I am wondering, and I don't have an exact answer - what is the optimal fraction that the characteristic function $\chi_\Omega$ does not lie in $H^s (\mathbb{R}^d)$ for any $\Omega$ of positive measure?
It is clear that $H^1 (\mathbb{R}^d)$ does not containt characteristic functions (these functions do not have jump-type discontinuities) for any $d \geq 1$. But what about eg. $H^{1/2} (\mathbb{R}^d)$?
Or, in a sense, a similar question; what is known about the Fourier transform of $\chi_\Omega$, I mean - the asymptotic bahviour (for a very general set $\Omega$)? Maybe it can give an answer for which $s$ there holds $$ \int_{\mathbb{R}^d} |\xi|^{s/2} | \mathcal{F}(\chi_\Omega) |^2 d\xi < \infty. $$