Convolution Powers of Salem Measures with Applications

TL;DR

This paper investigates the regularity properties of convolution powers of Salem measures supported on Salem sets, establishing new Fourier restriction results and extending constructions for measures with sharp regularity in harmonic analysis.

Contribution

It introduces novel constructions of $ ext{alpha}$-Salem measures with optimal regularity, and proves Fourier restriction theorems for their convolution powers, extending previous methods.

Findings

Existence of $ ext{alpha}$-Salem measures satisfying Fourier restriction in specific $p$ ranges

Sharp regularity results for $n$-fold convolutions of Salem measures

Extension of K"orner's ideas to construct measures with optimal regularity

Abstract

We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha$ of the form ${d}/{n}, n=2,3,\cdots$ there exist $\alpha$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-\alpha}$. The results rely on ideas of K\"orner. We extend some of his constructions to obtain upper regular $\alpha$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb N$.