Generalized Fractional Integral Operators on Generalized Local Morrey Spaces. (22nd April 2015) - Guliyev, V. S.; Ismayilova, A. F.; Kucukaslan, A.; Serbetci, A.
Abstract : We study the continuity properties of the generalized fractional integral operator I ρ on the generalized local Morrey spaces L M p, φ { x 0 } and generalized Morrey spaces M p, φ . We find conditions on the triple ( φ 1, φ 2, ρ ) which ensure the Spanne-type boundedness of I ρ from one generalized local Morrey space L M p, φ 1 { x 0 } to another L M q, φ 2 { x 0 }, 1 < p < q < ∞, and from L M 1, φ 1 { x 0 } to the weak space W L M q, φ 2 { x 0 }, 1 < q < ∞ . We also find conditions on the pair ( φ, ρ ) which ensure the Adams-type boundedness of I ρ from M p, φ 1 / p to M q, φ 1 / q for 1 < p < q < ∞ and from M 1, φ to W M q, φ 1 / q for 1 < q < ∞ . In all cases the conditions for the boundedness of I ρ are given in terms of Zygmund-type integral inequalities on ( φ 1, φ 2, ρ ) and ( φ, ρ ), which do not assume any assumption on monotonicity of φ 1 ( x, r ), φ 2 ( x, r ), and φ ( x, r ) in r .