Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in hardy type spaces

The 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.Russian Fund of Basic Research [15-01-02732]; SFEDU grant [07/2017-31]info:eu-repo/semantics/publishedVersio

LP → LQ - Estimates for the Fractional Acoustic Potentials and some Related Operators

Mathematics Subject Classification: 47B38, 31B10, 42B20, 42B15.We obtain the Lp → Lq - estimates for the fractional acoustic potentials in R^n, which are known to be negative powers of the Helmholtz operator, and some related operators. Some applications of these estimates are also given.* This paper has been supported by Russian Fond of Fundamental Investigations under Grant No. 40–01–008632 a

Hadamard-Bergman Convolution Operators

We 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.Russian Foundation for Fundamental ResearchRussian Foundation for Basic Research (RFBR) [18-01-00094-a]Russian Foundation for Basic ResearchRussian Foundation for Basic Research (RFBR) [18-01-00094-a, 19-01-00223-a]http://creativecommons.org/licenses/by/4.0

Variable order fractional integrals in variable generalized Hölder spaces of holomorphic functions

We 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.info:eu-repo/semantics/publishedVersio

Composition operators in generalized Morrey spaces

Let $\Omega$ be an open subset of $\mathbb{R}^n$. Let $f$ be a Borel measurable function from $\mathbb{R}$ to $\mathbb{R}$. We prove necessary and sufficient conditions on $f$ in order that the composite function $T_f[g]=f\circ g$ belongs to a generalized Morrey space ${\mathcal{M}}_p^w(\Omega)$ whenever $g$ belongs to ${\mathcal{M}}_p^w(\Omega)$. Then we prove necessary conditions and sufficient conditions on $f$ in order that the composition operator $T_f[\cdot ]$ be continuous, uniformly continuous, H\"{o}lder continuous and Lipschitz continuous in ${\mathcal{M}}_p^w(\Omega).$ We also consider its `vanishing' generalized Morrey subspace ${\mathcal{M}}_p^{w,0}(\Omega)$ and prove the related results for the composition operator as operator acting from ${\mathcal{M}}_p^{w,0}(\Omega)$ to ${\mathcal{M}}_p^{w}(\Omega)$ and also between the spaces ${\mathcal{M}}_p^{w,0}(\Omega)$. For the uniform, H\"{o}lder and Lipschitz continuity we also have conditions that are both necessary and sufficient. We also have both necessary and sufficient conditions for the continuity under certain additional natural assumptions. We also consider the most commonly used Morrey classes that are related to power-type weights in the context of a discussion of some of the conditions that we impose on the weights

A class of Hausdorff-Berezin operators on the unit disc

We 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.Fulbright Research Scholarship programJ. William Fulbright Research Scholarship Program [PS00267032]Russian Foundation for Fundamental ResearchRussian Foundation for Basic Research (RFBR) [18-01-00094]National Natural Science Foundation of ChinaNational Natural Science Foundation of China [11720101003]STU Scientific Research Foundation for Talents [NTF17009]info:eu-repo/semantics/publishedVersio

Generalized Hölder type spaces of harmonic functions in the unit ball and half space

summary:We study spaces of Hölder type functions harmonic in the unit ball and half space with some smoothness conditions up to the boundary. The first type is the Hölder type space of harmonic functions with prescribed modulus of continuity $\omega =\omega (h)$ and the second is the variable exponent harmonic Hölder space with the continuity modulus $|h|^{\lambda (\cdot )}$. We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary

Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis VIII

This proceedings volume gathers selected, peer-reviewed papers from the "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis VIII" (OTHA 2018) conference, which was held in Rostov-on-Don, Russia, in April 2018. The book covers a diverse range of topics in advanced mathematics, including harmonic analysis, functional analysis, operator theory, function theory, differential equations and fractional analysis – all fields that have been intensively developed in recent decades. Direct and inverse problems arising in mathematical physics are studied and new methods for solving them are presented. Complex multiparameter objects that require the involvement of operators with variable parameters and functional spaces, with fractional and even variable exponents, make these approaches all the more relevant. Given its scope, the book will especially benefit researchers with an interest in new trends in harmonic analysis and operator theory, though it will also appeal to graduate students seeking new and intriguing topics for further investigation