Generalized Fractional Integral Operators on Generalized Local Morrey Spaces

Abstract

We study the continuity properties of the generalized fractional integral operator Iρ on the generalized local Morrey spaces LMp,φ{x0} and generalized Morrey spaces Mp,φ. We find conditions on the triple (φ1,φ2,ρ) which ensure the Spanne-type boundedness of Iρ from one generalized local Morrey space LMp,φ1{x0} to another LMq,φ2{x0}, 1<p<q<∞, and from LM1,φ1{x0} to the weak space WLMq,φ2{x0}, 1<q<∞. We also find conditions on the pair (φ,ρ) which ensure the Adams-type boundedness of Iρ from Mp,φ1/p to Mq,φ1/q for 1<p<q<∞ and from M1,φ to WMq,φ1/q for 1<q<∞. In all cases the conditions for the boundedness of Iρ are given in terms of Zygmund-type integral inequalities on (φ1,φ2,ρ) and (φ,ρ), which do not assume any assumption on monotonicity of φ1(x,r), φ2(x,r), and φ(x,r) in r.