Given $g\in L^2(\mathbb{R}^3)$, consider the following function ( defined for $r>0$ ):
$$c(r):=\int\_{\mathbb{R}^3}\frac{g(x)}{|x|^2+r}dx$$
I'm interested in the behavior of $c(r)$ for large $r$. A simple application of Cauchy-Schwarz inequality gives
$$|c(r)|\lesssim r^{-\frac14}$$
However, I think that something better can be said about $c$. For example,


***is it true that $c\in L^4(1,+\infty)$?***


Thank you for any suggestions.



Asymptotic behavior of an integral transform
Once you asked, here goes. Let $h(r)=c(r^2)$. Then you need to look at the operator
$$
\Psi: f\mapsto \int\_0^\infty f(\rho)\frac{\rho}{\rho^2+r^2}\,d\rho.
$$
Note two things: 


1) The kernel is positive and dominated by $1/(r+\rho)$ (just consider the cases $\rho\le r$ and $\rho\ge r$ separately. Hence, it is not worse than the classical Hilbert operator and it is bounded in $L^2$ (by the Schur test with $\varphi(\rho)=\rho^{-1/2}$, say).


2) If $h$ is any non-negative $L^2$ function such that $h(r)r^2$ is non-decreasing, then
$$
(\Psi h)(r)\ge\int\_r^{2r}\frac{sh(s)}{s^2+r^2}\ge\int\_r^{2r}\frac{r^2h(r)/s}{s^2+r^2}\ge \frac 1{10}h(r)
$$
so nothing better can be said.


The remaining change of variable should be clear.


