Tb Theorems, Singular Integrals, Poisson Kernels, and Regularity of Boundaries

PI计划研究调和分析中的问题，这些问题与局部Tb定理、奇异积分估计、泊松核估计、平方函数估计和边界的正则性之间的相互作用有关。 拟研究的目标是：1）发展和应用"局部”Tb定理来研究自由边界的正则性和椭圆边值问题的可解性; 2）发展技术来研究复椭圆方程，或更一般地说，强椭圆系统的边值问题的可解性，具有有界可测系数; 3）研究层势的有界性、调和测度的性质和一致可求正性之间的关系;（4）继续发展适用于二阶散度型椭圆算子的哈代空间理论，特别是试图找到Kato平方根问题在临界指数以下的精确解.“该项目属于调和分析及其应用领域，并与几何测量理论和椭圆偏微分方程和系统的理论相互作用。 粗略地说，在调和分析中，人们研究函数和"算子”的性质（即，将一个函数转换为另一个函数的映射），方法是将它们分解为更小的、更容易理解的组成部分，然后重新组装这些部分。 这个名字本身是通过类比将音乐声音分解成各种频率成分（“谐波”）而产生的。 几何测度论涉及研究集合的几何性质与它们的“测度”（后者是长度、面积和体积概念的推广）之间的关系。 椭圆型偏微分方程和系统描述了真实的世界中的各种现象，包括静电、稳态温度分布和弹性变形。 在过去的十年中，这些不同的数学子领域之间的相互作用已经成为研究的沃土，还有许多令人兴奋的工作要做。 在有待审议的问题上取得进展，很可能会为这些领域的调查开辟进一步的途径。PI将通过会议、研讨会和研究生课程的讲座以及在其网站和ArXiv上发布的电子预印本传播所有这些进展。 PI计划让两名博士后和两名研究生参与与该项目相关的工作。

The PI plans to work on problems in harmonic analysis linked by the interplay among local Tb Theorems, singular integral estimates, Poisson kernel estimates, square function estimates, and the regularity of boundaries. The goals of the proposed research are:1) to develop and apply ``local" Tb theorems to study the regularity of free boundaries and the solvability of elliptic boundary value problems; 2) to develop techniques to study the solvability of boundary value problems for complex elliptic equations, or more generally, for strongly elliptic systems, with bounded measurable coefficients;3) to investigate the relationships among boundedness of layer potentials, properties of harmonic measure, and uniform rectifiability;4) to continue to develop the theory of Hardy spaces adapted to a second order divergence form elliptic operator; in particular, this work may be viewed as an attempt to find a sharp solution to the Kato square root problem ``below the critical exponent."The project lies within the field of harmonic analysis and its application to, and interaction with, geometric measure theory and the theory of elliptic partial differential equations and systems. Roughly speaking, in harmonic analysis one investigates properties of functions and ``operators" (i.e., mappings which transform one function into another) by decomposing them into smaller, constituent pieces, which are easier to understand, and then reassembling the pieces. The name itself arose by analogy to the decomposition of a musical sound into its various frequency components (``harmonics"). Geometric measure theory involves the study of the relationship between geometric properties of sets, and their ``measures" (the latter are generalizations of the notions of length, area, and volume). Partial differential equations and systems of elliptic type describe a wide variety of phenomena in the real world, including electrostatics, and steady-state temperature distributions and elastic deformations. In the last decade the interplay between these different subfields of mathematics has turned out to be a fertile ground for investigation, with much exciting work remaining to be done. Progress on the problems to be considered would in all likelihood open up further avenues of investigation in these areas. All such progress will be disseminated by the PI via lectures at conferences, seminars and graduate courses, and via electronic preprints posted on his website and on the ArXiv. The PI plans to involve two postdocs and two graduate students in work related to this project.