A Characterization of Some Function Classes

Correspondence should be addressed to M. T. Karaev, mubariztapdigoglu@sdu.edu.tr Received 12 February 2009; Accepted 2 December 2011 Academic Editor: Nicolae Popa Copyright q 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.


Introduction and Notations
In his book 1, page 96 , Pavlović proved the following characterization of functions belonging to the classical Hardy space: where : {z ∈ : |z| < 1} is the unit disc of the complex plain .
Theorem A. For a function f analytic in , the following assertions are equivalent: Here, P n f 1/a n n j 0 1/ j 1 s j f , where a n n j 0 1/ j 1 n 0, 1, 2, . . .and s j f are the partial sums of the Taylor series of f.
In the present paper, we give in terms of the so-called where dλ z dA z 1.5 is the M öbius invariant measure on .We refer to Duren 3 and Zhu 4 for the theory of these spaces.Let Ì ∂ , and let p p t , t ∈ Ì, be a bounded, positive, measurable function defined on it.Following by Kokilashvili and Paatashvili 5, 6 , we say that the analytic in the disc function f belongs to the Hardy class Recall that for any bounded linear operator A acting in the functional Hilbert space H H Ω over some set Ω with reproducing kernel k λ z , its Berezin symbol A is defined by where k λ : k λ / k λ is the normalized reproducing kernel of H.We mention 4, 7-11 as references for the Berezin symbols.

On the Membership of Functions in Hardy and Besov Classes with a Variable Exponent
In this section, we characterize the function classes H p • and B p • in terms of Berezin symbols.
For any bounded sequence {a n } n≥0 of complex numbers a n , let D {a n } denote the associate diagonal operator acting in the Hardy space H 2 by the formula D {a n } z n a n z n , n 0, 1, 2, . . . .

2.1
It is known that the reproducing kernel of the Hardy space H 2 has the form k λ z 1/ 1 − λz λ ∈ .Then, it is easy to show that see 11 that is, the Berezin symbol of the diagonal operator D {a n } on the Hardy space H 2 is a radial function.
Note that the inequality Proof.Indeed, by using the concept of Berezin symbols and formula 2.2 , let us rewrite the function f z ∞ n 0 f n z n ∈ Hol as follows: for every z re it ∈ , 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 f n O n −1 that the diagonal operator D {n f n } is bounded in H 2 and hence D { n 1 f n 1 e int } is bounded for every fixed t ∈ 0, 2π .Then, we have { n 1 f n 1 e int } √ r 1 − r .2.8 Berezin symbols a new characterization of analytic functions belonging to the Hardy class H p • and Besov class B p • with a variable exponent.Our results are new even for the usual Hardy and Besov spaces H p and B p .
p is defined to be the space of analytic functions f in such that 1 see, for instance, Duren 3 and Zhu 4 .Our main result is the following theorem.