Weighted norm inequalities for singular integrals

奇异积分的加权范数不等式

基本信息

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：
2021
资助国家：
加拿大
起止时间：
2021-01-01 至 2022-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741460
关键词：
Weighted norm inequalities singular integrals

项目摘要

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、分数积分和极大积分的加权范数不等式，以及在几乎所有研究的最关键情况下出现的简并椭圆偏微分方程的存在性和规律性。偏微分方程，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范数不等式的研究不仅扩展了候选人最近的研究中出现了许多新工具，例如具有更高消失矩的加权阿尔珀特小波和测试。仅限于加倍立方的特殊函数，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中的应用在短期内，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的局部 Tb 定理扩展到更高的维度，并寻求潜在的应用。关于简并偏微分方程，我们提出了一个扩展我们的广泛计划。最近关于无限简并算子的局部有界性、连续性和最大值原理的工作，这也可以帮助指出研究经典光滑情况的方法。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椭圆方程。无限堕落的政权。