Endpoint Fourier restriction and unrectifiability

TL;DR

This paper establishes a dichotomy for measures in  with Fourier restriction at endpoint exponents: such measures are either absolutely continuous or purely unrectifiable, revealing a fundamental geometric-analytic connection.

Contribution

It proves that measures with endpoint Fourier restriction are either absolutely continuous or 1-purely unrectifiable, linking harmonic analysis with geometric measure theory.

Findings

Measures with endpoint Fourier restriction are either absolutely continuous or unrectifiable.

The result characterizes the geometric nature of measures satisfying endpoint restriction.

Provides a new criterion connecting Fourier restriction and measure rectifiability.

Abstract

We show that if a measure of dimension $s$ on $\mathbb{R}^d$ admits $(p,q)$ Fourier restriction for some endpoint exponents allowed by its dimension, namely $q=\tfrac{s}{d}p'$ for some $p>1$, then it is either absolutely continuous or $1$-purely unrectifiable.