Research Article | Open Access

# A Characterization of Some Function Classes

**Academic Editor:**Nicolae Popa

#### Abstract

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], Pavloviฤ proved the following characterization of functions belonging to the classical Hardy space: where is the unit disc of the complex plain .

Theorem A. *For function analytic in , the following assertions are equivalent:*(a)*;
*(b)*;
*(c)*. **Here, , where and are the partial sums of the Taylor series of .*

Recently, Popa [2] 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 and Besov class with a variable exponent. Our results are new even for the usual Hardy and Besov spaces and .

Recall that the Hardy space is the collection of holomorphic functions in which satisfy the inequality Let be the area measure on normalized so that the area of is 1. In rectangular and polar coordinates, For , the Besov space is defined to be the space of analytic functions in such that where is the Mรถbius invariant measure on . We refer to Duren [3] and Zhu [4] for the theory of these spaces.

Let , and let , , 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 belongs to the Hardy class if where , .

For , the class coincides with the classical Hardy class .

Analogously, we say that the analytic in function belongs to the Besov class with a variable exponent if where , .

For , the class coincides with the Besov class .

Suppose that , . If , then it is obvious that

Recall that for any bounded linear operator acting in the functional Hilbert space over some set with reproducing kernel , its Berezin symbol is defined by where is the normalized reproducing kernel of . (We mention [4, 7โ11] 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 and in terms of Berezin symbols.

For any bounded sequence of complex numbers , let denote the associate diagonal operator acting in the Hardy space by the formula It is known that the reproducing kernel of the Hardy space has the form . Then, it is easy to show that (see [11]) that is, the Berezin symbol of the diagonal operator on the Hardy space is a radial function.

Note that the inequality , , is the necessary condition for the function to be in the spaces . Also note that if , then (see, for instance, Duren [3] and Zhu [4]).

Our main result is the following theorem.

Theorem 2.1. *Let be a function with the bounded sequence of Taylor coefficients . Then, the following are true:*(a)* if and only if
*(b)*if, in addition, as , then if and only if
*

*Proof. *Indeed, by using the concept of Berezin symbols and formula (2.2), let us rewrite the function as follows:
thus
for every , where, as usual, and . 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 that the diagonal operator is bounded in (and hence is bounded for every fixed ). Then, we have
thus
Therefore, by using formula (2.8), we have that
if and only if
that is,
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 [12] and Duren [3]): how an function can be recognized by the behavior of its Taylor coefficients?

#### References

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#### Copyright

Copyright ยฉ 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.