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

Abstract

In this paper, the fractional Hardy-type operator of variable order $\beta (x)$ is shown to be bounded from the Herz-Morrey spaces $M{\dot{K}}_{p_{{}_1},q_{{}_1}(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)$ with variable exponent $q_1(x)$ into the weighted space $M{\dot{K}}_{p_{{}_2},q_{{}_2}(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n,\omega )$, where $\omega =(1+|x|{)}^{-\gamma (x)}$ with some $\gamma (x)>0$ and $1/q_{{}_1}(x)-1/q_{{}_2}(x)=\beta (x)/n$ when $q_{{}_1}(x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_{{}_1}(x)$ satisfies the logarithmic continuity condition both locally and at infinity that $1<q_1(\infty )\le q_1(x)\le (q_1{)}_{+}<\infty (x\in {\mathbb{R}}^n)$.

13 pages