Geodesic
Convolution Powers of Salem MeasuresWith Applications
Canadian journal of mathematics, Tome 69 (2017) no. 2, pp. 284-320

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DOI

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 $n\,\in \,\mathbb{N}$ .
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. 
Chen, Xianghong; Seeger, Andreas. Convolution Powers of Salem MeasuresWith Applications. Canadian journal of mathematics, Tome 69 (2017) no. 2, pp. 284-320. doi: 10.4153/CJM-2016-019-6
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     author = {Chen, Xianghong and Seeger, Andreas},
     title = {Convolution {Powers} of {Salem} {MeasuresWith} {Applications}},
     journal = {Canadian journal of mathematics},
     pages = {284--320},
     year = {2017},
     volume = {69},
     number = {2},
     doi = {10.4153/CJM-2016-019-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2016-019-6/}
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