A Fourier restriction estimate for surfaces of positive curvature in \(\mathbb{R}^6\)
type: Item
instance of: scholarly article
title: A Fourier restriction estimate for surfaces of positive curvature in \(\mathbb{R}^6\)
zbMATH Open document ID: 1318.42010
DOI: 10.4171/RMI/805
author: Faruk Temur
published in: Revista Matemática Iberoamericana
publication date: 2014-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.4017
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 6353615
zbMATH Keywords: Fourier restriction estimate
zbMATH Keywords: compact hypersurface
zbMATH Keywords: positive curvature
zbMATH Keywords: multilinear restriction estimates
reviewed by: Raymond Johnson
MaRDI profile type: Publication
arXiv ID: 1108.4017
cites work: On the multilinear restriction and Kakeya conjectures
cites work: Besicovitch type maximal operators and applications to Fourier analysis
review text: Stein's restriction conjecture says that if \(S \subseteq R^n\) is a compact hypersurface with positive definite second fundamental form, and \(\sigma\) is the surface measure on \(S\), then for \(p > \frac{2n}{n -1}, f \in L^{\infty}(S_{\sigma})\), NEWLINE\[NEWLINE || \widehat{f d \sigma} ||_p \leq C_{p, S} || f ||_{\infty}. NEWLINE\]NEWLINENEWLINENEWLINE\noindent The result is known for \(n = 2\), and unknown for \(n \geq 3\) despite a number of general results beginning with the Tomas-Stein theorem that the estimate holds for \(p > \frac{2(n +1)}{n - 1}\), which was improved by Tao to \(p > \frac{2(n +2)}{n}\).NEWLINENEWLINE\noindent Bourgain and Guth used scaling in a multilinear estimate to improve this to \(p > \frac{8n +6}{4n -3}\) when \(n \equiv 0 \;\text{mod} \;3\) (with different lower bounds when \(n \equiv 1 \;\text{mod} \;3\) or \(n \equiv 2 \;\text{mod} \;3\)).NEWLINENEWLINE\noindent The author extends this result by improving the exponent by \(2/735\) for the case \(n = 6\). An improvement if also made when \(n \equiv 0 \;\text{mod} \;3\), but he does not calculate the precise improvement.
description: scientific article; zbMATH DE number 6353615