My question concerns estimates on the sum
$$\displaystyle \rho^{-1} \sum_{b,c \neq 0} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{ \cos (\rho \sqrt{x^2+y^2} - 3 \pi / 4)\cos(\rho \sqrt{(b-x)^2 + (c-y)^2} - 3 \pi / 4)}{(x^2+y^2)^{3/4} ((b-x)^2 + (c-y)^2)^{3/4}} \ \mathrm{d}x \ \mathrm{d}y.$$
In particular, I would like to know how this behaves as $\rho$ varies (particularly as $\rho$ gets very large). I am expecting that, as a function of $\rho$ (and assuming everything converges), then this expression would be about $O(\rho^{1/2}),$ though possibly with a larger power of $\rho$. (Here, $\rho$ is independent of $b,c$ and $x.$)
Now, it is tempting to try to bound the cosine terms in the integrand by $1$. However, I claim that the resulting sum will diverge if we use this primitive bound: the main reason is because we need the positive-negative cancellation of the cosine terms in order to guarantee convergence. So it seems that we lose too much information with such a bound.
Second of all, the terms that appear in this integrand suggest the use of something similar to polar co-ordinates. However, it does not seem obvious what sort of co-ordinate transform would be useful here, because the terms involving $b,c$ under the square roots prove to be very problematic.
Has anyone seen an integral like this before, and if so, could anyone give me any pointers on what would be the best way to proceed with this integral/sum? Is there a co-ordinate transform that would give me something more manageable?
[For some background on where this comes from, the original expression I was considering was:
$$\displaystyle \sum_{b,c \in \mathbb{Z} \backslash \{0\}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{J_{1}(\rho \sqrt{x^2 + y^2})J_{1}(\rho \sqrt{(b - x)^2 + (c - y)^2})}{\sqrt{x^2 + y^2} \sqrt{(b - x)^2 + (c - y)^2}} \ \mathrm{d}x \ \mathrm{d}y,$$
which can more compactly be written in terms of Euclidean norms and vectors (that is the reason for the square roots). Then, using the first term of the Bessel function asymptotic expansion (as seen in Gradshteyn-Ryzhik) given by:
$$J_{1}(x) = \cos(x - 3\pi / 4)\sqrt{\frac{2}{\pi x}} + O(x^{-3/2}),$$
we obtain the expression at the top of the question after omitting constants.]
Using trig identities, it is easy to turn the numerator into a sum of $\sin$ and $\cos$ terms which don't have an argument shifted by some factor of $\pi$. However, this doesn't seem to help deal with the $b$ and $c$ terms floating around, which make using a substitution problematic.