On the Fourier transform of the characteristic functions of domains with $C^1$ -smooth boundary

TL;DR

This paper investigates the conditions under which the Fourier transform of the characteristic function of domains with $C^1$-smooth boundaries belongs to $L^p$ spaces, enhancing understanding of Fourier analysis on smooth domains.

Contribution

It provides new criteria for the integrability of Fourier transforms of characteristic functions of $C^1$-smooth domains.

Findings

Identifies conditions for $ abla ext{hat}{1_D}$ to be in $L^p$

Establishes links between boundary smoothness and Fourier transform integrability

Provides bounds for Fourier transforms based on boundary regularity

Abstract

We consider domains $D\subseteq\mathbb R^n$ with $C^1$ -smooth boundary and study the following question: when the Fourier transform $\hat{1_D}$ of the characteristic function $1_D$ belongs to $L^p(\mathbb R^n)$?