A Fourier restriction estimate for surfaces of positive curvature in \(\mathbb{R}^6\)
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.
scientific article; zbMATH DE number 6353615