Maximal, potential and singular operators in the local “complementary” variable exponent Morrey type spaces

We consider local “complementary” generalized Morrey spaces ∁M{x0}p(⋅),ω(Ω) in which the p-means of function are controlled over Ω∖B(x0,r) instead of B(x0,r), where Ω⊂Rn is a bounded open set, p(x) is a variable exponent, and no monotonicity type condition is imposed onto the function ω(r) defining the “complementary” Morrey-type norm. In the case where ω is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case 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 type ∁M{x0}p(⋅),ω(Ω)→∁M{x0}q(⋅),ω(Ω)-theorem for the potential operators Iα(⋅), also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(r), which do not assume any assumption on monotonicity of ω(r).