Fourier transform and regularity of characteristic functions

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.