Boundedness for fractional Hardy-type operator on variable-exponent Herz–Morrey spaces

Abstract

In this article, the fractional Hardy-type operator of variable order $\beta \left(x\right)$ is shown to be bounded from the variable-exponent Herz–Morrey spaces $M{\dot{K}}_{p_1,q_1(\cdot )}^{\alpha (\cdot ),\lambda }\left({\mathbb{R}}^n\right)$ into the weighted space $M{\dot{K}}_{p_2,q_2(\cdot )}^{\alpha (\cdot ),\lambda }({\mathbb{R}}^n,\omega )$, where $\alpha \left(x\right)\in L^{\infty }\left({\mathbb{R}}^n\right)$ is log-Hölder continuous both at the origin and at infinity, $\omega =(1+|x|{)}^{-\gamma \left(x\right)}$ with some $\gamma \left(x\right)>0$, and $1/q_1\left(x\right)-1/q_2\left(x\right)=\beta \left(x\right)/n$ when $q_1\left(x\right)$ is not necessarily constant at infinity.