Loading...
20 results
Search Results
Now showing 1 - 10 of 20
- 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.
- 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.
- 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.
- 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.
- Herz spaces meet Morrey type spaces and complementary Morrey type spacesPublication . Rafeiro, Humberto; Samko, StefanWe introduce local and global generalized Herz spaces. As one of the main results we show that Morrey type spaces and complementary Morrey type spaces are included into the scale of these Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey type spaces and their complementary spaces, based on the mentioned inclusion.
- Addendum to “On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space”, Math Meth Appl Sci. 2020; 1–8Publication . Rafeiro, Humberto; Samko, StefanIn the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order 𝛼(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and 𝜆(x) of the Morrey space. Assumptions on the exponents were different depending on whether 𝛼(x)p(x)−n+𝜆(x) p(x) takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range 0 ⩽ 𝛼(x)p(x)−n+𝜆(x) p(x) ⩽ 1. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper.
- Fractional operators of variable order from variable exponent Morrey spaces to variable exponent Campanato spaces on quasi-metric measure spaces with growth conditionPublication . Rafeiro, Humberto; Samko, StefanWe study fractional potential of variable order on a bounded quasi-metric measure space (X, d, mu) as acting from variable exponent Morrey space L-p(center dot),L-lambda(center dot)(X) to variable exponent Campanato space L-p(center dot),L-lambda(center dot)(X). We assume that the measure satisfies the growth condition mu B(x,r) <= Cr-gamma, the distance is theta-regular in the sense of Macias and Segovia, but do not assume that the space (X, d, mu) is homogeneous. We study the situation when gamma - lambda(x) <= alpha(x)p(x) <= gamma-lambda(x) + theta p(x), and pay special attention to the cases of bounds of this interval. The left bound formally corresponds to the BMO target space. In the case of right bound a certain "correcting factor" of logarithmic type should be introduced in the target Campanato space.
- Variable exponent fractional integrals in the limiting case alpha(x)p(x) equivalent to n on quasimetric measure spacesPublication . Samko, StefanWe show that the fractional operator I-alpha(center dot), of variable order on a bounded open set in Omega, in a quasimetric measure space (X, d, mu) in the case alpha(x)p(x) = n (where n comes from the growth condition on the measure mu), is bounded from the variable exponent Lebesgue space L-p(center dot)(Omega) into BMO(Omega) under certain assumptions on p(x) and alpha(x).