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

Tb 定理、奇异积分、泊松核和边界正则性

• 批准号: 0801079
• 负责人: Steven Hofmann
• 金额: $ 27.11万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801079&HistoricalAwards=false
• 关键词: Tb; Theorems; Singular; Integrals; Poisson

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.

PI计划处理与本地结核病定理之间相互作用，奇异积分估计值，泊松内核估计值，平方函数估计以及边界规则性相关联的谐波分析问题。 拟议的研究的目标是：1）开发和应用``本地''结核病定理，以研究自由边界的规律性和椭圆边界价值问题的可溶性； 2）开发技术以研究为边界价值问题的解决性问题的解决性问题复杂的椭圆方程，或更普遍地用于强椭圆形系统，具有有界的可测量系数； 3）研究层势的界限之间的关系，谐波测量的特性和统一的重新讨论； 4）继续发展Hardy空间理论尤其是适应二阶差异的椭圆运算符；它的应用和与几何测量理论以及椭圆形偏微分方程和系统的理论相互作用。 粗略地说，在谐波分析中，一个人通过将它们分解为较小的组成部分，调查了功能的属性和``运算符''（即，将一个函数转换为另一个功能的映射），这些部分易于理解，然后重新组装零件。名称。名称。通过类比，音乐声音分解为各种频率组件（``谐波''）。 几何措施理论涉及集合几何特性之间的关系及其``措施''（后者是长度，面积和体积的概念的概括）。部分椭圆形类型的部分微分方程和系统描述了种类繁多的种类繁多现实世界中的现象，包括静电和稳态温度分布和弹性变形。在这些领域的可能性上，所有这些进度都将通过会议，研讨会和研究生课程在这些领域中揭示所有这些进度的进展。在Arxiv上。