Supercritical discrete restriction estimates for forms in many variables

Discrete Fourier restriction theorems in two dimensions

Consider the group ${\mathbb{R}}^2$ with the discrete topology, and denote its Fourier algebra by $A({\mathbb{R}_{\rm d}^2})$. We reformulate a theorem of V.A. Yudin as a statement about restrictions of functions in $A({\mathbb{R}_{\rm d}^2})$ to the boundary of a strictly convex domain when those functions vanish outside that boundary. We give visual proofs of that statement and a complementary one.

Discrete Fourier restriction associated with KdV equations

In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $ F(u)\p_x u$ provided $F\in C^5$ and the initial data $ϕ\in H^s$ with $s>1/2$.

Discrete restriction for $(x,x^3)$ and related topics

Defining the truncated extension operator $E$ for a sequence $a(n)$ with $n \in {\mathbb Z}$ by putting \[ E{a}(α,β):=\sum_{|n|\leq N}a(n) e(αn^3 + βn), \] we obtain the conjectured tenth moment estimate \[ \| E{a} \|_{L^{10}({\mathbb T}^2)}\lesssim_εN^{\frac{1}{10}+ε} \|a\|_{\ell^2({\mathbb Z})}. \] We obtain related conclusions when the curve $(x,x^3)$ is replaced by $(ϕ_1(x), ϕ_2(x))$ for suitably independent polynomials $ϕ_1(x),ϕ_2(x)$ having integer coefficients.

通过有效同余法实现离散傅里叶限制：基本原理

Discrete Fourier restriction via efficient congruencing: basic principles

本文将 Vinogradov 均值定理中的有效同余法推广至离散傅里叶限制问题，证明了 $s>k(k+1)$ 时的积分不等式，并由此导出周期 Strichartz 不等式与离散傅里叶限制问题中具有猜想量级的常数界。

On the restriction problem for discrete paraboloid in lower dimension

We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when $-1$ is not a square, we prove an $L^2 \to L^{\frac{32}{9} }$ extension estimate, improving the previously known exponent $\frac{68}{19}.$

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。

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…

三维Heisenberg幂零流形上的限制问题

Restriction problems on the three-dimensional Heisenberg nilmanifold

本文证明了三维Heisenberg幂零流形上的谱限制定理，作为平环面上Zygmund限制定理的次椭圆对应，并通过离散短时傅里叶变换证明了其最优性。

Incidence theory and restriction estimates

We investigate the interplay between the discrete restriction phenomenon and incidence theory. Two angles are explored. One is a refinement of the machinery developed by Thomas Wolff, which when combined with a recent subcritical estimate of Bourgain leads to new Strichartz estimates for irrational tori. The other one connects various additive energies with the Szemerédi-Trotter Theorem. The combination of the two approaches recovers the best known Strichartz estimates for the classical torus in dimensions $n\ge 3$, without any appeal to number theory.

$L^p$-加权锥限制估计

$L^p$-weighted cone restriction estimates

该研究针对截断锥的傅里叶延拓估计，在特定权重条件下，结合多线性精细Strichartz不等式等方法得到一系列界，并给出相关实例。

与归一化 Hermite 多项式相关的 Fourier-Hermite 变换的限制定理及 Ornstein-Uhlenbeck-Schrödinger 方程

Restriction theorem for the Fourier-Hermite transform associated with the normalized Hermite polynomials and the Ornstein-Uhlenbeck-Schrödinger equation

本文证明了归一化Hermite多项式相关的Fourier-Hermite变换在离散曲面上的Stein-Tomas和Strichartz限制定理，并获得了Ornstein-Uhlenbeck算子正交函数系统的Strichartz估计，证明了常数在函数数量趋于无穷时的渐近最优性。

Discrete gap probabilities and discrete Painleve equations

We prove that Fredholm determinants of the form det(1-K_s), where K_s is the restriction of either the discrete Bessel kernel or the discrete {}_2F_1 kernel to {s,s+1,...}, can be expressed through solutions of discrete Painleve II and V equations, respectively. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a poissonized Plancherel measure and a z-measure, or as normalized Toeplitz determinants with symbols exp(η(u+1/u)) and (1+u)^z(1+ξ/u)^{z'}. The proofs are based on a general formalism involving d…