Sharp bounds for Hardy type operators on higher-dimensional product spaces

TL;DR

This paper derives sharp bounds for fractional Hardy type operators on higher-dimensional product spaces, establishing their boundedness between weighted Lebesgue spaces and calculating their exact operator norms.

Contribution

The paper introduces novel methods to determine the sharp bounds and exact norms of Hardy type operators on product spaces, extending previous results to higher dimensions.

Findings

Operators are bounded between weighted Lebesgue spaces with explicit bounds.

Sharp bounds of the operators are obtained.

Exact operator norms are calculated for the unweighted case.

Abstract

In this paper, we investigate a class of fractional Hardy type operators $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ defined on higher-dimensional product spaces $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$. We use novel methods to obtain two main results. One is that we obtain the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is bounded from $L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^{\gamma})$ to $L^{q}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^{\alpha})$ and the bounds of the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is sharp worked out. The other is that when $\alpha=\gamma=(0,\cdots,0)$, the norm of the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is obtained.