I am looking at a multiple integral given in equation 3.1 of this paper, which is a Fourier transform, with $b = |\vec{b}|$ and $p = |\vec{p}|$:
$$ \hat{f}(b) \equiv \int d^{D-2} \vec{p} \, e^{i\vec{b}.\vec{p}} \, \frac{f(p)}{p^{D-3} \, \text{vol} S^{D-3}} = \Gamma \left( \frac{D-2}{2}\right) \int_0^{\infty} dp f(p) \frac{J_{\frac{D-4}{2}}(pb)}{(pb/2)^{\frac{D-4}{2}}}.$$ Here $J$ denotes the standard Bessel function. The factor $p^{D-3} \, \text{vol} S^{D-3}$ in the denominator is for convenience, it partially cancels with the measure. I substituted $e^{i\vec{b}.\vec{p}} = e^{i{b}{p} \cos \theta}$, but it did not give me a correct answer.
Additionally, in equation 3.2, they claim that the following equation follows from the above Fourier transform: $$ \int_0^{\infty} dp \, f(p) \, \frac{8\pi G}{p^2} = \int_0^{\infty} db \, \hat{f}(b)\, \frac{8\pi G b}{D-4}.$$ How do I see these two statements? Any help will be greatly appreciated.
I not really know the topics the paper talks about, but to me it seems the author used a common trick used in Fourier transforms when you have angular symetries: they made a change of variables to polar coordinates (which require the inclussion of a Jacobian and not just replacing $e^{i\vec{b}.\vec{p}} = e^{i{b}{p} \cos \theta}$ - in the standard transform is adding a radial term, but because of the volume included maybe other things could arise), and in the complex exponential kernel appears a term dependent in the angle, and if the integrand is independent of the angular coordinate (haves angular symetry), this complex exponential angular term becomes a Bessel function dependent only of the radial coordinate (actually, the Fourier Transform becomes a Hankel Transform, which resultant function depends only in the radial coordinate).
The derivation is not so short, but it isn't so hard anyway, I let you a link with the full explanation here.... I hope it was you were asking for.