Some recent progress on the Restriction conjecture

Slicing surfaces and Fourier restriction conjecture

We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in $\mathbb{R}^n$ implies that for the cone in $\mathbb{R}^{n+1}$. We also prove a new restriction estimate for any surface in $\mathbb{R}^3$ locally isometric to the plane and of finite type.

$\mathbb{R}^{3}$ 中限制猜想的证明

The Proof of restriction conjecture In $\mathbb{R}^{3}$

本文证明了 $\mathbb{R}^{3}$ 中的限制猜想，即对第二基本形式严格为正的光滑紧曲面建立了延拓算子的 $L^p$ 估计，并由此推出 n=3 时 Kakeya 集猜想成立。

Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces

In this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces $L^p_{rad} L^2_{ang} (\mathbb{R}^n)$. We produce sharp bounds for the restriction of the Fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier. We also present some results for the discrete restriction conjecture and state an intriguing open problem.

Improved Fourier restriction estimates in higher dimensions

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.

A note on the cone restriction conjecture in the cylindrically symmetric case

In this note, we present two arguments showing that the classical \textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The first is based on a dyadic restriction estimate, while the second follows from a strengthening version of the Hausdorff-Young inequality and the Hölder inequality in the Lorentz spaces.

Fourier restriction above rectangles

In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfa…

Extension and restriction principles for the HRT conjecture

The HRT (Heil-Ramanathan-Topiwala) conjecture asks whether a finite collection of time-frequency shifts of a non-zero square integrable function on $\mathbb{R}$ is linearly independent. This longstanding conjecture remains largely open even in the case when the function is assumed to be smooth. Nonetheless, the conjecture has been proved for some special families of functions and/or special sets of points. The main contribution of this paper is an inductive approach to investigate the HRT conjecture based on the following question. Suppose that the HRT is true for a given set of ($N$) points a…

任意余维二次流形的限制性估计

Restriction estimates for quadratic manifolds of arbitrary codimensions

本文将Bourgain-Guth多线性方法推广到任意余维二次流形，利用实代数几何中的覆盖引理与柱形分解算法，建立了统一的限制性估计并恢复了抛物面情形的主要定理。

A Fourier restriction estimate for surfaces of positive curvature in R^6

We improve the best known exponent for the restriction conjecture in R^6. Our idea is applicable to any dimension n satisfying n = 0 mod 3, though we do not explicitly calculate the improvement for n > 6. This improves the recent results of Bourgain and Guth.

Weighted restriction estimates and application to Falconer distance set problem

We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the Fourier transform of fractal measures, and therefore improve the results for the Falconer distance set conjecture in three and higher dimensions.

使用 decoupling 定理与 two-ends Furstenberg 不等式研究限制性估计

Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities

本文提出利用 decoupling 定理与 two-ends Furstenberg 不等式研究限制性猜想，并给出一个蕴含该猜想的 two-ends Furstenberg 猜想，在平面上予以证明，进而推导出三维 $p>22/7$ 的限制性估计及高维改进。

Recent progress on the Kakeya conjecture

The purpose of this article is to survey the developments on the Kakeya problem in recent years, concentrating on the period after the excellent 1999 survey of Wolff, and including some recent work by the authors. We will focus on the standard Kakeya problem for line segments and not discuss other important variants (such as Kakeya estimates for circles, light rays, or $k$-planes).

$\mathbb{R}^3$ 中抛物面的条带解耦不等式

Strip Decoupling Inequalities for the Paraboloid in $\mathbb{R}^3$

本文证明了三维空间中抛物面条带的尖锐解耦估计，通过归约为平面关联问题并利用两端Furstenberg估计，受Wang-Wu方法启发。

Finite field restriction estimates based on Kakeya maximal operator estimates

In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$ dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which $-1$ \emph{is} a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein-Tomas exponent in this setting.

Restrictions of smooth functions to a closed subset

We first provide an approach to the recent conjecture of Bierstone-Milman-Pawlucki on Whitney's old problem on smooth extendability of functions defined on a closed subset of a Euclidean space, using higher order paratangent bundle they introduced. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek's theorem on controllability of flatness by the values on a closed set. The multi-dimensional Vandermonde matrix plays an important role in both cases.

四维素数域球面限制猜想：突破Stein-Tomas指数及其应用

On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications

本文引入水平切片和平面-直线停止时间分解方法，通过将每个临界水平切片分解为富平面、富直线-贫平面和贫直线-贫平面分量，克服了球面Bochner-Riesz核中的Kloosterman障碍，证明了对于每个非零球面S_j和每个r>23/7，有R_{S_j}^*(2→r)≲1，从而在四维素数域Erdős-Falconer距离问题中首次改进了二十年前的(d+1)/2阈值。

The proof of the $l^2$ Decoupling Conjecture

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which (up to $N^ε$ losses) implies the full range of expected $L^p_{x,t}$ Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various applications in Additive Combinatorics, Incidence Geometry and Number Theory are also disc…

A dichotomy for Hörmander-type oscillatory integral operators

In this paper, we first generalize the work of Bourgain and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain's condition, they satisfy the same $L^p$ bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain's condition fails, then the $L^{\infty} \to L^q$ boundedness a…

环面本征函数的限制估计与球面区域中的格点问题

Restriction estimates for toral eigenfunctions and lattice points in spherical regions

本文通过切片与打包方法结合离散球面乘子逼近，建立了环面本征函数的新L^2限制估计，在某些情形下达到最优，从而证明了Huang-Zhang关于大余维光滑子流形的猜想，并为Bourgain-Rudnick猜想提供了新进展。

Lattice points on circles, squares in arithmetic progressions and sumsets of squares

Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number theory, arithmetic geometry, discrete geometry and additive combinatorics (some old and some new) which each, if true, would shed light on Rudin's conjecture.

Recent progress in the study of representations of integers as sums of squares

In this article, we collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. We start with a sketch of Milne's proof of two of these conjectures. We also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of $4s^2$, respectively $4s(s+1)$, triangular numbers. This approach is inspired by Zagier's proof of the Kac--Wakimoto formulas via modular forms. We end the survey with recent conjectures of the first author and Chua.