Maximal, Potential, and Singular Operators in the Generalized Variable Exponent Morrey Spaces on Unbounded Sets

Abstract

We consider generalized Morrey spaces Mp(·),?(·)(?) with a variable exponent p(x) and a general function ?(x, r) defining a Morrey type norm. We extend the results obtained earlier for bounded sets ? ? Rn by proving the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund singular operators with standard kernels in Mp(·),?(·)(?). We prove a Sobolev type Mp(·),?1(·)(?) › Mq(·),?2(·)(?)-theorem, both the Spanne and Adams versions, for potential operators I ?(·), where ?(x) can be variable even if ? is unbounded. The boundedness conditions are formulated either in terms of Zygmund type integral inequalities on ?(x, r) or in terms of supremal operators. Bibliography: 36 titles. © 2013 Springer Science+Business Media New York.