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

建议：DMS-9970352首席研究员：陆国珍摘要：将对海森堡群和分层群上的调和分析和次椭圆偏微分方程的各种问题进行研究，以及经典调和分析和偏微分方程组的交集上的许多问题。要研究的主要问题包括：(1)Heisenberg群和分层群内的多参数Hardy空间；(2)关于Heisenberg群和具有最优常数的Soblev不等式的CR-Yamabe问题，包括正解和对称性问题的分类；(3)Heisenberg群和分层群上的非各向同性Sobolev空间及其到度量空间的推广(度量空间上的高阶Sobolev空间是在非欧氏环境下进行位势理论的基础，如黎曼流形和分数形)；(4)薛定谔算子的Calderon-Zygmund估计及其对非线性偏微分方程的应用；(5)介绍引理及其在偏微分方程式和几何分析中的应用。本课程的专题涵盖许多当前感兴趣的课题，其中大部分属于“调和分析”这一宽泛的范畴，并且可以使用该学科经典工具的现代变体来接近。偏微分方程的定量和定性研究是该项目的主要重点之一，它构成了物理世界数学建模的很大一部分。这类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，它们在控制理论和动力系统(例如，物体在引力和其他力的影响下在空间中的运动)中有应用。通过本文提出的研究结果可以帮助揭示偏微分方程解的新性质，最终导致对这些方程所关联的物理过程的更好理解。

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.