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 $β(x)$ is shown to be bounded from the Herz-Morrey spaces $M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{α,λ}(\mathbb{R}^{n})$ with variable exponent $q_{1}(x)$ into the weighted space $M\dot{K}_{p_{_{2}},q_{_{2}}(\cdot)}^{α,λ}(\mathbb{R}^{n},ω)$, where $ω=(1+|x|)^{-γ(x)}$ with some $γ(x)>0$ and $ 1/q_{_{1}}(x)-1/q_{_{2}}(x)=β(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 $1< q_{1}(\infty)\le q_{1}(x)\le( q_{1})_{+}<\infty~(x\in \mathbb{R}^{n})$.