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 $1leq p< q<infty$ and from $M^{{x_0}}_{p,varphi_1}(w^{p})$ to another space $M^{{x_0}}_{q,varphi_{2}}(w^{q})$ for $1< p< q<infty$ with $w^{q} in A_{1+frac{q}{p'}}$. We also prove the Adams type boundedness of $M_{rho}$ from $M_{p,varphi^{frac{1}{p}}}(w)$ to the weighted weak space $WM_{q,varphi^{frac{1}{q}}}(w)$ for $1leq pIn all cases the conditions for the boundedness of the operator $M_{rho}$ are given in terms of supremal-type integral inequalities on the all $varphi$ functions and $r$ which do not assume any assumption on monotonicity of $varphi_1(x,r)$, $varphi_2(x,r)$ and $varphi(x,r)$ in $r$.