Convolution Powers of Salem Measures With Applications. (1st April 2017)

• Record Type: Journal Article
• Authors: Chen, Xianghong; Seeger, Andreas
• Is Part Of: Canadian journal of mathematics. Volume 69:Number 2(2017)
• Page Start: 284
• Page End: 320
• Publication Date: 2017-04-01
• Subjects: 42A85 -- 42B99 -- 42B15 -- 42A61; convolution powers -- Fourier restriction -- Salem sets -- Salem measures -- random sparse sets -- Fourier multipliers of Bochner-Riesz type.; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2016-019-6
• Languages: English
• ISSNs: 0008-414X

Abstract

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 $d\, /\, n, \, n\, =\, 2, 3, ...$ 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 $n\, \in \, \mathbb{N}$ .