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A sharp bilinear cone restriction estimate. (English) Zbl 1125.42302
The 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.

##### MSC:
 42B15 Multipliers for harmonic analysis in several variables 42B25 Maximal functions, Littlewood-Paley theory
##### Citations:
Zbl 0833.43008; Zbl 0949.42012; Zbl 0972.42005
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