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arXiv · 2404.09020

Sharp restriction estimates for some degenerate higher codimensional quadratic surfaces

Zhenbin Cao·Changxing Miao·Yixuan Pang
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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.

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Zhenbin Cao, Changxing Miao, Yixuan Pang. 2024-04-13. Sharp restriction estimates for some degenerate higher codimensional quadratic surfaces. https://arxiv.org/abs/2404.09020

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