Two type Estimates for the Boundedness of Generalized Riesz Potential Operator in the Generalized Weighted Local Morrey Spaces

Abstract

In this paper, we prove the Spanne-type boundedness of the generalized Riesz potential operator I ρ from the one generalized weighted local Morrey spaces Mp ′ for 1 < p < q < ∞ and from the generalized weighted local Morrey spaces M {x 0 } 1,ϕ 1 (w, R n ) to the weak generalized weighted local Morrey spaces W M {x 0 } q,ϕ 2 (w q , R n ) with w ∈ A 1,q for 1 < q < ∞. We also prove the Adams-type boundedness of the operator I ρ from the weighted spaces M p,ϕ 1 p (w, R n ) to the another one M q,ϕ 1 q (w, R n ) with w ∈ A p,q for 1 < p < q < ∞ and from the weighted spaces M 1,ϕ (w, R n ) to the weak weighted spaces W M q,ϕ 1 q (w, R n ) with w ∈ A 1,q for 1 < q < ∞.