Real-variable methods on symmetric spaces and Schrodinger operators

摘要：本项目主要涉及两个主题：非紧半单李群和对称空间上的调和分析，以及欧氏空间上薛定谔算子的唯一连续性问题和正本征值的缺失。作者的目标是发展可用于研究某些自然算子在非紧半单李群和对称空间上的有界性的实变量方法。需要研究的问题有：与昆泽-斯坦现象有关的精确估计，对称空间上波动方程解的大时间性态，对称空间上的奇异积分算子，对称空间上由傅立叶乘子定义的算子的转移原理，以及极大算子及其在遍历理论中的应用。在某些特殊情况下，主要是在李群和实一阶对称空间上，作者在这些问题上取得了进展。该项目的第二部分旨在了解薛定谔算子在适当的勒贝格空间中具有势并且在无穷远处具有一定的衰变性质时不存在正本征值。消除与一个或多个粒子相关的薛定谔算子的正本征值的可能性的问题来自数学物理。在物理基础上，人们认为这种正能量“束缚态”是不可能存在的。这确实是一大类与满足某些均匀衰减条件的势有关的薛定谔算符的情况，例如库仑势。T.Kato，S.Agmon，J.Weidman，B.Simon等人在60‘S中研究了这些势。在过去的二十年里，人们越来越有兴趣将这些经典结果推广到各种勒贝格空间中的更一般的位势。

ABSTRACT: This project concerns two main topics: harmonic analysis onnoncompact semisimple Lie groups and symmetric spaces, and uniquecontinuation problems and absence of positive eigenvalues of Schrodingeroperators on Euclidean spaces. The author's goal is to developreal-variable methods that could be used to study boundedness propertiesof certain natural operators on noncompact semisimple Lie groups andsymmetric spaces. Some of the problems to be investigated are thefollowing: sharp estimates related to the Kunze-Stein phenomenon,behaviour of the solution of the wave equation on symmetric spaces atlarge time, singular integral operators on symmetric spaces, transferenceprinciples for operators defined by Fourier multipliers on symmetricspaces, and maximal operators and applications in ergodic theory. Theauthor made progress on these problems in certain special cases, mostlyon Lie groups and symmetric spaces of real rank one. The second part ofthe project is aimed at understanding the absence of positive eigenvaluesfor Schrodinger operators with potentials in appropriate Lebesgue spacesand with certain decay properties at infinity.The problem of eliminating the possibility of positive eigenvalues for theSchrodinger operator associated to one or many particles comes frommathematical physics. One expects on physical grounds that such positiveenergy "bound states" cannot exist. This is indeed the case for a largeclass of Schrodinger operators associated to potentials which satisfycertain uniform decay conditions, such as the Coulomb potential.These potentials were studied in the 60's by T. Kato, S. Agmon, J.Weidman, B. Simon, and others. In the last twenty years there has beengrowing interest in extending these classical results to more generalpotentials in various Lebesgue spaces.