A sharp bilinear cone restriction estimate.

*(English)*Zbl 1125.42302The main result of this groundbreaking paper is a bilinear estimate for the adjoint of the Fourier restriction operator in \(\mathbb R^d\), where \(d\geq 3\). Consider two disjoint conical subsets \(\Gamma_1\), \(\Gamma_2\),
\[
\Gamma_i=\{x=(x',x_d)\colon x_d=|x'|, x'/|x_d|\in \Omega_i\}
\]
where \(\Omega_1\) and \(\Omega_2\) are separated closed subsets of the unit sphere. Then the estimate
\[
\|\widehat {fd\sigma} \widehat {gd\sigma}\|_{L^p(\mathbb R^d)} \lesssim \|f\|_2 \|g\|_2
\]
holds for two functions \(f\) and \(g\) supported in \(\Gamma_1\) and \(\Gamma_2\), respectively, provided that \(p>1+2/d\). In three dimensions this estimate provides an affirmative answer to a conjecture by S. Klainerman and M. Machedon, but Wolff actually proves a sharp bound (modulo endpoints) in all dimensions. The relevant range is \(p\in (1+2/d,1+2/(d-2))\) as for \(p\geq 1+2/(d-2)\) the bound follows from the Strichartz estimate. Only very partial results in three and four dimensions were previously known; see articles by J. Bourgain [Geometric aspects of functional analysis (Israel, 1992–1994), BirkhĂ¤user, Basel, Oper. Theory, Adv. Appl. 77, 41–60 (1995; Zbl 0833.43008)] and by T. Tao and A. Vargas [Geom. Funct. Anal. 10, No. 1, 185–215 (2000; Zbl 0949.42012)]. A corollary of Wolff’s theorem is a sharp \(L^p\) result for the Fourier restriction operator on the cone in \(\mathbb R^4\) (here \(p<3/2\)).

In a very interesting appendix the author establishes new \(L^p\to L^q\) results for families of X-ray transforms associated to cones, as well as improved mixed norm estimates. The bounds are essentially best possible in three or four dimensions, while in dimensions \(d\geq 5\) partial but sharp results are given in the range \(p\leq (d+1)/2\).

This article, and Geom. Funct. Anal. 10, No. 5, 1237–1288 (2000; Zbl 0972.42005), appear to be the last two journal publications by Thomas Wolff, who tragically died in July 2000. Both contributions are highly original masterpieces and should have a long lasting impact in analysis.

In a very interesting appendix the author establishes new \(L^p\to L^q\) results for families of X-ray transforms associated to cones, as well as improved mixed norm estimates. The bounds are essentially best possible in three or four dimensions, while in dimensions \(d\geq 5\) partial but sharp results are given in the range \(p\leq (d+1)/2\).

This article, and Geom. Funct. Anal. 10, No. 5, 1237–1288 (2000; Zbl 0972.42005), appear to be the last two journal publications by Thomas Wolff, who tragically died in July 2000. Both contributions are highly original masterpieces and should have a long lasting impact in analysis.

Reviewer: Andreas Seeger (MR1836285)