Marcinkiewicz integrals with rough kernel associated with Schrödinger operator on vanishing generalized Morrey spaces

Abstract

Let L = ? + V be a Schrödinger operator, where ? is the Laplacian on Rn, while nonnegative potential V belongs to the reverse Hölder class. In this paper, we study the boundedness of the Marcinkiewicz operator with rough kernels associated with Schrödinger operator µLj,? on vanishing generalized Morrey spaces V Mp,? (Rn). We find the sufficient conditions on the pair (?1, ?2) which ensure the boundedness of the operators µLj,? from one vanishing generalized Morrey space V Mp,?1(Rn) to another V Mp, ?2(Rn), 1 < p < ? and from the space V Mp, ?2(Rn) to the weak space W V M1, ?2(Rn). © 2010 AZJM All rights reserved.