JFSAJournal of Function Spaces and Applications0972-68022090-8997Hindawi Publishing Corporation79679810.1155/2012/796798796798Research ArticleA Characterization of Some Function ClassesKaraevM. T.1PopaNicolae1Isparta Vocational SchoolSuleyman Demirel University32260 IspartaTurkeysdu.edu.kz201225122011201212022009021220112012Copyright © 2012 M. T. Karaev.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give in terms of Berezin symbols a characterization of Hardy and Besov classes with a variable exponent.

1. Introduction and Notations

In his book [1, page 96], Pavlović proved the following characterization of functions belonging to the classical Hardy space: H1=H1(D):={fHol(D):f1=sup0<r<112π02π|f(reit)|dt<}, where 𝔻:={z:|z|<1} is the unit disc of the complex plain .

Theorem A.

For a function f analytic in 𝔻, the following assertions are equivalent:

fH1;

supn(1/an)j=0n(1/(j+1))sj(f)1<;

supnPnf1<.

Here, Pnf=(1/an)j=0n(1/(j+1))sj(f), where an=j=0n(1/(j+1))  (n=0,1,2,) and sj(f) are the partial sums of the Taylor series of f.

Recently, Popa  gave some generalization of this result by proving a similar characterization of upper triangular trace class matrices.

In the present paper, we give in terms of the so-called Berezin symbols a new characterization of analytic functions belonging to the Hardy class Hp(·) and Besov class Bp(·) with a variable exponent. Our results are new even for the usual Hardy and Besov spaces Hp and Bp.

Recall that the Hardy space Hp=Hp(𝔻)  (1p<) is the collection of holomorphic functions in 𝔻 which satisfy the inequalityfHp:=(sup0<r<112π02π|f(reit)|pdt)1/p<. Let dA(z) be the area measure on 𝔻 normalized so that the area of 𝔻 is 1. In rectangular and polar coordinates,dA(z)=1πdxdy=1πrdrdθ. For 1<p<+, the Besov space Bp=Bp(𝔻) is defined to be the space of analytic functions f in such thatfBp:=(D(1-|z|2)p|f(z)|pdλ(z))1/p<, wheredλ(z)=dA(z)(1-|z|2)2 is the Möbius invariant measure on 𝔻. We refer to Duren  and Zhu  for the theory of these spaces.

Let 𝕋=𝔻, and let p=p(t), t𝕋, be a bounded, positive, measurable function defined on it. Following by Kokilashvili and Paatashvili [5, 6], we say that the analytic in the disc 𝔻 function f belongs to the Hardy class Hp(·) ifsup0<r<102π|f(reiθ)|p(θ)dθ=C<+, where p(θ)=p(eiθ), θ[0,2π).

For p(θ)=p=const>0, the Hp(·) class coincides with the classical Hardy class Hp.

Analogously, we say that the analytic in 𝔻 function f belongs to the Besov class Bp(·) with a variable exponent if02π01(1-r2)p(t)|f(reit)|p(t)rdrdtπ<+, where p(t)=p(eit), t[0,2π).

For p(t)=p=const>0, the Bp(·) class coincides with the Besov class Bp.

Suppose that p̲:=inft𝕋p(t), p¯:=supt𝕋p(t). If p̲>0, then it is obvious thatHp¯Hp()Hp̲,Bp¯Bp()Bp̲.

Recall that for any bounded linear operator A acting in the functional Hilbert space H=H(Ω) over some set Ω with reproducing kernel kλ(z), its Berezin symbol Ã is defined byÃ(λ):=Ak̂λ,k̂λ(λΩ), where k̂λ:=kλ/kλ is the normalized reproducing kernel of H. (We mention [4, 711] as references for the Berezin symbols.)

2. On the Membership of Functions in Hardy and Besov Classes with a Variable Exponent

In this section, we characterize the function classes Hp(·) and Bp(·) in terms of Berezin symbols.

For any bounded sequence {an}n0 of complex numbers an, let D{an} denote the associate diagonal operator acting in the Hardy space H2 by the formulaD{an}zn=anzn,n=0,1,2,. It is known that the reproducing kernel of the Hardy space H2 has the form kλ(z)=1/(1-λ¯z)  (λ𝔻). Then, it is easy to show that (see )D̃{an}(λ)=(1-|λ|2)k=0an|λ|2k(λD), that is, the Berezin symbol of the diagonal operator D{an} on the Hardy space H2 is a radial function.

Note that the inequality |f̂(n)|const, n0, is the necessary condition for the function f(z)=n=0f̂(n)zn to be in the spaces Hp  (1p). Also note that if fBp, then f̂(n)=O(n-1/p)  (p1) (see, for instance, Duren  and Zhu ).

Our main result is the following theorem.

Theorem 2.1.

Let f(z)=n=0f̂(n)znHol(𝔻) be a function with the bounded sequence {f̂(n)}n0 of Taylor coefficients f̂(n)=f(0)(n)/n!  (n=0,1,2,). Then, the following are true:

fHp(·) if and only if sup0<r<112π02π|D̃{f̂(n)eint}(r)|p(t)dt(1-r)p(t)<+;

if, in addition, f̂(n)=O(n-1) as n, then fBp(·) if and only if 02π01|D̃{(n+1)f̂(n+1)eint}(r)|p(t)(1+r)p(t)-2(1-r)2rdrdt<+.

Proof.

Indeed, by using the concept of Berezin symbols and formula (2.2), let us rewrite the function f(z)=n=0f̂(n)znHol(𝔻) as follows: f(z)=f(reit)=n=0f̂(n)(reit)n=n=0f̂(n)eintrn=(1-r)n=0f̂(n)eintrn1-r=D̃{f̂(n)eint}(r)1-r, thus f(z)=D̃{f̂(n)eint}(r)1-r for every z=reit𝔻, where, as usual, r=|z| and t=arg(z). Now, assertion (a) is immediate from the definition of considering space and formula (2.6).

Let us prove (b). Indeed, it follows from the condition f̂(n)=O(n-1) that the diagonal operator D{nf̂(n)} is bounded in H2 (and hence D{(n+1)f̂(n+1)eint} is bounded for every fixed t[0,2π)). Then, we have f(z)=(n=0f̂(n)zn)=n=1nf̂(n)zn-1=n=0(n+1)f̂(n+1)zn=n=0(n+1)f̂(n+1)eintrn=(1-r)n=0(n+1)f̂(n+1)eintrn1-r=D̃{(n+1)f̂(n+1)eint}(r)1-r, thus f(z)=D̃{(n+1)f̂(n+1)eint}(r)1-r. Therefore, by using formula (2.8), we have that D(1-|z|2)p(t)|f(z)|p(t)dA(z)(1-|z|2)2<+ if and only if 02π01|D̃{(n+1)f̂(n+1)eint}(r)|p(t)(1-r2)p(t)(1-r)p(t)r(1-r2)2drdt<+, that is, 02π01|D̃{(n+1)f̂(n+1)eint}(r)|p(t)(1+r)p(t)-2(1-r)2rdrdt<+, as desired. The theorem is proved.

We remark that, in case of classical Hardy space, Theorem 2.1 shed some light on the following old problem for the Hardy space functions (see Privalov  and Duren ): how an Hp function can be recognized by the behavior of its Taylor coefficients?

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