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

Abstract

We consider generalized Morrey type spaces Mp(·), θ(·), ω(·) (ω)with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets ω ⊂ ℝn, we prove the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators with standard kernel. We prove a Sobolev-Adams type embedding theorem Mp(·), θ1(·), ω1(·) (ω) → Mp(·), θ2(·), ω2(·) (ω) for the potential type operator Iα(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles. © 2010 Springer Science+Business Media, Inc.