抛物面的混合范数 $l^2$ 解耦

$\mathbb{R}^3$ 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。

双曲抛物面的Mizohata--Takeuchi型估计

A Mizohata--Takeuchi-type estimate for the hyperbolic paraboloid

本文证明双曲抛物面傅里叶延拓算子的Mizohata--Takeuchi型加权$L^2$估计，通过多尺度平行六面体上的权重质量控制范数，并利用双线性精细解耦建立。核心贡献是给出该几何背景下加权估计的完整证明。

Extremizability of Fourier restriction to the paraboloid

In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in $\mathbb R^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result had previously been established for the case $q=2$. In the range where the boundedness of the restriction operator is still an open question, our result is conditional on improvements toward the restriction conjecture.

d维环面上薛定谔算子扰动的极大估计

Maximal estimates for perturbations of the Schrödinger operator on $\mathbb{T}^d$

该研究针对d维环面，证明周期薛定谔方程的猜想极大估计在抛物面小扰动下不成立，利用 incidence 估计新下界结合齐性动力学给出替代证明，且估计在抛物面解耦端点处本质最优。

A sharp inequality for the Strichartz norm

Let $u:\R \times \R^n \to \C$ be the solution of the linear Schrödinger equation $iu_t + Δu =0$ with initial data $u(0,x) = f(x)$. In the first part of this paper we obtain a sharp inequality for the Strichartz norm $\|u(t,x)\|_{L^{2k}_tL^{2k}_x(\R \times\R^n)}$, where $k\in \Z$, $k \geq 2$ and $(n,k) \neq (1,2)$, that admits only Gaussian maximizers. As corollaries we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi and Hundertmark - Zharnitsky) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper we expres…

$l^p$ decoupling for restricted $k$-broadness

To prove Fourier restriction estimate using polynomial partitioning, Guth introduced the concept of $k$-broad part of regular $L^p$ norm and obtained sharp $k$-broad restriction estimates. To go from $k$-broad estimates to regular $L^p$ estimates, Guth employed $l^2$ decoupling result. In this article, similar to the technique introduced by Bourgain-Guth, we establish an analogue to go from regular $L^p$ norm to its $(m+1)$-broad part, as the error terms we have the restricted $k$-broad parts ($k=2,\cdots,m$). To analyze the restricted $k$-broadness, we prove an $l^p$ decoupling result, which …

Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeg…