ROUGH SINGULAR INTEGRAL OPERATORS AND ITS COMMUTATORS ON GENERALIZED WEIGHTED MORREY SPACES

Abstract

Let 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 singular integral operators with rough kernels T-Omega and their commutators [b, T-Omega] 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, p not equal 1 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 T-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, p not equal 1, w is an element of A(p/q') or 1 < p < q, w(1-p'). A(p'/q') which ensures the boundedness of the operators [b, T-Omega] 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 T-Omega, [b, T-Omega] 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.