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扩展问题中的应用。当理解几乎所有描述我们物理世界的偏微分方程的最关键情况时，从量子力学中的薛定谔算子到流体流动中的Navier-Stokes方程，CZO似乎是一种天然的工具。 研究这些算子的两个权重范数不等式扩展了应用，并进一步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取得了重要的成功，在一篇由两部分组成的论文（申请人是第一部分的合著者）中证明了著名的Nazarov-Treil-Volberg T1猜想，用于希尔伯特变换（最简单的非平凡CZO）的30多年的两个权重问题。这个1D问题的解决方案打开了大门（更灵活的应用程序）本地Tb版本，和更高的维度版本。这些版本中缺少解决一维问题的关键工具，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。另一个最近的成功，其中加权范数不等式被证明是至关重要的是解决方案，由申请人和共同作者，16岁的猜想Astala关于扭曲（豪斯多夫措施）集下平面拟共形映射。这些地图是很好的弹性模型。Astala已经理解了这些地图如何扭曲较小维度（例如1D）的区域和平面集。但是，相应的较小度量（例如长度）的失真的精细性质仍然是一个谜，这是用上面描述的加权范数不等式解决的。不幸的是，对于申请人提出研究的物理相关3D情况，这些工具中的一些缺失。然而，2D中的加权不等式证明给出了足够的直观性，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扩展问题中产生了意想不到的应用。这些经典的“拟合函数的数据”的问题，问一个函数，先验只定义在一个小的集合，可以被视为（扩展）定义（良好的行为）在整个空间。这种扩展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。在理解Sobolev空间的Whitney延拓问题（这些是偏微分方程现代方法的自然设置）时，事实证明，申请人和合作者对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。

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.