Real-variable methods on symmetric spaces and Schrodinger operators

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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.