  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.
