Research Article | Open Access
A Characterization of Some Function Classes
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: where is the unit disc of the complex plain .
Theorem A. For function analytic in , the following assertions are equivalent:(a);
Here, , where and are the partial sums of the Taylor series of .
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 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 Möbius invariant measure on . We refer to Duren  and Zhu  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, 7–11] 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 ) that is, the Berezin symbol of the diagonal operator on the Hardy space is a radial function.
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:
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  and Duren ): how an function can be recognized by the behavior of its Taylor coefficients?
- M. Pavlović, Introduction to Function Spaces on the Disk, Matematički Institut SANU, Belgrade, Serbia, 2004.
- N. Popa, “A characterization of upper triangular trace class matrices,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 347, no. 1-2, pp. 59–62, 2009.
- P. L. Duren, Theory of Hp Spaces, Doveri Mineola, 2000.
- K. H. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, NY, USA, 1990.
- V. Kokilashvili and V. Paatashvili, “On Hardy classes of analytic functions with a variable exponent,” Proceedings of A. Razmadze Mathematical Institute, vol. 142, pp. 134–137, 2006.
- V. Kokilashvili and V. Paatashvili, “On the convergence of sequences of functions in Hardy classes with a variable exponent,” Proceedings of A. Razmadze Mathematical Institute, vol. 146, pp. 124–126, 2008.
- F. A. Berezin, “Covariant and contravariant symbols of operators,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 6, pp. 1117–1151, 1972.
- F. A. Berezin, “Quantization,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 8, pp. 1109–1163, 1974.
- H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000.
- E. Nordgren and P. Rosenthal, “Boundary values of Berezin symbols,” Operator Theory: Advances and Applications, vol. 73, pp. 362–368, 1994.
- M. T. Karaev, “Functional analysis proofs of Abel's theorems,” Proceedings of the American Mathematical Society, vol. 132, no. 8, pp. 2327–2329, 2004.
- I. I. Privalov, Graničnye Svoĭstva Analitičeskih Funkci\u\i, Deutscher, Berlin, Germany, 2nd edition, 1950.
Copyright © 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.