Fourier transform and regularity of characteristic functions

TL;DR

This paper investigates the regularity and integrability of characteristic functions of bounded domains in Euclidean space, especially at critical boundary cases, using advanced harmonic analysis tools like Lorentz spaces and Littlewood-Paley inequalities.

Contribution

It introduces refined techniques involving Lorentz and Lorentz-Sobolev spaces to analyze endpoint regularity and integrability properties of characteristic functions, extending previous results.

Findings

Established endpoint regularity conditions for characteristic functions.

Demonstrated the effectiveness of Lorentz spaces in harmonic analysis.

Extended known results to critical boundary cases.

Abstract

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.