Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey spaces

Abstract

We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz – Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathbb{R}^n, \omega )$, where $\alpha (x) \in L^{\infty} (\mathbb{R}^n) is log-Holder continuous both at the origin and at infinity, $\omega = (1+| x| ) \gamma (x)$ 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 and $1 < (q_1)_{\infty} \leq q_1(x) \leq (q_1)_+ < \infty, \;x \in \mathbb{R}$.