I am interested in the following multi-dimensional Fourier transform: $$ \int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}_{\parallel}\cdot\vec{r}_{\parallel}-\mathrm{i}\,\vec{p}_\perp\cdot\vec{r}_\perp}\,\frac{r_{\parallel}^{2\alpha}r_{\perp}^{2\beta}}{(r_{\parallel}^2+r_{\perp}^2)^{\gamma}}\ , $$ where: $$ \vec{p}_\perp,\vec{r}_\perp\in\mathbb{R}^p\ ,\qquad \vec{p}_\parallel,\vec{r}_\parallel\in\mathbb{R}^q\ ,\qquad r_{\perp}=|\vec{r}_{\perp}\cdot \vec{r}_{\perp}|^{1/2}\ ,\qquad r_{\parallel}=|\vec{r}_{\parallel}\cdot \vec{r}_{\parallel}|^{1/2}\ . $$ Furthermore, $\vec{w}_\parallel\cdot\vec{w}'_\parallel$ and $\vec{y}_\perp\cdot\vec{y}'_\perp$ are inner products in $\mathbb{R}^p$ and $\mathbb{R}^q$ respectively.
When $q=0$ or $p=0$, this integral is known to have the form: \begin{align} \begin{aligned} \int_{\mathbb{R}^d}\mathrm{d}^d \vec{r} e^{-\mathrm{i}\,\vec{p}\cdot\vec{r}}\frac{1}{|\vec{r}|^{2\Delta}}=\frac{\pi^{d/2} 2^{d-2\Delta}\Gamma(d/2-\Delta)}{\Gamma(\Delta)}|\vec{p}|^{\Delta-d/2}\ . \end{aligned} \end{align} However I want to know the form for general $p$ and $q$, in particular for $p=2$ and $\alpha=0,\beta\in\mathbb{Z}_{\geq0}$.
I would appreciate it if you tell me how to evaluate the above integral.
