Generalized Fractional Integral Operators on Generalized Local Morrey Spaces

We study the continuity properties of the generalized fractional integral operator I-rho on the generalized local Morrey spaces LMp,phi{x0} and generalized Morrey spaces M-p,M-phi. We find conditions on the triple (phi(1), phi(2), rho) which ensure the Spanne-type boundedness of I-rho from one generalized local Morrey space LMp,phi 1{x0} to another LMq,phi 2{x0}, 1 < p < q < infinity, and from LM1,phi 1{x0} to the weak space WLMq,phi 2{x0} 1 < q < infinity. We also find conditions on the pair (phi, rho) which ensure the Adams-type boundedness of I-rho from M-p,phi(1/p) to M-q,phi(1/q) for 1 < p < q < 8 and from M-1,M-phi to WMq,phi 1/q for 1 < q < infinity. In all cases the conditions for the boundedness of I-rho are given in terms of Zygmund-type integral inequalities on (phi(1), phi(2), rho) and (phi, rho), which do not assume any assumption on monotonicity of phi(1) (x, r), phi(2) (x, r), and phi(x, r) in r.