A comparison principle for convolution measures with applications

TL;DR

This paper introduces a geometric comparison principle for convolutions of singular measures, leading to new sharp Fourier extension inequalities and advancing Fourier restriction theory.

Contribution

It establishes a general comparison principle for convolutions of singular measures, enabling new sharp inequalities in Fourier analysis.

Findings

Proves a geometric comparison principle for n-fold convolutions.

Derives a new sharp Fourier extension inequality for convex perturbations of a parabola.

Provides applications to Fourier restriction theory.

Abstract

We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper.