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; 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.

申请提案要继续研究单数，分数和最大积分的加权规范不平等，并为退化椭圆形偏微分方程的存在和规律性。 Calderon-Zygmund单数积分和分数积分在实际上所有部分微分方程的研究的最关键案例中，从Schrodinger操作员到量子力学到流体流中的Navier-Stokes方程。对这些运营商的两个重量规范不平等的研究不仅扩大了应用的范围，而且揭示了正在考虑的个人操作员内核的重要特性，通常没有这种投资而隐藏了这种范围。候选人的最新研究已经出现了许多新工具，例如具有更高消失的力矩的加权Alpert小波，并且测试仅限于使立方体加倍的特殊功能，并且长期目标是将这些思想和其他人建立在现有理论的结构中，以便对两个重量的既定型既不相等，又对奇异的综合性以及其应用对数量的方程式提供了全面的治疗，并具有不同的测量方程。在短期内，我们将更具体地致力于（1）将解决方案扩展到NTV的猜想，这些猜想是希尔伯特（Hilbert）从一个L2空间转变为另一个L2空间的界限，再到更高的Calderon-Zygmund操作员在更一般的环境中，以及（2）扩展了我们最近的TB定理，以与更高的Dimensions合作，并与学生进行了更高的合作以及潜在的与学生的合作，并将其扩展。关于退化的部分微分方程，我们提出了一项广泛的计划，以扩大无限退化运算符的局部界限，连续性和最大原则的最大原则，这也可能有助于研究经典的平滑案例。这些投资的影响应有助于激发经典规律性的扩展结果对椭圆方程式的无限退化制度。