Harmonic Analysis and Subelliptic Partial Differential Equations

提案：DMS-9970660主要研究者：F.迈克尔克里斯特摘要：这项研究将集中在调和分析，次椭圆偏微分方程和谱理论的问题。一个循环的问题有关的相互作用之间的曲率，有限型条件，勒贝格空间范数不等式，精确光滑性质的Radon-like变换，振荡，大小的图像映射，和子水平集将进行调查。这些方面的偏微分方程进行调查关注subelliptical运营商，并顺利有界域的d-酒吧诺依曼问题。长期目标包括更深入地了解全球的规律性和不规则性，以及分析hypoellipticity。将进一步研究相空间中某个度量与这些问题的可能相关性。最后，我们将探讨具有慢衰减势的Schroedinger算子的绝对连续谱的存在性，以及大于1维的广义本征函数的渐近行为。多线性分析将被应用于这个问题，空间-时间的一个尚未确定的性质的估计将被寻求。基本的物理定律往往是制定在偏微分方程。数学家的任务是分析此类方程及其解，这些方程及其解通常无法显式计算，此外还要开发在此类分析中有用的通用技术。谐波分析师，特别是，试图结合联合收割机双重时间/空间和频率分析，以深入了解的问题，如可能的奇异性的解决方案或本征函数，如何反映这种奇异性的结构的基本微分方程或线性算子，以及什么性质的运营商的影响奇异性或渐近行为。特别是，各种形式的曲率在双曲型偏微分方程和复空间几何中起着重要的作用。这个研究项目的重点是一些相互关联的基本问题，调和分析及其应用偏微分方程和谱理论，其中曲率和时间/频率分析发挥主导作用。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，并最终理解数学中的任何物理定律。

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