In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel T on the generalized local Morrey spaces LM

We study the continuity properties of the generalized fractional integral operatorIρon the generalized local Morrey spacesLMp,φ{x0}and generalized Morrey spacesMp,φ. We find conditions on the triple(φ1,φ2,ρ)which ensure the Spanne-type boundedness ofIρfrom one generalized local Morrey spaceLMp,φ1{x0}to anotherLMq,φ2{x0},1<p<q<∞, and fromLM1,φ1{x0}to the weak spaceWLMq,φ2{x0},1<q<∞. We also find conditions on the pair(φ,ρ)which ensure the Adams-type boundedness ofIρfromMp,φ1/ptoMq,φ1/qfor1<p<q<∞and fromM1,φtoWMq,φ1/qfor1<q<∞. In all cases the conditions for the boundedness ofIρare given in terms of Zygmund-type integral inequalities on(φ1,φ2,ρ)and(φ,ρ), which do not assume any assumption on monotonicity ofφ1(x,r),φ2(x,r), andφ(x,r)inr.

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