So, I was doing some PDE related computations and I obtained the following integral $$ \iint \frac{(y-y')\,f(x',y')}{(|x-x'|^2+|y-y'|^2)^\frac{3}{2}}\, \left|\frac{x'}{x}\right|^2 \,\mathrm{d} x'\,\mathrm{d} y' $$ where $f$ is a continuous and compactly supported function, odd with respect to its first variable. Is this integral bounded (uniformly with respect to $x$ and $y$)?
I tried to do several changes of variable (for example one can replace $x'$ by $xx'$ to put the singularity in $x$ inside the part written $(..)^\frac{3}{2}$). I had a look on litterature on singular integrals but things seems usually more symmetric in these kind of theories.
Any ideas, examples, counterexamples or proof of a result are welcome!