Local maximizers of adjoint Fourier restriction estimates for the cone, paraboloid and sphere

Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maxi…

球面上Tomas-Stein不等式极大元的存在性条件

Conditional existence of maximizers for the Tomas-Stein inequality for the sphere

在球面与薛定谔方程Strichartz不等式最佳常数严格比较的条件下，利用精细Tomas-Stein不等式和剖面分解，证明了存在函数极大化该不等式，且任何极大化序列有子序列收敛到极大元。

Extremizers for adjoint Fourier restriction on hyperboloids: the higher dimensional case

We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The analogous result had been previously established in dimensions $d=1,2$ using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.

Strichartz不等式稳定性：极小值的存在性

Stability for Strichartz inequalities: Existence of minimizers

本文研究了Strichartz不等式稳定性问题，证明在特定条件下极小值存在并达到尖锐常数。

Bilinear restriction estimates for surfaces of codimension bigger than one

In connection with the restriction problem in $\mathbb R^n$ for hypersurfaces including the sphere and paraboloid, the bilinear (adjoint) restriction estimates have been extensively studied. However, not much is known about such estimates for surfaces with codimension (and dimension) larger than one. In this paper we show sharp bilinear $L^2 \times L^2 \to L^q$ restriction estimates for general surfaces of higher codimension. In some special cases, we can apply these results to obtain the corresponding linear estimates.

球面上 sharp Fourier restriction 的稳定性

Stability of sharp Fourier restriction to spheres

本文证明了在维度 $d \in \{3,4,5,6,7\}$ 中，单位球面上的常数函数在受有界扰动时仍最大化加权伴随 Fourier 限制不等式，并完整分类了复值最大化子，同时在 $\mathbb S^7$ 上导出了新的 sharp 伴随限制不等式。