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arXiv · 1509.00460

Convolution Powers of Salem Measures with Applications

Xianghong Chen·Andreas Seeger
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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 $n\in \mathbb N$.

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Xianghong Chen, Andreas Seeger. 2015-09-01. Convolution Powers of Salem Measures with Applications. https://doi.org/10.4153/cjm-2016-019-6

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