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

In this paper, we investigate a class of fractional Hardy type operators ${\mathcal{H}}_{{\beta }_1,\cdots ,{\beta }_m}$ 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 ${\mathcal{H}}_{{\beta }_1,\cdots ,{\beta }_m}$ is bounded from $L^p({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}\times \cdots \times {\mathbb{R}}^{n_m},\mathrm{\mid }x{\mathrm{\mid }}^{\gamma })$ to $L^q({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}\times \cdots \times {\mathbb{R}}^{n_m},\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })$ and the bounds of the operator ${\mathcal{H}}_{{\beta }_1,\cdots ,{\beta }_m}$ is sharp worked out. The other is that when $\alpha =\gamma =(0,\cdots ,0)$, the norm of the operator ${\mathcal{H}}_{{\beta }_1,\cdots ,{\beta }_m}$ is obtained.