相关论文: A refined trilinear Kakeya estimate in $\mathbb{R}…
We improve the Bennett--Carbery--Tao trilinear restriction estimate for subsets of the paraboloid in three dimensions, giving the sharp factor depending on the transversality.
We present a streamlined and simplified proof of the Kakeya set conjecture in $\mathbb{R}^3$.
We provide an alternative and self contained proof of the main result of Bennett, Carbery, Tao regarding the multilinear restriction estimate. The approach is inspired by the recent result of Guth about the Kakeya version of multilinear…
We prove the equivalence of two Kakeya conjectures: 1.The Kakeya maximal operator conjecture 2.The disjoint trilinear dual form of the Kakeya maximal operator conjecture
We obtain a sharp bilinear restriction estimate for the paraboloid in $\mathbb{R}^3$ for $q>3.25$.
We show that the Fourier extension conjecture on the paraboloid in three dimensions is equivalent to a local single scale smooth Alpert trilinear inequality, which is an improvement of an analogous multiscale trilinear inequality in…
We prove an x-ray estimate in general dimension which is stronger than the Kakeya estimates of Wolff. This generalizes an x-ray estimate in three dimensions which is also due to Wolff.
We investigate integral-geometric quantities arising from harmonic analysis which measure visibility and transversality. Motivated by their applications in multilinear Kakeya problems and affine-invariant measures on surfaces, we derive…
We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to…
We prove that a local trilinear extension inequality on the paraboloid in three dimensions is equivalent to the Fourier restriction conjecture, and then we prove a variant involving smooth Alpert wavelets that represents the weakest such…
In this paper, we establish a refined transversality theorem on linear perturbations from a new perspective of Hausdorff measures. Furthermore, we give its applications not only to singularity theory but also to multiobjective optimization.
We compare the smooth Alpert testing condition for the paraboloid Fourier extension conjecture in <cite>RiSa3</cite> to the modulated testing condition for the Kakeya conjecture in <cite>RiSa2</cite>. To this end, the modulated testing…
We prove that every compact K\"ahler threefold has arbitrarily small deformations to some projective manifolds, thereby solving the Kodaira problem in dimension 3.
This is a Seminaire Bourbaki survey of the proof of the Kakeya conjecture in three dimensions. The survey is written for a broad mathematical audience. We sketch all the ideas in the proof, with many pictures.
We obtain sharp estimates for certain trilinear oscillatory integrals. In particular, we extend Phong and Stein's seminal result to a trilinear setting. This result partially answers a question raised by Christ, Li, Tao and Thiele…
Bennett, Carbery and Tao considered the $k$-linear restriction estimate in $\mathbb{R}^{n+1}$ and established the near optimal $L^\frac2{k-1}$ estimate under transversality assumptions only. We have shown that the trilinear restriction…
We improve the $L^{p}\rightarrow L^p$ restriction estimate in $\mathbb{R}^3$ to the range $p>3+3/14$, based on some Kakeya type incidence estimates and the refined decoupling theorem.
We give a simple proof of a pointwise decay estimate in 3+1 dimensions stated in two versions, making advantage of a particular simplicity of inverting the spherically symmetric part of the wave operator and of the comparison theorem. We…
We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery, and Tao.
A quantitative version of the scalar lower bound under $C^0$ convergence was conjectured by Gromov. More recently, Mazurowski and Yao proved that a refined form of Gromov's conjecture holds in dimension three. Furthermore, they constructed…