A short proof of the multilinear Kakeya inequality

We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery, and Tao. The multilinear Kakeya inequality is a geometric estimate about the overlap pattern of cylindrical tubes in R pointing in different directions. This estimate was first proven by Bennett, Carbery, and Tao in [BCT]. Recently it has had some striking applications in harmonic analysis. Here is a short list of some applications. In [BCT], it was applied to prove a multilinear restriction estimate. In [BG], Bourgain and the author used this multilinear restriction estimate to make some progress on the original restriction problem, posed by Stein in [S]. In [B], Bourgain used it to prove new estimates for eigenfunctions of the Laplacian on flat tori. Most recently, in [BD], Bourgain and Demeter used the multilinear restriction estimate to prove the l decoupling conjecture. As a corollary of their main result, they proved essentially sharp Strichartz estimates for the Schrodinger equation on flat tori. The goal of this paper is to give a short proof of the multilinear Kakeya inequality. The original proof of [BCT] used monotonicity properties of heat flow and it is morally based on multiscale analysis. Later there was a proof in [G] using the polynomial method. The proof we give here is based on multiscale analysis. I think that the underlying idea is the same as in [BCT], but the argument is organized in a more concise way. Here is the statement of the multilinear Kakeya inequality. Suppose that lj,a are lines in R, where j = 1, ..., n, and where a = 1, ..., Nj . We write Tj,a for the characteristic function of the 1-neighborhood of lj,a. Theorem 1. Suppose that lj,a are lines in R, and that each line lj,a makes an angle of at most (10n)−1 with the xj-axis. Let QS denote any cube of side length S. Then for any > 0 and any S ≥ 1, the following integral inequality holds: