COMMUTATORS OF MARCINKIEWICZ INTEGRALS ASSOCIATED WITH SCHRODINGER OPERATOR ON GENERALIZED WEIGHTED MORREY SPACES

Abstract

Let L = -Delta + V be a Schrodinger operator, where Delta is the Laplacian on R-n, while nonnegative potential V belongs to the reverse Holder class. Let also Omega is an element of L-q(Sn-1) be a homogeneous function of degree zero with q > 1 and have a mean value zero on Sn-1. In this paper, we study the boundedness of the Marcinkiewicz operators mu(L)(j,Omega) and their commutators mu(L)(j,Omega,b) with rough kernels associated with Schrodinger operator on generalized weighted Morrey spaces M-p,M-phi(w). We find the sufficient conditions on the pair (phi(1), phi(2)) with q' < p < infinity and w is an element of A(p/q') or 1 < p < q and w(1-p') is an element of A(p'/q') which ensures the boundedness of the operators mu(L)(j,Omega) from one generalized weighted Morrey space M-p,M-phi 1(w) to another M-p,M-phi 2(w) for 1 < p < infinity. We find the sufficient conditions on the pair (phi(1), phi(2)) with b is an element of BMO(R-n) and q' < p < infinity, w is an element of A(p/q') or 1 < p < q w(1-p') is an element of A(p'/q') which ensures the boundedness of the operators mu(L)(j,Omega,b), j = 1, ... , n from M-p,M-phi 1(w) to M-p,M-phi 2(w) for 1 < p < infinity. In all cases the conditions for the boundedness of the operators mu(L)(j,Omega), mu(L)(j,Omega,b), j = 1, ... , n are given in terms of Zygmund-type integral inequalities on (phi(1), phi(2)) and w, which do not assume any assumption on monotonicity of phi(1)(x, r), phi(2)(x, r) in r.