Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

Abstract

We consider generalized Morrey spaces M p(·),ω ( ) with variable exponent p(x) and a general function ω(x, r) defining the Morrey-type norm. In case of bounded sets ⊂ R n we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(x, r), which do not assume any assumption on monotonicity of ω(x, r) in r.

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

“…In [20][21][22][23]25,33] and [35] there were obtained sufficient conditions on ϕ 1 and ϕ 2 for the boundedness of the maximal operator M and Calderón-Zygmund operator K from M p,ϕ1 to M p,ϕ2 , 1 < p < ∞ and of the fractional maximal operator M α and Riesz potential operator I α from M p,ϕ1 to M q,ϕ2 , 1 < p < q < ∞ (see also [3][4][5][6][7]). In [35,36] the following condition was imposed on ϕ(x, r):…”

Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces

“…In [12], [13], [14], [16], [20], [28] and [31], sufficient conditions on ϕ 1 and ϕ 2 for the boundedness of the maximal operator M and a Calderón-Zygmund operator K from the generalized Morrey spaces M p,ϕ1 to M p,ϕ2 for 1 < p < ∞ and from M 1,ϕ1 to W M 1,ϕ2 were obtained (see also [34], [2], [1]). In [9], the following condition was imposed on ϕ(x, r):…”

Sobolev's theorem and duality for Herz-Morrey spaces of variable exponent

“…In view of Guliyev, Hasanov and Samko [21,22], if (ω4.1) holds for all x 0 ∈ G with the same constant Q, then there exists a constant C > 0 such that…”