 )β dm(z) ¯w|1+−2α D D |1 − z |g(w) − f (z)|2 dm(z) (1 − |w|2 )β dm(w) × ¯w|2b−1−+2α (1 − |z|2 )4−2b+β D |1 − z |g(w) − f (z)|2 (1 − |w|2 )β− µ CMD1−2α dm(z)dm(w) ¯w|2b−1−+2α (1 − |z|2 )4−2b+β D D |1 − z µ CMD1−2α g 2D2,1−2α .

152

Chapter 6. As Symbols of Hankel and Volterra Operators ¯ set For γ > −1, z ∈ D and f, g ∈ H(D), z 1 f (w)g(w) dζ (1 − |w|2 )γ dm(w). Hf,γ (g)(z) = ¯ 2+γ 0 (1 − wζ) D

The forthcoming result, similar to Theorem 6.1.1, shows that each Wp exists as a symbol space of Hf,γ acting continuously on D2,p for a suitable γ. Theorem 6.4.2. Let f ∈ H, p ∈ [0, ∞) and γ > (p − 1)/2. Then the following statements are equivalent: (i) Hf,γ exists as a continuous operator from D2,p to D2,p . (ii) Vf is a continuous operator from D2,p to D2,p . (iii) f ∈ Wp .

Proof. (i)⇔(iii) If f ∈ Wp then g ∈ D2,p implies f¯ g ∈ L2 D, (1 − |z|2 )p dm(z) and hence Hf,γ g ∈ D2,p by Lemma 6.4.1 (i). Conversely, suppose Hf,γ extends to a continuous operator on D2,p . So its operator norm Hf,γ D2,p →D2,p < ∞. Furthermore, if g = 1 then Hf,γ (g) (z) =

πf (z) , 1+γ

¯ f ∈ H(D),

and hence

f D2,p Hf,γ D2,p →D2,p < ∞. To prove f ∈ Wp , we may in turn verify that 2 π f (w)g(w) − Hf,γ g (w) (1 − |w|2 )p dm(z) Hf,γ 2D2,p →D2,p g 2D2,p γ + 1 D holds for any g ∈ D2,p . Because of the formula f (z) g(w) − g(z) π (1 − |z|2 )γ dm(z), f (w)g(w) − Hf,γ g (w) = γ+1 (1 − z w) ¯ 2+γ D we can, in a similar manner to proving Lemma 6.4.1 (ii), derive that if p = 0, then 2 π f (w)g(w) − Hf,γ g (w) (1 − |w|2 )p dm(z) D γ+1 f 2D2,0 g 2D2,0 Hf,γ 2D2,0 →D2,0 g 2D2,0 ,

but also that if p > 0, then 2 π f (w)g(w) − Hf,γ g (w) (1 − |w|2 )p dm(z) f 2B g 2D2,p . D γ+1

6.4. Hankel and Volterra on Dirichlet Spaces

153

Of course, it remains to check that the boundedness of Hf,γ must yield

f B Hf,γ D2,p →D2,p < ∞ for p > 0. In fact, choose −n = [(1 − p)/2] — the greatest integer less than or equal to (1 − p)/2 — and put ga (z) =

z 1+n (1 − |a|2 )1+n−p/2 (1 − |a|2 )1+p/2 (1 − |z|2 )γ−p , h (z) = . a (1 − a ¯z)1+n (1 − a ¯z)2+γ

Then

ga D2,p 1,

D

|ha (z)|2 (1 − |z|2 )p dm(z) ≈ 1.

Meanwhile, using the reproducing formula for A2,γ , we ﬁnd Hf,γ ga (w)ha (w)(1 − |w|2 )p dm(w) D (1 − |w|2 )γ 2 1+p/2 f (z)ga (z) dm(w) dm(z) = (1 − |a| ) ¯ 2+γ (1 − z¯w)2+γ D D (1 − aw) π(1 − |a|2 )1+p/2 = f (z)ga (z)(1 − z¯a)−(2+γ) dm(z) γ+1 D π f (z)¯ z 1+n (1 − |a|2 )2+n = γ+1 D (1 − z¯a)3+γ+n π 2 (1 − |a|2 )n+2 f (n+2) (a) , = γ + 1 (2 + γ)(3 + γ) · · · (n + 2 + γ) so that sup (1 − |a|2 )n+2 |f (n+2) (a)| a∈D

Hf,γ D2,p →D2,p sup ga D2,p a∈D

D

12 |ha (z)| (1 − |z| ) dm(z) , 2

2 p

hence implying the desired assertion thanks to the higher derivative characterization of B (cf. Theorem 1.1.3 (v) and Remark 1.1.4). (ii)⇔(iii) This follows readily from the deﬁnitions of Vf , Wp and D2,p , and Theorem 6.2.3. Corollary 6.4.3. Let p ∈ [0, ∞). Then Wp is isomorphic to the dual of D2,p D2,2+p with respect to the pairing f (z)g (z)(1 − |z|2 )p dm(z), f ∈ D2,p D2,2+p , g ∈ Wp . f, gp = D

154

Chapter 6. As Symbols of Hankel and Volterra Operators

2,p D2,2+p and g ∈ Wp . Then Hg is bounded on D2,p Proof. Suppose ∞ f ∈ D ∞ and f = j=1 gj hj with j=1 gj D2,p hj D2,2+p < ∞. Hence by the Cauchy– Schwarz inequality,

|f, gp | =

∞ gj hj , gp j=1

=

∞ gj (z)hj (z)g (z)(1 − |z|2 )p dm(z)

=

∞ Hg,p (gj )hj (z)(1 − |z|2 )p dm(z)

j=1

D

j=1

D

Hg,p D2,p →D2,p

∞

gj D2,p hj D2,p+2 ,

j=1

hence g induces a continuous linear functional on D2,p D2,2+p under the pairing ·, ·p with |f, gp | Hg,p D2,p →D2,p f D2,p D2,2+p . Conversely, assume that L is a continuous linear functional on D2,p D2,2+p . Then its norm L < ∞ and hence for gj ∈ D2,p and hj ∈ D2,2+p we have |L(gj hj )| ≤ L

gj hj D2,p D2,p+2 ≤ gj D2,p hj D2,p+2 , thereby deriving that for ﬁxed gj ∈ D2,p , the mapping hj → L(gj hj ) is continuous from D2,p+2 to C. Using the Riesz–Fischer Theorem, we ﬁnd an element Gj ∈ A2,p such that L(gj hj ) = hj (z)Gj (z)(1 − |z|2 )p dm(z). D

Consequently, g is uniquely determined by gj and gj → Gj is bounded from D2,p to A2,p with Gj A2,p L

gj D2,p . Using the reproducing formula for A2,p , we ﬁnd p+1 p+1 Gj (w) = Gj (z)kp (z, w)(1 − |z|2 )p dm(z) = L gj kp (·, w) , π π D ¯ −2−p for z, w ∈ D. Note that if w is ﬁxed, then gj kp (·, w) where kp (z, w) = (1− wz) 2,p belongs to A . So there is g ∈ D2,p such that gj (z)kp (z, w)g (z)(1 − |z|2 )p dm(z). L gj kp (·, w) = D

Consequently,

Hg,p (gj ) D2,p Gj A2,p L

gj D2,p .

6.4. Hankel and Volterra on Dirichlet Spaces

155

This gives that Hg,p is bounded on D2,p , and so that g ∈ Wp due toTheorem 6.4.2. Now, it is easy to deduce that if f ∈ D2,p D2,p+2 , then f = ∞ j=1 gj hj 2,p 2,p+2 with gj ∈ D and hj ∈ D and hence L(f )

=

=

∞

L(gj hj )

j=1 ∞ j=1

D

j=1

D

gj (z)hj (z)Gj (z)(1 − |z|2 )p dm(z)

gj (z)kp (z, w)g (z) p+1 dm(w) = hj (w) dm(z) 2 −p π (1 − |z| ) (1 − |w|2 )−p D j=1 D

∞ kp (w, z) p+1 dm(z) = g (z)gj (z) hj (w) dm(w) 2 −p π (1 − |w| ) (1 − |z|2 )−p D j=1 D ∞ = gj (z)hj (z)g (z)(1 − |z|2 )p dm(z) ∞

=

D

f (z)g (z)(1 − |z|2 )p dm(z).

This completes the proof.

Corollary 6.4.4. Let p ∈ [0, 2) and f ∈ H. Then Mf is a continuous operator from D2.p to D2,p if and only if f ∈ H ∞ ∩ Wp . Proof. By Theorem 6.4.2, it suﬃces to prove that the continuity of Uf on D2,p implies f ∈ H∞ . To this end, given a ∈ D, let ⎧ ⎨ (1−|a|2 )1−p/2 , p ∈ (0, 2), 1−¯ az ga (z) = − log(1−¯ az) , p = 0, ⎩ √ 2 − log(1−|a| )

and evaluate

ga 2D2,p

⎧ 2 2−p ) (1−|z|2 )p ⎨ 1 + D (1−|a| |1−¯ dm(z) 1 , p ∈ (0, 2), 4 a z| −1 2 −log(1−|a| ) ⎩ 1+ D dm(z) 1 , p = 0. |1−¯ az|2

Using the proof of Corollary 6.1.2, we further derive

f H∞

sup a∈D

D

12 |f (z)|2 |ga (z)|2 (1 − |z|2 )p dm(z)

Uf D2,p →D2,p sup ga D2,p < ∞. a∈D

156

Chapter 6. As Symbols of Hankel and Volterra Operators

A quite natural question is: what condition should f have so that Mf is surjective from D2.p to D2,p ? This question can be solved via the following corona type decomposition. Theorem 6.4.5. Let p ∈ [0, 1) and n ∈ N. For (f1 , f2 , . . . , fn ) ∈ H × H × · · · × H, set n fk gk . M(f1 ,f2 ,...,fn ) (g1 , g2 , . . . , gn ) = k=1

Then the following two statements are equivalent: (i) M(f1 ,f2 ,...,fn ) maps D2,p × D2,p × · · · × D2,p onto D2,p . (ii) (f1 , f2 , . . . , fn ) ∈ (Wp ∩ H∞ ) × (Wp ∩ H∞ ) × · · · × (Wp ∩ H∞ ) with δ(f1 , f2 , . . . , fn ) = inf

n

z∈D

|fk (z)| > 0.

k=1

Proof. Suppose (i) holds. Then the ﬁrst result in (ii) follows right away from Corollary 6.4.4. To see the second assertion of (ii), we use the Open Map Theorem to derive that for each h ∈ D2,p there are g1 , g2 , . . . , gn ∈ Wp ∩ H∞ such that

gk D2,p gk H∞ + gk Wp h D2,p , k = 1, 2, . . . , n, and h =

n k=1

fk gk . Note that every f ∈ D2,p satisﬁes: (1 − |z|2 )p/2 |f (z)| f D2,p ,

and

f ∈ D2,p , p ∈ (0, 1)

− 1 − log(1 − |z|2 ) 2 |f (z)| f D2,0 ,

f ∈ D2,0 .

So, making use of these inequalities, and letting ! h(z) = in

n k=1

(1−|w|2 )1−p/2 1−wz ¯ −1

log(1 − wz) ¯

, p ∈ (0, 1), , p=0

fk gk = h, we obtain that if p ∈ (0, 1), then (1 − |w|2 )1−p/2 ≤ |fk (z)||gk (z)| (1 − |z|2 )−p/2 |fk (z)|, |1 − wz| ¯ n

n

k=1

k=1

6.4. Hankel and Volterra on Dirichlet Spaces

157

thus producing δ(f1 , f2 , . . . , fn ) > 0. Similarly, if p = 0, then log |1 − wz| ¯ −1

≤

n

|fk (z)|gk (z)|

k=1

n 12 2,0 |fk (z)| − log(1 − |z| ) h D

n − log(1 − |z|2 ) |fk (z)|,

2

k=1

k=1

and hence δ(f1 , f2 , . . . , fn ) > 0. Conversely, assume (ii) is true. If we can show D2,p ⊆ M(f1 ,f2 ,...,fn ) (D2,p × · · · × D2,p ), then we are done. For f ∈ D2,p choose f f¯k , hk = n 2 j=1 |fj |

k = 1, . . . , n.

n Then (h1 , h2 , . . . , hn ) is a solution to the equation k=1 fk hk = f. But this solution is not holomorphic, and thus must be modiﬁed. Without loss of generality, by ¯ Now, suppose the normal family principle we may assume that each fk is in H(D). ¯ we can ﬁnd functions bj,k , 1 ≤ j, k ≤ n, deﬁned on D such that ∂bj,k (z) ∂hk (z) = f hj , ∂ z¯ ∂ z¯

z ∈ D,

and such that the boundary value functions bj,k are of D2,p (T), namely, T

Then g k = f hk +

T

n j=1

|bj,k (ζ) − bj,k (η)|2 |dζ||dη| < ∞. |ζ − η|2−p

(bk,j − bj,k )fj ∈ D2,p

and

n

fk gk = f.

k=1

Accordingly, we have only to demonstrate that ∂∂bz¯ = f h is solvable in D2,p (T), ∞ 2 2 p k where b = bj,k and h = hj ∂h ∂ z¯ . Since each fk is in Wp ∩H , |fk (z)| (1−|z| ) dm(z) 2,p is a D -Carleson measure. Also since 2 nk=1 |fk (z)|2 2 |h(z)| ≤ 6 , δ(f1 , f2 , . . . , fn )

158

Chapter 6. As Symbols of Hankel and Volterra Operators

[Ga, p. 326], |h(z)|2 (1 − |z|2 )p dm(z) is a D2,p -Carleson measure. Note that a standard solution for the equation ∂∂bz¯ = f h is given by f (w)h(w) b(z) = π −1 dm(z) z−w D which is continuous on C and is C 2 on D. To prove that this function belongs to D2,p (T), we will show |∇b(z)|2 (1 − |z|2 )p dm(z) < ∞. D

To this end, we use |h(z)|2

n

|fk (z)|2 , f ∈ D2,p and fk ∈ Wp ∩ H∞

k=1

to get ∂b(z) 2 |f (z)h(z)|2 (1 − |z|2 )p dm(z) (1 − |z|2 )p dm(z) = ∂ z¯ D D n |(f fk ) (z)|2 (1 − |z|2 )p dm(z) k=1

D

n

+

k=1

D

|f (z)fk (z)|2 (1 − |z|2 )p dm(z)

< ∞.

Meanwhile, let B(F ) denote the Beurling transform of a function F ∈ L1loc (C, dm), that is, the following principal value integral: F (w) dm(w), z ∈ C. B(F )(z) = p.v. (z − w)2 C Then ∂b/∂z = π −1 B(F ) whenever f (z)h(z) , z ∈ D, F (z) = 0 , z ∈ C \ D. Observe that |1 − |z|2 |p is an A2 -weight [CuRu, p. 411] when p ∈ [0, 1), namely, 1 1 1 − |z|2 p dm(z) 1 − |z|2 −p dm(z) < ∞, sup m(∆) ∆ m(∆) ∆ ∆ wherethe supremum p is taken over all Euclidean disks ∆ in C. Thus B is bounded on L2 C, 1 − |z|2 dm(z) . Consequently, ∂b(z) 2 |F (z)|2 |1 − |z|2 |p dm(z) (1 − |z|2 )p dm(z) ∂z D C ≈ |f (z)h(z)|2 (1 − |z|2 )p dm(z) < ∞. D

6.5. Notes

159

The above-established estimates, along with |∇| ≤ 2 ∂/∂ z¯ + ∂/∂z , imply the required result.

6.5 Notes Note 6.5.1. Section 6.1 is produced via modifying and improving [AlCaSi, Corollary 4.1 and Theorem 5.2]. Note 6.5.2. Section 6.2 consists of Theorem 6.2.1 (from [Xi5]), Lemma 6.2.2 and Theorem 6.2.3 (from [Ste1]). From Theorem 6.2.1 it turns out that p |g|dµ g D1,−(1− p2 ) , g ∈ D1,−(1− 2 ) ⇔ µ ∈ CMp/2 . D

An analog and generalization of the Carleson-type embedding in Theorem 6.2.3 (ii) can be established by introducing the Besov capacities and their strong type inequalities through the boundary value functions of the Dirichlet spaces. In fact, for p ∈ (−1, 1) let L2p (T) be the space of all real-valued functions f ∈ L2 (T) with

f L2p (T) = f L2(T) +

T

T

12 |f (ζ) − f (η)|2 |dζ||dη| < ∞. |ζ − η|2−p

Given a compact subset K of T, deﬁne Cap K; L2p (T) = inf{ f 2L2p(T) : f ≥ 1K } as the Besov L2p (T)-capacity of K. This deﬁnition extends to any set E ⊆ T in terms of Cap E; L2p (T) = sup{Cap K; L2p (T) : compact K ⊆ E}. From [Wu3, Theorem 2.2] we see that the following capacitary strong type inequality holds for any f ∈ L2p (T), 0

∞

Cap {ζ ∈ T : |f (ζ)| > t}; L2p (T) dt2 f 2L2p (T) .

This estimate, plus the following two facts:

N(f ) L2p (T) f L2p (T) and {z ∈ D : |f (z)| > t} ⊆ T {ζ ∈ T : N(f )(ζ) > t} ; see also [Wu3, Lemmas 2.3 and 3.2], implies that a nonnegative Radon measure µ on D is D2,p -Carleson measure if and only if µ T (O) Cap O; L2p (T) holds for any open set O ⊂ T – see also [Ve, Theorem D].

160

Chapter 6. As Symbols of Hankel and Volterra Operators

Note 6.5.3. Section 6.3 is taken from [AlCaSi, Section 5]. The extreme case W0 = Q0 = D2,0 = D is a consequence of [Ste1, Theorem 4.2]. In addition, the following inequality for p ∈ (0, 1) is of independent interest:

g qDq,p f qQp ,1

⎧ ⎨

|g(z)|q |f (z)|q (1 − |z|2 )p dm(z) , q ∈ (1, 2), −1 2 ⎩ |g(z)|q |f (z)|q (1 − |z|2 )p log 1−|z| dm(z) , q = 2. D D

See also [AlCaSi, Theorem 5.2 (iii)-(iv)] for more details. Note 6.5.4. Section 6.4 is formed via modifying [RocWu, Lemmas A, 4 and 6; Theorem 2 ], [Wu1, Theorem 1] and [Xi2, Theorems 3.1 and 3.4]. There are many works on the Volterra operators acting on holomorphic spaces — see also [Sis1] and [Sis2]. A follow-up problem of Corollary 6.4.3 is to prove or disprove: Wp is isomorphic to D2,p D2,2−p , p ∈ [0, 2) with the Cauchy pairing — equivalently, Hf : D2,p → D2,2−p is continuous if and only if f ∈ Wp , since the Cauchy dual of D2,2−p is isomorphic to D2,p , but also since according to the well-known equivalent principle between the boundedness of the Hankel operator and the Cauchy duality of weak factorization (cf. [JaPeSe], [CohnVe]) one has Hf : D2,p → D2,p ⇔ f ∈ Θ (D2,p

D2,2−p )∗ .

Of course, the key point is to consider whether any element in the Cauchy dual of D2,p D2,2−p induces an element of Wp — here [Wu2] is maybe helpful. This is because: if f ∈ Wp

and g =

∞

uj vj ∈ D2,p

D2,2−p

j=1

where uj ∈ D2,p

and vj ∈ D2,2−p ,

then f, g = 2πf (0)g(0) + 2 lim

r→1

D

f (rz)g (rz)(− log |z|2 )dm(z)

6.5. Notes and

161

(uj vj ) (z)g (z)(− log |z|2 )dm(z) D ≤ |uj (z)||vj (z)||g (z)|(− log |z|2 )dm(z) D + |uj (z)||vj (z)||g (z)|(− log |z|2 )dm(z) D 12 |vj (z)|2 |g (z)|2 (1 − |z|2 )2−p dm(z) uj D2,p D 12 + vj D2,2−p |uj (z)|2 |g (z)|2 (1 − |z|2 )p dm(z) D

uj D2,p vj D2,2−p g B + uj D2,p vj D2,2−p g Wp uj D2,p vj D2,2−p g Wp . Here we have used the inclusion: Wp ⊆ Qp ⊆ B, p ∈ [0, 2). Moreover, Theorem 6.4.5 describes the corona structure of Wp ∩H∞ , p ∈ [0, 1) which says: Mf1 ,f2 ,...,fn : (Wp ∩ H∞ ) × (Wp ∩ H∞ ) × · · · × (Wp ∩ H∞ ) → Wp ∩ H∞ is onto if and only if (f1 , f2 , . . . , fn ) ∈ (Wp ∩ H∞ ) × (Wp ∩ H∞ ) × · · · × (Wp ∩ H∞ ) with δ(f1 , f2 , . . . , fn ) = inf

z∈D

n

|fk (z)| > 0.

k=1

In particular, the case p = 0 gives an aﬃrmative answer to [BroSh, Question 18]. This assertion can be proved by using Lemma 6.4.1 (ii) and an explicit L∞ -solution to the ∂¯ equation — see [Jo] and [Xi2, Theorem 3.4]. Additionally, for a discussion on the stable rank of Qp ∩ H∞ , p ∈ (0, 1), see also [PauS].

Chapter 7

Estimates for Growth and Decay We present in this chapter several of the size estimates that arise in the process of determining growth and decay of a Qp -function. The details are included in the following four sections: • • • •

Convexity Inequalities; Exponential Integrabilities; Hadamard Convolutions; Characteristic Bounds of Derivatives.

7.1 Convexity Inequalities For convenience, we begin by recalling some basic concepts on Calder´on’s complex interpolation. Suppose (X0 , X1 ) is a compatible couple of complex Banach spaces with norms · X0 and · X1 , i.e., there is a Hausdorﬀ topological vector space V containing both X0 and X1 . Then X0 ∩ X1 and X0 + X1 are two subspaces of V, but also Banach spaces equipped with norms below:

f X0 ∩X1 = max f X0 , f X1 and

f X0 +X1 = inf f0 X0 + f1 X1 , f = f0 + f1 , f0 ∈ X0 , f1 ∈ X1 . A Banach space X is called an intermediate space between X0 and X1 provided X0 ∩ X1 ⊆ X ⊆ X0 + X1 .

164

Chapter 7. Estimates for Growth and Decay

Given S = {z ∈ C : 0 ≤ z ≤ 1} and S ◦ = {z ∈ C : 0 < z < 1}. Let F (X0 , X1 ) be a family of mappings F : S → X0 + X1 such that (i) F is holomorphic on S ◦ and continuous on S; (ii) supz∈S F (z) X0 +X1 < ∞; (iii) F (iy) ∈ X0 and F (1 + iy) ∈ X1 for all y ∈ R; (iv) y → F (iy) and y → F (1 + iy) are continuous and bounded on R. Then F (X0 , X1 ) becomes a Banach space once it is equipped with the norm

F F (X0,X1 ) = max sup F (iy) X0 , sup F (iy) X1 . y∈R

y∈R

And hence, for s ∈ (0, 1), the complex interpolation space [X0 , X1 ]s is deﬁned by [X0 , X1 ]s = f ∈ X0 + X1 : f = F (s), F ∈ F(X0 , X1 ) with the norm

f [X0,X1 ]s = inf F F (X0,X1 ) : f = F (s) . When regarding D, BMOA and B as the endpoint spaces in the chain of Qp , p ∈ [0, ∞), we naturally see the following result. Example 7.1.1. Let s ∈ (0, 1). Then 2 consisting of all f ∈ H with (i) [D, B]s equals the holomorphic Besov space B 1−s

f [D,B]s =

D

1−s 2 2 2s |f (z)| 1−s (1 − |z|2 ) 1−s dm(z) < ∞.

(ii) [D, BMOA]s comprises all f ∈ H with

f [D,BMOA]s =

T

0

1

2

2 s

|f (rζ)| (1 − r ) rdr

1 1−s

1−s 2 |dζ| < ∞.

(iii) [D, BMOA]s ⊂ [D, B]s ⊂ Qs . Proof. (i) See [Zhu1, Theorem 5.3.8]. (ii) As for this assertion, we note the boundary behavior of functions in D and BMOA respectively, and use [Tr, p. 45, Theorem (i) (4) where s0 = 12 , p0 = 2 ] to derive that f ∈ 2, p1 = 2, p = 2; s1 = 0, q0 = 2, q1 = ∞, q = 1−s [D, BMOA]s if and only if f is holomorphic on D and its boundary value function f on T obeys 1 1−s |f (ζ) − f (η)|2 |dζ| |dη| < ∞. |ζ − η|2−s T T

7.1. Convexity Inequalities

165

This ﬁniteness condition amounts to (cf. e.g. [Ve, Theorem F]) T

1

0

2

|f (rζ)| (1 − r) rdr

1 1−s

s

|dη| < ∞.

2 (iii) The ﬁrst strict inclusion follows from (i) and (ii). Since B 1−s ⊂ Qs , we conclude [D, B]s ⊂ Qs .

Motivated by Example 7.1.1, we can establish some convexity inequalities and their corresponding dual forms which can be viewed as the improved isoperimetric inequalities without sharp constants. Theorem 7.1.2. For p1 , p2 ∈ [0, ∞) and s ∈ [0, 1] let p = sp1 + (1 − s)p2 . Then s 1−s Ep (f, w) ≤ Ep1 (f, w) Ep2 (f, w) and

s 1−s Fp (f, w) ≤ Fp1 (f, w) Fp2 (f, w)

hold for f ∈ H and w ∈ D. Consequently: p (i) f Qp ,1 ≤ f 1−p D f BMOA , p (ii) f D f 1−p Pp f H ,

(iii) f Qp ,1 f sD f 1−s B , p (iv) f D f 1−p Pp f B1 ,

f ∈ D, p ∈ [0, 1].

f ∈ H , p ∈ (0, 1]. f ∈ D, s ∈ (0, 1], 1 − s < p ≤ 1. f ∈ B1 , p ∈ (0, 1].

Proof. When p1 , p2 ∈ [0, ∞) and s ∈ (0, 1) satisfy p = sp1 + (1 − s)p2 , H¨ older’s inequality implies

=

Ep (f, w) 1 sp1 +(1−s)p2 2 2s+2(1−s) |f (z)| − log |σw (z)| D

≤ =

s2

1−s 2 |f (z)|2 dm(z) dm(z) −p1 −p2 D − log |σw (z)| D − log |σw (z)| s 1−s Ep1 (f, w) Ep2 (f, w) , f ∈ H, w ∈ D. |f (z)|2

This argument also works for Fp (·, ·). (i) and (iii) follow from the second convexity inequality above with p = s, p1 = 0, p2 = 1 and p = (1 − s)p2 , p2 > 1 (which ensures Qp2 = B), p1 = 0 respectively.

166

Chapter 7. Estimates for Growth and Decay

p (ii) Suppose ω is a nonnegative function on D such that ω LN 1 (H∞ ) ≤ 1. Then ω(z) (1 − |z|2 )−p . By H¨older’s inequality, we have that if f ∈ H , then f ∈ Pp ⊂ D with f D f H and hence

f 2D |(1 − |z|2 ) zf (z) |2 dm(z)

D

1−p −1 |(1 − |z|2 ) zf (z) |2 ω(z) (1 − |z|2 )−p dm(z) D p 1−p 2 2 2 1−p p × |(1 − |z| ) zf (z) | ω(z) (1 − |z| ) dm(z)

1−p −1 |(1 − |z|2 ) zf (z) |2 ω(z) (1 − |z|2 )−p dm(z) D p × |(1 − |z|2 ) zf (z) |2 dm(z)

D

D

D

D

1−p −1 2 2 −p |(1 − |z| ) zf (z) | ω(z) (1 − |z| ) dm(z)

f 2p D 2

1−p −1 |(1 − |z|2 ) zf (z) |2 ω(z) (1 − |z|2 )−p dm(z)

f 2p H .

By taking the inﬁmum over all ω in the above estimates, we derive 2(1−p)

f 2D f Pp

f 2p H .

(iv) Observe that (cf. Note 4.6.4) B1 ⊂ H ⊂ Pp ⊂ D, argument for (ii) gives (iv) right away.

p ∈ (0, 1). So the

In view of Theorem 7.1.2 we can next handle Qp -seminorms of the dilated functions fr (z) = f (rz), r ∈ (0, 1) via the product between an f -free quantity and a convex multiplication of both Qq -seminorm and B-seminorm of f ∈ H. Theorem 7.1.3. Let p, q ∈ [0, ∞) and s ∈ [1, ∞]. If f ∈ Qq and r ∈ (0, 1), then 1 1− 1 C1 (p, q, r, s) f Qq s f Bs

fr Qp ≤ where ⎧ ⎪ ⎨ C1 (p, q, r, s) =

2r 1+r2

0

⎪ ⎩

sup

(− log t)ps (− log t)q(s−1) p

(− log t) (− log t)q

dt2 (1−t2 )2

: 0
1s

2r 1+r 2

Proof. Case 1: s ∈ (1, ∞). For r ∈ (0, 1) and w ∈ D let Drw = σrw ({z ∈ D : |z| < r}).

,

s ∈ [1, ∞),

,

s = ∞.

7.1. Convexity Inequalities Then

167

Drw ⊂ {z ∈ D : |z| < 2r(1 + r2 )−1 }.

Applying the maximum principle to the function z over Drw , σw r−1 σrw (z) we derive |z| ≤ |σw r−1 σrw (z) | and consequently, p

fr 2Qp = sup |f (z)|2 − log |σw (r−1 z)| dm(z) w∈D |z|≤r p = sup |(f ◦ σrw ) (z)|2 − log σw r−1 σrw (z) dm(z) w∈D Drw p ≤ sup |(f ◦ σrw ) (z)|2 − log |z| dm(z). z →

w∈D

Next, writing

Drw

g(z) = (1 − |z|2 )2/s (− log |z|)q(s−1)/s−p ,

and using the H¨ older inequality plus the estimate (1 − |z|2 )|(f ◦ σrw ) (z)| ≤ f B , we compute

fr 2Qp

≤

sup w∈D

Drw

⎛

1s |g(z)|−s dm(z)

⎞ s−1 s q |(f ◦ σrw ) (z)| − log |z| ⎝ ⎠ × sup 2 dm(z) w∈D Drw (1 − |z|2 )|(f ◦ σrw ) (z)| 1−s 2r 1s 1+r 2 2/s 2(s−1)/s |g(t)|−s dt2 f B f Qq

2

0

2/s

2(s−1)/s

C1 (p, q, r, s) f B f Qq

,

hence getting the desired inequality. Case 2: s = 1. This follows from the limit of Case 1 as s → 1. 2/s Case 3: s = ∞. This can be checked directly because the factor f B is replaced by 1. The following two extreme cases of Theorem 7.1.3 are quite interesting. First, if s = ∞ then !

f Qp , p ∈ [0, ∞), p−1

fr Qp ≤ 1+r 2 2

f Q1 , p ∈ [0, 1], log 2r

168

Chapter 7. Estimates for Growth and Decay

and

fr D ≤

Second, if s = 1 then 2r 1+r 2

fr BMOA 0

1 + r 2 − 2

f Qp , 2r p

log

p ∈ [0, ∞).

1 1 − log t 2 2 2 −1 2 dt

f

log(1 − r )

f B . B (1 − t2 )2

We demonstrate next that the last estimate can be improved under a superharmonic condition. Theorem 7.1.4. Let p, q ∈ [0, ∞), s ∈ [1, ∞) and ps − q(s − 1) > 0. If f ∈ Qq and r ∈ (0, 1), then 1 1− 1

fr Qp ≤ C2 (p, q, r, s) f Qq s f Bs , where

C2 (p, q, r, s) = π

r

0

r ps−q(s−1) dt2 log t (1 − t2 )2

1s .

Proof. Since ps − q(s − 1) > 0 ensures that z → (− log |z|)ps−q(s−1) is a superharmonic function on D and vanishes on T, we conclude that the radial function ps−q(s−1) r2 dm(z) − log |σw (z)| L(p, q, r, s; w) = sup (1 − r2 |z|2 )2 w∈D D is superharmonic on D and so assumes the maximum at the origin; this is, r2 sup L(p, q, r, s; w) = dm(z) (− log |z|)ps−q(s−1) (1 − r2 |z|2 )2 w∈D D s = C2 (p, q, r, s) . Next we employ H¨older’s inequality with s ∈ (1, ∞) to derive 2s 2 2 p 2(s−1) r s Ep (fr , w) − log |σw (z)| dm(z) ≤ f B |fr (z)| s 2 1 − |rz| D s−1 s 2 q 2 s |fr (z)| − log |σw (z)| dm(z) ≤ C2 (p, q, r, s) f B D

≤

2 s

2(s−1)

C2 (p, q, r, s) f B f Qqs

,

hence establishing the desired inequality. By a slight change of the preceding argument, we can readily verify that the case s = 1 is also true. Remark 7.1.5. Here it is worthy of special remark that Theorem 7.1.4 with p = 1, q = 2 and s = 1 goes back to the Korenblum inequality below: 1

fr Q1 ≤ f B 2−1 π| log(1 − r2 )| 2 .

7.2. Exponential Integrabilities

169

7.2 Exponential Integrabilities In this section we concern ourselves with the majorization and length-area principle via the exponential growth of a holomorphic function. First of all, we compare the quadratic Dirichlet spaces with the Bergman and Hardy spaces. Theorem 7.2.1. Let p ∈ [0, 2). Then ! 2,p

D ⊂

A2,0 q 0
p ∈ (1, 2), p ∈ [0, 1].

, ,

p

Furthermore, if f H2 > 0, p ∈ [0, 1] and ∈ (−p, 1 − p), then 1

f D2,p 1−p− |f (ζ)| 2 |f (ζ)| log |dζ| . exp

f H2

f H2 T f H2 ∞ Proof. Case 1: p ∈ (1, 2). If f (z) = j=0 aj z j lies in D2,p , then

f A2,0 ≈

∞

(1 + j)−1 |aj |2

12

∞ 12 1−p (1 + j 2 ) 2 |aj |2 .

j=0

j=0

j/2 2j z . This function To see the strictness of the inclusion, we take f (z) = ∞ j=0 2 2,0 2,p is in A \ D for p ∈ (1, 2). Case 2: p ∈ [0, 1]. We make the following consideration. If p = 0, then D ⊂ BMOA ⊂ ∩q<∞ Hq , as desired. If p = 1, then D2,p = H2 which is contained properly in ∩0
f 2D2,p ≈

∞

(1 + j 2 )

1−p 2

|aj |2 .

j=0

Accordingly, a combination of the Hausdorﬀ–Young Theorem (cf. [Du, Theorem 6.1]) and the Cauchy–Schwarz inequality implies that if q = α/(α − 1) ∈ (0, 2/p), α ∈ (1, 2] and α > 2p, then

f qHq

∞

|aj |α

j=0

∞

|aj |2 (1 + j 2 )

1−p 2

α2

≈ f α D 2,p ,

j=0

as required. Of course, the fact that f (z) =

∞ j=1

shows the strict inclusion.

j

j −2 z 2

belongs to

+ 2 0
Hq \ D2,p

170

Chapter 7. Estimates for Growth and Decay

To verify the second assertion, we use the H¨older inequality and the ﬁrst inclusion, and we derive that for 2 < q < 2/(p + ) and 0 < < 1 − p, 2−q(p+) q−2 |f (ζ)|q |dζ| = |f (ζ)| 1−p− |f (ζ)| 1−p− |dζ| T

T

≤

T

f

|f (ζ)|

q−2 1−p− D 2,p

2 p+

(p+)(q−2) 1− (p+)(q−2) 2(1−p−) 2(1−p−) 2 |dζ| |f (ζ)| |dζ|

T

T

1− (p+)(q−2) 2(1−p−) 2 |f (ζ)| |dζ| .

Accordingly, if T |f (ζ)|2 |dζ| = 1, then |f (ζ)|2 |dζ| can be treated as a probability measure on T and hence T

1 1 q−2 q−2 1 1−p− |f (ζ)|q−2 |f (ζ)|2 |dζ| = |f (ζ)|q |dζ| f D . 2,p

T

Letting q → 2, we ﬁnd exp T

1 1−p− |f (ζ)|2 log |f (ζ)||dζ| f D . 2,p

The general case follows from considering f f −1 H2 .

As a matter of fact, the ﬁrst part of Theorem 7.2.1 is a sort of holomorphic Sobolev inequality — its conformally invariant form appears to be the well-known embedding Qp ⊂ BMOA for p ∈ [0, 1). Meanwhile, the second part of Theorem 7.2.1 may be viewed as the logarithmic Sobolev inequality associated with the weighted Dirichlet space. When exploring its conformally invariant form, we feel that it is necessary to investigate certain exponential integrals of functions in Qp through a dual form of L2α -Bessel capacity. To be more speciﬁc, for a compact set K ⊂ C let µ be a positive measure on C that takes 1 on K and 0 on C \ K. Then we introduce the potential log |z − ζ|−1 dµ(ζ) , α = 0, K P(µ, α)(z) = |z − ζ|−α dµ(ζ) , α ∈ (0, 2), K and then deﬁne the α-capacity of K as −1 . capα (K) = inf sup P(µ, α)(z) µ

z

The compact sets K satisfying capα (K) = 0 have the property that a countable −1 < ∞, then there is a union of them has α-capacity 0. Of course, if capα (K) unique µα such that the corresponding potential P(µα , α)(z) is not greater than

7.2. Exponential Integrabilities

171

−1 −1 for all z ∈ C but also is equal to capα (K) for all z ∈ K except capα (K) perhaps for a set with α-capacity 0. For the general set E, we deﬁne capα (E) = sup capα (K) K⊂E

where the supremun is taken over all compact subsets K of E. We need the following three auxiliary results. The ﬁrst one is the Beurling weak-type estimate for the extreme case cap0 (·) — the logarithmic capacity. ¯ For r ∈ (0, 1] let M = sup Theorem 7.2.2. Given f ∈ H(D). |z|≤r |f (z)| < ∞. Then

t ds cap0 {ζ ∈ T : |f (ζ)| > t} ≤ r−1/2 exp −π . M {z∈D:|f (z)|=s} |f (z)||dz| Proof. For a domain Ω ⊂ C and two subsets E1 , E2 of the boundary ∂Ω, suppose ρΩ (E1 , E2 ) is the extremal distance between E1 and E2 with respect to Ω. For example, if Tr1 and Tr2 are the circles centered at the origin with radii r1 and r2 and if 0 < r1 < r2 and Ar1 ,r2 denotes the annulus bounded by Tr1 and Tr2 , then r2 ρAr1 ,r2 Tr1 , Tr2 = (2π)−1 log . r1 For the sake of simplicity, we will remove the subscript Ω in the sequel. According to [Ahl, Theorems 2-4 and 4-9], we have that if γs = {z ∈ D : |f (z)| = s}, then using the hypothesis M = sup|z|≤r |f (z)| < ∞ for r ∈ (0, 1], log cap0 {ζ ∈ T : |f (ζ)| > t} ≤ −π lim ρ(Ts , γt ) + (2π)−1 log s s→0 ≤ −π lim ρ(Ts , Tr ) + ρ(Tr , γM ) + ρ(γM , γt ) + (2π)−1 log s s→0 −1 ≤ − 2 log r + πρ(γM , γt ) . To complete the argument, we next deﬁne a metric on D by −1 ρ(z) = |f (z)| |f (z)||dz| , z ∈ γs γs

and let ΩM,t = {z ∈ D : M < |f (z)| < t}. Then

2 ρ(z) dtdy

t

|f (z)|

=

ΩM,t

M

t

γs

= M

γs

γs

−2 |f (w)||dw| |dz|ds

−1 |f (z)||dz| ds.

172

Chapter 7. Estimates for Growth and Decay

Consequently, if γ is a curve in ΩM,t such that γ(0) ∈ γM and γ(1) ∈ γt , then

−1 |f (z)||dz| |dw| ≥

ρ(z)|dz| = f (γ)

γ

t

M

γs

−1 |f (z)||dz| ds.

γs

Taking the inﬁmum in the last estimate over all the possible curves γ, we derive −1 t ρ(γM , γt ) ≥ |f (z)||dz| ds, M

γs

hence deducing the desired inequality. The second one is Marshall’s Lemma as follows.

Lemma 7.2.3. There exists a numerical constant r > 0 with the following property: if f ∈ D is normalized by f (0) = 0 and f 2D ≤ π, then there exists a constant M depending only on f , with M ∈ (0, 1]; {z ∈ D : |z| < r} ⊆ {z ∈ D : |f (z)| < M }; and

M

0

{z∈D: |f (z)|=s}

|f (z)||dz| ds ≥

πM 2 . 3

Proof. Since f ∈ D has the growth 1 1 |f (z)| ≤ π − 2 f D − log(1 − |z|2 ) 2 , √ we conclude from the hypothesis that |z| < 1 − e−M 2 implies |f (z)| < M . In the meantime, t 2 πt ≤ |f (z)||dz| ds as t ∈ (0, 1) is small. 0

γs

Clearly, if 2−1 πt2 ≤

t 0

|f (z)||dz| ds

for all t ∈ (0, 1),

γs

√ then we choose M = 1 and r = 1 − e−1 . If this is not the case, then we may suppose that there is an M ∈ (0, 1] obeying M −1 2 3 πM ≤ |f (z)||dz| ds ≤ 2−1 πM 2 . 0

γs

Consequently, f (D) omits at least two points w1 , w2 ∈ C with 2−1 M ≤ |w1 | ≤ |w2 | ≤ M

and |w2 − w1 | ≥ 2−1 M.

7.2. Exponential Integrabilities

173

Now, let h be the covering map from D onto C \ {0, 1} and h−1 its inverse. Then f (z) − w 1 g(z) = h−1 w2 − w1 deﬁnes an element of H∞ with g H∞ ≤ 1. Note that w w 1 2 2−2 ≤ ≤ 2 and 2−2 ≤ . w2 − w1 w2 − w1 So, it follows that |g(0)| < c < 1 for a numerical constant c. An application of Schwarz’s Lemma to g yields that if r > 0 (independent of f and M ) is suﬃciently small, then (f (z) − w1 )/(w2 − w1 ) must be in a disk of radius at most 2−3 and hence |f (z)| < M whenever |z| < r. We are done. The third one is the well-known Moser’s Theorem. Theorem 7.2.4. There exists a constant κ > 0 such that t ∞ ∞ 2 2 φ(t) dt ≤ 1 ⇒ exp φ(s)ds − t dt ≤ κ. 0

0

0

Proof. Assume t 2 φ(s)ds ≤ s , Es = t ≥ 0 : t −

s > 0.

0

To check the theorem, we just verify that t1 , t2 ∈ Es and 2s ≤ t1 < t2 imply t2 − t1 ≤ 20s. For then we have t ∞ ∞ 2 exp φ(s)ds − t dt = |Es |e−s ds ≤ 22. |Es | ≤ 22s and 0

0

0

Needless to say, in the above and below, |E| stands for the Lebesgue measure of a set E ⊆ (−∞, ∞). Now the Cauchy–Schwarz inequality gives: t ∞ t 2 2 2 t ∈ Es ⇒ t − s ≤ φ(x) dx ≤ t − t φ(x) dx, φ(x)dx ≤ t 0

0

t

2 ∞ and so t φ(x) dx ≤ s/t. Using this and the Cauchy-Schwarz inequality again, we further derive t1 2 t2 φ(x)dx + φ(x)dx t2 − s ≤ 0

t1

∞ 1 2 12 2 1 2 φ(x) dx ≤ t1 + (t2 − t1 ) 2 t1 ≤ t1 + 2 (t2 − t1 )s + 2−1 (t2 − t1 ), hence getting t2 − t1 < 20s.

174

Chapter 7. Estimates for Growth and Decay

Now, we are at a position to state the Chang–Marshall Theorem for the exponential growth of the Dirichlet space. Theorem 7.2.5. There exists a constant κ > 0 such that if f ∈ D with f (0) = 0 2 and f 2D ≤ π, then T e|f (ζ)| |dζ| ≤ κ. ¯ and set Proof. Without loss of generality, we may assume that f ∈ H(D) |f (z)||dz|. L(f, s) = {z∈D: |f (z)|=s}

Using [Ahl, Theorem 2-7], Theorem 7.2.2 and Lemma 7.2.3 we ﬁnd sin 2−2 |{z ∈ T : |f (ζ)| > t}| ≤ r−1/2 exp −π

t

−1 L(f, s) ds .

M

This yields 2 e|f (ζ)| |dζ|

∞

=

2π +

≤

2π 1 +

0

T

2

|{z ∈ T : |f (ζ)| > t}|et dt2 M

2

et dt2

0

+ 2−1 πr−1/2

f H∞

exp t2 − π

−1 ds dt2 . L(f, s)

t

M

M

For our purpose, set ⎧ −1 ⎪ M ⎨ πM x 0 L(f, s)ds −1 ψ(x) = ⎪ ⎩ π x L(f, s) −1 ds + πM 2 M L(f, s)ds M 0

, x ∈ [0, M ], , x ∈ [M, f H∞ ).

Then Lemma 7.2.3 tells us that ψ(M ) ≤ 3,

ψ ∞ =

sup x∈[0, f H∞ )

|ψ(x)| < ∞ and M ≤ 1.

Accordingly, it is enough to control the integral 0

f H∞

exp x2 − ψ(x) dx2

from above. In so doing, we apply Theorem 7.2.4 to the function x , y = ψ(x), φ(y) =

f H∞ , y > ψ ∞ .

7.2. Exponential Integrabilities

175

Of course, this function φ satisﬁes ∞ 2 φ (y) dy = π −1 φ(0) = 0 and 0

Therefore 0

f H∞

f H∞

0

exp x2 − ψ(x) dx2 = 2

∞

exp 0

L(f, s)ds = π −1 f 2D ≤ 1.

2 φ(t) − t φ (t)dx.

From this formula and an approximation we may assume that φ is continuous and has compact support in (0, ∞). Integrating by parts, we know that it is enough to prove that there exists a constant κ > 0 such that if φ is absolutely continuous 2 ∞ with φ(0) = 0 and 0 φ (t) dt ≤ 1, then ∞ ∞ t 2 2 exp φ(t) − t dt = exp φ (s)ds − t dt ≤ κ, 0

0

0

but this has been veriﬁed via Theorem 7.2.4. So we are done.

The foregoing discussion leads to the following exponential integral estimates. Theorem 7.2.6. Let p ∈ [0, ∞). Then: π(1 − r2 )p |f (rζ) − f (0)|2 (i) sup exp |dζ| < ∞.

f 2D2,p r∈(0,1), f D2,p >0 T

π(− log r)p |f ◦ σw (rζ) − f (w)|2 exp |dζ| < ∞. (ii) sup

f 2Qp r∈(0,1), w∈D, f Qp >0 T Proof. Note that for any r ∈ (0, 1) and w ∈ D,

f − f (0) r 2D ≤ (1 − r2 )−p |f (z)|2 (1 − |z|2 )p dm(z) D

and

f ◦ σw − f (w) r 2D ≤ (− log r)−p

D

|(f ◦ σw ) (z)|2 (− log |z|)p dm(z).

So, the desired assertions follow from Theorem 7.2.5.

As an immediate consequence of Theorem 7.2.6, we obtain an interesting exp-characterization of the Bloch space. Corollary 7.2.7. Let p ∈ (1, ∞) and f ∈ H. Then f ∈ B if and only if there is a constant c(f ) > 0 depending only on f such that exp c(f )(− log r)p |f ◦ σw (rζ) − f (w)|2 |dζ| < ∞. sup r∈(0,1),w∈D

T

176

Chapter 7. Estimates for Growth and Decay

Proof. Suppose f ∈ B. If f B = 0, then there is nothing to argue. Hence, assume

f B > 0, i.e., f Qp > 0 due to p > 1. Taking c(f ) = π f −2 Qp and employing Theorem 7.2.6, we derive the desired ﬁniteness. Conversely, if a constant c(f ) > 0 (depending only on f ) obeys C(f ) = sup exp c(f )(− log r)p |f ◦ σw (rζ) − f (w)|2 |dζ| < ∞, T

r∈(0,1),w∈D

then T

(1 − r)p |f ◦ σw (rζ) − f (w)|2 |dζ| ≤

C(f ) , c(f )

r ∈ (0, 1),

w ∈ D.

Multiplying the last inequality by rdr and integrating over (0, 1), we ﬁnd C(f ) , |f ◦ σw (z) − f (w)|2 (1 − |z|)p dm(z) ≤ c(f ) D and consequently, sup w∈D

D

|(f ◦ σw ) (z)|2 (1 − |z|2 )p+2 dm(z)

C(f ) ; c(f )

that is, f ∈ Qp+2 = B thanks to p + 2 > 1.

In view of Theorem 7.2.6, we can get the following exponential estimate of Korenblum type regarding the Bloch space. Theorem 7.2.8. There exist numerical constants κ > 0 such that

κ|f (rζ) − f (0)| sup |dζ| < ∞. exp

f B | log(1 − r2 )| r∈(0,1), f B >0 T Moreover lim sup sup r→1

f B >0

f B

|f (rζ) − f (0)| < ∞. log | log(1 − r)| | log(1 − r2 )|

Proof. Because the well-known John–Nirenberg distribution theorem on BMOA implies that there is a numerical constant κ > 0 such that κ|f (ζ) − f (0)| sup exp |dζ| < ∞,

f BMOA

f BMOA >0 T the ﬁrst assertion of the theorem follows from an application of the just-mentioned exponential estimate for BMOA and Remark 7.1.5 to (f − f (0))r , r ∈ (0, 1).

7.3. Hadamard Convolutions

177

To see the second assertion, we use the ﬁrst part to get another numerical constant κ1 > 0 such that for any f ∈ B with f B > 0,

1 κ|f (rζ) − f (0)| dr exp |dζ| 2 < κ1 . 2

f B | log(1 − r )| 0 T (1 − r) 1 − log(1 − r) Consequently, for almost all ζ ∈ T, we have

1 κ|f (rζ) − f (0)| dr exp 2 < ∞ 2

f B | log(1 − r )| 0 (1 − r) 1 − log(1 − r) thereupon producing 1+r 2 lim exp r→1

r

Writing

κ|f (tζ) − f (0)| dt 2 < ∞. 2

f B | log(1 − t )| (1 − t) 1 − log(1 − t)

1 + r (f, r, ζ) = min |f (rζ) − f (0)| : r ≤ t ≤ , 2

we derive

lim

r→1

κ(f, r, ζ) − 2 log | log(1 − r)| = −∞.

f B | log(1 − r)|

Accordingly, for almost all ζ ∈ T, (f, r, ζ) f B | log(1 − r)| log | log(1 − r)|

as r → 1.

This estimate, together with the inequality |f (t2 ζ) − f (t1 ζ)| ≤ (log 2) f B

for all t1 , t2 ∈ [r,

1+r ], 2

yields the above-desired limit.

7.3 Hadamard Convolutions We have seen from the last section that the dilation fr plays an important role in dealing with the exponential decay of f ∈ Qp . Therefore, ∞another look at fr would reveal a few more properties on Qp . Let now f (z) = j=0 aj z j be in H. Then for ¯ one has fr (z) = ∞ aj rj z j . If g(z) = ∞ z n = (1 − z)−1 r ∈ (0, 1) and z ∈ D j=0 j=0 for z ∈ D, then gr (z) = (1 − rz)−1 and hence fr (z) = f gr (z). Here and hereafter, ∞ denotes the Hadamard convolution; that is, if f, g ∈ H with f (z) = j=0 aj z j ∞ j and g(z) = j=0 bj z , then f g(sζ) =

∞ j=0

j

−1

aj bj (sζ) = (2π)

√ √ f ( sη)g( sζ η¯)|dη|, T

s ∈ (0, 1), ζ ∈ T.

178

Chapter 7. Estimates for Growth and Decay

Recall that for q ∈ (0, ∞] and α ∈ (0, 1), Λ(q, α) stands for the mean Lipschitz class of f ∈ H with

f Λ(q,α) = |f (0)| + sup (1 − r)1−α (f )r Hq < ∞. r∈(0,1)

It is not hard to see that f ∈ Λ(q, α) when and only when sup (1 − r)2−α (f )r Hq < ∞. r∈(0,1)

Using this fact and the monotonic property of (f )r Hq in r ∈ (0, 1), we can easily check that if Df (z) = (zf (z)) , then

f Λ(q,α) ≈ |f (0)| + sup (1 − r)2−α (Df )r Hq . r∈(0,1)

At the same time, denote by Λ(q, p, α), where q, p ∈ (0, ∞] and α ∈ (0, 1), the integrated Lipschitz class of f ∈ H obeying

f Λ(q,p,α) = |f (0)| +

1

0

1−α

(1 − r)

(f )r Hq

p

−1

(1 − r)

p1 dr

< ∞.

Similarly, we have that f ∈ Λ(q, p, α) if and only if 1 p (1 − r)2−α (f )r Hq (1 − r)−1 dr < ∞. 0

Accordingly,

f Λ(q,p,α) ≈ |f (0)| +

0

1

p (1 − r)2−α (Df )r Hq (1 − r)−1 dr

p1 .

Clearly, Λ(q, α) can be treated as the limit space of Λ(q, p, α) as p → ∞. Moreover, if q = ∞ then Λ(q, q −1 ) = Λ(∞, 0) = B. Taking its Qp -setting into account, we ﬁnd the following assertion. Lemma 7.3.1. Let p ∈ (0, 1]. (i) If 2 < q <

2 1−p ,

then f ∈ Λ(q, q −1 ) implies f ∈ Qp with

f Qp,2 sup (1 − r)2−q

−1

(Df )r Hq .

r∈(0,1)

(ii) If q =

2 1−p ,

then f ∈ Λ(q, 2, q −1 ) implies f ∈ Qp with

f Qp ,2

0

1

−1

(1 − r)

2−q−1

(1 − r)

(Df )r

12

2 Hq

dr

.

7.3. Hadamard Convolutions

179

Proof. To see f ∈ Qp provided that f obeys the conditions in (i) and (ii), we ﬁx a Carleson box S(I) based on a subarc I of T, and then use H¨ older’s inequality with 2 < q ≤ 2/(1 − p) to obtain

2

|f (z)| (1 − |z|) dm(z) p

1− 2q

|I|

2

|I|1− q

1

0

2 1−p ,

(i) Let f ∈ Λ(q, q −1 ). If 2 < q <

1

1−|I|

S(I)

T

T

2q |f (rζ)| |dζ| (1 − r)p dr

q

q2 |f (rζ)|q |dζ| (1 − r)p dr.

the last estimates give

2

2 1−q−1

|f (z)| (1 − |z|) dm(z) |I| p

sup (1 − r)

p

(f )r Hq

r∈(0,1)

S(I)

≈ |I|

p

2 2−q−1

sup (1 − r)

(Df )r Hq

,

r∈(0,1)

hence giving f ∈ Qp with the required seminorm inequality. 2 (ii) Let f ∈ Λ(q, 2, q −1 ). If 2 < q = 1−p , we similarly get

2

|f (z)| (1 − |z|) dm(z) |I| p

1

2 −1 (1 − r)−1 (1 − r)1−q (f )r Hq dr

1

2 −1 (1 − r)−1 (1 − r)2−q (Df )r Hq dr,

p

S(I)

0

≈ |I|p

0

as required.

The above lemma founds a basis of the Hadamard convolution inequalities below. Theorem 7.3.2. Let p ∈ (0, 1]. (i) If 2 < q <

2 1−p ,

then f ∈ Λ(q, q −1 ) and g ∈ Λ(1, 0) imply f g ∈ Qp with

f g Qp ,2 f Λ(q,q−1 ) g Λ(1,0).

(ii) If q =

2 1−p ,

then f ∈ Λ(q, q −1 ) and g ∈ H1 imply f g ∈ Qp with

f g Qp ,2 f Λ(q,q−1 ) g H1 .

(iii) If q =

2 2−p ,

then f ∈ D2,p and g ∈ Λ(q, 0) imply f g ∈ Qp with

f g Qp ,2 f D2,p g Λ(q,0).

180

Chapter 7. Estimates for Growth and Decay

Proof. The starting point is the identity D(f g) = f g

for f, g ∈ H.

(i) By Lemma 7.3.1 (i) and the Minkowski inequality with q > 2, we get that if f ∈ Λ(q, q −1 ) and g ∈ Λ(1, 0), then

f g Qp ,2

sup (1 − r)2−q

−1

r∈(0,1)

sup (1 − r)2−q

D(f g) r Hq

−1

(f g )r Hq

−1

(f )√r Hq (g )√r H1

r∈(0,1)

sup (1 − r)2−q

r∈(0,1)

f Λ(q,q−1 ) g Λ(1,0) .

(ii) When f ∈ Λ(q, q −1 ) and g ∈ H1 , we use the following Hardy–Littlewood inequality 2 1 (1 − r) |g (rζ)||dζ| dr g 2H1 0

T

and Lemma 7.3.1 (ii) to get

f g 2Qp ,2

f 2Λ(q,q−1 )

1

0

(1 − r) (g )r 2H1 dr

f 2Λ(q,q−1 ) g 2H1 , hence producing the desired inequality. (iii) Suppose f ∈ D2,p and g ∈ Λ(q, 0). Then these two functions enjoy Young’s inequality (more general than Minkowski’s inequality):

(f g)r Hq3 ≤ f√r Hq1 f√r Hq2

where

1 1 1 + − = 1 and q1 , q2 , q3 ≥ 1. q1 q2 q3

2 2 Using this inequality with q1 = 2, q2 = 2−p , q3 = 1−p , plus Lemma 7.2.8 (ii), we get 1 2 2 (1 − r)p+2 (f )√r H2 (g )√r 2−p dr

f g 2Qp ,2 0

g 2Λ(

2 2−p ,0)

g 2Λ(

2 2−p ,0)

0

H

1

(1 − r)p (f )√r 2H2 dr

f 2D2,p ,

thereupon establishing the required inequality. Corollary 7.3.3. Let p ∈ [0, ∞). Then Qp Qp ⊆ Qp .

7.3. Hadamard Convolutions

181

Proof. If p ∈ (0, 1], then the assertion follows immediately from Theorem 7.3.2 (iii) and two inclusions Qp ⊂ D2,p

and B ⊆ Λ(q, 0),

q > 0.

In the case of p ∈ (1, ∞), we have Qp = B. So if f, g ∈ B then

f g B

(1 − r)2 |D(f g)(rζ)|

sup

r∈[0,1), ζ∈T

sup r∈[0,1), ζ∈T

T

√ √ (1 − r)2 |f ( rη)||g ( rζ η¯)||dη|

f B g B , as required. In the case of p = 0, we get that if f, g ∈ D, then by D ⊂ B and the Cauchy–Schwarz inequality,

f g 2D

1

0

|D(f g)(rζ)|2 |dζ| (1 − r)2 dr

1

T

0

T

g 2B

1

T

0 T 2 2

g D f D ,

2 √ √ |f ( rη)||g ( rζ η¯)||dη| |dζ|(1 − r)2 dr

√ |f ( rη)|2 |dη|dr

as desired.

In fact, Lemma 7.3.1 (ii) can tell us a forward and backward estimate for the Qp seminorm. ∞ 2 . For n ∈ N and f (z) = j=0 aj z j ∈ Theorem 7.3.4. Given p ∈ [0, 1] and q = 1−p H let I0 = {0}, In = {k ∈ N : 2n−1 ≤ k < 2n } and (∆n f )(z) = k∈In ak z k . Then

∞

∆n f 2L2 (T)

n=1

12 f D2,p f Qp,2

2(p−1)n

∞

∆n f 2Lq (T) n=1

2(p−1)n

12 .

∞ Proof. Note that f ∈ Λ(q, 2, q −1 ) amounts to n=1 2(1−p)n ∆n f 2Lq (T) < ∞. So, from Lemma 7.3.1 (ii) it turns out that we only need to check the left-hand-side of the above-desired inequalities. Since f ∈D

2,p

⇔

0

1

(1 − r)p (f )r 2H2 dr < ∞,

182

Chapter 7. Estimates for Growth and Decay

we conclude that f ∈ Qp ⊆ D2,p gives f ∈ Λ(2, 2, 1−p 2 ) with

f Λ(2,2, 1−p ) − |f (0)| f D2,p f Qp . 2

But, the most left quantity of the last inequalities dominates ∞

2(1−p)n ∆n f 2L2 (T) ,

n=1

and thus this completes the proof.

Corollary 7.3.5. Given p ∈ [0, 1]. Suppose {n } is a sequence of natural numk∞ bers with inf n /n > 1. Then f (z) = ak z nk is in Qp if and only if k∈N k+1 k k=0 ∞ 2 2k−1 ≤nj <2k |aj | is convergent. k=1 Proof. Theorem 7.3.4, along with

∆n f Lq (T) ≤ ∆n f L∞ (T) ≤

|aj |

2n−1 ≤nj <2n

and the fact that the cardinal of {j ∈ N : 2k−1 ≤ nj < 2k } is not greater than

1+

log 2

log inf k∈N

nk+1 nk

,

yields the desired assertion.

7.4 Characteristic Bounds of Derivatives For f ∈ H and r ∈ [0, 1), let −1

T (r, f ) = (2π)

T

max log |f (rζ)|, 0 |dζ|

be the Nevanlinna characteristic of f . It is easy to see that if f ∈ B then T (r, f ) ≤ − log(1 − r) + O(1) as

r → 1.

The estimate is sharp in the sense that there exists an f ∈ B such that − log(1 − r) − T (r, f ) = O(1) as which gives

0

1

r → 1,

(1 − r) exp 2T (r, f ) dr = ∞.

Since B = Qp for p > 1, a natural question is: how about Qp , p ∈ [0, 1]? Below is the answer.

7.4. Characteristic Bounds of Derivatives

183

Theorem 7.4.1. Let p ∈ [0, 1]. Then f ∈ Qp implies 1 (1 − r)p exp 2T (r, f ) dr < ∞. 0

Furthermore, if φ : (0, 1) → (0, ∞) is an increasing function and satisﬁes the following three conditions: 1+p (i) r → (1 − r) 2 exp φ(r) is a decreasing function on (0, 1); (ii) φ(r) − φ(s) → ∞ as (1 − r)/(1 − s) → 0; 1 (iii) 0 (1 − r)p exp 2φ(r) dr < ∞, then there is an f ∈ Qp such that T (r, f ) > φ(r) as r → 1. Proof. Suppose f ∈ Qp and r ∈ (0, 1). Using Jensen’s inequality, we derive 1 exp 2T (r, f ) (1 − r)p dr 0 1 ≤ log 1 + |f (rζ)|2 |dζ| exp (2π)−1 (1 − r)p dr 0

≤

(2π)−1

0

1

≤

1+

1 + f 2D2,p

T

0

T

2 1 + |f (rζ)| |dζ| (1 − r)p dr

1

|f (rζ)| |dζ| (1 − r)p dr

T

2

1 + f 2Qp ,1 .

To check the second part of the theorem, we assume that φ obeys those hypotheses. Since φ is increasing on (0, 1), the condition (iii) implies 1 ∞ > (1 − r)p exp 2φ(r) dr 0

≥

∞ k=1

≥

1−2−k−1

1−2−k

2−1−p

∞

(1 − r)p exp 2φ(r) dr

2−k(1+p) exp 2φ(1 − 2−k ) .

k=1

This ﬁniteness, along with the Dini Theorem in [Kno, p. 297], generates an increasing sequence {ck } of natural numbers greater than 2 such that limk→∞ ck = ∞, limk→∞ ck+1 /ck = 1 and ∞ k=1

∞ −k(1+p) −k c1+p 2 exp 2φ(1 − 2 ) ≤ c2k 2−k(1+p) exp 2φ(1 − 2−k ) < ∞. k k=1

184

Chapter 7. Estimates for Growth and Decay

Choosing now n1 = 1 and nk+1 = ck nk for k ∈ N, we have nk+1 > 2k for k ∈ N, and ∞

c1+p n−1−p k k+1

∞ −1 −k(1+p) exp 2φ(1 − nk+1 ) ≤ c1+p exp(2φ(1 − 2−k ) < ∞, k 2

k=1

k=1

thanks to the condition (i). If f (z) =

∞

ak z nk

−1 ak = 10n−1 k exp 2φ(1 − nk+1 ) ,

with

k=1

then it is easy to ﬁnd f ∈ Qp due to Corollary 7.3.5. The argument will end up with verifying T (r, f ) ≥ φ(r). In so doing, we note that for k − 1 ∈ N and |z| = 1 − n−1 k , |f (z)| ≥

∞ |zf (z)| = aj nj z nj j=1 k−1

≥

ak |z|nk −

aj nj |z|nj −

=

(I) − (II) − (III).

j=1

1 nj aj n j 1 − nk j=1

k−1

For the ﬁrst term, we trivially have (I) ≥ 2−2 ak nk for nk ≥ 2. For the second term, we use the condition (ii) to get ak n k −1 = lim exp φ(1 − n−1 lim k+1 ) − φ(1 − nk+2 ) = 0 k→∞ ak+1 nk+1 k→∞ which readily deduces limk→∞ (II)/(ak nk ) = 0. For the third term, we employ the elementary inequality (1 − t)n < 2(nt)−2 for t ∈ (0, 1) and n ∈ N ∞ to derive (III) ≤ 2n2k j=k+1 aj n−1 j . By the condition (i), we ﬁnd 1+p p−3 c exp φ(1 − n−1 nk a−1 2 k+1 k+2 ) k 2 lim ≤ lim = lim c = 0, k→∞ nk+1 a−1 k→∞ c2 exp φ(1 − n−1 ) k→∞ k c k k+1 k+1 k hence we get limk→∞ (III)/(ak nk ) = 0 via the technique in the ratio test for positive series. All together, we can select a natural number k0 so that k ≥ k0 implies |f (z)| > 2−3 ak nk > exp φ(1 − n−1 for |z| = 1 − n−1 k+1 ) k . Consequently, T (1 −

n−1 k ,f )

−1

= (2π)

T

−1 max f (1 − n−1 k )ζ , 0 |dζ| > φ(1 − nk+1 ).

7.4. Characteristic Bounds of Derivatives

185

−1 −1 As for r ≥ 1 − n−1 k0 , we take k ≥ k0 to ensure 1 − nk ≤ r < 1 − nk+1 . Because T (r, f ) and φ(r) increase with r, we ﬁnally reach −1 T (r, f ) ≥ T (1 − n−1 k , f ) > φ(1 − nk+1 ) ≥ φ(r),

as required.

On the basis of the radial growth of f for f ∈ B: (1 − |z|)|f (z)| ≤ f B ∞ j which is sharp in the sense that if f (z) = j=1 z n with n ∈ N being big enough, then f ∈ B with 1 ≤ (1 − |z|2 )|f (z)| for 1 − n−k ≤ |z| ≤ 1 − n−k−1/2 and so

1 lim sup(1 − r)|f (rζ)| r→1

for all ζ ∈ T.

We next deal with the radial growth of the derivative of a function in Qp for p ∈ [0, 1]. Theorem 7.4.2. Let p ∈ [0, 1]. If f ∈ Qp , then lim (1 − r)

1+p 2

r→1

|f (rζ)| = 0

for a.e. ζ ∈ T.

2 1 Furthermore, if φ : (0, 1) → (0, ∞) increases and satisﬁes 0 (1 − r)p φ(r) dr < ∞, then there exists f ∈ Qp such that −1 lim sup φ(r) |f (rζ)| = ∞ for all ζ ∈ T. r→1

Proof. Assume f ∈ Qp . From the Hardy–Littlewood maximal estimate for H2 it follows that 1 max |f (tζ)|2 |dζ| (1 − r2 )p dr2 0

1

T t∈[0,r]

0

T

|f (rζ)|2 |dζ| (1 − r2 )p dr2

f 2Qp ,1 .

Accordingly, for a.e. ζ ∈ T, we have 1 max |f (tζ)|2 (1 − r2 )p dr2 < ∞, 0 t∈[0,r]

hence we get 1+p

lim (1 − r)

r→1

2

|f (rζ)|

≤

2

1

(1 + p) lim max |f (tζ)| r→1 t∈[0,r]

≤

(1 + p) lim

=

0.

r→1

1

(1 − s)p ds r

max |f (tζ)|2 (1 − s)p ds

r t∈[0,s]

186

Chapter 7. Estimates for Growth and Decay

This proves the radial growth of f . As for the second part of the theorem, we may assume without loss of generality that limr→1 φ(r) = ∞ and prove −1 1 lim sup φ(r) |f (rζ)|

for every ζ ∈ T

r→1

since otherwise it is possible to choose an increasing function φ1 : (0, 1) → (0, ∞) with 1 2 lim φ1 (r) = ∞ and (1 − r)p φ(r)φ1 (r) dr < ∞, r→1

0

thereby yielding that for all ζ ∈ T, −1 −1 ∞ = lim sup φ1 (r) φ(r)φ1 (r) |f (rζ)| = lim sup φ(r) |f (rζ)|. r→1

r→1

Under this assumption, we choose an increasing sequence {rk } such that limk→∞ rk = 1 and 3−p

r1 > 2

−2

rk+1 − rk φ(rk+1 ) (1 − rk+1 ) 2 = ∞; lim sup ; > 2−1 ; lim k→∞ φ(rk ) 1 − rk (1 − rk )2 k→∞

< ∞.

With these conditions, we make a simple calculation to get ∞

1+p

(1 − rk )

2 φ(rk )

k=1

∞

k=1 rk ∞ rk+1 k=1 1

<

rk+1

0

2 (1 − r)p φ(rk ) dr 2 (1 − r)p φ(r) dr

rk

2 (1 − r)p φ(r) dr

∞.

Now, for k ∈ N, suppose that nk is the unique natural number ensuring nk ≤ (1 − rk )−1 < nk + 1. So we have −1 , 1 − n−1 k ≤ rk < 1 − (1 + nk )

2−2 < nk (1 − rk ) ≤ 1

and nk+1 /nk > 5/4.

∞ If f (z) = k=1 (1 − rk )φ(rk )z nk , then f ∈ H. Our aim is to verify f ∈ Qp . In so doing, for j ∈ N let k(j) ∈ N be the unique number obeying 2k(j) ≤ nj < 21+k(j) .

7.4. Characteristic Bounds of Derivatives

187

Then the requirements on {rk } imply ∞

2k(1−p)

2 (1 − rj )2 φ(rj )

∞

≤

2k ≤nj <2k+1

k=0

2 n1−p (1 − rj )2 φ(rj ) j

j=1 ∞

≤

2 (1 − rj )1+p φ(rj )

j=1

∞,

< hence giving f ∈ Qp . Finally, we are about to demonstrate −1 1 lim sup φ(r) |f (rζ)| r→1

for all ζ ∈ T.

If k − 1 ∈ N and |z| = 1 − n−1 k , then |f (z)| ≥

∞ |zf (z)| = nj (1 − rj )φ(rj )z nj j=1

n

≥

nk (1 − rk )φ(rk )rknk −

nj (1 − rj )φ(rj )rk j

≥

k−1 ∞ 1 nk 1 nj 2−2 φ(rk ) 1 − − φ(rj ) − φ(rj ) 1 − nk 1 + nk j=1

=

(IV ) − (V ) − (V I).

j =k

j=k+1

In a similar manner to controlling (I) and (II) in the argument for Theorem 7.4.1, we easily obtain φ(rk ) (IV ) and limk→∞ (V )/φ(rk ) = 0. Moreover, the deﬁnitions of rk and nk yield 24 (1 − r )2 φ(r ) n2j /φ(rj ) j+1 j+1 ≤ 2 nj+1 /φ(rj+1 ) φ(2−2 ) (1 − rj )2 24 (1 − r ) 3−p 1+p j+1 2 (1 − rj+1 ) 2 φ(rj+1 ) = −2 2 φ(2 ) (1 − rj ) → 0 as j → ∞. Accordingly, we achieve ∞ (V I) (nk + 1)2 φ(rj ) ≤ 2 lim sup lim sup = 0, φ(rk ) n2j k→∞ φ(rk ) k→∞ j=k+1

hence getting a constant c > 0 and a natural number k0 such that −1 |f (rk ζ)| ≥ c whenever k ≥ k0 and ζ ∈ T. φ(rk ) Clearly, the desired estimate for the limit superior follows.

188

Chapter 7. Estimates for Growth and Decay

7.5 Notes Note 7.5.1. Section 7.1 is taken from [Xi4, Section 4] and [AlSi, Section 3]. An interesting problem related to Example 7.1.1 is to give certain function-theoretic characterizations of [BMOA, B]s and even [D, Qp ]s for p ∈ (0, 1). /n 1 Observe that k=1 σwk D ≈ n 2 for any ﬁnite subset {wk }nk=1 of D (see e.g. [ArFiPe3, Theorem]). Thus, Theorem 7.1.2 (i) yields " " "1−p " "p " n n n "0 " " "0 " "0 1−p " " " " " " σwk " ≤" σwk " σwk " n 2 . " " " " " " " " k=1

D

k=1

Qp ,1

BMOA

k=1

On the other hand, we know from [Bo, Theorem 4.1 with s = 12 ] that 1

1

1−p 1−p

f D f D 2,p f Q ,1 p

holds for any inner function f on D. Thus, we achieve " " n " "0 1−p " " n 2 " σwk " , " " k=1

hence producing

" " n " "0 " " σwk " " " " k=1

≈n

Qp ,1

1−p 2

,

n ∈ N.

Qp ,1

Theorems 7.1.3 and 7.1.4 are the special cases of the Korenblum-type estimates in [AlSi, Theorems 3.1 and 3.2] which are more general than [Kor, Theorem 1]. Note 7.5.2. Section 7.2 comes from [Xi5] and [BetBrJa, Proposition 1] plus a combination of [ChaSMar], [Mar] and [Kor]. In fact, there are two facts related to Theorem 7.2.5: (i) If 1 < α < ∞, then sup exp α|f (ζ)|2 |dζ| : f ∈ D with f (0) = 0 and f 2D ≤ π = ∞. T

(ii) If 0 < α < 1, then exp α|f (ζ)|2 |dζ| : f ∈ D with f (0) = 0 and f 2D ≤ π < ∞. sup T

To check (i), we simply take (cf. [ChaSMar]) − log(1 − az) f (z) = , − log(1 − a2 )

a ∈ (0, 1).

7.5. Notes

189

And, we may employ the Littlewood–Paley form of Green’s Theorem (cf. [Koo, p. 224] to get exp α|f (ζ)|2 |dζ| T = 2π + 4α (− log |z|) 1 + α|f (z)|2 ) exp α|f (z)|2 |f (z)|2 dm(z) D

hence proving (ii) — see [PaVu]. The proofs of Theorems 7.2.4 and 7.2.5 are taken from [Mar] which extracts the technique in both Adams’ Lemma in [Ad1] and Moser’s Theorem in [Mos]. Here it is also worth pointing out that [ChaSMar] provides a capacity-free proof via the well-known Beurling inequality: |{ζ ∈ T : |f (ζ)| > t}| ≤ exp(−t2 + 1) for t > 0 and f 2D ≤ π. Furthermore, [Es] shows some sharp inequalities on the uniform harmonic majorants that extend Chang–Marshall’s Theorem. In addition, some weighted area integral estimates of the exponential type for the diagonal Besov spaces can be found in [BuFeVu]. Theorem 7.2.8 reveals the law of the iterated logarithm for the Bloch space — see also [Mak] for more information. Note 7.5.3. Section 7.3 is an adaptation of [Pa2]. It is worth comparing Theorem 7.3.1 with [Xi3, Theorem 4.4.2] — both may be regarded as mutually reversed propositions in some sense. Note 7.5.4. Section 7.4 is taken from the major part of [GoMa]. For the Bloch part of Theorem 7.4.1 see also [Gir3]. Beyond the radial growth of derivatives we can say something on the radial variation of functions in Qp . For f ∈ H and ζ ∈ T let 1

V (f, ζ) = 0

|f (rζ)|dr

be the radial variation of f along the segment from 0 to ζ ∈ T. Then the exceptional set of f is deﬁned by E(f ) = {ζ ∈ T : V (f, ζ) = ∞}. As an immediate consequence of Theorem 7.4.2, we get that if f ∈ Qp , p ∈ (0, 1), then |E(f )| = 0. But this result cannot extend to p ≥ 1 since f (z) =

k

k −1 z 2

k=1

is an element of Q1 = BMOA and has the property V (f, ζ) = ∞ for all ζ ∈ T.

190

Chapter 7. Estimates for Growth and Decay

On the other hand, to Beurling–Zygmund’s Theorem: if f ∈ D2,p , p ∈ according [0, 1), then capp E(f ) = 0 (cf. [Beu] for p = 0 and [KahSa, Chapter 4]), we can trivially get f ∈ Qp , p ∈ [0, 1) ⇒ capp E(f ) = 0. In addition to this, we can also derive that if f ∈ Qp where p ∈ [0, 1), then limr→1 (1 − r)|f (rζ)| = 0 for every ζ ∈ T except perhaps for a set with p-capacity 0 – see [GoMa, Section 4]. Here, it is worthy of special remark that [JoMu] gives an aﬃrmative answer to Anderson’s conjecture: for any conformal map f ∈ H there is ζ ∈ T such that 1 |f (rζ)|dr < ∞. 0

Chapter 8

Holomorphic Q-Classes on Hyperbolic Riemann Surfaces Up to this point we have dealt with many essential properties of the holomorphic Q functions on the open unit disk which is the canonical example of all hyperbolic Riemann surfaces. We broaden our perspective now to consider their generalizations on hyperbolic Riemann surfaces. Since this study is far from complete, the very ﬁrst step is for us to introduce such a Qp class, naturally and geometrically. This idea leads to the main topic of this chapter and will be detailed through the following ﬁve sections: • • • • •

Basics about Riemann Surfaces; Area and Seminorm Inequalities; Intermediate Setting – BMOA Class; Sharpness; Limiting Case – Bloch Classes.

8.1 Basics about Riemann Surfaces In complex analysis, a Riemann surface is a one-dimensional complex manifold. Riemann surfaces may be thought of as deformed versions of the complex plane — locally near every point they look like patches of the complex plane, but the global topology can be quite diﬀerent. For example, they can look like a sphere or a torus or a couple of sheets glued together. Below is the precise deﬁnition. Deﬁnition 8.1.1. A Riemann surface is a connected Hausdorﬀ space X together with a collection of charts {Uα , zα } satisfying the following three conditions: (i) The Uα form an open covering of X. (ii) Each zα is a homeomorphic mapping of Uα onto an open subset of C.

192

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

(iii) If Uα ∩ Uβ = ∅, then zαβ = zβ ◦ zα−1 is holomorphic on zα (Uα ∩ Uβ ). In order to get a better understanding of Riemann surfaces, we consider some typical examples as follows. Example 8.1.2. (i) C is the most trivial Riemann surface. The identity map zα (z) = z deﬁnes a chart for C, but also the conjugate map wβ (z) = z¯ deﬁnes another chart on C. These two charts are not compatible, so this endows C with two distinct Riemann surface structures. (ii) In an analogous fashion, every open subset of C can be viewed as a Riemann surface in a natural way. More generally, every open subset of a Riemann surface is a Riemann surface. (iii) Let S = C ∪ {∞} and let zα (z) = z where z ∈ S \ {∞} and wβ (z) = 1/z where z ∈ S\ {0} and 1/∞ is deﬁned to be 0. Then we obtain two compatible charts, making S into a Riemann surface which is called the Riemann sphere because it can be interpreted as wrapping C around the sphere. Unlike C, it is compact. On the other hand, it is worth remarking a few more facts. First of all, the system {Uα , zα } deﬁnes a conformal structure on X, and if it is understood which conformal structure we are referring to, then we will directly speak of the Riemann surface X. Secondly, the topology of X is completely determined via the mappings {zα }, and a point u ∈ Uα is uniquely determined by the complex number zα (u) — due to this, zα is regarded as a local variable. The subscript is often dropped and z(u) is identiﬁed with u — for example, {z ∈ C : |z − z0 | < r} can refer either to an open disk on C or to its inverse image on X. Thirdly, the identiﬁcation of a point on the Riemann surface with the corresponding value of a local variable produces no diﬃculty whenever one deals with concepts such as holomorphic/harmonic/subharmonic functions and analytic arcs that are invariant under conformal mappings. Finally, we want to say that the main point of Riemann surfaces is that holomorphic functions may be deﬁned between them. So Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the logarithm. In the meantime, we know that geometrical facts about Riemann surfaces are as nice as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties Here we recall the following deﬁnition. Deﬁnition 8.1.3. (i) A mapping f : X → Y between two Riemann surfaces X and Y is called holomorphic/harmonic/subharmonic provided for every chart {Uα , zα } of X and every chart {Vβ , wβ } of Y, the map wβ ◦ f ◦ zα−1 is holomorphic/harmonic/subharmonic as a function from C to C wherever it is deﬁned. In particular, if Y = C, then the corresponding f is called a holomorphic/harmonic/subharmonic function on X.

8.1. Basics about Riemann Surfaces

193

(ii) Two Riemann surfaces X and Y are called conformally equivalent provided there exists a bijective holomorphic function from X to Y whose inverse is also holomorphic. The well-known uniformization theorem for Riemann surfaces says that every simply connected Riemann surface is conformally equivalent to C or to the Riemann sphere Ce or to the open disk D. From this, Riemann surfaces can be classiﬁed into compact and noncompact ones. Traditionally, a noncompact Riemann surface is also said to be open. But, the open Riemann surfaces can be compactiﬁed by adding a single point — the ideal boundary: if un lies outside any given compact set for all suﬃciently large n ∈ N, then we say the sequence {un } of points tends to ∞. To establish a further classiﬁcation of Riemann surfaces, we need to discuss the existence of the Green function on a Riemann surface via the so-called Perron family of subharmonic functions. To be more speciﬁc, given a Riemann surface X, let F be a family of real-valued subharmonic functions on X satisfying the following two conditions: (i) If f1 , f2 ∈ F then max{f1 , f2 } ∈ F. (ii) Let J be a Jordan region on X. Assume f ∈ F and suppose F is a real-valued harmonic function on J with the same boundary values as f and F = f on X \ J. Then F ∈ F. Such an F is called a Perron family on X. The essential property of the Perron families is that sup{f : f ∈ F } is either harmonic or identical with ∞. Given a point x0 ∈ X, let z be a local variable at x with z(x0 ) = 0 and Fx0 be the family of real-valued functions f with the following three properties: (i) f is deﬁned and subharmonic on X \ {x0 }; (ii) f is identically 0 outside a compact set; (iii) lim supx→x0 f (x) + log |z(x)| < ∞. Clearly Fx0 is a Perron family. Moreover, if gX (x, x0 ) = sup{f (x) : f ∈ Fx0 } < ∞,

x ∈ X,

then we say that X has the Green function gX (x, x0 ) with a pole at x0 . Clearly, this function is harmonic on X \ {x0 } and independent of the choice of local variable z(x) at x0 . It is not hard to see that limx→x0 gX (x, x0 ) = ∞ and so that gX (x, x0 ) is not constant. Note that if X is compact, then it has no Green function because otherwise it follows that gX (x, x0 ) would have a minimum, contradicting the previous properties of the Green function. So, a further classiﬁcation of Riemann surfaces follows. Deﬁnition 8.1.4. An open Riemann surface is called hyperbolic if it has the Green function; otherwise it is called parabolic.

194

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

For example, D is hyperbolic and C is parabolic. Of course, S is neither hyperbolic nor parabolic. Parabolic surfaces share many properties with compact Riemann surfaces — in particular, a positive harmonic function on a parabolic Riemann surface is a constant. Given a hyperbolic Riemann surface X we can introduce two conformally invariant metrics. First, we can use the Green function to deﬁne the Robin function and hence the logarithmic capacitary density as follows: ﬁx z0 ∈ X and let γX (z0 ) = lim gX (z, z0 ) + log |z − z0 | , z→z0

where z and z0 also represent the values of local variable at the points z and z0 . The number γ(z0 ) is the well-known Robin’s constant at z0 with respect to the local variable z. This generates a deﬁnition of the logarithmic capacitary density at z0 via cX (z0 ) = exp − γX (z0 ) . Naturally, cX (z)|dz| is called the logarithmic capacity metric. Note that if X = D then cD (z)|dz| = (1 − |z|2 )−1 |dz|, z ∈ D. This actually suggests a consideration of the hyperbolic or Poincar´e metric on X. To be more speciﬁc, let X and Y be two Riemann surfaces, and consider a holomorphic mapping τ : Y → X. We say that τ is a local homeomorphism if every point on Y has a neighborhood V such that the restriction τ |V of τ to V is a homeomorphism. In this case, the pair (Y, τ ) is called a covering surface of X, τ (y) is the projection of y ∈ Y and y is said to lie over τ (y). As is well known, a hyperbolic Riemann surface X can be modeled by a Fuchsian model D/F uc(D) where F uc(D) is a Fuchsian group — a discrete subgroup of Aut(D). In this case, the open unit disk D is called the universal covering surface of X and so there is a holomorphic mapping, i.e., universal covering mapping, τ from D onto X. Now, the Poincar´e metric density of a hyperbolic Riemann surface X is deﬁned by λX (z) = inf{(1 − |w|2 )−1 : τ (w) = z}, which certainly does not depend on the choice of a universal covering mapping since τ ◦ γ = τ for all γ ∈ F uc(D). The hyperbolic metric λX (z)|dz| on X is real-analytic and has constant Gaussian curvature −4. It is the unique metric on X obeying τ ∗ λX (z)|dz| = λD (w)|dw| = (1 − |w|2 )−1 |dw|, where the left-hand-side is the pull-back metric. Moreover, there is associated to F uc(D) the Poincar´e normal polygon determined below. Fix z0 ∈ D with γ(z0 ) = z0 unless γ = id — the identity element of F uc(D). Recall that dD (z1 , z2 ) = log

1 + |σ (z )| 12 z1 2 , 1 − |σz1 (z2 )|

z1 , z2 ∈ D

8.2. Area and Seminorm Inequalities

195

is the hyperbolic distance on D. And let Ωz0 = z ∈ D : dD (z, z0 ) < dD z, γ(z0)

for all γ ∈ F uc(D) \ {id}

be a fundamental region for F uc(D) which is an open domain Ω ⊂ D satisfying γ(Ω) ∩ Ω = ∅ for all γ ∈ F uc(D) \ {id} and D = γ(Ω). γ∈F uc(D)

Nevertheless, the important properties are the facts that any fundamental region Ω is a copy of the hyperbolic Riemann surface X, any universal covering mapping τ : Ω → X is surjective and the boundary ∂Ω has zero area. Furthermore, the Green function of Ω is deﬁned by the well-known Myrberg’s formula: gΩ (z, w) = gX τ (z), τ (w) = gD z, γ(w) , z, w ∈ Ω. γ∈F uc(D)

8.2 Area and Seminorm Inequalities A careful look at the a priori estimate in Example 1.1.1, along with the foregoing introduction of Green functions, leads to a consideration of the so-called holomorphic Q classes over general hyperbolic Riemann surfaces. Deﬁnition 8.2.1. Let p ∈ [0, ∞) and X be a hyperbolic Riemann surface. We say that a holomorphic function f : X → C belongs to Qp (X) provided

f Qp (X) = sup Ep (f, w, X) < ∞, w∈X

where

12 p i 2 |f (z)| gX (z, w) dz ∧ d¯ z Ep (f, w, X) = 2 X and dz ∧ d¯ z = 2idxdy for a local variable z = x + iy. In particular, we denote Q0 (X) and Q1 (X) by D(X) — the Dirichlet space on X and BMOA(X) — the BMOA space on X, respectively. Here and henceforth, the derivative of a function f on X is determined in the following way: if z ∈ X and t is a local parameter in a neighborhood of z such that t(z) = 0, then f (z) represents the usual derivative of f ◦ t−1 at 0. A very basic problem is to clarify the relationship among these Q classes. To do so, we need to have a deep understanding of the isoperimetric inequality for Riemann surfaces.

Lemma 8.2.2. Suppose X is a Riemann surface. Assume Ω is a relatively compact subdomain of X and has a piecewise smooth boundary ∂Ω. If f : X → C is holomorphic, then the following isoperimetric inequality holds: i |f (z)|2 dz ∧ d¯ z ≤ (4π)−1 Length f (∂Ω) = (4π)−1 |dz|. 2 Ω f (∂Ω)

196

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Proof. Assuming f (∂Ω) is positively oriented, we use the Green formula to derive Ω

|f (z)|2 dz ∧ d¯ z=

Ω

df (z) ∧ df (z) =

f (z)df (z) =

f (∂Ω)

wdw. ¯ f (∂Ω)

Suppose that Length(C) stands for the length of a curve C. By the polygonal approximation, we may assume that f (∂Ω) is a polygonal closed curve. Accordingly, f (∂Ω) may be written as a sum of ﬁnite polygonal Jordan closed curves {Cj }nj=1 so that n

Length(Cj ) ≤ Length(C)

and

wdw ¯= C

j=1

n j=1

wdw. ¯

Cj

If Ωj represents the Jordan region bounded by Cj , then it follows from the Green formula and the elementary isoperimetric inequality that wdw ¯ = dwdw¯ ≤ (2π)−1 Length(Cj ), j = 1, . . . , n. Ωj

Cj

Consequently, i 2

Ω

|f (z)|2 dz ∧ d¯ z ≤

2−1

n j=1

≤

(4π)−1

wdw¯

Cj

n

Length(Cj )

j=1

≤

(4π)−1 Length(C),

as desired.

This lemma is used to derive the following result. Theorem 8.2.3. Let X be a hyperbolic Riemann surface and gX (z, z0 ) be its Green function with pole at z0 ∈ X. For t ≥ 0 let Xt = {z ∈ X : gX (z, z0 ) > t}. If f : X → C is holomorphic, then the function i A(t) = |f (z)|2 dz ∧ d¯ z 2 Xt has the following three properties: (i) A(t) is continuous and decreasing with increasing t ≥ 0. (ii) e2s A(s) ≤ e2t A(t) for s ≥ t ≥ 0.

8.2. Area and Seminorm Inequalities

197

(iii) For p, t ∈ [0, ∞), 2 Ep (f, z0 , Xt ) =

0

∞

A(s)dsp = −

∞

sp dA(s).

t

Here and hereafter, the right-hand-side integral (as well as its like) will be understood under Riemann–Stieltjes integration. Proof. First, we prove that those three properties are valid under the assumption ¯ of X. that X is a ﬁnite Riemann surface and f is holomorphic on the closure X For the sake of simplicity, we write ∗ (z, z0 ) = g(z) + ig ∗ (z), G(z) = gX (z, z0 ) + igX ∗ where g ∗ (z) = gX (z, z0 ) is the harmonic conjugate of g(z) = gX (z, z0 ) and is locally deﬁned up to an additive constant. For t ≥ 0, set f (z) 2 ∂g(z) ¯ : gX (z, z0 ) = t} and A0 (t) = ds. Γt = {z ∈ X ∂n Γt G (z)

Here and afterwards, ds denotes the arc length measure on Γt and tive in the inner normal direction with respect to Xt . Under p, t ∈ [0, ∞), we make the substitutions dg(z) ∧ dg ∗ (z) = |G (z)|2 dz ∧ d¯ z,

λ = g(z) and dλ =

∂g(z) ∂n

∂g(z)

to get a chain of equalities: p i |f (z)|2 gX (z, z0 ) dz ∧ d¯ z 2 Xt p i f (z) 2 = z gX (z, z0 ) |G (z)|2 dz ∧ d¯ 2 Xt G (z) p i f (z) 2 = gX (z, z0 ) dg(z) ∧ dg ∗ (z) 2 Xt G (z) ∞ p ∂g(z) 2 f (z) 2 dsdn = gX (z, z0 ) G (z) ∂n t Γ ∞ λ p ∂g(z) f (z) 2 = dsdλ gX (z, z0 ) G (z) ∂n t Γ ∞ λ λp A0 (λ)dλ. = t

In particular,

A(t) =

∞

A0 (λ)dλ, t

t ≥ 0.

the deriva-

∂n

dn

198

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Thus, (i) and (iii) follow. As before, let Length f (Γt ) be the length of f (Γt ). Then by the Cauchy– Schwarz inequality, we get 2 2 Length f (Γt ) ≤ |f (z)||dz| Γt

f (z) ∂g(z) 2 ds G (z) ∂n Γt ∂g(z) f (z) 2 ∂g(z) ds ds ∂n ∂n Γt G (z) Γt 2πA0 (t).

= ≤ =

Note that the isoperimetric inequality in Lemma 8.2.2 ensures 2 A(t) ≤ (4π)−1 Length f (Γt ) . So it follows that 2A(t) ≤ A0 (t)

and

d e2t A(t) = e2t 2A(t) − A0 (t) ≤ 0, dt

and consequently, e2t A(t) decreases with increasing t ≥ 0. This proves (ii). Now, let us handle the case that X is a general hyperbolic Riemann surface. For this, let {Xj }j∈N be a regular exhaustion of X and Aj (t) be deﬁned for each Xj . If f and z0 are given as above, then from what we have veriﬁed for ﬁnite Riemann surfaces, we can read oﬀ s ≥ t ≥ 0 ⇒ e2s Aj (s) ≤ e2t Aj (t) and p, t ≥ 0 ⇒

i 2

Xjt

p |f (z)|2 gXj (z, z0 ) dz ∧ d¯ z=−

∞

λp dAj (λ).

t

Here, gXj (z, z0 ) is the Green function of Xj and Xjt = {z ∈ X : gXj (z, z0 ) > t}. Clearly, limj→∞ Aj (t) = A(t) for every t ≥ 0. Accordingly, (ii) follows. Furthermore, via integrating by parts we derive that for p, t ≥ 0, ∞ 2 = tp Aj (t) + p λp−1 Aj (λ)dλ. Ep f, z0 , Xjt t

This, together with letting j → ∞ and integrating by parts once again, produces: ∞ ∞ 2 Ep f, z0 , Xt = tp A(t) + p λp−1 A(λ)dλ = − λp dA(λ), t > 0, t

t

thereupon proving (iii) for p ≥ 0 and t > 0. Finally, letting t → 0 completes the argument.

8.2. Area and Seminorm Inequalities

199

To compare one Q class with another, we need the following elementary lemma. Lemma 8.2.4. Given a nonnegative function A(t) on (0, ∞) with the following two properties: (i) A(t) is continuous and decreasing with increasing t > 0. (ii) e2t2 A(t2 ) ≤ e2t1 A(t1 ) when t2 ≥ t1 > 0. For p, t ∈ [0, ∞) let Bp (t) = −

∞

sp dA(s). If p ≥ q ≥ 0, then

t

Bp (0) ≤

2q Γ(p + 1) Bq (0) 2p Γ(q + 1)

for which Bp (0) =

2q Γ(p + 1) Bq (0) < ∞ 2p Γ(q + 1)

if and only if A(0) = lim A(t) < ∞ t→0

A(t) = e−2t A(0),

and

t > 0.

Proof. Suppose p ≥ q ≥ 0 and Bq (0) < ∞. Integrating by parts and using the above condition (ii), we have that for t > 0, Bq (t)

∞ = tq A(t) + q sq−1 A(s)ds t ∞ q sq−1 e−2s ds ≤ A(t) t + qe2t t ∞ 2t q −2s = 2A(t)e s e ds. t

Using again (ii) above, we also ﬁnd 2tq A(t)dt tq e−2t dt dBq (t) ≤− ≤ − ∞ q −2s , Bq (t) Bq (t) s e ds t

t > 0.

Integrating this inequality from 0 to t, we get a further inequality 2q+1 Bq (0) Bq (t) ≤ Γ(q + 1)

t

∞

sq e−2s ds,

t ≥ 0.

200

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Consequently, Bp (0) = −

∞

tp−q tq dA(t) ∞ tp−q−1 Bq (t)dt = (p − q) 0 2q+1 (p − q)Bq (0) ∞ p−q−1 ∞ q −2s ≤ t s e dsdt Γ(q + 1) 0 t 2q Γ(p + 1) Bq (0). = 2p Γ(q + 1) 0

This proves the ﬁrst part of the lemma. Below is the argument for the second part. Suppose A(0) = lim A(t) t→0

and A(t) = A(0)e−2t

for

t > 0.

Then Bp (0) = 2−p Γ(p + 1)A(0)

for p ≥ 0,

and hence the desired equality holds with p ≥ q ≥ 0. Conversely, suppose Bp (0) =

2q Γ(p + 1) Bq (0) < ∞, 2p Γ(q + 1)

p > q ≥ 0,

but there exists s0 > t0 > 0 such that A(s0 ) < e−2(s0 −t0 ) A(t0 ). Then the continuity of A(·) implies that there is a δ > 0 such that A(s0 ) < e−2(s0 −t0 ) A(t)

as t ∈ (t0 − δ, t0 ].

Hence by the condition (ii), A(s) < e−2(s−t) A(t)

as t ∈ (t0 − δ, t0 ] and s ≥ s0 .

As a consequence, we obtain Bq (t) < 2A(t)e2t

∞

sq e−2s ds

as t ∈ (t0 − δ, t0 ],

t

and so Bp (0) <

2q Γ(p + 1) Bq (0) < ∞, 2p Γ(q + 1)

Clearly, this contradicts the hypothesis.

p > q ≥ 0.

Now, we can establish the following area and seminorm inequalities associated with the holomorphic Q classes over hyperbolic Riemann surfaces.

8.3. Intermediate Setting – BMOA Class

201

Theorem 8.2.5. Let 0 ≤ q < p and X be a hyperbolic Riemann surface with w ∈ X. Then: q 1 2 Γ(p + 1) 2 (i) Ep (f, w, X) ≤ Eq (f, w, X) for any holomorphic f : X → C. 2p Γ(q + 1) (ii) Qq (X) ⊆ Qp (X) with

f Qp (X) ≤

2q Γ(p + 1) 2p Γ(q + 1)

12

f Qq (X) ,

f ∈ Qq (X).

Proof. It follows immediately from Theorem 8.2.3 and Lemma 8.2.4.

8.3 Intermediate Setting – BMOA Class The case q = 0 and p = 1 of Theorem 8.2.5 is of independent interest since it is a critical case and indeed has a root in the BMOA-theory on hyperbolic Riemann surfaces. Theorem 8.3.1. Let X be a hyperbolic Riemann surface and f : X → C be holomorphic. (i) Given w ∈ X, let hX (z, w) be the least harmonic majorant of |f (z) − f (w)|2 on X, then 2 π E1 (f, w, X) = hX (w, w). 2 . (ii) f BMOA(X) ≤

Area f (X) . 2

Proof. (i) Taking a regular exhaustion of X into account, we see that it is enough to check the formula under the hypothesis that X is a ﬁnite Riemann surface and f is holomorphic on the closure X of X. Suppose W is the interior of a compact bordered Riemann surface W and ∂W stands for the boundary of W. If u and v are C 2 functions on W, then the Green formula just says ∂v ∂u i u −v ds, (v∆u − u∆v)dz ∧ d¯ z= 2 W ∂n ∂n ∂W where ∆, ∂/∂n and ds stand for the Laplacian, diﬀerentiation in the inner normal direction and arc length measure on ∂W, respectively. Applying this formula to u(z) = |f (z) − f (w)|2 and v(z) = gX (z, w) in the domain obtained by removing from X a small open disk centered at w, shrinking

202

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

the disk, and noticing ∆u(z) = 4|f (z)|2 and ∆v(z) = 0, we derive the formula in (i) since the integrals along the boundary of the small disk approach zero and ∂gX (z, w) ds = 2π. ∂n ∂X (ii) The argument for the inequality is split into three steps. Step 1. Suppose τ1 : D → X is a universal covering mapping. Then the least harmonic majorant of a subharmonic function is preserved by τ1 , that is to say, if h is the least harmonic majorant of a subharmonic function u on X, then h ◦ τ1 is the least harmonic majorant of u ◦ τ1 on D. This general fact implies

f BMOA(X) = f ◦ τ1 BMOA(D) . In fact, for ζ, η ∈ D let hD (ζ, η) be the least harmonic majorant of the subharmonic function |f ◦ τ1 (ζ) − f ◦ τ1 (η)|2 on D, then the above-mentioned fact gives hD (ζ, η) = hX τ1 (ζ), τ1 (η) , ζ, η ∈ D. This, along with (i), implies 2π −1 f 2BMOA(X)

=

sup hX (w, w) sup hX τ1 (η), τ1 (η)

w∈X

=

η∈D

=

sup hD (η, η)

η∈D

= 2π −1 f ◦ τ1 2BMOA(D) , as required. Step 2. In order to verify the inequality in (ii), we choose a universal covering mapping τ2 from D onto the range set f (X) of f , and show

f BMOA(X) ≤ τ2 BMOA(D) . Clearly, it suﬃces to prove

f ◦ τ1 BMOA(D) ≤ τ2 BMOA(D) . Note that τ1−1 ◦ (f ◦ τ1 ) has a single-valued branch, say, φ on D. So, it follows that φ ∈ H∞ with φ H∞ ≤ 1 and so that f ◦ τ1 = τ2 ◦ φ is valid. With this, we see that if b = φ(a), then f ◦ τ1 (a) = τ2 (b). Now, let hD,1 (z, a) and hD,2 (z, a) be the least harmonic majorants of u(z) = |f ◦ τ1 (z) − f ◦ τ1 (a)|2

and v(z) = |τ2 (z) − τ2 (b)|2

8.3. Intermediate Setting – BMOA Class

203

respectively. Then u(z) is subordinate to v(z) and hence by the Littlewood Subordination Principle [Hil, p. 421], hD,1 (z, a) ≤ hD,2 φ(z), b , z ∈ D. Using this inequality as well as (i), we derive 2 2

f ◦ τ1 2BMOA(D) = sup hD,1 (a, a) ≤ sup hD,2 (b, b) ≤ τ2 2BMOA(D) , π π a∈D b∈D hence reaching the desired inequality. Step 3. To complete the proof, let I be the identity mapping from f (X) onto itself. Substituting f (X) and I for X and f in Step 1, we get

I

BMOA f (X)

= I ◦ τ2 BMOA(D) = τ2 BMOA(D) .

This, together with Step 2 and Theorem 8.2.5, implies

f 2BMOA(X)

≤ τ2 2BMOA(D)

= I 2

BMOA f (X)

≤ 2−1 I 2 D f (X) = 2−1 Area I f (X) = 2−1 Area f (X) ,

as required. To handle equality of Theorem 8.3.1 and to check when √

f BMOA(X) = 2−1 f D(X) happens, we need the following lemma. Lemma 8.3.2. Let X ⊂ C be a hyperbolic Riemann surface with ﬁnite area.

(i) Given w ∈ X, let 1X (z, w) be the least harmonic majorant of |z − w|2 on X, then Area(X) 1X (w, w) ≤ π for which equality holds if and only if X is a domain of the form X = {z ∈ C : |z − w| < r} \ E, where r > 0 and E is a closed set with cap0 (E) = 0. (ii) If I(z) = z for z ∈ X, then

I BMOA(X) ≤

1

Area(X) 2

for which equality holds if and only if X is a domain of the form X = {z ∈ C : |z − w| < r} \ E, where r > 0 and E is a closed set with cap0 (E) = 0.

204

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Proof. (i) Step 1. Suppose that X is a domain in C with smooth boundary ∂X. Using the Green formula, we derive ∂|z − w|2 2 ds. ∆(|z − w| )dm(z) = 4Area(X) = ∂n X ∂X If u(z) = |z − w|2 exp 2gX (z, w) , then another application of the Green formula produces ∂u(z) ds ∆u(z)dm(z) = X ∂X ∂n ∂|z − w|2 ∂gX (z, w) = ds + 2 ds |z − w|2 ∂n ∂n ∂X ∂X = 4Area(X) − 4π1X (w, w), and so the inequality in (i) follows. Step 2. If X is a general domain in C, then there is a smooth exhaustion {Xj } of X such that w ∈ ∩∞ j=1 Xj . Writing 1Xj (z, w) and uj (z) respectively for the functions for Xj that correspond to 1X (z, w) and u(z) as in Step 1, we get from Step 1 that 1 Area(Xj ) , j ∈ N. 1Xj (w, w) + ∆uj (z)dm(z) = 4π Xj π Letting j → ∞ and using the Lebesgue Monotone Convergence Theorem and Fatou’s Lemma, we ﬁnd 1 Area(X) , 1X (w, w) + ∆u(z)dm(z) = 4π X π thus establishing the required inequality. Step 3. We check the equivalent condition for equality. Assuming 1X (z, w) =

Area(X) , π

we work out from Step 2 that ∆u(z) = 0 on X and so that r gX (z, w) = log |z − w| for some constant r > 0. This forces that X is a domain as determined in (i). Conversely, if X is such a domain, then 1X (z, w) = r2 on X and hence equality occurs since cap0 (E) = 0 implies Area(E) = m(E) = 0. (ii) The inequality is a special case of Theorem 8.3.1 (ii). It remains to check the equality condition. On the one hand, if 1 Area(X) ,

I BMOA(X) = 2

8.3. Intermediate Setting – BMOA Class

205

then there is a sequence of points {wj } in X such that

I 2BMOA(X) =

π lim 1X (wj , wj ). 2 j→∞

Without loss of generality, we may assume that wj is convergent to some point w ∈ X and gX (z, wj ) converges uniformly on compact subsets of X \ {w}. If vj (z) = |z − wj |2 exp 2gX (z, wj ) , then an application of Fatou’s Lemma yields 2 1 Area(X)

I 2BMOA(X) + . lim ∆vj (z)dm(z) ≤ π 4π X j→∞ π Consequently, limj→∞ ∆vj (z) = 0 for all z ∈ X thanks to the above-assumed equality. Therefore we obtain lim gX (z, wj ) = log

j→∞

r |z − w|

for some constant r > 0. This in turns implies that X is contained in the open disk {z ∈ C : |z − w| < r}. If E = {z ∈ C : |z − w| < r} \ X and cap0 (E) > 0, then we can choose a small number > 0 such that cap0 (E ) = cap0 E ∩ {z ∈ C : |z − w| ≥ } > 0. Letting X = X ∪ {z ∈ C : |z − w| < }, we read X ⊆ X ⊆ {z ∈ C : |z − w| < r} and consequently, log

r r = lim gX (z, wj ) ≤ lim gX (z, wj ) = gX (z, w) ≤ log . j→∞ |z − w| j→∞ |z − w|

This tells us that cap0 (E ) = 0, a contradiction. Accordingly, cap0 (E) = 0 and X must be a domain as described in the equality condition of (ii). Conversely, if X is as above, then it is easy to calculate 1 Area(X) 2 . lim 1X (z, z) = r as well as I BMOA(X) = z→w 2 We are done.

Theorem 8.3.3. Let X be a hyperbolic Riemann surface and f : X → C be holomorphic.

206

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

(i) If

.

f BMOA(X) =

Area f (X) , 2

then f (X) = {w ∈ C : |w − c| < r} \ E with c ∈ C, r > 0 and cap0 (E) = 0, but not conversely.

f D(X) √ if and only if there is a simply connected hyper2 bolic Riemann surface Y such that X equals Y set-minus at most E with cap0 (E) = 0 and f is extended to a conformal mapping from Y onto an open disk in C.

(ii) f BMOA(X) =

Proof. (i) This follows from the argument for Theorem 8.3.1 (ii) and the equality condition of Lemma 8.3.2 (ii). Concerning the converse, we select the conformal f1(0) = 0 and put f = f12 . mapping f1 : D → D ∩ {z ∈ C : z > −2−1 } with Clearly, we have f (D) = D which yields Area f (D) = π. However, for any

∈ (0, 1) there is a δ ∈ (0, 1) such that |f ◦ σ−w (ζ)|2 |dζ| ≤ 1 − δ and |w| > ⇒ |f (w)| ≥ δ. |w| ≤ ⇒ T

Accordingly, it follows from Theorem 8.3.1 (i) and a routine computation that if hX (z, w) still stands for the least harmonic majorant of |f (z) − f (w)|2 on D, then 2π −1 f 2BMOA(X)

=

sup hX (w, w) −1 sup (2π) |f ◦ σ−w (ζ)|2 |dζ| − |f (w)|2

w∈X

=

T

w∈X

≤ 1 − δ, and hence the desired equality is not valid. (ii) If

f BMOA(X) =

√ 2−1 f D(X),

then by Theorem 8.3.1 we have √

f BMOA(X) = 2−1 Area f (X) = 2−1 f D(X). This, together with (i), implies that f (X) = {w ∈ C : |w − c| < r} \ E for some c ∈ C, some r > 0 and some E with cap0 (E) = 0, but also that f is univalent on X. Note that cap0 (E) = 0 implies that E is removable for BMOA functions — see also the forthcoming Lemma 8.4.1. So f and X must be as above. The converse follows from the fact that hX (w, w) tends to r2 as w → ∂X and f (w) → c.

8.4. Sharpness

207

8.4 Sharpness In this section, we determine when two equalities in Theorem 8.2.5 occur. To this end, we recall that the Green function is a conformal invariant — this means that if f : X → Y is a conformal mapping from the hyperbolic Riemann surface X to the other hyperbolic Riemann surface Y, then gY f (z), f (w) = gX (z, w) for all (z, w) ∈ X × X. This result can be generalized in the following form. Lemma 8.4.1. Let X, Y ⊂ C be two hyperbolic Riemann surfaces and f : X → Y holomorpic. If f is injective and Y \ f (X) is a closed set of logarithmic capacity zero, then gY f (z), f (w) = gX (z, w) for all (z, w) ∈ X × X. Conversely, if there are two points z, w ∈ X such that gY f (z), f (w) = gX (z, w), then f is injective and Y \ f (X) is a closed set of logarithmic capacity zero. Proof. It suﬃces to check the second part because the ﬁrst part follows from the fact that Y and Y \ f (X) has the same Green function in this situation. In so doing, we assume gY f (z), f (w) = gX (z, w) for distinct points z, w ∈ X. Then the Lindel¨ of Principle gives n(η, f )gX (ζ, w) + uf (w) (ζ), ζ ∈ X, gY f (ζ), f (w) = f (η)=f (w)

where n(η, f ) is the order of f at η ∈ X and uy is a nonnegative harmonic function attached to y ∈ Y and deﬁned on X. For ζ = z the last formula yields gY f (z), f (w) ≥ n(w, f )gX (z, w) + uf (w) (z) ≥ gX (z, w). This, along with the assumption, implies the equality holds throughout, so n(w, f ) = 1, Consequently,

uf (w) (z) = 0 and f (ζ) = f (w) gY f (ζ), f (w) = gX (ζ, w)

for all ζ ∈ X \ {w}.

for all ζ ∈ X.

Furthermore, by the symmetry of the Green function, it follows that f (η) = f (ζ) for all η ∈ X \ {ζ}, n(η, f ) = 1 and gY f (η), f (ζ) = gX (η, ζ) for all η ∈ X. But this also holds for all ζ ∈ X. As a consequence, f is injective. Moreover, from the last equation we conclude that uf (ω) = 0 for all ω ∈ X. According to

208

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

the dichotomy associated with the Lindel¨of Principle (cf. [Hei]), we must have uy (x) = 0 for all x ∈ X and y ∈ Y. Since f is injective, we ﬁnd that sup y∈Y

and so that

n(x, f ) = 1,

f (x)=y

Y \ f (X) = y ∈ Y :

n(x, f ) < 1

f (x)=y

is a closed set of logarithmic capacity zero.

In what follows, we say that a subset E of a simply connected hyperbolic Riemann surface X is of logarithmic capacity zero provided the conformal map from X onto D sends E onto a set of logarithmic capacity zero on D. Theorem 8.4.2. Let X be a hyperbolic Riemann surface, w ∈ X, and f be a nonconstant holomorphic function on X. Then the following two statements are equivalent: (i) Ep (f, w, X) =

2q Γ(p + 1) 2p Γ(q + 1)

12

Eq (f, w, X) < ∞,

p > q ≥ 0.

(ii) X is obtained from a simply connected hyperbolic Riemann surface Y by removing at most a set of logarithmic capacity zero and f is extended onto a conformal mapping from Y to an open disk in C centered at f (w) ∈ C. Proof. Suppose (i) is valid. Upon continuing to use the notation in Theorem 8.2.3 −2t and Lemma8.2.4, we ﬁnd that A(t) = e A(0) for any t > 0, and that G(z) and F (z) = exp − G(z) are multiple-valued but G(z)and |F (z)| are single-valued. Then F (z) is single-valued near w and F (w) = exp − γX (w) = 0 where γX (w) is the Robin constant. Taking ξ = ζ + iη = F (z) as a local variable near ∞w, we have ξ(w) = 0 and gX (z, w) = − log |ξ| near w. Upon writing f (ξ) = j=0 bj ξ j where b0 = f (z0 ), we get that for a large t > 0, e

−2t

A(0) = A(t) = |ξ|<e−t

|f (ξ)|2 dm(ξ) = π|b1 |2 e−2t + 2π|b2 |2 e−4t + · · · ,

so that A(0) = π|b1 |2 and bj = 0 for j − 1 ∈ N. This actually means that f (ξ) = b0 + b1 ξ = b0 + b1 F (z) near z0 and then f = b0 + b1 F — where F is singlevalued on X thanks to the uniqueness theorem. Consequently, f (X) is contained in the open disk D(b0 , |b1 |) centered at b0 = f (z0 ) with radius |b1 | = A(0)/π due to |F | < 1 on X.

8.4. Sharpness

209

Using Theorem 8.3.1 and making a simple calculation with e−2t A(0) = A(t) and Theorem 8.2.3 (iii), we derive 2−1 Area f (X) ≥

2 E1 (f, z0 , X) 2−1 A(0) i |f (z)|2 dz ∧ d¯ z 4 X 2−1 Area f (X) ,

= = ≥

hence ﬁnding that inequalities become equalities, and consequently f is univalent with f −1 as its inverse. Note again that the Green function is conformal invariant. So the above argument yields gf (X) w, f (z0 ) gX f −1 (w), z0 − log |F f −1 (w) |

gf (X) (w, b0 ) = = =

|b1 | |w − b0 | gD(b0 ,|b1 |) (w, 0),

=

log

=

w ∈ f (X).

These equalities plus Lemma 8.4.1 imply that D(b0 , |b1 |) \ f (X) is of logarithmic capacity 0. This proves (ii). Conversely, if (ii) maps X holds. Then we may assume that f conformally into an open disk D f (z0 ), r0 with radius r0 about f (z0 ) and D f (z0 ), r0 \ f (X) has logarithmic capacity 0. Thus, for p ≥ 0 we have 2 Ep (f, z0 , X)

= f (X)

p gf (X) (ξ, f (z0 ) dm(ξ)

=

D f (z0 ),r0

p gDf (z0 ),r0 (ξ, f (z0 ) dm(ξ)

log

= D(b0 ,|b1 |)

=

p r0 dm(ξ) |ξ − f (z0 )|

πΓ(p + 1) , 2p

hence verifying (i). So, we are done.

To deal with equality in Theorem 8.2.3 (ii), we need to have an insight into the Green function near the ideal boundary of a Riemann surface.

210

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Lemma 8.4.3. Let X be a hyperbolic Riemann surface and ∂X its ideal boundary; let zj → ∂X and gX (z, zj ) converge to a harmonic function gX (z) locally uniformly in X, and let f : X → C be a nonconstant holomorphic function with f Q0 (X) < ∞. For t ≥ 0 and j ∈ N, deﬁne i Xjt = {z ∈ X : gX (z, zj ) > t} and Aj (t) = |f (z)|2 dz ∧ d¯ z. 2 Xjt Then the following statements are valid: Case 1 : gX ≡ 0 implies (i) limj→∞ Aj (t) = 0 for t > 0; (ii) limj→∞ Ep (f, zj , X) = 0 for p > 0. Case 2 : gX ≡ 0 implies that if p, t ≥ 0,

Xt = {z ∈ X : gX (z) > t}

and

A(t) =

i 2

Xt

|f (z)|2 dz ∧ d¯ z,

then (iii) limj→∞ Aj (t) = A(t) for t ≥ 0; (iv) A(t) is a continuous and decreasing function with e2t2 A(t2 ) ≤ e2t1 A(t1 ) for t2 ≥ t1 ≥ 0; (v)

lim

j→∞

0

∞

p

t dAj (t) =

and lim Ep (f, zj , X) =

j→∞

i 2

X

∞

tp dA(t)

0

p |f (z)|2 gX (z) dz ∧ d¯ z.

Proof. Case 1: gX ≡ 0. Fix t > 0. Since f Q0 (X) < ∞, for any > 0 there is a compact set K ⊂ X such that i |f (z)|2 dz ∧ d¯ z < . 2 X\K Using the hypothesis that gX (z, zn ) tends to 0 uniformly on K, we get an N ∈ N such that n − N ∈ N, z ∈ K ⇒ gX (z, zn ) ≤ t. Thus, n − N ∈ N implies Xnt ⊆ X \ K and then i i |f (z)|2 dz ∧ d¯ z≤ |f (z)|2 dzd¯ z < , An (t) = 2 Xnt 2 X\K

8.4. Sharpness

211

proving (i). To see (ii), we follow the second part of the argument for Theorem 8.2.3 to get ∞ 2 lim Ep (f, zj , Xnt ) = lim tp Aj (t) + p λp−1 Aj (λ)dλ j→∞ j→∞ ∞t 2t sp e−2s ds ≤ 2e lim Aj (t) j→∞

t

≤ 2Γ(p + 1)e2t lim Aj (t) j→∞

= 0. In the meantime, for p, t > 0, 2 tp i Ep f, zj , X \ Xjt ≤ |f (z)|2 dz ∧ d¯ z < tp f 2Q0 (X) . 2 X\Xjt The foregoing two estimates imply (ii). Case 2: g ≡ 0. In this case, we have gX (z) > 0 for z ∈ X, but we claim that gX (z) takes arbitrarily small value. Assume, on the contrary, that there is a positive number t0 such that gX (z) ≥ t0 > 0 for z ∈ X. Then, for 0 < t < t0 , we have Xt ⊆ X = lim inf Xjt = j→∞

∞ ∞ + k=1

Xjt .

j=k

Consequently,

f 2Q0 (X) ≤ lim inf Aj (t) ≤ lim sup Aj (t) ≤ f 2Q0 (X) , j→∞

j→∞

a contradiction to the inequality e2t2 Aj (t2 ) ≤ e2t1 Aj (t1 ) for t2 ≥ t1 ≥ 0. This veriﬁes the claim. Accordingly, gX (z) cannot be a constant and A(t) is continuous. Note that if Xt stands for the closure of Xt for t > 0, then Xt ⊆ lim inf Xjt = j→∞

∞ ∞ + k=1

∞ ∞ + Xjt ⊆ lim sup Xjt = Xjt ⊆ Xt

j=k

j→∞

k=1

j=k

implies that A(t) ≤ lim inf j→∞ Aj (t) and i i 2 |f (z)| dz ∧ d¯ z= |f (z)|2 dz ∧ d¯ z. lim sup Aj (t) ≤ 2 Xt 2 Xt j→∞ Thus, limj→∞ Aj (t) = A(t) for which e2t2 A(t2 ) ≤ e2t1 A(t1 ) for t2 ≥ t1 ≥ 0.

212

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

This veriﬁes (iii). Of course, (iv) follows from (iii). It is clear that (v) is valid for p = 0. When p > 0, integrating by parts twice, we obtain ∞ ∞ ∞ ∞ tp dAj (t) = p tp−1 Aj (t)dt → p tp−1 A(t)dt = tp dA(t), 0

0

0

0

hence obtaining the ﬁrst limit of (v). To check the second one, let p > 0 and > 0 be given. Then Theorem 8.2.5 (i) gives 2 2 Ep (f, zj , X) ≤ 2−p Γ(p + 1) E0 (f, zj , X) < ∞. And hence Fatou’s Lemma asserts that p i |f (z)|2 gX (z) dz ∧ d¯ z<∞ 2 X so that, by the absolute continuity of integrals, there is a δ > 0 such that p i i 2 |f (z)| dz ∧ d¯ z<δ⇒ |f (z)|2 gX (z) dz ∧ d¯ z < /3 2 E 2 E holds for any measurable set E. Taking t > 0 such that ∞ 2 sp e−2s ds + 2tp e−2t < /3, 2A(t) < δ and f Q0 (X) t

and letting ∞

Yjt =

Xkt

t and Yt = lim sup Xjt = ∩∞ j=1 Yj ,

k=j

j→∞

we have A(t) =

i 2

|f (z)|2 dz ∧ d¯ z=

Yt

i lim 2 j→∞

Yjt

|f (z)|2 dz ∧ d¯ z

hence getting an N ∈ N such that i |f (z)|2 dz ∧ d¯ z < 2A(t) < δ 2 YN implies i 2

YN

p |f (z)|2 gX (z) dz ∧ d¯ z < /3.

8.4. Sharpness

213

From the preceding estimates we can conclude that for j > N , 2 Ep (f, zj , YN )

2 2 Ep (f, zj , Xjt ) + Ep (f, zj , YN \ Xjt ) ∞ 2t < Aj (t)e sp e−2s ds + 2tp A(t) t ∞ 2 p −2s p −2t ≤ f Q0 (X) s e ds + 2t e =

t

< /3. On the other hand, since gX (z, zj ) ≤ t for j ≥ N and z ∈ X \ YN , by the Bounded Convergence Theorem, there is an N1 > N such that as j > N1 , 2 i p |f (z)|2 gX (z) dz ∧ d¯ z < /3. Ep (f, zj , X \ YN ) − 2 X\YN Putting together the last three /3-estimates and using the triangle inequality, we conclude that if j > N1 , then 2 i p |f (z)|2 gX (z) dz ∧ d¯ z < , Ep (f, zj , X) − 2 X and the second limit of (iv) follows.

The following assertion characterizes when equality in Theorem 8.2.5 (ii) happens. Theorem 8.4.4. Let X be a hyperbolic Riemann surface, z0 ∈ X, and f : X → C a nonconstant holomorphic function with f Q0 (X) < ∞. Then the following two statements are equivalent: q 1 2 Γ(p + 1) 2

f Qq (X) , p > q ≥ 0. (i) f Qp (X) = 2p Γ(q + 1) (ii) X is obtained from a simply connected hyperbolic Riemann surface Y by removing at most a set of logarithmic capacity zero and f is extended onto a conformal mapping from Y to an open disk in C centered at f (w) ∈ C. Proof. (i)⇒(ii) Suppose (i) is valid. Then there exists a sequence {zj } of points in X, which converges to a point z0 in either X or ∂X, such that 2 lim Ep (f, zj , X) = f 2Qp (X) .

j→∞

If z0 ∈ X, then 2 2q Γ(p + 1)

f 2Qq (X) , Ep (f, z0 , X) = f 2Qp (X) = p 2 Γ(q + 1)

214 and hence

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces 2 2 2q Γ(p + 1) Eq (f, z0 , X) Ep (f, z0 , X) = p 2 Γ(q + 1)

which produces (ii) thanks to Theorem 8.4.2. If z0 ∈ ∂X, then there exists zj in X such that zj → z0 as j → ∞. Since limj→∞ supz∈E gX (z, zj ) < ∞ for any compact set E ⊂ X, an application of the Montel Theorem produces a subsequence of {gX (z, zj )} converging to a harmonic function gX (z) locally uniformly in X. This gX is not identical with 0. In fact, if not, then by Lemma 8.4.3 (ii), limj→∞ Ep (f, zj , X) = 0 and so f Qp (X) = 0 which forces that f is a constant — this is a contradiction to the hypothesis. Now, by Lemma 8.4.3, we have ∞ ∞ 2 p

f Qp (X) = − lim t dAj (t) = − tp dA(t). j→∞

0

0

Meanwhile, we also have

f 2Qq (X) ≥ − lim

j→∞

0

∞

tq dAj (t) = −

∞

tq dA(t).

0

These, plus Lemma 8.4.1, yield ∞ ∞ 2q Γ(p + 1) ∞ q p − t dA(t) ≥ − p t dA(t) ≥ − tp dA(t), 2 Γ(q + 1) 0 0 0 and so equality holds and implies A(t) = A(0)e−2t for t > 0. Consequently, ∞ ∞ 2 A(0) A(0) ≥ E1 (f, z0 , X) ≥ lim . tdAj (t) = tdA(t) = j→∞ 2 2 0 0 This implies (ii) thanks to Theorem 8.3.3. (ii)⇒(i) If (ii) holds, then by the uniformization theorem we may assume that X is obtained from the unit disk by removing at most a set of logarithmic capacity zero and f is the identity. Now the calculation in Example 1.1.1 implies

f 2Qp(X) = 2−p πΓ(p + 1), giving (i). We are done. Looking over Theorem 8.4.4, we have a natural question below: is the hypothesis f Q0 (X) = f D(X) < ∞ necessary? The forthcoming example shows that we cannot drop this condition at least for p ≥ 1. Example 8.4.5. Let {zj } be an increasing sequence/on [0, 1) with z1 = 0 and ∞ |σzj−1 (zj )| = 1 − j −2 for j − 1 ∈ N, and let B(z) = j=1 σzj (z). Then 1

B D = ∞ and B Qp = π2−p Γ(p + 1) 2 , but Theorem 8.4.4 (ii) is not valid for such D and B.

p≥1

8.5. Limiting Case – Bloch Classes

215

Proof. Clearly, the above-deﬁned function B is not conformal on D. So Theorem 8.4.4 (ii) fails in this case. Now, for j ∈ N and z ∈ D, let 0 σzk σzj (z) . Bj (z) = B σzj (z) = z k =j

Then, Bj (0) = 0 and Bj (D) ⊂ D for j ∈ N. A simple calculation gives 2j 0 (0) ≥ 1 − (2j + 1)−2 1 > B2j+1 1 − k −2 → 1 as j → ∞, k>2j+1 (0) = 1. This, along with a normal family argument and and so limj→∞ B2j+1 an application of the Schwarz Lemma, implies that B2j+1 (z) converges z locally uniformly in D. Now, for p > 0 and 0 < r < 1, we have p lim |B (z)|2 gD (z, z2j+1 ) dm(z) = (− log |z|)p dm(z). j→∞

|σz2j+1 (z)|
|z|
This implies that if p ≥ 1, then 12 π2−p Γ(p + 1) ≤ B Qp 12 ≤ 2−p Γ(p + 1) B BMOA(D) 12 12 2−1 Area f (D) ≤ 2−p Γ(p + 1) 12 ≤ π2−p Γ(p + 1) and hence the equalities hold. Note also that B is an inﬁnite Blaschke product. So it cannot belong to D, i.e., B D = ∞. We are done.

8.5 Limiting Case – Bloch Classes In this section we deal with the limiting case p → ∞ of Theorem 8.2.3. Lemma 8.5.1. Given a nonnegative function A(t) on (0, ∞) with the following two properties: (i) A(t) is continuous and decreasing with increasing t > 0. (ii) e2t2 A(t2 ) ≤ e2t1 A(t1 ) when t2 ≥ t1 > 0. ∞ Furthermore, for p, t ∈ [0, ∞) let Bp (t) = − t sp dA(s). If there exists p0 ∈ [0, ∞) such that Bp0 (0) < ∞, then 2p Bp (0) = lim e2t A(t). p→∞ Γ(p + 1) t→∞ lim

216

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Proof. By Lemma 8.2.4 we know that Bp0 (0) < ∞ ensures Bp (0) < ∞ as p ≥ p0 . Moreover, integrating by parts twice and using the condition (ii) we have Bp (0)

∞

= p 0 ∞ = p

tp−1 A(t)dt e2t A(t)tp−1 e−2t dt

0

= =

∞ ∞ t pe A(t) s e ds − p sp−1 e−2s ds d(e2t A(t)) 0 0 0 0 ∞ t Γ(p + 1) lim e2t A(t) − p sp−1 e−2s ds d(e2t A(t)). t→∞ 2p 0 0

2t

t

p−1 −2s

Therefore, it suﬃces to check the limit Ip =

2p p Γ(p + 1)

∞

0

t 0

sp−1 e−2s ds d(e2t A(t)) → 0

as p → ∞.

To this end, we observe that the condition (ii) yields that for any > 0 there is a ∞ t0 > 0 such that − < t0 d(e2t A(t)) ≤ 0, and consequently,

Ip,1 =

2p p Γ(p + 1)

∞

t0

0

t

sp−1 e−2s ds d e2t A(t) ≥

∞

d e2t A(t) > − .

t0

At the same time, integrating by parts we get Ip,2

=

2p p Γ(p + 1)

≥

2p Γ(p + 1) p

t0

sp−1 e−2s ds d e2t A(t)

0

0

t

t0

tp d e2t A(t)

0

≥

2 Γ(p + 1)

≥

0 2p e2t0 tp−p 0 Γ(p + 1)

0

t0

tp e2t dA(t)

t0

tp0 dA(t)

0

0 2p e2t0 tp−p Bp0 (0) 0 Γ(p + 1) → 0 as p → ∞.

≥

−

Accordingly, −2 < Ip ≤ 0 for suﬃciently large p. This proves the lemma.

Theorem 8.5.2. Let X be a hyperbolic Riemann surface, z0 ∈ X, and f : X → C holomorphic and nonconstant.

8.5. Limiting Case – Bloch Classes

217

(i) If Xt = {z ∈ X : gX (z, z0 ) > t} and A(t) = then lim e2t A(t) = π

t→∞

i 2

Xt

|f (z)|2 dz ∧ d¯ z for

t > 0,

|f (z )| 2 0 . cX (z0 )

(ii) If Ep0 (f, z0 , X) < ∞ for some p0 ∈ [0, ∞), then 2 |f (z0 )| 2 −1 . lim 2p πΓ(p + 1) Ep (f, z0 , X) = p→∞ cX (z0 ) (iii)

|f (z )| 2 −1 2 0 Ep (f, z0 , X) , p ∈ [0, ∞), ≤ 2p πΓ(p + 1) cX (z0 ) where equality holds if and only if X is obtained from a simply connected hyperbolic Riemann surface Y by removing at most a set of logarithmic capacity zero and f is extended onto a conformal mapping from Y to an open disk in C centered at f (w) ∈ C.

(iv) |f (w)| ≤ w∈X cX (w)

f CB(X) = sup

−1 2p πΓ(p + 1)

f Qp (X) ,

p ∈ [0, ∞),

where equality holds under f D(X) < ∞ if and only if X is obtained from a simply connected hyperbolic Riemann surface Y by removing at most a set of logarithmic capacity zero and f is extended onto a conformal mapping from Y to an open disk D f (w) centered at f (w) ∈ C. Proof. (i) Note that

∗ (z, z0 )) ξ = exp − (gX (z, z0 ) + igX

may be taken as a local variable at z0 . So it follows that ξ(z0 ) = 0, γX (0) = 0 and cX (0) = 1 under the variable ξ. For a suﬃciently large t > 0, Xt is the parameter disk {ξ ∈ D : |ξ| < e−t }, and therefore 2t 2t e A(t) = e |f (ξ)|2 dm(ξ) → π|f (0)|2 as t → ∞, |ξ|<e−t

as desired. (ii) follows from (i) and Lemma 8.5.1. (iii) follows from (ii), Theorem 8.2.5 (i), Theorem 8.4.2 and the uniformalization theorem. (iv) follows from (iii), Theorem 8.2.5 (ii), Theorem 8.4.4 and the uniformalization theorem.

218

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

This theorem suggests that we introduce a Bloch-type space based on the capacitary density. More explicitly, we say that a holomorphic function f on a given hyperbolic Riemann surface X is of the capacitary Bloch class, written f ∈ CB(X) provided f CB(X) < ∞. It is clear that if X is simply connected and τ : D → X is surjectively conformal, then

f CB(X) = f ◦ τ B and hence CB(X) is a natural generalization of the classical Bloch space from the open unit disk D to the hyperbolic Riemann surface X. Meanwhile, we can deﬁne another type of Bloch space through the hyperbolic metric. Given a hyperbolic Riemann surface X with the Poincar´e metric λX , we say that a holomorphic function f : X → C is of Bloch class, written f ∈ B(X), provided |f (z)|

f B(X) = sup < ∞. z∈X λX (z) It is interesting to compare the above two Bloch classes. Theorem 8.5.3. Let X be a hyperbolic Riemann surface, F uc(D) a Fuchsian group such that D/F uc(D) is conformally equivalent to X, and Ω the fundamental region of F uc(D). Then: (i) CB(X) ⊆ B(X). But there is a hyperbolic Riemann surface Y such that CB(Y) = B(Y). (ii) δ(X) = inf w∈Ω conversely.

/ γ∈F uc(D)

|σw γ(w) | > 0 implies CB(X) = B(X), but not

Proof. (i) The ﬁrst part just follows from the inequality cX (z) ≤ λX (z) for any hyperbolic Riemann surface X. So, it is enough to check this inequality. To this end, given z ∈ X and a local variable t in a neighborhood of z with t(z) = 0, let A∗z (X) be the family of all multiple-valued holomorphic functions F : X → C with |F | being single-valued, F (z) = 0 and F (z) = 1 for one of the branches. From [SariO, pp. 177–178] it turns out cX (z) = max sup |F (w)| : F ∈ A∗z (X) > 0. w∈X

Assume FX ∈ A∗z (X) is the unique extremal function with respect to the local coordinate t, and τ : D → X is a holomorphic universal covering mapping such that τ (0) = z. Then cX (FX ◦ τ ) is a single-valued holomorphic self-map of D that has 0 as its ﬁxed point. Note that this function vanishes at each point of the set τ −1 (z). So, the Schwarz Lemma is employed to deduce cX (z)|FX (z)||τ (0)| ≤ 1,

8.5. Limiting Case – Bloch Classes

219

where τ (0) stands for the derivative of t ◦ τ at 0. Consequently, cX (z) ≤ |τ (0)|−1 = λX (z), as required. As for the second part, let f (z) = (1 + z)/(1 − z) and L = {m + in : m, n ∈ Z}, where Z stands for the set of all integers. Then f ∈ B and f −1 (L) is a countable set and hence a set of logarithmic capacity zero. If Y = D \ f −1 (L), then gD (z, w) = gY (z, w) for z, w ∈ Y and consequently, cD (z) = cY (z) for each z ∈ Y. Putting h = f |Y , i.e., the restriction of f to Y, we see that h ∈ B(Y) since h omits all values in L. But, because of f ∈ B, it follows that f ∈ CB(D). Also, since f (z) = h (z) and cD (z) = cY (z) for each z ∈ Y, we conclude that h ∈ CB(Y) amounts to f ∈ CB(D), thereupon getting h ∈ B(Y) \ CB(Y), as desired. (ii) We begin with claiming that if τ : D → X is a universal covering mapping, then 0 cX τ (w) = |σw γ(w) |, w ∈ Ω. λX τ (w) γ∈F uc(D)

Since gΩ (z, w) =

− log σz γ(w) ,

γ∈F uc(D)

it is easy to derive lim gΩ (z, w) + log |z − w| = log(1 − |w|2 ) + − log σw γ(w) .

z→w

γ =id

This, plus the conformal invariance of cX (·) and λX (·), yields cX τ (w) |τ (w)| = |τ (w)| exp − lim gΩ (z, w) + log |z − w| z→w 0 σw γ(w) = |τ (w)|cD (w) = λX τ (w)

γ∈F uc(D)

0

σw γ(w) .

γ∈F uc(D)

Note once again that τ (w) denotes the derivative of t ◦ τ at w where t is a local variable at τ (w) ∈ X with t τ (w) = 0. Thus, the last sequence of equalities veriﬁes the desired claim. Next, it follows from the above claim that cX τ (w) ≤ 1, w ∈ Ω. 0 < δ(X) ≤ λX τ (w)

220

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Since τ : Ω → X is injective and τ : Ω → X is surjective and ∂Ω has area zero, we conclude that CB(X) = B(X). To check the converse, we consider the punctured unit disk Y = D \ {0} which is certainly a hyperbolic Riemann surface. In this case, we have gY (z, w) = gD (z, w) for all z, w ∈ Y, ensuring cY (w) = cD (w) for each w ∈ Y, which in turn means that CB(Y) = CB(D). Since B(Y) = B, it follows that CB(Y) = B(Y). Now, if z+1 and τ (z) = exp φ(z), φ(z) = z−1 then φ and τ are a conformal mapping and a universal covering mapping from D onto LH = {ζ ∈ C : ζ < 0} and Y respectively. Hence, if ψ(ζ) = eζ , then ψ(ζ + 2kπi) = ψ(ζ) for each ζ ∈ LH and each k ∈ Z, and hence, if Γ is a Fuchsian group such that D/Γ is conformally equivalent to Y, then for γ ∈ Γ we have φ γ(z) + 2kπi for some integer k. For the deﬁniteness, we use the = φ(z) notation φ γk (z) = φ(z) + 2kπi for each integer k. Nevertheless, the hyperbolic distance dLH (ζ, ζ + 2πi) on LH approaches 0 as ζ → −∞. Thus, we get lim dD x, γ1 (x) = 0 whence lim |σx γ1 (x) | = 0. 0<x→1

0<x→1

Since ∞ 0 0 |σx γ1 (x) | ≥ |σx γk (x) | ≥ |σx γ(x) | for

x ∈ (0, 1),

γ∈Γ

k=1

we must have δ(Y) = 0. This proves the desired assertion.

Finally, we compare Qp (X), p > 1 with B(X). Theorem 8.5.4. Let X be a hyperbolic Riemann surface, τ : D → X a universal covering mapping, and dX (z, w) = inf{dD (ζ, η) :

z = τ (ζ), w = τ (η)}

the hyperbolic distance between z, w ∈ X. Then: (i) Qp (X) ⊆ B(X) for p ∈ [0, ∞), but there is a hyperbolic Riemann surface Y such that Qp (X) = B(X) for p ∈ (1, ∞). (ii) A holomorphic function f : X → C belongs to B(X) if and only if f is Lipschitz continuous in the hyperbolic distance; that is, |f (z) − f (w)| < ∞. dX (z, w) z,w∈X

f B(X),1 = sup (iii) If

C(X) = sup dX (z, w) log z,w∈X

.

−1 exp gX (z, w) + 1 < ∞, exp gX (z, w) − 1

8.5. Limiting Case – Bloch Classes

221

then B(X) = Qp (X) for any p ∈ (1, ∞). But there is a hyperbolic Riemann surface Y such that C(Y) = ∞ and B(Y) = Qp (Y). Proof. (i) The inclusion follows from Theorems 8.5.2 and 8.5.3. On the other hand, let again f (z) = (1 + z)/(1 − z) and L = {m + in : m, n ∈ Z}. Then f ∈ B = Qp for p > 1, and f −1 (L) is a countable set and hence a set of logarithmic capacity zero. If Y = D \ f −1 (L), then gD (z, w) = gY (z, w) for z, w ∈ Y. Putting h = f |Y , i.e., the restriction of f to Y, we see that h ∈ B(Y) since h omits all values in L. However, f ∈ Qp as p > 1 and Ep (h, w, Y) = Ep (f, w, D) = Ep (f, w),

w∈Y

lead to h ∈ Qp (Y). This proves B(Y) = Qp (Y). (ii) Suppose that τ is a universal covering mapping from D onto X. If f ∈ B(X), then f ◦ τ ∈ B and hence |f τ (ζ) − f τ (η) | |f τ (ζ) − f τ (η) | = dD (ζ, η) f ◦ τ B dD (ζ, η), dD (ζ, η) thus giving f B(X),1 < ∞. Conversely, since |f τ (ζ) − f τ (η) | |f (z) − f (w)| ≤ , dD (ζ, η) dX (z, w)

z = τ (ζ), w = τ (η),

we conclude that f B(X),1 < ∞ implies f ◦ τ ∈ B, namely, f ∈ B(X). (iii) Suppose {Xj } is a regular exhaustion of X and ∗ (z, w) hj (z, w) = gXj (z, w) + igX j ∗ (z, w) is a harmonic conjugate of gXj (z, w), being locally deﬁned up to where gX j an additive constant. Applying the formulas p−2 ∆u(z) = 4|f (z)|2 and ∆gXj (z, w) = p(p − 1) gXj (z, w) |∇gXj (z, w)|2

to the Green formula ∂v ∂u i u −v ds v(z)∆u(z) − u(z)∆v(z) dz ∧ d¯ z=2 2 Xj ∂n ∂n ∂Xj with

u(z) = |f (z) − f (w)|2

p and v(z) = gXj (z, w) ,

and taking the limit as the radius of the disk at w shrinks to 0, we derive 2 2 ip(p − 1) 2 |∇gXj (z, w)| Ep (f, w, Xj ) = |f (z) − f (w)| z. 2−p dz ∧ d¯ 22 Xj gXj (z, w)

222

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces Let d(j) (z, w) be the restriction of dX (z, w) to Xj and . exp gXj (z, w) + 1 Lj (z, w) = log . exp gXj (z, w) − 1

Then the inequality gXj (z, w) ≤ gX (z, w) for z, w ∈ Xj implies C(Xj )

= sup ≤

d(j) (z, w) Lj (z, w)

: z, w ∈ Xj

sup dX (z, w) log

z,w∈Xj

.

−1 exp gX (z, w) + 1 exp gX (z, w) − 1

≤ C(X). Now if f ∈ B(X) and f B(Xj ),1 = sup

z,w∈Xj

|f (z) − f (w)| , d(j) (z, w)

then by (i),

≤ ≤

2 Ep (f, w, Xj ) 2 ip(p − 1) (j) 2 |∇gXj (z, w)|2 d

f B(Xj ),1 (z, w) z 2−p dz ∧ d¯ 22 Xj gXj (z, w) 2 2 |∇gXj (z, w)|2 2 ip(p − 1) (z, w) z

f B(Xj ),1 L C(Xj ) 2−p dz ∧ d¯ j 22 Xj gXj (z, w) ∂gXj (z,w) ∞ ds 2 2 ∂n {z∈Xj :gXj (z,w)=t}

f B(Xj ),1 C(Xj ) dtp−1 −2 t e +1 0 log et −1 2 ∞ 2 2 et + 1 C(Xj ) dtp−1 log t

f B(Xj ),1 e −1 0

2 2 |f (z) − f (w)| C(Xj ) sup dX (z, w) z,w∈Xj 2 C(Xj ) f 2B(X),1 .

Note that gXj (z, w) → gX (z, w) as j → ∞. So we have 2 2 Ep (f, w, Xj ) C(Xj ) f 2B(X),1 . This implies f ∈ Qp (X), p > 1. Consequently, B(X) = Qp (X), p > 1 thanks to Qp (X) ⊆ CB(X).

8.6. Notes

223

For the converse, let Y = D \ {0}, then gY (z, w) = gD (z, w) for z, w ∈ Y and hence Qp (Y) = Qp . Note that limz→0 gY (z, 2−1 ) = log 2. So . lim log

z→0

√ exp gY (z, 2−1 ) + 1 2. = log exp gY (z, 2−1 ) − 1

However, by using the mapping z = eζ from LH = {ζ ∈ C : ζ < 0} onto Y, it is not hard to see that dY (z, 2−1 ) grows asymptotically to log(− log |z|) as z → 0, so C(Y) = ∞. But, we will see that B(Y) = Qp (Y) for p > 1. In fact, suppose f ∈ B(Y). If F (ζ) = f (eζ ) belongs to B(LH) then sup |z|(− log |z|)|f (z)| = sup | ζ||F (ζ)| < ∞. z∈Y

ζ∈LH

But f ∈ B(Y) forces that it has an isolated singularity at z = 0 which means that either f is bounded near z = 0 or lim supz→0 |z|−2 |f (z)| > 0. From the last equality it turns out that for f ∈ B(Y) we must have that f is bounded near z = 0 and so that we can deﬁne f (0) and f (0) as ﬁnite values. Consequently, f ∈ B = Qp when p > 1. This gives B(Y) ⊆ Qp (Y) ⊆ CB(Y) ⊆ B(Y), as desired.

8.6 Notes Note 8.6.1. Section 8.1 is a small (but necessary for our aim) portion of [Ahl, Chapters 9 and 10]. Some basic properties on the hyperbolic and capacitary metrics can be found in [Min1], [Min2] and [Min3]. Note 8.6.2. Section 8.2 is a combination of [AuCh, Sections 2 and 3]. Here, it is worth pointing out that the inclusion chain: Q0 (X) ⊆ Qq (X) ⊆ Qp (X),

for p > q > 0,

was ﬁrst veriﬁed by [AuHRZ] in a way extending [Met] and [Kob1] which proved Q0 (X) ⊆ BMOA(X). In addition, the left-hand-side inclusion can be improved under the condition that the hyperbolic Riemann surface X is regular; that is, lim gX (z, w) = 0,

w→∂X

z ∈ X.

More precisely, if f ∈ Q0 (X) and X is regular, then f ∈ Qp,0 (X),

i.e.,

lim Ep (f, w, X) = 0;

w→∂X

see also [AuHRZ, Theorem 3.2]. Theorem 8.2.5 has been extended to Qp spaces on the unit ball of Cn — see also [Che].

224

Chapter 8. Holomorphic Q-Classes on Hyperbolic Riemann Surfaces

Note 8.6.3. Section 8.3 just describes the main ideas and results in [Kob1] and [Kob2]. A survey on the study of BMOA on Riemann surfaces is given by [Met]. Here it is perhaps appropriate to mention that Theorem 8.3.1 (ii) cannot extend to p ∈ (0, 1). This is because exp (z + 1)/(z − 1) does not belong to Qp where p ∈ [0, 1) while its image area is clearly less than π. On the other hand, we can obtain an interesting consequence of Theorem 8.3.1 as follows: if G is a hyperbolic subdomain of C, then G

gG (z, w)dm(z) ≤ 2−1 Area(G);

see also [Kob1, Corollary 7]. This leads to the concept of a BMOA-domain G which means that any holomorphic function sending D into G lies in BMOA. According to [Met, Theorem 7.3], we have that G is a BMOA-domain if and only if the identity map I belongs to BMOA(G), namely, sup w∈G

G

gG (z, w)dm(z) < ∞.

Furthermore, it has been proved in [HayPo] and [Ste2] that G is a BMOA-domain when and only when there are positive constants κ1 and κ2 such that cap0 (C \ G) ∩ D(w, κ1 ) ≥ κ2 ,

w ∈ G,

where D(w, κ1 ) stands for the open disk centered at w with radius κ1 . Of course, there are many examples to show that this condition does not characterize the range of a function in BMOA — see [Met, p. 90]. Naturally, we can deﬁne a Qp -domain, but ﬁnd that the cases p ≥ 1 produce the BMOA-domains and Blochdomains while the cases p ∈ [0, 1) do not make sense — in other words, there are no Qp -domains whenever p ∈ [0, 1) — see [EsXi] — but we can talk about conformal domains in this case — see Section 1.3. In addition, a John–Nirenberg type estimate for BMOA on the hyperbolic Riemann surface was established in [Zha1] — more explicitly, a holomorphic function f on the hyperbolic Riemann surface X belongs to BMOA(X) if and only if there are positive constants κ1 and κ2 such that for every w ∈ X and t > 0, i 2

{z∈X:|f (z)−f (w)|>t}

|f (z)|2 gX (z, w)dz ∧ d¯ z ≤ κ1 exp(−κ2 t).

Furthermore, for another investigation of holomorphic maps between Riemann surfaces which preserve BMO, see also [Got1], [Got2], and [Got3]. Note 8.6.4. Section 8.4 is a modiﬁed combination of [AuCh, Sections 4 and 5]. Lemma 8.4.1 is a sort of special formulation of [Min3, Theorem 1] and its proof.

8.6. Notes

225

Note 8.6.5. Section 8.5 is taken from [AuCh, Section 6] and [AuLXZ]. The argument for cX (z)|dz| ≤ λX (z)|dz| in Theorem 8.5.3 is extracted from [Min3, Theorem 2] which also tells us that both metrics coincide at a single point if and only if X is simply connected. Furthermore, it would be of interest to study when CB(X) is contained in Qp (X) whenever p > 1. In [AuCh, Section 7] the following area inequality has been proved: if X is a ﬁnite Riemann surface with func the Green −1 tion gX (·, ·) and f : X → C is holomorphic, then f CB(X) ≤ π Area f (X) , which is established via verifying Area f ({z ∈ X : gX (z, w) > t0 }) 2(t−t0 ) , t > t0 ≥ 0. e ≤ Area f ({z ∈ X : gX (z, w) > t})

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Index Besov space, 14 Bessel capacity, 139, 140 Beurling inequality, 189 transform, 158 weak-type estimate, 171 Beurling–Zygmund’s Theorem, 190

John–Nirenberg distribution, 176

Chang–Marshall theorem, 174 Choquet integral, 74 complex interpolation, 164 conditional expectation, 26 corona, 156 curvature, 3

Marshall’s Lemma, 172 Melnikov’s Covering Lemma, 75 M¨ obius transform, 2 Moser’s Theorem, 173 Myrberg’s formula, 195

dualities Anderson–Clunie–Pommerenke, 108 Feﬀerman–Sarason, 108 dyadic subarc, 74 extremal distance, 171 fundamental region, 195 Green function, 3, 193, 195 Hardy space, 20 Hardy–Littlewood identity, 27 Hardy–Stein identity, 36 Harnack principle, 16 hyperbolic surface, 193 integrated Liptschitz class, 178 interpolating sequence, 144 isoperimetric, 165, 195

Korenblum inequality, 168 Liouville equation, 2 Littlewood–Payley inequalities, 36 logarithmic capacity, 171

Nevanlinna characteristic, 182 nontangential maximal function, 79 outer function, 31 parabolic surface, 193 Poincar´e metric, 194 pointwise multipliers, 137 radius conformal, 2 harmonic, 3 hyperbolic, 2 reproducing formula, 43, 57 Robin’s constant, 194 Schwarz derivative, 23 Sobolev space, 89 strong type inequality, 139 symmetrization, 7 total variation, 81, 83

240 universal covering surface, 194 vanishing Q class, 96 weak factorization, 133 weighted Bergman space, 32

Index