I'm investigating the following double integral:
$$\displaystyle \int_{|k| \geqslant \rho} \int_{|\alpha| \geqslant \rho} \frac{J_{1}^{2}(|k - \alpha|)}{|k-\alpha|^2} \ \mathrm{d}\alpha \ \mathrm{d}k,$$
where $\rho \gg 1$ is some (very) large constant, $k \neq 0$ is a vector in $\mathbb{R}^2$, $\alpha \in \mathbb{R}^2$, $|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^2$, and $J_{\nu}$ denotes the Bessel function of the first kind.
Note that, for sufficiently large, positive $z$, we have $|J_{\nu}(z)| \leqslant C_{\nu}|z|^{-1/2}$. This lets us bound the integral by
$$\displaystyle \int_{|k| \geqslant \rho} \int_{|\alpha| \geqslant \rho} \frac{1}{|k-\alpha|^3} \ \mathrm{d}\alpha \ \mathrm{d}k.$$
I would like to know whether or not this integral converges for any $\rho$ sufficiently large. I suppose this could be done by making the substitution $\tau = |k - \alpha|$, but I'm not sure how the limits of the integrals (particularly the lower limits) behave under this change of coordinates. Can anyone help?