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On the convergence of sequences of functions in Hardy classes with a variable exponent. (English) Zbl 1166.47033

This paper is a continuation of the authors’ earlier paper [V. Kokilashvili and V. Paatashvili, Proc. A. Razmadze Math. Inst. 142, 134–137 (2006; Zbl 1126.47031)], where they introduced the Hardy class of analytic functions with a variable exponent and announced some results on the properties of functions from that class. In the present paper, the authors describe some results (not supplying any proofs) concerning the convergence of sequences of functions in these classes. In particular, imposing supplementary conditions on the function \(p(t)\), the authors generalize the well-known theorem of Tumarkin for the usual Hardy class \(H^{p}\) over the unit disk \(\mathbb{D}\subset\mathbb{C}\) to the classes \(H^{p(\cdot)}\). Such a supplementary condition is either the Log-Hölder condition or the Hölder condition. These results are the natural generalizations of the well-known results in the case of Hardy and Smirnov classes.

MSC:

47B38 Linear operators on function spaces (general)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
45P05 Integral operators

Citations:

Zbl 1126.47031
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