Convolution Powers of Salem MeasuresWith Applications

Chen, Xianghong ; Seeger, Andreas

Canadian journal of mathematics, Tome 69 (2017) no. 2, pp. 284-320

Résumé

We study the regularity of convolution powers for measures supported on Salem sets, andprove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha $ of the form $d\,/\,n,\,n\,=\,2,3,...$ 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 $n$ -foldconvolutions for all $n\,\in \,\mathbb{N}$ .

Détail

DOI : 10.4153/CJM-2016-019-6

Mots-clés : 42A85, 42B99, 42B15, 42A61, convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz type.

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