Multiparameter Hardy Space, CR-Yamabe Problems and Nonisotropic Sobolev spaces on the Heisenberg and stratified groups, L^p estimates, unique continuation and covering lemmas

摘要：将研究海森堡和分层群上有关调和分析和亚椭圆型偏微分方程的各种问题，以及经典调和分析和偏微分方程交叉的许多问题。主要研究的课题有：(1)Heisenberg群和分层群内的多参数Hardy空间；(2)具有最优常数的Heisenberg群和Sobolev不等式的CR-Yamabe问题，包括正解的分类和对称性问题；(3)海森堡群和分层群上的非各向同性Sobolev空间及其对度量空间的推广（度量空间上的高阶Sobolev空间是在黎曼流形和分形等非欧几里德环境中进行势理论的基础）；(4) Schrodinger算子的Calderon-Zygmund估计及其在非线性偏微分方程中的应用；(5)介绍引理及其在偏微分方程和几何分析中的应用。这里进行的项目涵盖了当前感兴趣的许多主题，其中大多数都属于“谐波分析”的广泛保护伞之下，并且可以使用该主题的经典工具的现代变体。偏微分方程的定量和定性研究是该项目的主要焦点之一，它构成了物理世界数学建模的很大一部分。这些方程的解在物理科学、生命科学、工程和经济学中有着重要的应用。特别是，它们可以应用于控制理论和动力系统（例如，物体在重力和其他力影响下的空间运动）。通过这里提出的研究获得的结果可以帮助揭示偏微分方程解的新性质，最终导致更好地理解与这些方程相关的物理过程。

Proposal: DMS-9970352Principal Investigator: Guozhen LuAbstract: Research will be conducted on a variety of questions concerning harmonic analysis and subelliptic partial differential equations on the Heisenberg and stratified groups, and numerous questions lying in the intersection between classical harmonic analysis and partial differential equations. Among the principal topics to be investigated are: (1) multiparameter Hardy spaces within the Heisenberg group and stratified groups; (2) CR-Yamabe problems on the Heisenberg group and Sobolev inequalities with optimal constants, including the classification of positive solutions and symmetry problems; (3) nonisotropic Sobolev spaces on the Heisenberg group and stratified groups and their generalizations to metric spaces (high order Sobolev spaces on metric spaces are fundamental to doing potential theory in non-Euclidean settings such as Riemannian manifolds and fractals); (4) Calderon-Zygmund estimates for Schrodinger operators and their applications to nonlinear partial differential equations; (5) covering lemmas and their applications to partial differential equations and geometric analysis.Projects conducted here cover many topics of current interest, most of them falling under the broad umbrella of "harmonic analysis" and approachable using modern variants of the classical tools of that subject. The quantitative and qualitative study of partial differential equations, one of the principal focal points of the project, constitutes a large part of the mathematical modeling of the physical world. Solutions of such equations have important applications in the physical sciences, the life sciences, engineering, and economics. In particular, they have applications to control theory and dynamical systems (e.g., the motion of objects in space under the influence of gravitational and other forces). Results obtained through the research proposed here could help to uncover new properties of solutions of partial differential equations, ultimately leading to a better understanding of the physical processes with which these equations are associated.