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 $1< q{1}(\infty)\le q_{1}(x)\le( q_{1})_{+}<\infty~(x\in \mathbb{R}^{n})$.