The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on the Generalized Weighted Local Morrey Spaces

Abstract

In this paper, we study two-type estimates which are the Spanne and Adams type estimates for the continuity properties of the generalized fractional maximal operator Mρ on the generalized weighted local Morrey spaces M{x0}p,ϕ (wp) and generalized weighted Morrey spaces M p,ϕ1/p (w), including weak estimates. We prove the Spanne type boundedness of the generalized fractional maximal operator Mρ from generalized weighted local Morrey spaces M{x0}p,ϕ1 (wp) to the weighted weak space WM{x0}q,ϕ2 (wq) for 1 ≤ p < q < ∞ and from M{x0}p,ϕ1 (wp) to another space M{x0}q,ϕ2 (wq) for 1 < p < q < ∞ with wq ∈ A q.We also prove the Adams type boundedness of Mρ from Mp,ϕ1/p(w) to the weighted weak space WMq,ϕ1/q (w)for 1 ≤ p < q < ∞ and from Mp,ϕ1/p (w) to Mq,ϕ1/q (w) for 1 < p < q < ∞ with w ∈ Ap,q. The all weight functions belong to Muckenhoupt-Weeden class Ap,q. In all cases the conditions for the boundedness of the operator Mρ are given in terms of supremal-type integral inequalities on the all ϕ functions and r which do not assume any assumption on monotonicity of ϕ1 (x, r), ϕ2 (x, r) and ϕ(x, r) in r. © MatDer.