Boundedness Of The Maxımal, Potentıal And Sıngular Operators In The Generalızed Varıable Exponent Morrey Spaces

Abstract

We consider generalized Morrey spaces M(P(.),omega)(Omega) with variable exponent p(x) and a general function omega(x, r) defining the Money-type norm. In case of bounded sets Omega subset of 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 Sob lev-Adams type M(p(.),omega)(Omega)-> M(q(.),omega)(Omega)-theorem for the potential operators I(alpha(.)), also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on omega(x, r), which do not assume any assumption on monotonicity of omega(x, r) in r