Problems in complex and harmonic analysis related to weighted norm inequalities

申请人建议继续研究Calderon-Zygmund算子(CZO)的加权范数不等式及其在拟共形映射和Whitney扩张问题中的应用。在理解描述我们物理世界的几乎所有偏微分方程的最关键情况时，从量子力学中的薛定谔算符到流体流动中的纳维斯托克斯方程，CZO似乎是一种自然的工具。研究这些算子的两个加权范数不等式扩展了我们的应用，并进一步加深了我们对所考虑的单个算子的理解，在没有这项研究的情况下，这些方式仍然是完全模糊的。更具体地说，申请者和合作者最近取得了重要的成功，他们在一篇由两部分组成的论文(申请者是第一部分的合著者)中证明了著名的Nazarov-Treil-Volberg T1猜想，该猜想针对30多年前的希尔伯特变换(最简单的非平凡CZO)的两个权重问题。这个1D问题的解决为(在应用中更灵活的)本地TB版本和更高维度版本打开了大门。这些版本中缺少解决1D案例的关键工具，为进一步研究留下了足够的空间，其中一些研究将在学生中进行。最近证明加权范数不等式至关重要的另一个成功是申请人和合著者解决了16年来Astala关于平面拟共形映射下的(Hausdorff测度)集的扭曲的猜想。这些地图是很好的弹性模型。阿斯塔拉已经了解了这些地图如何扭曲较小维度的面积和平面集合(例如1D)。但相应的较小度量(例如长度)的扭曲的精细性质仍然是一个谜，这一点通过上述类型的加权范数不等式得到了解决。不幸的是，对于申请人提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来适应3D中的攻击计划。申请者和合作者对加权范数不等式的研究在Whitney扩张问题中产生了意想不到的应用。这些经典的“将函数与数据相匹配”的问题是，一个仅在一个小集合上定义的先验函数何时可以被视为(扩展的)整个空间的定义(具有良好的行为)。这一推广是许多相关问题的关键一步，以便能够应用现代工具来分析控制物理现象的偏微分方程组。虽然理解了Sobolev空间的Whitney延拓问题(这些是现代偏微分方程方法的自然背景)，但事实证明，申请者和合作者关于加权范数不等式的研究的某些工具似乎非常有希望和相关。申请人建议继续这一研究方向。

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.