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- 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.
- 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).
- On embeddings of Morrey type spaces between weighted Lebesgue or Stummel spaces with application to Herz spacesPublication . Rafeiro, Humberto; Samko, StefanWe study embeddings of Morrey type spaces M-p,M-q,M-omega(R-n), 1 <= p < infinity, 1 <= q < infinity, both local and global, into weighted Lebesgue spaces L-p(R-n, w), with the main goal to better understand the local behavior of functions f is an element of M-p,M-q,M-omega(R-n) and also their behavior at infinity. Under some assumptions on the function omega, we prove that the local Morrey type space is embedded into L-p(R-n, w), where w(r) = omega(r) if q = 1, and w(r) is "slightly distorted" in comparison with omega(r) if q > 1. In the case q > p we show that the embedding, in general, cannot hold with omega = w. For global Morrey type spaces we also prove embeddings into Stummel spaces. Similar embeddings for complementary Morrey type spaces are obtained. We also study inverse embeddings of weighted Lebesgue spaces L-p(R-n, w) into Morrey type and complementary Morrey type spaces. Finally, using our previous results on relations between Herz and Morrey type spaces, we obtain "for free" similar embeddings for Herz spaces.
- Coincidence of variable exponent Herz spaces with variable exponent Morrey type spaces and boundedness of sublinear operators in these spacesPublication . Rafeiro, Humberto; Samko, StefanWe introduce generalized local and global Herz spaces where all their characteristics are variable. As one of the main results we show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized variable Herz spaces with application to variable Morrey type spaces and their complementary spaces, based on the mentioned inclusion.
- On a class of sublinear operators in variable exponent Morrey-type spacesPublication . Rafeiro, H.; Samko, Stefan G.For a class of sublinear operators, we find conditions on the variable exponent Morrey-type space L-p(.),L-q,L-omega(.,L-.)(R-n) ensuring the boundedness in this space. A priori assumptions on this class are that the operators are bounded in L-p(.)(R-n) and satisfy some size condition. This class includes in particular the maximal operator, singular operators with the standard kernel, and the Hardy operators. Wealso prove embedding of variable exponent Morrey-type spaces into weighted L-p(.)-spaces.