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

TL;DR

This paper proves the boundedness of a fractional Hardy-type operator with variable order on variable exponent Herz-Morrey spaces, extending the understanding of such operators in variable exponent function spaces.

Contribution

It establishes the boundedness of the fractional Hardy-type operator on variable exponent Herz-Morrey spaces, considering variable order and weights, which is a novel extension in this area.

Findings

Boundedness of the operator is established under specified conditions.

Results extend classical boundedness to variable exponent and variable order settings.

The paper introduces new conditions on the exponents and weights for boundedness.

Abstract

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 $M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{\alpha(\cdot),\lambda}(\R^{n})$ into the weighted space $M\dot{K}_{p_{_{2}},q_{_{2}}(\cdot)}^{\alpha(\cdot),\lambda}(\R^{n},\omega)$, where $\alpha(x)\in L^{\infty}(\mathbb{R}^{n})$ be log-H\"older 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.