On linear operators and their applications in complex function spaces of the unit disc

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uef.fi

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-3103-0

Dissertations in Forestry and Natural Sciences

DISSERTATIONS | TANELI KORHONEN | ON LINEAR OPERATORS AND THEIR APPLICATIONS IN... | No 344

TANELI KORHONEN

ON LINEAR OPERATORS AND THEIR APPLICATIONS IN COMPLEX FUNCTION SPACES OF THE UNIT DISC

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

This thesis contains several new results on linear operators acting on spaces of complex-

valued functions and their applications to the study of analytic functions. Two weight inequalities for maximal Bergman projection and a radial averaging operator are discussed.

We apply Carleson measures and boundedness of certain operators in the study of weighted Bergman spaces. Operator theoretic approach

yields results on the growth of solutions of linear differential equations in the unit disc.

TANELI KORHONEN

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PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND DISSERTATIONS IN FORESTRY AND NATURAL SCIENCES

N:o 344

Taneli Korhonen

ON LINEAR OPERATORS AND THEIR APPLICATIONS IN COMPLEX FUNCTION

SPACES OF THE UNIT DISC

ACADEMIC DISSERTATION

To be presented by the permission of the Faculty of Science and Forestry for public examination in the Auditorium M100 in Metria Building at the University of Eastern Finland, Joensuu, on June 15th, 2019, at 12 o’clock.

University of Eastern Finland Department of Physics and Mathematics

Joensuu 2019

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Grano Oy Jyväskylä, 2019

Editors: Pertti Pasanen, Jukka Tuomela, Matti Tedre, and Raine Kortet

Distribution:

University of Eastern Finland Library / Sales of publications julkaisumyynti@uef.fi

http://www.uef.fi/kirjasto

ISBN: 978-952-61-3103-0 (print) ISSNL: 1798-5668

ISSN: 1798-5668 ISBN: 978-952-61-3104-7 (pdf)

ISSNL: 1798-5668 ISSN: 1798-5676

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Author’s address: University of Eastern Finland

Department of Physics and Mathematics P.O.Box 111

80101 JOENSUU FINLAND

email: taneli.korhonen@uef.fi Supervisors: Professor Jouni Rättyä

University of Eastern Finland

Department of Physics and Mathematics P.O.Box 111

80101 JOENSUU FINLAND

email: jouni.rattya@uef.fi

Reviewers: Professor Oscar Blasco

Universitat de València

Department of Mathematical Analysis Avda. Doctor Moliner, 50

46100 BURJASSOT SPAIN

email: oscar.blasco@uv.es

University Lecturer Jari Taskinen University of Helsinki

Department of Mathematics and Statistics P.O.Box 68

00014 HELSINKI FINLAND

email: jari.taskinen@helsinki.fi

Opponent: Professor Igor Chyzhykov

Ivan Franko National University of Lviv Faculty of Mechanics and Mathematics 1 Universytetska Street, 1

79000 LVIV UKRAINE

email: chyzhykov@yahoo.com

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Taneli Korhonen

On linear operators and their applications in complex function spaces of the unit disc

Joensuu: University of Eastern Finland, 2019 Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences N:o 344

ABSTRACT

This thesis contains several new results on linear operators acting on spaces of complex-valued functions and their applications to the study of analytic functions.

Two weight inequality for maximal Bergman projection induced by a regular weight is characterized. Conditions for boundedness of a radial averaging operator acting on Bergman and Lebesgue spaces are studied and compared to those of the Bergman projection.

Operator theoretic approach is applied to study the growth of solutions of linear differential equations in the unit disc. The method used is based on successive integration and yields results that force the solutions of the differential equation, or their derivatives, to the general growth spaceH⁸_ωpDq.

Carleson measures and boundedness of certain operators are used to study weighted Bergman spaces A^pω induced by a doubling weight or a Bekollé-Bonami- type weight. A characterization for zero sets of Bergman spaces is given, and Horowitz’s factorization result from 1977 is generalized. Results on dominating sets and sampling measures are also presented.

MSC 2010:30H20, 47G10, 34M10, 47B34.

Keywords:Bekollé-Bonami weight, Bergman projection, Bergman space, Carleson measure, differential equation, dominating set, doubling weight, factorization, growth space, Hardy spaces, Muckenhoupt weight, radial averaging operator, sampling measure, two weight in- equality, unit disc, zero sequence.

Library of Congress Subject Headings:Functions of complex variables; Differential equa- tions, Linear; Function spaces; Bergman spaces; Operator theory; Linear operators; Factor- ization (Mathematics).

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ACKNOWLEDGEMENTS

First of all, I wish to express my gratitude to my supervisor, Professor Jouni Rättyä, for his valuable advice during the four years of working on this thesis.

I give my warmest thanks to the faculty and staff of the Department of Physics and Mathematics of the University of Eastern Finland for providing me with a friendly and comfortable working environment. I also wish to thank Professor Brett Wick for his kind guidance and the rest of the staff of the Department of Mathemat- ics and Statistics of the Washington University in St. Louis for their hospitality and welcoming atmosphere during my visit in the fall of 2016.

For financial support, I am indebted to the Faculty of Science and Forestry of the University of Eastern Finland, projects #4900024, #931351 and #23376, and to the Academy of Finland through the research projects #268009 of Professor Rättyä and

#286877 of Professor Risto Korhonen.

I am grateful to Professor Oscar Blasco and University Lecturer Jari Taskinen for the review of my thesis. I also wish to thank Professor Igor Chyzhykov for acting as my opponent.

Finally, I want to thank my parents Marjut and Jukka, the rest of my family and all my friends for their constant support and encouragement during my studies.

You bring soul and meaning to my life and help pull my head out of the sometimes confusing world of math when it starts to be too much for my sanity. Special thanks to Jouko and Johannes for their friendship and rewarding discussions. Thanks also to Joensuun Mieslaulajat, in particular Kari, Timo, Perttu, Jarmo, Olli, Ossi, Niklas and Leevi, for the ageless brotherhood of artists and jokers.

Joensuu, February 27, 2019 Taneli Korhonen

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LIST OF PUBLICATIONS

This thesis consists of the present review of the author’s work in the field of mathematical analysis and the following selection of the author’s publications:

I J.-M. Huusko, T. Korhonen and A. Reijonen, "Linear differential equations with solutions in the growth spaceH⁸_ω,"Ann. Acad. Sci. Fenn. Math.41(2016), 399–

416.

II T. Korhonen and J. Rättyä, "Zero sequences, factorization and sampling measures for weighted Bergman spaces,"Math. Z.291(2019), no. 3-4, 1145–1173.

https://doi.org/10.1007/s00209-019-02243-7

III T. Korhonen, J. Á. Peláez and J. Rättyä, "Radial two weight inequality for maximal Bergman projection induced by a regular weight," submitted

https://arxiv.org/abs/1805.01256

IV T. Korhonen, J. Á. Peláez and J. Rättyä, "Radial averaging operator acting on Bergman and Lebesgue spaces,"Forum Math.(2019)

https://doi.org/10.1515/forum-2018-0293

Throughout the overview, these papers will be referred to by Roman numerals.

AUTHOR’S CONTRIBUTION

The publications selected in this dissertation are original research papers on mathematical analysis.

PaperIis a continuation of research done in Joensuu. All authors have made an equal contribution.

The research for PaperIIwas done in Joensuu, both authors have made an equal contribution.

In PapersIIIandIV, all authors have made an equal contribution.

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1 Introduction

Linear operators are an important and much studied topic in the field of mathematical analysis. In addition to being interesting in their own right, they are useful tools in the study of many other properties of functions and function spaces. In this overview part of the thesis, we consider questions related to boundedness of several linear operators acting on L^p spaces, weighted Bergman spaces and some other spaces of analytic functions. Some of the operators considered are also applied in the study of properties of these function spaces. In PapersIandII, linear operators appear mostly as a tool in achieving other goals, while PapersIIIandIV are dedicated more to the study of the operators themselves.

The simplest example of a linear operator between two linear spaces is the identity operatorId that maps an element to itself,Id:x ÞÑx. At least in the context of spaces of analytic functions on a domainDof the complex placeC, even this operator can be of interest. Indeed, Carleson measures for a spaceXof analytic functions onDare defined as positive Borel measuresµonDfor which the identity operator is bounded from the spaceXto the Lebesgue spaceL^pµpDq, 0 p 8, meaning that there exists a constantCCpp,µ,Xq ¡0 such that}f}_L^p

µpDq ¤C}f}X for all f PX.

Here and in the remainder of this overview,CCpqdenotes an absolute constant whose value depends on parameters indicated in the parentheses and may change from one occurrence to another. Moreover, inequalities of the type above are often written as aÀb, meaning that there exists an absolute constantC Cpq ¡0 such thata¤Cb. The notationaÁbis understood in an analogous manner, and ifaÀb andaÁb, we writeab.

In this thesis, Carleson measures are mainly considered in the case of weighted Bergman spaces A^pω, 0 p 8, induced by a radial weight ω in the unit disc D tzPC:|z| 1uadmitting a certain doubling property. They will be applied in the proofs of different integral estimates and boundedness of operators throughout PapersII–IV. Especially in PaperII, many of these estimates and operators are then used to prove other properties of the space. We consider, for example, zero sets and factorization of functions in Aω^p, and dominating sets and sampling measures for Aω^p. Some of these topics are also considered in a case where the weight ω is a certain type of non-radial weight. Certain operators are an important tool also in this non-radial case but, due to the lack of characterizations for Carleson measures, their boundedness is determined by other means.

The non-radial weights considered in this thesis are defined either as an analogue for the radial doubling weights mentioned above or via the Bergman projection. The boundedness of the Bergman projection Pη, η ¡ 1, was characterized by Bekollé and Bonami [6, 7], who showed thatPηis bounded inL^pω, 1 p 8, if and only if

»

SpaqωpzqdApzq »

Spaqωpzq^p1¹ p1 |z|²q^ηp¹dApzq _p₁

À p1 |a|q^p^η ^2qp

for all a P Dzt0u. Here and throughout this overview, dApzq ^{dx dy}_π denotes the

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element of the normalized Lebesgue measure onD,p¹ P p1,8qis such that ¹_p _p¹₁ 1 when 1 p 8, and Spaq  D denotes the Carleson square determined by the point a P Dzt0u. In addition to applying this classical result in introducing a new class of non-radial weights in D, PaperIIIhas been dedicated to the study of the boundedness of the maximal Bergman projectionPω, induced by a slightly more general radial weightω, between two Lebesgue spaces induced by different weights.

Due to its many applications in the theory of Bergman spaces, characterizing this two weight inequality for the Bergman projection is a famous and, in general, still unsolved problem.

The boundedness of the Bergman projection is, in some cases, related to the boundedness of a Hardy type integral operatorTω measuring theω-average over a line segment from a pointzPD to the boundary of the unit disc. This operator is a generalization of the classical Hardy operator which has been a topic of extensive study for decades. In PaperIV, we study the operatorTωand compare the obtained conditions for its boundedness to those of the Bergman projection.

Linear operators can also be used to study ordinary linear differential equations.

For example in PaperI, we discuss the growth of solutions of the equation

f^pnq A_n1f^pn1q A1f¹ A0f An, n¥2, (1.1) where the coefficients A0, . . . ,Anare analytic in a simply connected domainDC, denoted by A0, . . . ,An P HpDq. Our method in this study is based on successive integration and proves to be quite efficient when forcing the solutions of (1.1) to the growth space H⁸ωpDqby restricting the coefficientsAk. The use of operators in this approach is more easily visible when considering the important second order case

f² A f 0

in the unit disc, when it can be seen that the solutions f belong to the space H⁸_ω whenever the operator

SApfqpzq

»_z

0

»_ζ

0 fpwqApwqdw

dζ, zPD,

is bounded in H⁸_ω with }S_A}_H⁸_ω_ÑH_ω⁸ 1. This operator theoretic method in the study of differential equations was introduced by Pommerenke in [59] and also yields concrete estimates for the norm of the solution f.

The remainder of this overview is organized as follows. In Chapter 2, we introduce several function spaces and related topics, most of which are considered in Paper II. Chapter 3 is devoted to a discussion of linear differential equations and growth of their solutions along with the introduction of the methods considered in Paper I. Linear operators, in particular Hardy operators and the Bergman pro- jection, are introduced in Chapter 4. Finally, Chapter 5 contains the summaries of PapersI–IV.

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2 Function spaces

In this chapter we define several function spaces that are of interest considering the scope of this thesis. Most of the time we are interested in functions that are analytic in a domain Dof the complex planeC. With that in mind, letHpDqdenote the algebra of analytic functions in a domain D. In what follows, the domain Dis usually the unit disc D tzPC :|z| 1u, but we will also briefly consider cases where Dis either the whole planeCor some other starlike domain. We say that a simply connected domainDis starlike if it contains the origin and all line segments connecting it to any other point in the domain, that is, r0,zs  DwheneverzP D, wherer0,zsdenotes the line segment connecting the pointzand the origin.

We begin by introducing several classes of weights that are necessary to define some of the spaces we are interested in. We then define Hardy, Bloch andQKspaces before moving on to Lebesgue and (weighted) Bergman spaces which are at the center of this thesis. Finally, we discuss some topics, such as zero sets, factorization and Carleson measures, related to the spaces in question. We will also introduce results of interest regarding these topics in some of the spaces mentioned above, mainly the weighted Bergman space.

2.1 CLASSES OF WEIGHTS

Let Dbe a domain in the complex planeC. A functionω : DÑ r0,8q, integrable over D, is called a weight. Such a function also naturally induces a new measure for the domain, and thus we write ωpEq ³

EωpzqdApzqfor measurable subsets E of the domain D. For the remainder of this section, let D Dand ω be a weight in D. We say that ω is radial if ωpzq ωp|z|q for all z P D. Whenever dealing with analytic functions in a space induced by a radial weight ω, we assume that ωpzq p ³₁

|z|ωpsqds¡0 for allzPD, since otherwise the space would, in most cases, contain all analytic functions inD.

A radial weightωbelongs to the classDp if there exists a constantCCpωq ¡1 such that the doubling condition ωprq ¤p Cωpp ¹₂^rq is satisfied for all 0 ¤ r 1.

Moreover, if there existKKpωq ¡1 andCCpωq ¡1 such that p

ωprq ¥Cωp

11r K

, 0¤r 1,

we then writeω P qD. The intersectionDpX qD is denoted byD. The definitions of these classes of weights are geometric in nature, and the classes themselves arise naturally in the study of classical operators, see for example [53]. A radial weightω is regular if ωprq p ωprqp1rqfor all 0 ¤ r 1. The class of regular weights is denoted byR^{, and}RD. The difference between the classesR^andD is that the weights in Ddo not need to have any local smoothness and can even vanish on a set of positive measure.

We now present two lemmas on the classesDp^andD^qin order to give some insight into their properties and for further reference. For a proof of a more expansive

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version of the first lemma, see [49, Lemma 2.1]; the second lemma can be proved by similar arguments.

Lemma 2.1. Letωbe a radial weight inD. Then the following statements are equivalent:

(i) ωP pD;

(ii) There exist constants CCpωq ¡0andββpωq ¡0such that ωprq ¤p C

1r 1t

_β

ωptq, 0p ¤r¤t 1;

(iii) There exist constants CCpωq ¡0andγγpωq ¡0such that

»_t

0

1t 1s

_γ

ωpsqds¤Cωptq, 0p ¤t 1;

(iv) There exists a constant CCpωq ¡0such that ωx

»₁

0 s^xωpsqds¤Cωp

11 x

, xP r1,8q.

Lemma 2.2. Let ωbe a radial weight in D. Thenω P qD if and only if there exist C Cpωq ¡0andγγpωq ¡0such that

ωptq ¤p C 1t

1r γ

ωprq, 0p ¤r¤t 1.

Lemma 2.1(ii) shows that if ω P pD, then there exists β βpωq ¡ 0 such that

p ωprq

p1rq^β is essentially increasing onr0, 1q. Similarly, ifωP qD^{, then} _p1rq^ω^p^p^r^qγ is essentially decreasing on r0, 1q for γ γpωq ¡ 0 sufficiently small by Lemma 2.2. Thus, it is evident that belonging to classes Dp ^or D^q puts restrictions on the growth of the weights when approaching the boundary of the disc: the weights in the class Dp cannot decrease much faster than the weights of the type p1rq^α, 1 α 8, and the weights in Dq cannot grow much faster than these weights. We say that the weights ωpzq p1 |z|²q^α, 1 α 8, are of a standard type as they all belong toRand are the most natural weights to consider in the unit disc. For more information on the classesDp^,D^q^,D^andR, see [49–52].

Let us now consider some non-radial weights inD. We begin with a generalization of the class Dp, and write ω P pDpDqif there existsC Cpωq ¡0 such that ωpSpaqq ¤ CωpSp¹ ₂^|â^|eⁱârgâqq for all a P Dzt0u. Here, Spaq  D is the Carleson square determined by the point aPDzt0uand defined by

Spaq

"

re^iθPD :|a| r 1,|argae^iθ| 1 |a|

2

*

, aPDzt0u.

EachωP pDpDqsatisfiesωpSpa¹qq ¤CpC 1qωpSpaqqfor alla,a¹ PDzt0uwith|a¹|

|a|and arga¹ arga p1 |a|q, and thusωpSpaqq À ωpSpbqqwhenever|b| ¹ ₂^|a|

and Spbq  Spaq. Furthermore, it is obvious that radial weights in DppDqform the classDp. To provide more insight into the behavior of weights in the classDp^{pDq, we} present the following lemma which has been proven in PaperII.

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Lemma 2.3. [35, Lemma 14] Let ω be a weight on D such that ωpSpaqq ¡ 0 for all aPDzt0u. Then the following statements are equivalent:

(i) ωP pDpDq;

(ii) There existββpωq ¡0and CCpωq ¥1such that ωpSpaqq

p1 |a|q^β ¤C ωpSpa¹qq

p1 |a¹|q^β, 0 |a| ¤ |a¹| 1, argaarga¹; (iii) For some (equivalently for each) K¡0there exists CCpω,Kq ¡0such that

ωpSpaqq ¤Cω

S

K |a|

K 1 eⁱ^arg^a

, aPDzt0u;

(iv) There existηηpωq ¡0and CCpη,ωq ¡0such that

»

D

ωpzq

|1az|^η dApzq ¤C ωpSpaqq

p1 |a|q^η, aPDzt0u.

We now proceed to introduce a class of weights with more local regularity. To do this, let ϕapzq _1az^az, a,z P D, be the automorphism of the unit disc that in- terchanges the origin and the point a P D, and let ρpz,wq |ϕwpzq| be the pseudohyperbolic metric in the unit disc. Denote the pseudohyperbolic disc centered at aPD and of radiusrP p0, 1qby ∆pa,rq tzPD :ρpz,aq ru. The weightω inD is called invariant if for some (equivalently for all)r P p0, 1qthere exists a constant CCpω,rq ¥1 such thatC¹ωpaq ¤ωpzq ¤Cωpaqfor allaPDandzP∆pa,rq. For more information on invariant weights, see [50].

Finally, we define a class of weights in a manner similar to the classical Bekollé- Bonami class [6,7] which will be briefly discussed later in Section 4.2. For 1 q 8, writeωPBq if the weightωis almost everywhere strictly positive and

Bqpωq sup

S

1

|S|²

»

Sωpzqp1 |z|²q^2qdApzq 1

|S|²

»

Sωpzq^q1¹ dApzq _q1

8,

where the supremum is taken over all Carleson squares S D and |S| denotes the Euclidean area of S. Moreover, we denote B8

q¡1Bq. We will discuss the relation of the classes Bq to the Bekollé-Bonami classes in Section 4.2. For more on the properties of the classesBq, see Proposition 12 in PaperII.

2.2 HARDY, BLOCH AND QK SPACES

In this section we briefly define several function spaces for later discussion. We begin with the Hardy spaces by defining the integral means as follows. For a function f, measurable inD, and 0 r 1, set

Mppr,fq 1

2π

»_2π

0 |fpre^itq|^pdt _1{p

, 0 p 8,

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and M₈pr,fq sup_|z|r|fpzq|. For 0 p ¤ 8, the Hardy spaceH^pconsists of f P HpDqsuch that}f}_H^p sup₀_r₁Mppr,fq 8. We will discuss some properties of the Hardy spaces later and refer to [14] for further information.

Let us build on the definition of the spaceH⁸. This space consists of bounded analytic functions in the unit disc. However, if we wish to allow a specific rate of growth for the functions in the space, we may add a suitable weight function inside the supremum. Hence, we define the growth space as a sort of weighted H⁸ as follows. Letωbe a weight in a domain Dof the complex plane. The growth space H⁸_ωpDqconsists of functions f PHpDqsuch that

}f}H_ω⁸ :sup

zPD|fpzq|ωpzq 8.

IfDD, we simply writeH⁸_ω. If, in addition,ωpzq p1 |z|q^pfor somepP p0,8q, we write H⁸_ω H⁸_p . Here, we have put ωpzq p1 |z|q^p instead of the usual p1 |z|²q^p in order to simplify calculations in PaperI. This choice, of course, does not change the space generated because 1r¤1r²¤2p1rqfor allrP r0, 1q.

A similar modification can also be made for the H^p spaces: For 0 α 8, the space Hα^p consists of functions f, analytic in the unit disc, such that }f}_H^p

α

sup_0 r 1Mppr,fqp1r²q^α 8. Of course, this definition can also be generalized for weights other than the standard ones, but we will not need such spaces in this overview.

In order for the space Hω⁸pDqto be non-trivial, the weightω must satisfy some conditions. Firstly, ω cannot have a limit supremum tending to infinity on a set of positive measure at the boundary, otherwise the space would contain only the function f 0. Secondly, ω also should not vanish on an ”annulus” next to the boundary because otherwise the space would contain all analytic functions inD.

Let 0 α 8. The α-Bloch space B^α consists of functions f P HpDq such that }f}_B^α : sup_zP_D|f¹pzq|p1 |z|²q^α 8. In the case α 1, we write B¹ B and simply speak of the Bloch space. It is worth noting that } }_B^α, as defined above, is not a norm in B^α since it does not distinguish functions that differ from each other by an added constant. To obtain a norm inB^α, one may instead define }f}_Bαsup_zP_D|f¹pzq|p1 |z|²q^α |fp0q|.

We now turn our attention toQKspaces. Letgpz,wq log^1wz_w_zbe the Green’s function in the unit disc, and let K :r0,8q Ñ r0,8qbe non-decreasing. The space QKconsists of functions f PHpDqsuch that

supaPD

»

D

|f¹pzq|²Kpgpz,aqqdApzq 8.

Respectively,QK,0is the space of functions f PHpDqsuch that

|a|Ñ1lim

»

D|f¹pzq|²Kpgpz,aqqdApzq 0.

IfK1, thenQKis the Dirichlet spaceD.

Let us now briefly introduce some properties of the spacesQKandQK,0. Firstly,

if »₈

1 Kprqe^2rdr 8 (2.1)

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does not hold, then QKcontains only constant functions. Thus, we shall here after assume that (2.1) is true. If, in addition,Kis continuous, thenQKB^and^DQK. Furthermore,QKBif and only if

»₁

0

Kplogrq

p1rq² r dr 8, (2.2)

D QKif and only if Kp0q ¡ 0, andD QK,0if and only ifKp0q 0. Finally, the conditionsB^αQK,0,B^αQKand

»₁

0

Kplogrq

p1rq^2α r dr 8 (2.3)

are equivalent. For more information on theQKspaces and proofs of the facts above, see [15].

2.3 BERGMAN AND LEBESGUE SPACES

Let 0 p 8, and letpX,µqbe a measure space, where the measureµis positive.

TheL^pspace of the setXconsists ofµ-measurable functions f :XÑCsuch that }f}_L^p_µ_pXq

»

X|fpxq|^pdµpxq ¹_p

8.

In this overview, the set Xis usually the unit discD, in which case we writeLµ^p L^pµpDq. If, in addition, the measure µ is a weighted Lebesgue measure, that is, dµpzq ωpzqdApzq, we write Lω^p Lµ^p. Finally, in the case whereω is of the form p1 |z|²q^ηforηPR, we use the notationL^pη.

We will also consider the so-called weak L^p-spaces. For a measure spacepX,µq with a positive measure, the weak L^p space Lµ^p,8pXq is the space of measurable functions f such that

}f}_L^p,8

µ pXqsup

λ¡0λpµtxPX:|fpxq| ¡λuq¹^{^p 8.

In special cases, we make the same simplifications to our notations as with the standard L^p spaces. In particular, we write L^p,µ⁸ when X is the unit disc D. In relation to the standard L^p spaces, we find that if f P L^pµpXq, then }f}_L^p,8

µ pXq ¤ }f}_L^p

µpXq. In particularLµ^ppXq Lµ^p,8pXq. Finally, we note that the quantity} }_L^p,8

µ pXq

is not a norm since it does not satisfy the triangle inequality.

We now move on to define the weighted Bergman spaces which are the center of attention for a large part of this thesis. Let 0 p 8andωbe a weight inD. The weighted Bergman space A^pωinduced by the weightωconsists of analytic functions in the space Lω^p, that is,A^pωL^pωXHpDq. The norm is adopted from the Lebesgue spaces, that is,

}f}_A^p_ω

»

D|fpzq|^pωpzqdApzq ¹_p

.

Bergman spaces, as well as L^pand weakL^pspaces, are complete metric spaces. For theL^pcase, we of course consider two functions to be the same when they coincide almost everywhere.

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Assume for a moment thatωis a radial weight inD. Then the norm convergence in the Hilbert space A²ω implies uniform convergence on compact subsets ofD, and thus the point evaluations Lz : f ÞÑ fpzq, z P D, are bounded linear functionals on A²ω. Therefore, the Riesz representation theorem guarantees the existence of reproducing kernelsB^ω_z PA²ω such that

fpzq xf,B_z^ωy_A²_ω

»

D fpζqB^ω_zpζqωpζqdApζq, zPD, f PA²ω. (2.4) These reproducing Bergman kernels have the representation

B^ω_zpζq ¸⁸

n0

pzζqⁿ

2ω_2n ₁, z,ζPD, whereω_x³₁

0s^xωpsqdsfor all1 x 8. In the case of a standard weightωpzq p1 |z|²q^α, α ¡ 1, the right-hand side becomes B^α_zpζq pα 1qp1zζq^p² ^α^q. Moreover, the reproducing formula (2.4) actually holds for all f P A¹_ω A²_ω. The reproducing formula also gives rise to the Bergman projection, the definition of which we leave to Section 4.2. Let us, however, note one recent result on the integral means and norms of the Bergman kernels.

Theorem 2.4. [52, Theorem 1] Let0 p 8,ω,ν P pD and N PNY t0u. Then the following statements hold:

(i) M^pp

r,pB^ω_aq^p^N^q 1 2π

»_2π

0 |B^ω_apre^iθq|^pdθ

»_|a|r

0

dt

ωptqp ^pp1tq^ppN ^1q, r,|a| Ñ 1;

(ii) }pB^ω_aq^pNq}^p_Ap

ν

»_|a|

0

p νptq

ωptqp ^pp1tq^p^p^N ¹^qdt, |a| Ñ1. It is worth noting that the upper estimate

}B^ω_a}^p_Ap ν À

»_|_a_|

0

pνptq

ωptqp ^pp1tq^pdt, |a| Ñ1, holds forωP pDand any radial weightν[52, p. 106].

To end this section, let us define the mixed norm spaces and point out their relation to Bergman and Lebesgue spaces. While the mixed norm spaces could, of course, also be defined for a general weightω, we shall restrict ourselves to the case of standard weights.

Let 0 p ¤ 8andα P R. The mixed norm space is the space of functions f, measurable inD, such that

}f}_Lpp,qq

α

»₁

0 M^qppr,fqp1r²q^αdr ¹_q

8,

for 0 q 8, and}f}_Lpp,8q

α sup_0 r 1Mppr,fqp1r²q^α ¹ 8forq 8. For the analytic case, we write A^pp,qqα L^pp,qqα XHpDqand note that this space is nontrivial

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only when either 0 q 8and α ¡ 1, orq 8and α ¥ 1. If p q, these spaces become the regular Lebesgue and Bergman spaces, that is, L^pp,pqα L^pα and A^pp,pqα A^pα. The analytic case for q 8gives the Hardy spaces: A^pp,8qα Hα^pfor α ¡ 1, and A^pp,8q₁ H^p. If both p and q are infinite, we obtain the previously defined growth space, A^p8,8qα H⁸α .

2.4 ZERO SETS AND FACTORIZATION

Let Xbe a linear space of analytic functions in the unit disc and f P X such that f 0. We say that a sequenceZDis the zero set (or sequence) of f if fpaq 0 for allaPZ, counting multiplicities, and fpaq 0 for allaPDzZ, that is, the function f vanishes precisely on the points of the sequence Z, counting multiplicities, and nowhere else. We denote the zero set of a function f by Zpfq. A set Z D is called a zero set for the space Xif there exists a nonzero function f P X such that Z Z^pfq. The set of all zero sets ofX is denoted byZpXq. If a set Z D is not a zero set for the space X, we say that it is a set of uniqueness for X, as any two distinct functions in the space must differ at a point in Z.

Zeros of functions in Hardy spaces are neatly characterized by the Blaschke condition: for 0 p 8 and a sequence Z D, Z P ZpH^pq if and only if

°aPZp1 |a|q 8. For Bergman spaces, the situation is more complicated since the geometric distribution of zeros is not very well understood. Some of the most commonly known results on zeros of functions in Bergman spaces are by Luecking, Korenblum, Hedenmalm, Horowitz and Seip. Horowitz studied unions and subsets of zero sets of functions in Aα^p, as well as their dependence on the parameter p, see [25, 27, 28]. Some of these results were later generalized to certain Aω^p by Peláez and Rättyä in [50]. Korenblum [32], Hedenmalm [18] and Seip [61,62] obtained more information on the geometric properties of the zero sets by applying methods based on densities defined via Blaschke sums, Stolz star domains and Beurling-Carleson characteristic of the corresponding boundary set. Although the necessary and suffi- cient geometric conditions obtained via this method do not quite coincide, the gap between them is small. Luecking [44] described the zero sets of Aα^pvia an auxiliary function generated by the zero set, which yielded accurate information on subsets of zero sets. We will now take a closer look at some of Luecking’s results.

LetZbe a sequence in the unit disc, and define ψ_Zpzq ¹

aPZ

a az 1azexp

1a az 1az

, zPD,

WZpzq e^k^Z^pzq, kZpzq |z|² 2

¸

aPZ

1 |a|²₂

|1az|² , zPD.

Theorem 2.5. [44, Theorem 3] Let 0 p ¤ 8, and either0 q 8 andα ¡ 1, or q 8andα 1. Let Z be a sequence inD. Then the following statements are true.

(a) Z P ZpA^p,qα q if and only if °

aPZp1 |a|q² 8 and there exists a nowhere zero analytic function F such that FWZPL^p,qα .

(b) If ZPZpA^p,qα q, then the same is true for any subsequence of Z.

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(c) Z PZpA^p,qα qif and only if°

aPZp1 |a|q² 8and there exists a nonzero analytic function F such that FW_ZPLα^p,q.

(d) If Z P Z^pAα^p,qq, then the mapping f ÞÑ f{ψ_Z is a continuous isomorphism from tf PAα^p,q:ZZ^pfquontotFPH^pDq^:FWZPL^p,qα u.

Theorem 2.5 can be used to describe the zero sets in terms of the existence of a certain harmonic function and also to obtain information on the effect the parameter phas on zeros, see [44, 45] for details.

A key point in the proof of Theorem 2.5 is factoring out the zeros of the function f, which is done by finding a nowhere zero analytic function F such that f FψZ, when Z Zpfq. The following result is key in keeping this factorization under control with respect to the norm of the space.

Theorem 2.6. [44, Theorem 2] Let 0 p ¤ 8, and either0 q 8 andα ¡ 1, or q 8andα 1. Let f PHpDq, and let Z be a sequence inDwith ZZpfq. Then the function

gpzq |fpzq|

±aPZ

"

_1az^a^zexp 1

2

1_1az^a^z²

*, zPD,

belongs to L^pp,qqα if and only if f PA^pp,qqα . Moreover, there exists a constant CCpp,q,αq ¡ 0such that

}f}_App,qq

α ¤ }g}_Lpp,qq

α ¤C}f}_App,qq

α , f PHpDq.

Like the description of zeros, factorization in Hardy spaces is also quite simple since each function f P H^p has the inner-outer factorization f BSF, where B is a Blaschke product containing all zeros of f,Sis a singular inner function and the outer functionFhas the same norm as f, see [14] for details. In the case of Bergman spaces, the best-known result is probably that of Horowitz.

Theorem 2.7. [26, Theorem 1] Let0 p 8, 0 ¤ α 8and n P Nzt1u. Then there exists a constant C Cpp,α,nq ¡0 such that, for all f P Aα^p, there exist functions

f1, . . . ,fnPAα^pnwith f ±_n

j1fjand

¹n j1

}fj}_A^pn_α ¤C}f}_A^p_α.

The defect of Theorem 2.7 with respect to the factorization in Hardy spaces is that it does not necessarily allow one of the factors to be non-vanishing, and thus it does not behave as well as the inner-outer factorization in Hardy spaces. In [29], Horowitz and Schnaps generalized this result to Bergman spaces induced by slightly more general radial weights. Theorem 2.7 has also been generalized toAω^p, whereω is invariant with a hypothesis on the density of polynomials, by Peláez and Rättyä in [50].

Theorem 2.8. [50, Theorem 3.1] Let0 p 8andω PInv such that polynomials are dense in A^pω. Let f PAω^p, and let0 p1,p2 8such that p¹p¹₁ p¹₂ . Then there exist f1PAω^p¹ and f2PAω^p² such that f f1f2and

}f1}^p

Aω^p¹}f2}^p

A^pω² ¤ p p1}f1}^p¹

Aω^p¹

p p2}f2}^p²

A^pω² ¤C}f}^p_Ap ω

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for some constant CCpp1,p2,ωq ¡0.

The main argument used to prove Theorems 2.7 and 2.8 is probabilistic in that the zeros of the function f P Aω^p are randomly distributed to the factors f_k. The L¹ω-norms of the expected values of f_k^p^k can then be controlled by the norm of f, which yields the factorization together with the norm estimates.

2.5 DOMINATING SETS

Let 0 p 8, δ P p0, 1q and ω be a weight inD. We say that a measurable set GGpfq Dis aδ-dominating set for f PA^pω if

»

G|fpzq|^pωpzqdApzq ¥δ}f}^p_Ap

ω. (2.5)

Of course, the interesting case here is when the parameterδis large. A setGDis called a dominating set forAω^p if there existsδP p0, 1qfor which (2.5) is valid for all

f PAω^p.

The idea of dominating sets is trying to observe how large a part of the disc can be ignored without losing much information about the function or the space. This idea will be later used when studying sampling.

Dominating sets for Bergman spaces have been studied earlier by Luecking in [40, 41, 43, 45]. Let us now discuss some of these results.

Theorem 2.9. [45, Lemma 2] Let0 q p 8,1 α 8and f PAα^p. Then

|fpzq|^q¤ α 1 π

»

D

|fpζq|^qp1 |z|²q² ^α

|1ζz|^{4 2α}p1 |ζ|²q^αdApζq, zPD.

Moreover, there exists C Cpp{q,αq ¡ 0such that if EEpε,q,fqis the set of points in Dfor which

|fpzq|^q¤ε

»

D|fpζq|^qp1 |z|²q² ^α

|1ζz|^{4 2α}p1 |ζ|²q^αdApζq,

then »

E|fpζq|^pp1 |ζ|²q^αdApζq ¤ pCεq^p{q}f}^p_Ap α.

Therefore,ε¡0may be chosen independent of f such that GDzE satisfies

»

G|fpζq|^pp1 |ζ|²q^αdApζq ¥ 1 2}f}^p_Ap

α, that is, GGpq,fq Dis a ¹₂-dominating set for f .

Luecking used this result to prove a characterization for sampling measures ofA^pα. Sadly, while Theorem 2.9 is generalized forωP pD^{in Paper}II, Luecking’s ap- proach to studying sampling measures is, in other parts, not applicable to the case of weights with as little local regularity; see Paper II or the discussion following Theorem 2.13 in the next section for more information.

Next, we consider dominating sets for the whole spaceA^pω, and start by defining one more class of weights. Let 1 q 8, and let ω be an almost everywhere

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positive weight in D. If, for some (equivalently for each)r P p0, 1q, there exists a constantCCpq,r,ωq ¡0 such that

»

∆pz,rqωpzqdApzq ¹_q »

∆pz,rqωpzq^q^q¹ dApzq _q1¹

¤C|∆pz,rq|, zPD, we writeωPCq. Additionally, we setC₈ Yq¡1Cq.

Theorem 2.10. [43, Theorem 3.9] Let0 p 8,ωPC₈and GDbe measurable. If

|GX∆pz,rq| ¥δ|∆pz,rq|, zPD, (2.6) for someδ¡0and rP p0, 1q, then G is a dominating set for Aω^p.

The condition (2.6) is equivalent to the existence of a constantδ₀ δ₀ ¡0 such that|GXS| ¡δ0|S|for all Carleson squaresS. Unsurprisingly, the pseudohyperbolic disc∆pa,rqcan also be replaced by a suitable Euclidean disc such asDpa,ηp1 |a|qq with a fixed 0 η 1. The proofs of these equivalences can be found in [40], where Luecking showed that ifωis a standard weight, then (2.6) is also a necessary condition for Gto be a dominating set for A^pω.

2.6 CARLESON AND SAMPLING MEASURES

In his famous proof of the Corona theorem [10], Lennart Carleson needed a continuous embedding of the Hardy spaces to L^p spaces of the disc and proved a result on measures of the disc. These measures have since been defined and studied for many spaces of analytic functions and are called Carleson measures.

Let 0 q 8andXbe a space of analytic functions in the unit disc. A positive Borel measureµonDis aq-Carleson measure for the spaceXif the identity operator is bounded fromXtoL^qµ, that is,

}f}_L^q_µ À }f}X, f PX.

In the following, the space X in the general definition above will be the weighted Bermgan space A^pω induced byωP pD. These measures have been studied in [50, 51, 54] and can be characterized via the properties of the following weighted maximal function. For a weightω, a positive Borel measureµonDandα¡0, set

Mω,αpµqpzq sup

zPSpaq

µpSpaqq

pωpSpaqqq^α, zPD.

Ifα1, we simply writeMωpµq. We also use the notationΓpzqfor the non-tangential approach region defined by

Γpzq

"

ζPD:|θargpζq| 1 2

1|ζ|

r

*

, zre^iθ PDzt0u,

and denote the related tent by Tpζq tz P D : ζ P Γpzqu. The following theorem by Peláez and Rättyä gives the above mentioned characterization ofq-Carleson measures for weighted Bergman spaces induced by a weight in the classDp^.

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Theorem 2.11. [51, Theorem 1] Let 0 p,q 8, ω P pD and µ be a positive Borel measure onD.

(a) If p¡q, the following statements are equivalent:

(i) µis a q-Carleson measure for A^pω; (ii) The function

Bµpzq

»

Γpzq

dµpζq

ωpTpζqq, zPDzt0u, belongs to L

pqp

ω ; (iii) Mωpµq PL

pqp

ω .

(b) µis a p-Carleson measure for Aω^p if and only if Mωpµq PL⁸. (c) If q¡p, the following statements are equivalent:

(i) µis a q-Carleson measure for A^pω; (ii) M_ω,q{ppµq PL⁸;

(iii) The function zÞÑ _p_ω^µ_p^p∆pz,rqq_S_p_z_qqq_q{p belongs to L⁸ for any fixed rP p0, 1q.

Carleson measures for different spaces of analytic functions have many applications in function and operator theory. In particular, they are often useful in proving the boundedness of linear operators acting on spaces of analytic functions. The q-Carleson measures for Aω^p and their characterization above are used frequently throughout the proofs in PapersII–IV.

For 0 p 8 and a weight ω in D, a positive Borel measure µ on D is a sampling measure forAω^p if

»

D|fpzq|^pdµpzq }f}^p_Ap

ω, f PAω^p.

Note that the measuresµsatisfying the inequality ”À” are thep-Carleson measures for A^pω defined and characterized above.

Luecking characterized the sampling measures for Bergman spaces induced by a standard weight in [45]. To state the result in question, we need some additional notation and the concept of weak convergence of measures. Beginning with the lat- ter, we say that a sequence of measurespµnqonDconverges weakly to a measureµ, denoted byµnáµ, if

»

D

hpzqdµnpzq Ñ

»

D

hpzqdµpzq, hPCcpDq,

whereCcpDqis the class of nonnegative, continuous and compactly supported functions inD. The following theorem is one of the steps in Luecking’s characterization of sampling measures.

On linear operators and their applications in complex function spaces of the unit disc

uef.fi

Dissertations in Forestry and Natural Sciences

TANELI KORHONEN

ON LINEAR OPERATORS AND THEIR APPLICATIONS IN COMPLEX FUNCTION SPACES OF THE UNIT DISC

TANELI KORHONEN

Taneli Korhonen

ON LINEAR OPERATORS AND THEIR APPLICATIONS IN COMPLEX FUNCTION

SPACES OF THE UNIT DISC

TABLE OF CONTENTS

1 Introduction

2 Function spaces