I have a question which I'd be greatly happy to hear an answer for (because it sounds really cool!). Say I'm given some "nice" region $\Omega$ in $[0,1]^{2}$, how can one determine all the $p>0$ such that $\hat{f} \in L^{p}(\mathbb{R}^{2})$? Here $f$ denotes the characteristic function of $\Omega$ and $\hat{f}$ denotes its Fourier transform.
I'm being a bit vague about the region on purpose, since I have no intuition what would be some good properties. For concreteness, can someone solve it when $\Gamma$ is the unit ball or the unit square $[0,1]^{2}$? I believe at least one of these two should be known/doable.