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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
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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 ${d}/{n}$, $n=2,3,\dots$ 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 K├Ârner. 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 convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz type
MSC Classifications: 42A85, 42B99, 42B15, 42A61 show english descriptions Convolution, factorization
None of the above, but in this section
Multipliers
Probabilistic methods
42A85 - Convolution, factorization
42B99 - None of the above, but in this section
42B15 - Multipliers
42A61 - Probabilistic methods
 

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