Assume that $D$ is the unit ball in $\mathbb{R}^n$ and let $f\in C\_0^\infty(D)$. Let $a>0$ and let $F$ be the Fourier transform. Define $$g(x)= F^{-1} (|x|^{-2a} \cdot (F f)(x)).$$ My question is for which parameters $a$ we have $$g\in L^2(\mathbb{R}^n).$$



Fourier transform (their inverse)
$F(f)$ is analytic and tends to $0$ faster than any power because $f$ is $C\_0^\infty$.
$F$ and $F^{-1}$ map $L^2$ onto itself. So the question is when $|x|^{-2a}$ belongs
to $L^2$ locally (near the origin). Normally this happens when $a<n/4$.
This is necessary and sufficient unless $Ff(0)$ accidentally equals to $0$.


Edit.
Can we relax the condition $f\in C\_0^\infty$ ? Yes, we can. Compact support was not really used much. Fast enough decrease will be enough. On the other hand, smoothness was
used substantially, to ensure that the behavior at infinity is good for every $a$.


