Sublinear operators with rough kernel generated by Calderon-Zygmund operators and their commutators on generalized local Morrey spaces

Abstract

In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel T-Omega on the generalized local Morrey spaces LM rho,phi{x0}, for s' <= p, p not equal 1 or p < s, where Omega is an element of L-s(Sn-1) with s > 1 are homogeneous of degree zero. In the case when b is an element of LCp,lambda{x0} is a local Campanato spaces, 1 <p < infinity, and T-Omega,T-b be is a sublinear commutator operator, we find the sufficient conditions on the pair (phi(1),phi(2)) which ensures the boundedness of the operator T-Omega,T-b, from one generalized local Morrey space LMp,phi 1{x0} to another LMp,phi 2{x0}. In all cases the conditions for the boundedness of T-Omega are given in terms of Zygmund-type integral inequalities on (phi(1), phi(2)), which do not make any assumptions on the monotonicity of phi(1), phi(2) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudo-differential operators, Littlewood-Paley operators, Marcinkiewicz operators, and Bochner-Riesz operators.