Functions whose Fourier transform vanishes on a surface

TL;DR

This paper investigates the structure of functions in $L_p(R^d)$ whose Fourier transforms vanish on a smooth surface, revealing conditions under which these functions form the entire space or are dense, with implications for harmonic analysis.

Contribution

It establishes the precise range of $p$ for which the subspace of functions with Fourier transforms vanishing on a surface equals $L_p$, and provides density results and differential operator characterizations.

Findings

For $p > 2d/(d+1)$, the subspace equals $L_p$.

For $p < 2d/(d+1)$, smooth functions are dense in the subspace for certain surfaces.

Provides an equivalent differential operator-based definition of the subspace.

Abstract

We study the subspaces of $L_p(\mathbb{R}^d)$ that consist of functions whose Fourier transforms vanish on a smooth surface of codimension $1$. We show that a subspace defined in such a manner coincides with the whole $L_p$ space for $p > \frac{2d}{d+1}$. We also prove density of smooth functions in such spaces when $p < \frac{2d}{d+1}$ for specific cases of surfaces and give an equivalent definition in terms of differential operators.