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 $1\leq p< q<\infty$ 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 $1\leq 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 $r$.

Keywords

• Generalized fractional maximal operator
• Generalized weighted local Morrey spaces
• Generalized weighted Morrey spaces
• Muckenhoupt-Weeden classes

Project Number

1059B191600675

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