抛物面ℓ²解耦定理的学习指南

关于解耦与限制估计

On decoupling and restriction estimates

本文证明限制猜想蕴含高维解耦定理，并给出三维情形下的简单证明。

壳型限制下抛物面的解耦不等式及其在周期Zakharov系统中的应用

Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system

本文通过改进抛物面在壳型限制下的解耦不等式，建立新的三线性离散Fourier限制估计，从而在低正则性条件下证明了环面上Zakharov系统的局部适定性，并改进了Kishimoto的结果。

有限域上抛物面的猜想和改进的扩张定理

Conjecture and improved extension theorems for paraboloids in the finite field setting

研究了q元有限域F_q上d维向量空间抛物面的扩张估计。我们利用抛物面的L ^2基约束估计和L^p \to L^r扩张估计之间的联系。因此，我们改进了A. Lewko和M. Lewko在偶数维度d \ge 6和奇数维度d=4 \ell +3中对于\ell \in \mathbb N。我们的结果推广了M.向更高维度进发。本文还阐明了抛物面的有限域扩张问题。

A decoupling proof of the Tomas restriction theorem

We give a new proof of a classic Fourier restriction theorem for the truncated paraboloid in $\mathbb{R}^n$ based on the $l^2$ decoupling theorem of Bourgain-Demeter. Focusing on the extension formulation of the restriction problem (dual to the original restriction formulation), we find that the $l^2$ decoupling theorem directly implies a local variant of the desired extension estimate incurring an $\varepsilon$-loss. To upgrade this result to the desired global extension estimate, we employ some $\varepsilon$-removal techniques first introduced by Tao. By adhering to the extension formulation…

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2-surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or "saddle" z = xy. The bilinear method gave here sharp results for p > 10/3 (Lee 05, Vargas 05, Stovall 17), and this result was recently improved to p > 3.25 (Cho-Lee 17, Kim 17). Th…

$\ell^2$ Decoupling in $\mathbb R^2$ for curves with Vanishing Curvature

We expand the class of curves $(φ_1(t),φ_2(t)),\ t\in[0,1]$ for which the $\ell^2$ decoupling conjecture holds for $2\leq p\leq 6$. Our class of curves includes all real-analytic regular curves with isolated points of vanishing curvature and all curves of the form $(t,t^{1+ν})$ for $ν\in (0,\infty)$.

复曲线上去耦及改进的三次矩曲线上去耦

Decoupling for complex curves and improved decoupling for the cubic moment curve

本文通过双线性方法证明了复曲线上尖锐的ℓ²去耦不等式，并对三次矩曲线去耦不等式进行了对数修正。

凸曲线的均匀解耦

Uniform decoupling for convex curves

本文通过高/低论证证明了一般平面凸曲线的普适\u2113²L⁶解耦估计，且常数对所有此类曲线一致。

一个加权解耦不等式及其在极大Bochner-Riesz问题中的应用

A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

该研究证明了p=2n/(n-1)时的加权L^pℓ^p解耦估计，将其应用于极大Bochner-Riesz问题，得到三维空间中超出实插值指数的结果，并改进了平面情形的相关结论。

Sharp superlevel set estimates for small cap decouplings of the parabola

We prove sharp bounds for the size of superlevel sets $\{x\in \mathbb{R}^2:|f(x)|>α\}$ where $α>0$ and $f:\mathbb{R}^2\to\mathbb{C}$ is a Schwartz function with Fourier transform supported in an $R^{-1}$-neighborhood of the truncated parabola $\mathbb{P}^1$. These estimates imply the small cap decoupling theorem for $\mathbb{P}^1$ of Demeter, Guth, and Wang, and the canonical decoupling theorem for $\mathbb{P}^1$ of Bourgain and Demeter. New $(\ell^q,L^p)$ small cap decoupling inequalities also follow from our sharp level set estimates.

关于矩曲线 $\ell^2$ decoupling 的简短证明

A short proof of $\ell^2$ decoupling for the moment curve

本文利用受 Wooley 嵌套高效同余论证启发的双线性方法，为 $\mathbb{R}^k$ 中的矩曲线 $\ell^2$ decoupling 不等式提供了一个简短且初等的证明。

Decoupling for moment manifolds associated to Arkhipov--Chubarikov--Karatsuba systems

We prove $\ell^{p}L^{p}$ decoupling inequalities for a class of moment manifolds. These inequalities imply optimal mean value estimates for multidimensional Weyl sums of the kind considered by Arkhipov, Chubarikov, and Karatsuba and by Parsell. In our proofs we take a new point of view on the Bourgain--Demeter--Guth induction on scales argument. This point of view substantially simplifies even the proof of $\ell^{2}L^{p}$ decoupling for the moment curve.

量子调和分析框架下Schatten类算子的解耦

Decoupling for Schatten class operators in the setting of Quantum Harmonic Analysis

本研究在量子调和分析框架下提出算子解耦概念，证明函数经典ℓ^qL^p解耦与ℝ^{2d}有界集上算子的ℓ^q𝒮^p解耦等价，且等价性仅依赖有界集本身，证明基于量子版Wiener除引理。

Decoupling for certain quadratic surfaces of low co-dimensions

We prove sharp $\ell^{p}L^{p}$ decoupling inequalities for $2$ quadratic forms in $4$ variables. We also recover several previous results (arXiv:1409.1634, arXiv:1501.07224, arXiv:1609.02022, arXiv:1609.04107) in a unified way.

Decoupling inequalities for quadratic forms

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.

Decoupling for fractal subsets of the parabola

We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly co…

修复5/4平面 pinned Falconer定理的精细去耦证明

Repairing the refined-decoupling proof of the 5/4 planar pinned Falconer theorem

本文指出Guth等人的平面pinned Falconer定理证明中精细去耦估计的任意包形式有误，构造反例后通过加权ℓ²去耦的归纳法修复了证明，使α>5/4时的能量论证闭合，且与后续波包处理无重叠。