Fourier transform and regularity of characteristic functions

Abstract

Let E E be a bounded domain in R d \mathbb R^d . We study regularity property of χ E \chi _E and integrability of χ E ^ \widehat {\chi _E } when its boundary ∂ E \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.