Weighted $L^2$ restriction and comparison of nondegeneracy conditions for quadratic manifolds of arbitrary codimensions

TL;DR

This paper advances the understanding of weighted L^2 restriction estimates for quadratic manifolds of arbitrary codimension, establishing a comprehensive relation diagram among various nondegeneracy conditions using diverse mathematical tools.

Contribution

It provides a systematic comparison of nondegeneracy conditions for quadratic manifolds and introduces sharp Fourier decay estimates with a refined analytical method.

Findings

Established a nearly complete relation diagram for nondegeneracy conditions.

Proved that optimal Stein-Tomas restriction implies optimal decoupling.

Unified multiple analytical techniques to analyze quadratic manifolds.

Abstract

We systematically study weighted $L^2$ restriction for quadratic manifolds of arbitrary codimensions by sharp uniform Fourier decay estimates and a refinement of the Du-Zhang method. Comparison with prior results is also discussed. In addition,we obtain an almost complete relation diagram for all existing nondegeneracy conditions for quadratic manifolds of arbitrary codimensions. These conditions come from various topics in harmonic analysis related to "curvature": Fourier restriction, decoupling, Fourier decay, Fourier dimension, weighted restriction, and Radon-like transforms. The diagram has many implications, such as "best possible Stein-Tomas implies best possible $\ell^pL^p$ decoupling". The proof of the diagram requires a combination of ideas from Fourier analysis, complex analysis, convex geometry, geometric invariant theory, combinatorics, and matrix analysis.