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.