Bulletin of the
Korean Mathematical Society BKMS

In this paper, the fractional Hardy-type operator of variable order $\beta(x)$ is shown to be bounded from the Herz-Morrey spaces $M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{\alpha,\lambda}(\R^{n})$ with variable exponent $q_{1}(x)$ into the weighted space $M\dot{K}_{p_{_{2}},q_{_{2}}(\cdot)}^{\alpha,\lambda}(\R^{n},\omega)$, 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 $1< q_{1}(\infty)\le q_{1}(x)\le( q_{1})_{+}<\infty~(x\in \R^{n})$.

Keywords: Herz-Morrey space, Hardy operator, Riesz potential, variable exponent, weighted estimate

MSC numbers: Primary 42B20; Secondary 47B38