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

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 $L_{|x|}^p{L}_{\theta }^{\bar{p}}({\mathbb{R}}^n)$ to $L_{|x|}^q{L}_{\theta }^{\bar{q}}({\mathbb{R}}^n)$, 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 ${\mathcal{H}}^{\ast }$ and ${\mathcal{H}}_{\beta }^{\ast }$ are also established. In addition, the sharp weak-type estimate for $\mathcal{H}$ is also considered.

13 pages