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

In this paper, the fractional Hardy-type operator of variable order $\beta (x)$ is shown to be bounded from the variable exponent Herz-Morrey spaces $M{\dot{K}}_{p_{{}_1},q_{{}_1}(\cdot )}^{\alpha (\cdot ),\lambda }({\mathbb{R}}^n)$ into the weighted space $M{\dot{K}}_{p_{{}_2},q_{{}_2}(\cdot )}^{\alpha (\cdot ),\lambda }({\mathbb{R}}^n,\omega )$, where $\alpha (x)\in L^{\mathrm{\infty }}({\mathbb{R}}^n)$ be log-H"older continuous both at the origin and at infinity, $\omega =(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-\gamma (x)}$ with some $\gamma (x)>0$ and $1\mathrm{/}{q}_{{}_1}(x)-1\mathrm{/}{q}_{{}_2}(x)=\beta (x)\mathrm{/}n$ when $q_{{}_1}(x)$ is not necessarily constant at infinity.