Uniform rectifiability, Singular Integrals and Harmonic Measure

研究者计划研究几个与局部TB定理的共同主题相关的问题，以及奇异积分估计、泊松核估计、平方函数估计和边界正则性之间的相互作用。本文的主要研究方向包括：1.研究了层电位的有界性、调和测度性质和一致整流性之间的关系。2.研究一致可整集的结构。3.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。4.发展研究复椭圆型方程边值问题的可解性的方法，或更一般地，研究具有有界可测系数的强椭圆型方程组的边值问题的可解性。该项目属于调和分析及其在几何测度论和椭圆型偏微分方程组理论中的应用及其相互作用的领域。粗略地说，在调和分析中，人们通过将它们分解成更小的、更容易理解的组成部分，然后重新组装，来研究函数和“运算符”(即，将一个函数转换为另一个函数的映射)的性质。这个名字本身源于对音乐声音分解成各种频率成分(“谐音”)的类比。几何测度论涉及研究集合的几何性质和它们的“度量”(后者是长度、面积和体积概念的推广)之间的关系。椭圆型偏微分方程组描述了现实世界中各种各样的现象，包括静电学、稳态温度分布和弹性变形。本提案的一个特别重点是进一步探讨几何和“调和度量”性质之间的关系。后者的研究领域已经在最近的声学工程工作中找到了应用，特别是在具有理想声学特性的房间的设计中。在将要审议的问题上取得进展，很可能会开辟进一步的调查途径。调查员将通过在会议、研讨会和研究生课程上的讲座以及在他的网站和Arxiv网站上张贴的电子预印本来传播所有这些进展。研究人员将邀请博士生和一名博士后研究与拟议工作相关的问题。两名前博士后已经参与了这个项目。

The investigator plans to investigate several questions linked by the common themes of local Tb Theorems, and the interplay between singular integral estimates, Poisson kernel estimates, square function estimates, and the regularity of boundaries. Among the main directions of the proposed research are: 1. To investigate the relationships among boundedness of layer potentials, properties of harmonic measure, and uniform rectifiability. 2. To investigate the structure of uniformly rectifiable sets. 3. To develop and apply "local" Tb theorems to study the regularity of free boundaries and the solvability of elliptic boundary value problems. 4. 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.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. 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 phenoma in the real world, including electrostatics, and steady-state temperature distributions and elastic deformations. A particular focus of the present proposal is to explore further the relationship between geometry and properties of "harmonic measure." The latter area of investigation has already found application in recent work in acoustical engineering, in particular in the design of a room with desirable acoustic properties. Progress on the problems to be considered would in all likelihood open up further avenues of investigation. All such progress will be disseminated by the investigator via lectures at conferences, seminars and graduate courses, and via electronic preprints posted on his website and on the ArXiv web site. The investigator will involve Ph.D. students and a postdoc on problems related to the proposed work. Two former postdocs are already involved in the project.