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Variable Exponent Spaces of Analytic Functions

By Gerardo A. Chacón and Gerardo R. Chacón

Submitted: December 13th 2019Reviewed: April 21st 2020Published: May 25th 2020

DOI: 10.5772/intechopen.92617

Abstract

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

Keywords

• variable exponents
• Hardy spaces
• Bergman spaces
• Carleson measures
• analytic functions

1. Introduction

In recent years the interest for nonstandard function spaces has risen due to several considerations including their need in some fields of applied mathematics, differential equations, and simply the possibility to study extensions and generalizations of classical spaces.

One of such type of spaces is the variable exponent Lebesgue spaces. A first introduction to them was due to Orlicz in 1931. He considered measurable functions usuch that

01uxpxdx<,

the usual exponent pfrom the classical theory of Lebesgue spaces is replaced by a suitable function p. Orlicz was interested in the summability of Fourier series which led him to consider spaces of sequences anso that

n=1λanpn<,

for λ>0and pn1.

Later generalizations consisted on considering real functions uon a domain Ωsuch that

Ωφxuxdx<,

where

φ:Ω×00

is a suitable function. Such spaces are known nowadays as Musielak-Orlicz spaces. We are interested in the case

φxy=ypx,

which corresponds to variable exponent Lebesgue spaces, with some adequate restrictions for the exponentp.

The study of problems in spaces with nonstandard growth has been mainly related to Lebesgue spaces with variable order px, to the corresponding Sobolev spaces, and to generalized Orlicz spaces (see surveys [1, 2]). The investigation of structural properties of variable exponent spaces and of operator theory in such spaces is of interest not only due to their intriguing mathematical structure—also worthy of investigation—but because such spaces appear in applications. One of these applications is the mathematical modeling of inhomogeneous materials, like electrorheological fluids. These are special viscous liquids that have the ability to change their mechanical properties significantly when they come in contact with an electric field. This can be explored in technological applications, like photonic crystals, smart inks, heterogeneous polymer composites, etc. [3, 4, 5, 6].

Another known application is related to image restoration. Based on variable exponent spaces, the authors of [7] proposed a new model of image restoration combining the strength of the total variation approach and the isotropic diffusion approach. This model uses one value of pnear the edge of images and another one near smooth regions, which leads to the variable exponent setting of the problem. Other references discussing image processing are [8, 9, 10, 11, 12, 13]. This theory has also been applied to differential equations with nonstandard growth (see [14, 15]).

The theory of spaces of analytic functions on domains of the complex plane, or in general of Cn, happens in parallel. Starting from the classical Hardy HpDand Bergman ApDspaces, the functional Banach spaces have had a preferential study.

A Banach space Xof complex-valued functions on a domain Ωis said to be a functional Banach space (cf. [16]) if the following hold:

• Vectorial operations are precisely the pointwise operations on Ω.

• If fx=gxfor every xΩ, then f=g.

• If fx=fyfor every fX, then x=y.

• Evaluation functionals xfxare continuous on X.

Classical Hardy and Bergman spaces are examples of functional Banach spaces. In the case of spaces of analytic functions, the evaluation functionals Kxf=fx(xΩ) are an extension of the well-known reproducing kernels in Hilbert spaces of analytic functions. The analyticity property of the functions in the space interacts with the Banach space structure by means of the continuity of evaluation functionals raising a fruitful relation between function theory and operator theory.

The setting of nonstandard spaces of analytic functions is a young area of research; a lot is yet to be explored. In this chapter, an introduction to the subject of variable exponent spaces of analytic functions will be given, making emphasis in the special place that evaluation functionals have in the theory.

2. Variable exponents

Perhaps the most known function spaces in mathematical analysis are the Lebesgue spacesLp. These are a fundamental basis of measure theory, functional analysis, harmonic analysis, and differential equations, among other areas. A recent generalization is the so-called variable exponent Lebesgue spaces. In this case, the defining parameter pis allowed to vary.

The basics on variable Lebesgue spaces may be found in the monographs [1, 17, 18]. For ΩRdwe put pΩ+esssupxΩpxand pΩessinfxΩpx; we use the abbreviations p+=pΩ+and p=pΩ. For a measurable function p:Ω1, we call it a variable exponent and define the set of all variable exponents with p+<as PΩ.

For a complex-valued measurable function φ:ΩC, we define the modularρp,μby

ρp,μφΩφxpxdμx

and the Luxemburg-Nakano norm by

φLpΩμinfλ>0:ρp,μφλ1.E1

Given pPΩ, the variable Lebesgue spaceLpΩμis defined as the set of all complex-valued measurable functions φ:ΩCfor which the modular is finite, i.e., ρp,μφ<.Equipped with the Luxemburg-Nakano norm (1) is a Banach space.

Most of the research being done on variable exponent spaces makes use of a regularity condition in the variable exponent in order to have a “fruitful” theory.

Definition 1. A function p:ΩRis said to be log-Hölder continuous—or satisfy the Dini-Lipschitz condition on Ωif there exists a positive constant Clogsuch that

pxpyCloglog1/xy,

for all x,yΩsuch that xy<1/2, from which we obtain

pxpy2Cloglog2xy

for all x,yΩsuch that xy<. We will write pPlogΩwhen pis log-Hölder continuous.

This condition has proven to be very useful in the theory since, among other things, it implies the boundedness of the Hardy-Littlewood maximal operator in LpΩ. Such operator is defined as

Mfxsupr>01BxrBxrfydy

and represents an important tool in harmonic analysis. We used it in [19] to show that the Bergman projection is bounded on variable exponent Bergman spaces.

Another consequence of the log-Hölder condition is the following inequality

BpBp+BC,

that hold for all balls BC. Here Bstands for the Lebesgue measure of B, p+B=esssupzBpz, and pB=essinfzBpz. This will be a useful tool when studying complex function spaces since it allows to pass from an exponent that varies over a ball to a constant exponent at the center of such ball.

One interesting point of the theory of variable exponents is that, in general, the classical approach to Bergman spaces seems to fail in the variable framework. For example, in [19] it is shown that the Bergman projection is bounded from ApDto ApD. This is done by using the theory of Békollé-Bonami weights and a theorem that extends Rubio de Francia’s extrapolation theory to variable Lebesgue spaces. This method is quite different from the usual ways of showing that the Bergman projection is bounded, which use either Schur’s test or Calderón-Zygmund theory.

3. Variable exponent Hardy spaces

In this section we will give an introduction to the variable exponent Hardy spaces and the results and techniques that are usually found. We will follow the presentation as in [20].

Definition 2. For each zin the unit disk D, the Poisson kernel Pzζis defined as

Pzζ=1z2zζ2,

and the Poisson transform of a function fLpTis defined as

Pfz=TPzζfζmζ,

where mdenotes the normalized Lebesgue measure on T.

We will use the following result from [21]:

Theorem 3.Suppose thatpislog-Hölder continuous. For each0r<1, andfLpT, the linear operatorPr:LpTLpTdefined as

Prfζ=Pf

is uniformly bounded on LpT, PrfLpTfLpT, and for every fLpT,

fPrfLpT0,asr1.

We are now ready to define the harmonic Hardy spaces with variable exponents. Given f:DCand 0r<1, the dilationsfr:TCof fare defined as frζ=f.

Definition 4. The variable exponent harmonic Hardy spacehpDis defined as the space of harmonic functions f:DCsuch that

fhpD=sup0r<1frLpT<.

Notice that, since Pfis harmonic for fLpT, then Theorem 3 shows that the Poisson transform P:LpThpDis bounded. Moreover, we have the following theorem analogous to the constant exponent context.

Theorem 5.Suppose thatp:02π1islog-Hölder continuous. Then for everyUhpD, there existsuLpTsuch thatPu=Uand moreover,UhpDuLpT.

Theorem 6.The following chain of inclusions holds

hp+DhpDhpD,

and moreover, the inclusions are continuous.

Definition 7. The variable exponent Hardy space HpDis defined as the space of analytic functions f:DCsuch that fhpD.

In an analogous way to the classical setting HpDcan be identified with the subspace of functions in LpTwhose negative Fourier coefficients are zero, and as such, HpDis a Banach space.

Recall that for all functions fH1D, we have the reproducing formula:

fz=Tfζ1ζ¯zdmζ.

For each zD, the functions Kz:DCdefined as

Kzw=11z¯w,

are called reproducing kernels. They are bounded on Dand consequently belong to every space HpD. Moreover, as a consequence of the reproducing formula and the Hahn-Banach theorem, the linear span of Kz:zDis dense in HpD. Consequently, the set of polynomials is also dense in HpD.

Theorem 8.For eachzDconsider the linear functionalγz:HpDCdefined asγzf=fz. Thenγzis a bounded operator for everyz=zeDand

γz11z1/pθ.

Consequently, the convergence in theHpD-norm implies the uniform convergence on compact subsets ofD.

The main tool for proving the previous theorem is following the version of a Forelli-Rudin inequality, adapted to the case of variable exponents. Here, the Log-Hölder continuity plays a key role.

Lemma 9.Supposep:02π1andq:02π1are such that1pθ+1qθ=1 for everyθ02π. Let1/2<r<1andz=ze. Define the functionφ:02πR+as

φt=1z1/qθ1zreitθ.

Then ifφt>1, it holds that

φtptφtpθ.

4. Variable exponent Bergman spaces

The theory of Bergman spaces was introduced by S. Bergman in [22] and since the 1990s has gained a great deal of attention mainly due to some major breakthroughs at the time. For details on the theory of Bergman spaces, we refer to the books [5, 23].

Definition 10. Given a measurable function pPD, we define the variable exponent Bergman spaceApDas the space of all analytic functions on Dthat belong to the variable exponent Lebesgue space LpDwith respect to the area measure dAon the unit disk D, i.e.,

ApD=fis an analytic function andDfzpzdAz<.

One of the first results proven in [19] about this theory was the boundedness of the evaluation functionals that, as we saw before, will allow us to conclude that convergence in ApDimplies uniform convergence on compact subsets of Dand consequently Lp-limits of sequences in ApDare analytic. Hence ApDis a closed subspace of LpD(and hence a Banach space). The first approach to such result used the following interpolation result based on Forelli-Rudin estimates:

Lemma 11.LetpPD, then for every functionfLpDwe have

fLpDfL1D1/p+fLD11/p+.

Using different techniques, the following sharp estimate was obtained in [24]:

Theorem 12.LetpPlogD. Then for everyzDwe have that

γzp11z22/pz.E2

The proof of this fact relies on the Log-Hölder condition and on the following version of a Jensen-type inequality for variable exponent spaces (see [17]).

Lemma 13.Suppose thatpPlogDand let

1qzymax1pz1py0.

Then for0<γ<1,

BxrfydAypzBxrfypydAy+BxrγqzydAy

for all zBxrprovided that fLpD1.The constant depends on γand p.

The additional term that appears in the previous inequality makes the boundedness of the domain a necessary condition to use such technique. One question to address in future investigations is how the situation is for Bergman spaces defined on unbounded domains. In such case, the error term accumulates despite the regularity condition on p, and consequently a different technique is needed.

4.1 Mollified dilations

One technique developed to study variable exponent Bergman spaces is a combination of mollifiers and dilations. Such concept was developed for studying the density of the set of polynomials in Bergman spaces. In the classical case, this is a consequence of the fact that convergence in ApDimplies convergence on compact subsets of Dand that for any fApD, its radial dilations

frz=frz

converge in norm to ffor r1. This is a simple fact that depends on the dilation-invariant nature of Bergman Spaces, something that does not hold in the case of variable exponent Bergman spaces. This is one of the main difficulties for generalizing the classical theory.

One way of overcoming this problem in the case of variable exponent Lebesgue spaces is the use of a mollification operator. But in the case of the unit disk, changes need to be made. That is, when we had the idea to introduce a mollified dilation, given a function fApDand r121, define the mollified dilationfr:1+r2rDCof fas

frzDfrwηrzwdAw.

where ρDstands for the complex disk with radius ρ,

ηrz4r21r2η2rz1r

and

ηzexp1z21,ifz<1,0,ifz1,

Mollified dilations allow us to approximate analytic functions in the variable exponent Bergman spaces by functions that are analytic on a neighborhood of the unit disk.

Theorem 14.LetpPlogDand letfApD. Then for½r<1,frApD, and

sup½r<1frApDfApD.

Moreover,frfApD0asr1.

As a consequence of the previous theorem, we can approximate a function fApDby a function frwhich is analytic in 1+r2rD. Hence, the sequence of Taylor polynomials associated with frconverges uniformly on Dto fr. Therefore, given ε>0, there exists 1/2r<1such that ffrApD<ε/2. And there exists a polynomial psuch that pfrApD<ε/2.

4.2 Carleson measures

Another classical problem that has a variable exponent equivalent consists on finding a geometrical characterization of Carleson measures in variable exponent Bergman spaces. Given a positive Borel measure μdefined on the unit disk D, we say that μis a Carleson measure for the variable exponent Bergman space ApDif there exists a constant C>0such that

fLpDμCfApD.

In other words, μis a Carleson measure on ApDif ApDis continuously embedded in LpDμ.

Lennart Carleson gave a geometric characterization of Carleson measures on Hardy spaces Hpand used this result in his proof of the Corona theorem and some interpolation problems (cf. [25]). After that, Carleson measures have gained interest, and a geometric characterization is usually important in spaces of analytic functions.

In the case of the classical Bergman spaces ApD, Carleson measures are characterized in terms of the measure on pseudo-hyperbolic disks (see [26]). An interesting fact is the independence of the characterization on the exponent p. In the case of variable exponent Bergman spaces, the independence of pdoes not hold unless the psatisfies the Log-Hölder condition. The counterexample for the general case and the theorem for the restricted case were found in [24].

Theorem 15.Letμbe a finite, positive Borel measure onDandpPlogD. Thenμis a Carleson measure forApDif and only if there exists a constantC>0such that for every0<r<1and everyaD,μDraCDra, whereDradenotes pseudo-hyperbolic disk with center ataDand radiusr:

DrazD:az1a¯z<r.

The main tool for the proof of the previous theorem is Eq. (2) that implies the following estimate for the reproducing kernels.

Theorem 16.LetpPlogD. Then for everyzDwe have that

kzApD11z2211/pz.

5. Open problems

5.1 Zero sets

An interesting area will be to study some analytic properties of functions belonging to variable exponent Bergman spaces. One starting point will be to study the structure of zero sets in the space.

A sequence of points znDis a zero set for the Bergman space ApDif there exists fApDsuch that fzn=0for all nand fz0if zzn.

Although the problem of completely describing the zero sets in ApDis still open, some information is available. For example, if nrdenotes the number of zeroes of a function fApDhaving modulus smaller than r>0, then it is known that

nr=O11rlog11r,r1.

This result depends on studying the radial growth of functions in the Bergman space and on Jensen’s formula for the sequence zkof zeroes of fsuch that zk)<r.

logf0=k=1nrlogzkr+12π02πlogfredθ.

This equation holds for every analytic function on D. Consequently, if we want to find out whether a similar result about the growth of the zeroes holds in the variable exponent case, then it is necessary to study the radial growth of functions in ApD. Such results, in the case of a constant exponent again, depend on the subharmonicity of fp.

Another result involving zero sets is that if f00, then

k=1nr1zk=Onr1/p,r1.

Finding an extension of such result to variable exponents would include making sense of the right-hand side term that is not even clear in this context.

5.2 Sampling sequences

A related concept that we would like to address in variable exponent Bergman spaces is that of sampling sequences. A sequence of points zkDis sampling in ApDif there exists constants c>0and C>0such that for all fApD

cfppk=11zk22fzkpCfpp.

One difference that is expected to occur in the case of variable exponents is that sampling sequences will probably not be invariant under automorphism of the unit disk. This is true for a constant exponent, and it will be interesting to find counterexamples if the exponent varies. Sampling sequences are related to the concept of “frames” in Hilbert spaces and can be thought as sequences that contain great information of the space. It is expected that a sampling sequence is a somehow “big” and “spread out” subset of the unit disk. The notion is clearly related in Apto the concept of Carleson measures.

If we define a discrete measure μas

μ=k=11zk22δzk

where δzkis the Dirac measure with a point mass at zk, then the sequence zkis sampling if μis a Carleson measure and satisfies a “reverse” Carleson condition. Then it becomes relevant to find a geometric characterization of the measures that satisfy a reverse Carleson condition (see [27, 28, 29]). The characterization depends on the concept of dominant sets and it is independent of p. The methods used by Luecking for Bergman spaces rely on a calculation of the norm of evaluation functionals and a submean value property. Different techniques must be developed for the variable exponent counterpart.

5.3 Operators in variable exponent Bergman spaces

The characterization of Carleson measures obtained in [24] opens the possibility of studying the boundedness and compactness of certain operators acting on variable exponent Bergman spaces.

Among those operators worth studying are multiplication operators. Those are formally defined on a functions space Fas

Mfg=fg

where gF. One natural question is to find the set of symbols fthat make the operator Mfmap the space ApDto itself.

Such question has been addressed in the case of weighted Bergman spaces in [23]. The proof relies on a geometric characterization of Carleson measures, a composition with a disk isomorphism, and a theorem of change of variables. Such tool is difficult to use in the setting of variable exponents since any change of variables will also affect the exponent and consequently the spaces are not necessarily invariant under composition with isomorphisms. This makes the situation different from the case of a constant exponent.

Other operators related to multiplication are Toeplitz operators. Those have already being studied in [30] for the case of weighted Bergman spaces with non-radial weights. Given a function φL1D, the Toeplitz operator Tφis defined on the set of polynomials as

Tφpz=DpwkwzφwdAw.

One natural question is to ask whether this operator can be extended boundedly to the space ApD. This question is addressed in [31] were a characterization of the boundedness and compactness of Tφis found in terms of Carleson measures and the Berezin transform associated with Tφ. This type of results are sometimes referred to as Axler-Zheng theorems, and similar questions looking to establish the relation between the Berezin transform of a finite product of Toeplitz operators with its compactness have been addressed by several authors; a survey of this type of results is given in [32]. It is then reasonable to ask weather an Axler-Zheng type result for products of Toeplitz operators holds in the context of variable exponents.

The composition operators are another type of operators to be studied. Given an analytic self-map φof the unit disk, the composition operator Cφcan be formally defined as

Cφf=fφ.

A natural question is to find function-theoretic conditions on φthat guarantee the boundedness and/or compactness of its associated composition operator acting on ApD. One approach to follow is to address the boundedness problems of linear operators acting on ApDthrough the use Rubio de Francia extrapolation. Given an open set ΩC, we denote by Fa family of pairs of non-negative, measurable functions defined on Ω, and by A1we denote the class of Muckenhoupt weights w. That is, the non-negative functions wsuch that

esssupzΩMwzwz<

where Mdenotes the Hardy-Littlewood maximal operator. Rubio de Francia extrapolation in the setting of variable exponent spaces can be stated as follows.

Proposition 17 (Thm. 5.24 in [1]). Suppose that for somep01the family of pairs of functionsFis such that for allwA1,

ΩFxp0wxdxC0ΩGxp0wxdx,FGF.

GivenpPΩ, ifp0pp+<and the maximal operator is bounded onLp/p0Ω, then

FLpΩCpGLpΩ.

This result allows, under certain conditions, to pass from the of studying operators defined on variable exponent spaces, to study operators defined on weighted constant exponent spaces, and therefore it is possible to use known results from the Muckenhoupt weights theory.

5.4 Analytic Besov spaces

Another type of spaces we are interested is the class of analytic Besov spaces Bp. These are the spaces of all analytic functions in the unit disk such that

Dfzp1z2p2dAz<.

For each pthis is a normed space with norm

fBp=f0+Dfzp1z2p2dAz1/p.

One particular property of this space is that they are Möbius invariant in the sense that fφBp=fBpfor every automorphism of the unit disk φ. This is probably not true in general for a variable exponent version due to the problems with the change of variables formula. A first question to address is to characterize the exponents that leave the space invariant under disk automorphisms.

One useful technique to study such spaces is a derivative-free characterization of functions in Bp. It can be seen (see, e.g., [33]) that fBpif and only if

DDfzfwp1zw¯4dAzdAw<.

A similar result in the case of variable exponents could serve as a starting point for investigating other properties of the space.

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Gerardo A. Chacón and Gerardo R. Chacón (May 25th 2020). Variable Exponent Spaces of Analytic Functions [Online First], IntechOpen, DOI: 10.5772/intechopen.92617. Available from: