Harmonic Analysis and Subelliptic Partial Differential Equations

建议：DMS-9970660首席研究员：F.Michael Christ摘要：这项研究将集中在调和分析、次椭圆型偏微分方程和谱理论中的问题。我们将研究一系列与曲率、有限类型条件、勒贝格空间范数不等式、类Radon变换的精确光滑性、振动性、映象的大小和子水平集之间的相互作用有关的问题。要研究的偏微分方程的那些方面涉及次椭圆算子，以及光滑有界域上的d-bar Neumann问题。长期目标包括更深入地理解全球的规律性和不规则性，以及解析的亚椭圆性。将进一步研究相空间中某一度量与这些问题的可能相关性。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，以及大于一维的广义本征函数的渐近行为。我们将对这个问题进行多线性分析，并对尚未确定的性质进行时空估计。基本的物理定律通常是用偏微分方程式表示的。数学家的任务是分析这样的方程及其解，这些方程和它们的解通常不能被显式计算，而且还开发出在这种分析中有用的一般技术。尤其是调和分析者，寻求结合双重时间/空间和频率分析来洞察诸如解或特征函数的可能奇点、这种奇点如何反映基本的微分方程式或线性算子的结构、以及该算子的什么性质影响奇点或渐近行为等问题。特别是，各种形式的曲率对于双曲型偏微分方程和复杂空间中的几何都起着重要的作用。本文主要研究了调和分析及其在偏微分方程组和谱理论中的应用，其中曲率分析和时频分析在其中起着主导作用。其目的是通过对几个具体问题的调查，发展基本的分析方法，这可能会导致更好地理解潜在的数学现象，并最终导致对数学

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