location:  Publications → journals → CJM
Abstract view

# Convolution Powers of Salem Measures with Applications

Published:2016-09-14
Printed: Apr 2017
• Xianghong Chen,
Department of Mathematics , University of Wisconsin-Madison , Madison, WI 53706, USA
• Andreas Seeger,
Department of Mathematics , University of Wisconsin-Madison , Madison, WI 53706, USA
 Format: LaTeX MathJax PDF

## 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 ${d}/{n}$, $n=2,3,\dots$ 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 Körner. We extend some of his constructions to obtain upper regular $\alpha$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb{N}$.
 Keywords: convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz type
 MSC Classifications: 42A85 - Convolution, factorization 42B99 - None of the above, but in this section 42B15 - Multipliers 42A61 - Probabilistic methods

 top of page | contact us | privacy | site map |