Boundedness of some sublinear operators and their commutators on generalized local Morrey spaces

In this paperwe prove the boundedness of certain sublinear operators T , a, a. 0, n , generated by fractional integral operators with rough kernels . Ls( Sn- 1), s > 1, from one generalized local Morrey space LM {x0} p,.1 to another LM {x0} q,.2, 1 < p < q < 8, 1p - 1q = a n, and from the space LM {x0} 1,.1 to the weak space WLM {x0} q,.2, 1 < q < 8, 1 - 1q = a n. In the case b belongs to the local Campanato space LC {x0} p2,. and T , b, a is a linear operator, we find the sufficient conditions on the pair (.1,.2) which ensures the boundedness of the commutator operators T , b, a from LM {x0} p1,.1 to LM {x0} q,.2, 1 < p < 8, 1p = 1 p1 + 1 p2, 1q = 1p - a n, 1 q1 = 1 p1 - a n. In all cases the conditions for the boundedness of T , a are given in terms of Zygmund- type integral inequalities on (.1,.2), which do not assume any assumption on monotonicity of.1,.2 in r.