A Fourier restriction estimate for surfaces of positive curvature in $\mathbb{R}^6$

Abstract

We improve the best known exponent for the restriction conjecture in R 6 , improving the recent results of Bourgain and Guth. The proof is applicable to any dimension n satisfying n ≡ 0 mod 3.

Average decay of the Fourier transform of measures with applications

“…The improvement manifests itself in the fact that the functions Ψ τ have better integrability properties and so we pay less while removing them. This kind of thing was first observed by Temur in the context of the linear restriction problem [39]. Here, the reduced integrability in the estimates leads to the estimate (5.13) having a worse dependency on R (this produces the second term in the minimum), however the improved properties of Ψ τ lead to both (5.13) and (5.11) having a better dependency on δ, and together they would yield (5.17) after choosing the limiting scale λ in an optimal fashion.…”

Aspects of multilinear harmonic analysis related to transversality

“…For further applications of Theorem 4.1 and the Bourgain-Guth method, see [20], [21], [23], [52] and [36].…”

Improved Fourier restriction estimates in higher dimensions

“…There is certainly a precedent for the recursive approach: for instance, in the fourth section of [6], Bourgain and Guth reformulate their key induction-on-scale argument as a recursive procedure to allow for the use of additional information coming from X-ray transform estimates (see also [20,36] for an elaboration of this argument). Similarly, in a recent article of Wang [38], the induction-on-scale procedure of [11] was rewritten as a recursion; this permitted a more detailed analysis of the underlying geometry of the extension operator and led to the current best known bounds for the restriction conjecture in R 3 .…”

“…(since it has been shown that all other conditions for [tang] are met). For a suitable choice of constant C tang (recalling (36)), this implies (38). The functions f B,trans and sets B are further decomposed so as to ensure favourable tangency properties with respect to translates of the variety Z at the new scale ρ j+1 .…”

“…holds for all p > 2n n−1 , where 1/p+1/p ′ = 1. This was proved by Fefferman and Stein in two dimensions [10], but remains open in higher dimensions despite extensive study; see, for example, [1,2,6,8,9,11,12,22,26,31,33,34,35,36,37,38,42] and the references therein.…”