Sharp restriction estimates for some degenerate higher codimensional quadratic surfaces
Abstract
The Fourier restriction conjecture is a fundamental problem in harmonic analysis. In this paper, we investigate restriction estimates for degenerate higher codimensional quadratic surfaces and obtain sharp results for some types of degenerate cases. A major obstacle in establishing sharp restriction estimates is the failure of rescaling invariance, which is crucial for induction on scale to be effective. Motivated by the work of Guo and Oh (2022), we introduce a method, building on an iterative variant of the broad-narrow analysis, that does not heavily rely on induction on scale. To obtain suitable transversality conditions for this analysis and to derive desirable bounds for the broad part, we define a generalized notion of Jacobian, and establish its structural properties. These properties are proved using tools and techniques from both algebra and graph theory.
Cite
@article{arxiv.2404.09020,
title = {Sharp restriction estimates for some degenerate higher codimensional quadratic surfaces},
author = {Zhenbin Cao and Changxing Miao and Yixuan Pang},
journal= {arXiv preprint arXiv:2404.09020},
year = {2026}
}
Comments
34 pages, 10 figures. This version unifies the treatment of monomial cases in the main theorem, yielding a more general result. It also introduces new combinatorial arguments from graph theory to study the generalized Jacobian. The title has been updated to reflect the expanded scope