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 $M_{rho}$ on the generalized weighted local Morrey spaces $M^{{x_0}}_{p,varphi}(w^{p})$ and generalized weighted Morrey spaces $M_{p,varphi^{frac{1}{p}}}(w)$, including weak estimates. We prove the Spanne type boundedness of the generalized fractional maximal operator $M_{rho}$ from generalized weighted local Morrey spaces $M^{{x_0}}_{p,varphi_{1}}(w^{p})$ to the weighted weak space $WM^{{x_0}}_{q,varphi_2}(w^{q})$ for $1leq p< q