Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces

Vagifs Guliyev

doi:10.1155/2009/503948

Abstract

We consider generalized Morrey spaces with a general function defining the Morrey-type norm. We find the conditions on the pair which ensures the boundedness of the maximal operator and Calderon-Zygmund singular integral operators from one generalized Morrey space to another , , and from the space to the weak space . We also prove a Sobolev-Adams type -theorem for the potential operators . In all the cases the conditions for the boundedness are given it termsof Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in . As applications, we establish the boundedness of some Schrodinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Holder class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.

Highlights

For x ∈ Rn and r > 0, let B x, r denote the open ball centered at x of radius r and B x, r denote its complement.Let f ∈ Ll1oc Rn
We study the boundedness of maximal operator M and Calderon-Zygmund singular integral operators T from one generalized Morrey space Mp,ω1 to another Mp,ω2, 1 < p < ∞, and from the space M1,ω1 to the weak space WM1,ω2
We study the boundedness of fractional maximal operator Mα and Riesz potential operators Mα from Mp,ω1 to Mq,ω2, 1 < p < q < ∞, and from the space M1,ω1 to the weak space WM1,ω2, 1 < q < ∞

Summary

Introduction

For x ∈ Rn and r > 0, let B x, r denote the open ball centered at x of radius r and B x, r denote its complement. We study the boundedness of maximal operator M and Calderon-Zygmund singular integral operators T from one generalized Morrey space Mp,ω1 to another Mp,ω2 , 1 < p < ∞, and from the space M1,ω1 to the weak space WM1,ω2. We study the boundedness of fractional maximal operator Mα and Riesz potential operators Mα from Mp,ω1 to Mq,ω2 , 1 < p < q < ∞, and from the space M1,ω1 to the weak space WM1,ω2 , 1 < q < ∞. We establish the boundedness of some Schodinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials

Morrey Spaces

Generalized Morrey Spaces

Spanne Type Result

Some Applications