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

Abstract

We consider generalized Morrey spaces Mp,w (R(n)) with a general function. w(x, r) defining the Morrey- type norm. We find the conditions on the pair (w(1), w(2)) which ensures the boundedness of the maximal operator and Calderon-Zygmund singular integral operators from one generalized Morrey space M(p,w1) (R(n)) to another M(p,w2)(R(n)), 1 < p < infinity, and from the space M(1,w1)(R(n)) to the weak space W M(1,w2) (R(n)). We also prove a Sobolev- Adams type M(p,w1)(R(n)) -> M(q,w2) (R(n))-theorem for the potential operators I(alpha). In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on (w(1),w(2)), which do not assume any assumption on monotonicity of w(1),w(2) in r. 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. Copyright (C) 2009 Vagif S. Guliyev.