Loading...
6 results
Search Results
Now showing 1 - 6 of 6
- Hadamard-Bergman Convolution OperatorsPublication . Karapetyants, Alexey; Samko, StefanWe introduce a convolution form, in terms of integration over the unit disc D, for operators on functions f in H(D), which correspond to Taylor expansion multipliers. We demonstrate advantages of the introduced integral representation in the study of mapping properties of such operators. In particular, we prove the Young theorem for Bergman spaces in terms of integrability of the kernel of the convolution. This enables us to refer to the introduced convolutions as Hadamard-Bergman convolution. Another, more important, advantage is the study of mapping properties of a class of such operators in Holder type spaces of holomorphic functions, which in fact is hardly possible when the operator is defined just in terms of multipliers. Moreover, we show that for a class of fractional integral operators such a mapping between Holder spaces is onto. We pay a special attention to explicit integral representation of fractional integration and differentiation.
- Variable order fractional integrals in variable generalized Hölder spaces of holomorphic functionsPublication . Karapetyants, Alexey; Samko, StefanWe introduce and study the variable generalized Holder spaces of holomorphic functions over the unit disc in the complex plane. These spaces are defined either directly in terms of modulus of continuity or in terms of estimates of derivatives near the boundary. We provide conditions of Zygmund type for imbedding of the former into the latter and vice versa. We study mapping properties of variable order fractional integrals in the frameworks of such spaces.
- Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in hardy type spacesPublication . Karapetyants, Alexey; Samko, StefanThe aim of the paper is twofold. First, we present a new general approach to the definition of a class of mixed norm spaces of analytic functions A(q;X)(D), 1 <= q < infinity on the unit disc D. We study a problem of boundedness of Bergman projection in this general setting. Second, we apply this general approach for the new concrete cases when X is either Orlicz space or generalized Morrey space, or generalized complementary Morrey space. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having l(q) summable Taylor coefficients.
- Boundedness of the Bergman projection and some properties of Bergman type spacesPublication . Karapetyants, Alexey; Rafeiro, Humberto; Samko, StefanWe give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139-142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.
- A class of Hausdorff-Berezin operators on the unit discPublication . Karapetyants, Alexey; Samko, Stefan; Zhu, KeheWe introduce and study a class of Hausdorff-Berezin operators on the unit disc based on Haar measure (that is, the Mobius invariant area measure). We discuss certain algebraic properties of these operators and obtain boundedness conditions for them. We also reformulate the obtained results in terms of ordinary area measure.
- Aggrandization of spaces of holomorphic functions reduces to aggrandization on the boundaryPublication . Karapetyants, Alexey; Samko, StefanWe show that grand spaces of holomorphic functions may be equivalently defined in terms of aggrandization related only to the boundary. We base ourselves on recent studies of the so-called local aggrandization of Lebesgue spaces and extent this approach to the case of arbitrary Banach spaces of functions on metric spaces. We apply this approach to prove, in the case of Bergman and Bergman-Morrey spaces on the unit disk, that these grand spaces may be equivalently defined as grand spaces with weighted aggrandization on the boundary.