Problems in complex and harmonic analysis related to weighted norm inequalities

与加权范数不等式相关的复数和调和分析问题

• 批准号: RGPIN-2021-03545
• 负责人: UriarteTuero, Ignacio
• 金额: $ 1.53万
• 依托单位: University of Toronto
• 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=758703
• 关键词: Problems; complex; harmonic; analysis; related

The applicant proposes to continue researching weighted norm inequalities for Calderon-Zygmund operators (CZOs), and their applications in quasiconformal maps, and Whitney extension problems. CZOs appear as natural tools when understanding the most critical cases of virtually all partial differential equations that describe our physical world, from Schrodinger operators in quantum mechanics to Navier-Stokes equations in fluid flow. Researching into two weight norm inequalities for these operators extends the applications and furthers our understanding of the individual operators under consideration, in ways that would remain completely obscure without this research. More specifically, the applicant and collaborators recently attained an important success by proving in a two-part paper (the applicant is a coauthor in part one) the well-known Nazarov-Treil-Volberg T1 conjecture for the 30+ year-old two weight problem for the Hilbert transform (the simplest non-trivial CZO). The resolution of this 1D problem opens the door for the (more flexible in applications) local Tb version, and higher dimensional versions. Crucial tools in the solution of the 1D case are missing in these versions, leaving enough room for further research, some of it to be performed with students. Another recent success in which weighted norm inequalities proved to be crucial was the solution, by the applicant and coauthors, of the 16-year-old conjecture of Astala regarding distortion of (Hausdorff measure of) sets under planar quasiconformal maps. These maps are good models for elasticity. Astala had understood how these maps distort area and planar sets of smaller dimension (e.g. 1D). But the fine properties of the distortion of the corresponding smaller measures (e.g. length) remained a mystery, which was solved with the type of weighted norm inequalities described above. Unfortunately, some of these tools are missing for the physically relevant 3D case, which the applicant proposes to study. However, enough intuition is given by the weighted inequality proof in 2D to adapt into a plan of attack in 3D. The applicant and collaborators' research on weighted norm inequalities gave rise to an unexpected application in Whitney extension problems. These classical "fitting a function to data" problems ask when a function, a priori only defined on a small set, can be viewed (extended) as defined (with good behaviour) in the whole space. This extension is a key step in many relevant problems to be able to apply the modern tools for the analysis of the partial differential equations governing physical phenomena. While understanding the Whitney extension problems for Sobolev spaces (these are the natural setting for the modern approach for partial differential equations), it turned out that certain tools from the applicant and collaborators' research on weighted norm inequalities appear as very promising and relevant. The applicant proposes to continue this line of research.

申请人建议继续研究卡尔德隆-齐格蒙德算子（CZO）的加权范数不等式及其在拟共形映射和惠特尼扩展问题中的应用。在理解描述我们物理世界的几乎所有偏微分方程（从量子力学中的薛定谔算子到流体流动中的纳维-斯托克斯方程）的最关键情况时，CZO 似乎是自然的工具。 对这些算子的两个权重范数不等式的研究扩展了应用，并加深了我们对所考虑的各个算子的理解，如果没有这项研究，这些方法将完全模糊。更具体地说，申请人和合作者最近在一篇由两部分组成的论文中（申请人是第一部分的合著者）证明了著名的 Nazarov-Treil-Volberg T1 猜想，针对 30 多年前的两个权重，取得了重要的成功。希尔伯特变换（最简单的非平凡 CZO）问题。这个一维问题的解决为（应用更灵活）本地 Tb 版本和更高维度的版本打开了大门。这些版本中缺少一维案例解决方案中的关键工具，为进一步研究留下了足够的空间，其中一些需要与学生一起进行。最近另一个被证明加权范数不等式至关重要的成功是申请人和合著者解决了 16 年前的阿斯塔拉猜想，该猜想涉及平面拟共形映射下集合的畸变（豪斯多夫测度）。这些图是很好的弹性模型。阿斯塔拉已经了解这些地图如何扭曲较小尺寸（例如一维）的面积和平面集。但是相应的较小度量（例如长度）的扭曲的精细特性仍然是一个谜，这可以通过上述加权范数不等式的类型来解决。不幸的是，对于申请人建议研究的物理相关的 3D 案例，其中一些工具缺失。然而，2D 中的加权不等式证明给出了足够的直觉，可以适应 3D 中的攻击计划。申请人和合作者对加权范数不等式的研究在惠特尼可拓问题中产生了意想不到的应用。这些经典的“将函数拟合到数据”问题询问何时可以将一个先验仅在小集合上定义的函数视为（扩展）为在整个空间中定义（具有良好的行为）。这种扩展是许多相关问题的关键一步，能够应用现代工具来分析控制物理现象的偏微分方程。在理解索博列夫空间的惠特尼可拓问题（这是现代偏微分方程方法的自然设置）的同时，事实证明，申请人和合作者关于加权范数不等式的研究中的某些工具似乎非常有前途和相关。申请人提议继续这一研究方向。