Weighted norm inequalities for singular integrals

奇异积分的加权范数不等式

• 批准号: RGPIN-2020-06829
• 负责人: Sawyer, Eric
• 金额: $ 1.75万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759118
• 关键词: Weighted; norm; inequalities; singular; integrals

The applicant proposes to continue investigations into weighted norm inequalities for singular, fractional and maximal integrals, and into existence and regularity for degenerate elliptic partial differential equations. Calderon-Zygmund singular integrals and fractional integrals arise in the most critical cases of the study of virtually all partial differential equations, from Schrodinger operators to quantum mechanics to Navier-Stokes equations in fluid flow. The study of two weight norm inequalities for these operators not only extends the scope of application, but reveals important properties of the kernels of the individual operators under consideration, that often remain hidden without such investigation. A number of new tools have arisen from the candidate's recent research, such as weighted Alpert wavelets with higher vanishing moments and testing restricted to special functions for doubling cubes, and a long term goal is to build these ideas and others into the fabric of existing theory to provide a comprehensive treatment of two weight norm inequalities for singular integrals, and their application to geometric measure theory and partial differential equations. In the short term, we will work more specifically to (1) extend our solution to the NTV conjecture characterizing boundedness of the Hilbert transform from one L2 space to another one, to higher dimensional Calderon-Zygmund operators in more general settings, and (2) to extend our recent local Tb theorem for the Hilbert transform to higher dimensions in collaboration with students, and pursue potential applications. In regard to degenerate partial differential equations, we propose a broad program of extending our recent work on local boundedness, continuity and maximum principles for infinitely degenerate operators, which could help point the way to investigating the classical smooth case as well. The impact of these investigations should help motivate the extension of the classical regularity results on elliptic equations to the infinitely degenerate regime.

申请人建议继续研究奇异积分、分数积分和最大积分的加权范数不等式，以及简并椭圆偏微分方程的存在性和正则性。卡尔德隆-齐格蒙德奇异积分和分数积分出现在几乎所有偏微分方程研究的最关键情况下，从薛定谔算子到量子力学再到流体流动中的纳维-斯托克斯方程。对这些算子的两个权重范数不等式的研究不仅扩展了应用范围，而且揭示了所考虑的各个算子的核的重要属性，这些属性在没有这种研究的情况下通常是隐藏的。候选人最近的研究产生了许多新工具，例如具有更高消失矩的加权阿尔珀特小波和仅限于加倍立方的特殊函数的测试，长期目标是将这些想法和其他想法构建到现有理论的结构中，以提供奇异积分的两个权范数不等式的综合处理，以及它们在几何测度理论和偏微分方程中的应用。短期内，我们将更具体地致力于 (1) 将我们的解决方案扩展到表征希尔伯特变换从一个 L2 空间到另一个 L2 空间的有界性的 NTV 猜想，以及更一般设置中的更高维 Calderon-Zygmund 算子，以及 (2) 与学生合作将我们最近的希尔伯特变换局部 Tb 定理扩展到更高维度，并寻求潜在的应用。关于简并偏微分方程，我们提出了一个广泛的计划，扩展我们最近在无限简并算子的局部有界性、连续性和最大值原理方面的工作，这也有助于为研究经典光滑情况指明方向。这些研究的影响应该有助于推动椭圆方程的经典正则性结果扩展到无限简并状态。