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主要研究者：陆国珍摘要：本论文将研究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，以及经典调和分析和偏微分方程之间的许多交叉问题。主要研究内容包括：（1）Heisenberg群和分层群中的多参数哈代空间，（2）Heisenberg群上的CR-Yamabe问题和具有最优常数的Sobolev不等式，包括正解的分类和对称性问题;（3）Heisenberg群和分层群上的各向异性Sobolev空间及其对度量空间的推广（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）涵盖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许多当前感兴趣的主题，他们中的大多数属于广泛的保护伞下的“谐波分析”和平易近人使用现代变体的经典工具的主题。偏微分方程的定量和定性研究是该项目的主要重点之一，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、工程学和经济学中有着重要的应用。特别是，它们应用于控制理论和动态系统（例如，物体在重力和其他力的影响下在空间中的运动）。通过这里提出的研究所获得的结果可以帮助发现新的性质的偏微分方程的解决方案，最终导致更好地理解这些方程相关联的物理过程。

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.