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

Authors : Abdulhamit KÜÇÜKASLAN

Pages : 56-65

View : 18 | Download : 4

Publication Date : 2020-06-29

Article Type : Research Paper

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 $M^{\{x_0\}}_{p,\varphi}insert ignore into journalissuearticles values(w^{p});$ and generalized weighted Morrey spaces $M_{p,\varphi^{\frac{1}{p}}}insert ignore into journalissuearticles values(w);$, including weak estimates. We prove the Spanne type boundedness of the generalized fractional maximal operator $M_{\rho}$ from generalized weighted local Morrey spaces $M^{\{x_0\}}_{p,\varphi_{1}}insert ignore into journalissuearticles values(w^{p});$ to the weighted weak space $WM^{\{x_0\}}_{q,\varphi_2}insert ignore into journalissuearticles values(w^{q});$ for $1\leq p< q<\infty$ and from $M^{\{x_0\}}_{p,\varphi_1}insert ignore into journalissuearticles values(w^{p});$ to another space $M^{\{x_0\}}_{q,\varphi_{2}}insert ignore into journalissuearticles values(w^{q});$ 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 $M_{p,\varphi^{\frac{1}{p}}}insert ignore into journalissuearticles values(w);$ to the weighted weak space $WM_{q,\varphi^{\frac{1}{q}}}insert ignore into journalissuearticles values(w);$ for $1\leq p In 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 $\varphi_1insert ignore into journalissuearticles values(x,r);$, $\varphi_2insert ignore into journalissuearticles values(x,r);$ and $\varphiinsert ignore into journalissuearticles values(x,r);$ in $r$.
Keywords : Generalized fractional maximal operator, Generalized weighted local Morrey spaces, Generalized weighted Morrey spaces, Muckenhoupt Weeden classes