Generalized Fractional Integral Operators on Generalized Local Morrey Spaces

Abstract

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.

Generalized fractional integral operators on Orlicz–Hardy spaces

“…In this paper, using the operator , we extend Stein and Weiss' result to Orlicz–Hardy spaces. For more information of this type of operators, see [3, 6–8], etc, as well as [5] including an application.…”

Generalized fractional integral operators on vanishing generalized local Morrey spaces

“…The following is a result of Adams type for generalized fractional integral operator I ρ in generalized Morrey spaces which was proved in [12].…”

“…The following theorem is the second main result of our paper in which we prove the Adams-Guliyev type boundedness of the operator I ρ in vanishing generalized Morrey spaces and 25) where C 1 , C 2 does not depend on x ∈ R n and r > 0. Then the operator I ρ is bounded from vanishing generalized Morrey spaces Under the conditions (3.2), (3.23) and (3.25) we know that (see [12]) for all (B(x,r)) ϕ(x, r) 1/q < ε for small r, we use the estimates (3.28) and (3.19) where we split the right-hand side:…”

“…The generalized fractional integral operator I ρ was initilally investigated in [9,13,16]. Nakai [16] proved the boundedness of I ρ from the generalized Morrey spaces M 1,ϕ to the spaces M 1,ψ for suitable functions ϕ and ψ. Nowadays many authors have been culminating important observations about I ρ especially in connection with Morrey spaces (see [6,12,14]).…”

Necessary and sufficient conditions for the boundedness of the anisotropic Riesz potential in anisotropic modified Morrey spaces

“…Later on Chiarenza and Frasca [9] was reproved boundedness of the Riesz potential I α in these spaces. By more general results of Guliyev [13] (see also [14,17]) one can obtain the following generalization of the results in [1,9,33] to the anisotropic case. Theorem A.…”