Fourier transform and regularity of characteristic functions

Let E be a bounded domain in đ^d. We study regularity property of χE and integrability of χE when its boundary ∂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.