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- Boundary value problems for analytic functions in the class of Cauchy-type integrals with density inPublication . Kokilashvili, V.; Paatashvili, V.; G. Samko, StefanWe study the Riemann boundary value problem , for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces with variable exponent. We consider both the case when the coefficient is piecewise continuous and it may be of a more general nature, admitting its oscillation. The explicit formulas for solutions in the variable exponent setting are given. The related singular integral equations in the same setting are also investigated. As an application there is derived some extension of the Szegö-Helson theorem to the case of variable exponents.
- Grand lebesgue spaces with mixed local and global aggrandization and the maximal and singular operatorsPublication . Rafeiro, H.; Samko, Stefan; Umarkhadzhiev, S.The approach to "locally" aggrandize Lebesgue spaces, previously suggested by the authors and based on the notion of "aggrandizer", is combined with the usual "global" aggrandization. We study properties of such spaces including embeddings, dependence of the choice of the aggrandizer and, in particular, we discuss the question when these spaces are not new, coinciding with globally aggrandized spaces, and when they proved to be new. We study the boundedness of the maximal, singular, and maximal singular operators in the introduced spaces.
- On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operatorsPublication . Alabalik, Aysegul C.; Almeida, Alexandre; Samko, StefanWe consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of C-0(infinity) (R-n) in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.
- Approximation in Morrey spacesPublication . Almeida, Alexandre; Samko, StefanA new subspace of Morrey spaces whose elements can be approximated by infinitely differentiable compactly supported functions is introduced. Consequently, we give an explicit description of the closure of the set of such functions in Morrey spaces. A generalisation of known embeddings of Morrey spaces into weighted Lebesgue spaces is also obtained. (C) 2016 Elsevier Inc. All rights reserved.
- Preservation of certain vanishing properties of generalized Morrey spaces by some classical operatorsPublication . Alabalik, Aysegul C.; Almeida, Alexandre; Samko, StefanWe show that certain vanishing properties defining closed subspaces of generalized Morrey spaces are preserved under the action of various classical operators of harmonic analysis, such as maximal operators, singular-type operators, Hardy operators, and fractional integral operators. Those vanishing subspaces were recently used to deal with the delicate problem on the description of the closure of nice functions in Morrey norm.
- On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato spacePublication . Rafeiro, Humberto; Samko, StefanFor the Riesz potential of variable order over bounded domains in Euclidean space, we prove the boundedness result from variable exponent Morrey spaces to variable exponent Campanato spaces. A special attention is paid to weaken assumptions on variability of the Riesz potential.
- Maximal operator with rough kernel in variable musielak-morrey-orlicz type spaces, variable herz spaces and grand variable lebesgue spacesPublication . Rafeiro, Humberto; Samko, StefanIn the frameworks of some non-standard function spaces (viz. Musielak-Orlicz spaces, generalized Orlicz-Morrey spaces, generalized variable Morrey spaces and variable Herz spaces) we prove the boundedness of the maximal operator with rough kernel. The results are new even for p constant.
- A note on vanishing Morrey -> VMO result for fractional integrals of variable orderPublication . Rafeiro, Humberto; Samko, StefanIn the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.
- Morrey spaces are closely embedded between vanishing stummel spacesPublication . Samko, StefanWe prove a new property of Morrey function spaces by showing that the generalized local Morrey spaces are embedded between weighted Lebesgue spaces with weights differing only by a logarithmic factor. This leads to the statement that the generalized global Morrey spaces are embedded between two generalized Stummel classes whose characteristics similarly differ by a logarithmic factor. We give examples proving that these embeddings are strict. For the generalized Stummel spaces we also give an equivalent norm.
- 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.