  Let $E$ be a bounded domain in $\mathbb R^d$. We study regularity property of
$\chi_E$ and integrability of $\widehat {\chi_E }$ when its boundary $\partial
E$ satisfies some conditions. At the critical case these properties are
generally known to fail. By making use of Lorentz and Lorentz-Sobolev spaces we
obtain the endpoint cases of the previous known results. Our results are based
on a refined version of Littlewood-Paley inequality, which makes it possible to
exploit cancellation effectively.
