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Zhenbin Cao

Publications and source records attributed to Zhenbin Cao.

5 recordsLinked to original sources
arXiv2025

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

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.

math.CA↗
arXiv2024

Sharp restriction estimates for some degenerate higher codimensional quadratic surfaces

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.

math.CA↗
arXiv2024

Fourier decay of fractal measures on surfaces of co-dimension two in $\mathbb{R}^5$

Fourier decay of fractal measures on surfaces plays an important role in geometric measure theory and partial differential equations. In this paper, we study the quadratic surfaces of high co-dimensions. Unlike the case of co-dimension 1, quadratic surfaces of high co-dimensions possess some special scaling structures and degenerate characteristics. We will adopt the strategy from Du and Zhang, combined with the broad-narrow analysis with different dimensions as divisions, to obtain a few lower bounds of Fourier decay of fractal measures on quadratic surfaces of co-dimension two in $\mathbb{R}^5$.

math.CA↗
arXiv2023

$L^{p}$-estimate of Schrödinger maximal function in higher dimensions

Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by building the sharp $L^p$-estimate of Schrödinger maximal function. Du-Guth-Li in \cite{DGL} proved the sharp $L^p$-estimates for all $p \geq 2$ in $\mathbb{R}^{2+1}$. Du-Zhang in \cite{DZ} proved the sharp $L^2$-estimate in $\mathbb{R}^{n+1}$ with $n \geq 3$, but for $p>2$ the sharp $L^p$-estimate of Schrödinger maximal function is still unknown. In this paper, we obtain partial results on this problem by using polynomial partitioning.

math.AP↗