Harmonic Analysis and Subelliptic Partial Differential Equations

摘要：本课题主要研究谐波分析、亚椭圆型偏微分方程和谱理论等问题。一系列与曲率、有限型条件、勒贝格空间范数不等式、类radon变换的精确平滑特性、振荡、映射图像的大小和子水平集之间的相互作用有关的问题将被研究。所研究的偏微分方程涉及亚椭圆算子和光滑有界区域的d-bar Neumann问题。长期目标包括更深入地理解全球规则性和不规则性，以及解析的半椭圆性。相空间中某个度量与这些问题的可能相关性将进一步研究。最后，讨论了具有慢衰减势的薛定谔算子的绝对连续谱的存在性，以及广义本征函数在大于1维上的渐近行为。多线性分析将应用于这个问题，并且将寻求尚未确定性质的时空估计。基本的物理定律通常用偏微分方程来表述。数学家的任务是分析这类通常不能明确计算的方程及其解，此外还要发展在这类分析中有用的一般技术。特别是谐波分析，寻求将对偶时间/空间和频率分析结合起来，以深入了解诸如解或特征函数的可能奇点，这些奇点如何反映底层微分方程或线性算子的结构，以及算子的哪些性质影响奇点或渐近行为等问题。特别是，各种形式的曲率在双曲型偏微分方程和复空间中的几何中起着重要的作用。本研究项目主要关注谐波分析及其在偏微分方程和谱理论中的应用，其中曲率和时频分析起主导作用。其目的是通过对几个具体问题的研究，发展基本的分析方法，从而更好地理解潜在的数学现象，并最终理解数学所涉及的任何物理定律

Proposal: DMS-9970660Principal Investigator: F. Michael ChristAbstract: This research will focus on problems in harmonic analysis, subelliptic partial differential equations, and spectral theory. A circle of problems related to the interplay between curvature, finite type conditions, Lebesgue-space norm inequalities, precise smoothing properties of Radon-like transforms, oscillation, sizes of images of mappings, and sublevel sets will be investigated. Those aspects of partial differential equations to be investigated concern subelliptic operators, and the d-bar Neumann problem for smoothly bounded domains. Long-term goals include a deeper understanding of global regularity and irregularity, and of analytic hypoellipticity. The possible relevance of a certain metric in phase space to these issues will be further studied. Lastly, the existence of absolutely continuous spectra for Schroedinger operators with slowly decaying potentials, and the asymptotic behavior of generalized eigenfunctions in dimensions greater than one will be explored. Multilinear analysis will be applied to this problem, and space-time estimates of an as yet undetermined nature will be sought.Fundamental physical laws are often formulated in terms of partial differential equations. The mathematician's task is to analyze such equations and their solutions, which generally cannot be computed explicitly, and moreover to develop general techniques useful in such analyses. The harmonic analyst, in particular, seeks to combine dual time/space and frequency analysis to gain insight into issues such as the possible singularities of solutions or eigenfunctions, how such singularities reflect the structure of the underlying differential equation or linear operator, and what properties of the operator influence singularities or asymptotic behavior. In particular, curvature in various guises plays an important role for hyperbolic partial differential equations, and for geometry in complex space. This research project focuses on a number of interrelated fundamental questions concerning harmonic analysis and its applications to partial differential equations and to spectral theory, in which curvature and time/frequency analysis play leading roles. The aim is to develop basic methods of analysis, through the investigation of several concrete problems, that may lead to a better understanding of underlying mathematical phenomena and, ultimately, of any physical laws that the mathematics