Real-variable methods on symmetric spaces and Schrodinger operators

摘要：本项目涉及两个主要课题：非紧半单李群和对称空间上的调和分析，以及Euclidean空间上Schrodinger算子的连续问题和不存在正特征值。作者的目标是发展可用于研究非紧半单李群和对称空间上某些自然算子的有界性的实变量方法。一些问题要调查如下：尖锐的估计有关的Kunze-斯坦现象，行为的解决方案的波动方程的对称空间在大的时间，奇异积分算子的对称空间，transferenceprinciples运营商定义的傅立叶乘子在duplex空间，最大运营商和应用遍历理论。作者在某些特殊情况下，主要是在李群和真实的秩为1的对称空间上，对这些问题进行了研究。第二部分的目的是了解薛定谔算符在适当的勒贝格空间中的势和在无穷远处的某些衰减性质不存在正本征值的问题，消除与一个或多个粒子相关联的薛定谔算符正本征值的可能性的问题来自数学物理。从物理学的角度来看，这种正能量的“束缚态”是不可能存在的。对于一类与满足某些一致衰变条件的势有关的薛定谔算符，如库仑势，情况确实如此。Kato，S. Agmon，J.魏德曼，B.西蒙和其他人。在过去的20年里，人们对将这些经典结果推广到更一般的各种Lebesgue空间的势有着浓厚的兴趣。

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.