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

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 $L_{\mathrm{\mid }x\mathrm{\mid }}^p{L}_{\theta }^{\bar{p}}({\mathbb{R}}^n)$ to $L_{\mathrm{\mid }x\mathrm{\mid }}^q{L}_{\theta }^{\bar{q}}({\mathbb{R}}^n)$, where $0<\beta <n$, $1<p,q,\bar{p},\bar{q}<\mathrm{\infty }$ and $1\mathrm{/}p-1\mathrm{/}q=\beta \mathrm{/}n$. 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.