Sharp bounds for Hardy-type operators on mixed radial-angular spaces

TL;DR

This paper determines the exact bounds for Hardy and fractional Hardy operators on mixed radial-angular spaces, extending understanding of their behavior and establishing duality and weak-type estimates.

Contribution

It provides the first sharp bounds for Hardy operators on mixed radial-angular spaces and explores duality and weak-type estimates, advancing the theoretical framework.

Findings

Sharp bounds for $n$-dimensional Hardy operator $\\mathcal{H}$ on mixed spaces.

Sharp bounds for fractional Hardy operator $\mathcal{H}_\beta$ between specific mixed Lebesgue spaces.

Duality results and weak-type estimates for the operators.

Abstract

In this paper, by using the rotation method, we calculate that the sharp bound for $n$-dimensional Hardy operator $\mathcal{H}$ on mixed radial-angular spaces. Furthermore, we also obtain the sharp bound for $n$-dimensional fractional Hardy operator $\mathcal{H}_\beta$ from $L^p_{|x|}L_{\theta}^{\bar{p}}({\Bbb R}^n)$ to $L^q_{|x|}L_{\theta}^{\bar{q}}({\Bbb R}^n)$, where $0<\beta<n$, $1<p,q,\bar{p},\bar{q}<\infty$ and $1/p-1/q=\beta/n$. By using duality, the corresponding results for the dual operators $\mathcal{H}^*$ and $\mathcal{H}^*_\beta$ are also established. In addition, the sharp weak-type estimate for $\mathcal{H}$ is also considered.