This project concerns two main topics: harmonic analysis on noncompact semisimple Lie groups and symmetric spaces, and unique continuation problems and absence of positive eigenvalues of Schrodinger operators on Euclidean spaces. The author's goal is to develop real-variable methods that could be used to study boundedness properties of certain natural operators on noncompact semisimple Lie groups and symmetric spaces. Some of the problems to be investigated are the following: sharp estimates related to the Kunze-Stein phenomenon, behaviour of the solution of the wave equation on symmetric spaces at large time, singular integral operators on symmetric spaces, transference principles for operators defined by Fourier multipliers on symmetric spaces, and maximal operators and applications in ergodic theory. The author made progress on these problems in certain special cases, mostly on Lie groups and symmetric spaces of real rank one. The second part of the project is aimed at understanding the absence of positive eigenvalues for Schrodinger operators with potentials in appropriate Lebesgue spaces and with certain decay properties at infinity.
The problem of eliminating the possibility of positive eigenvalues for the Schrodinger operator associated to one or many particles comes from mathematical physics. One expects on physical grounds that such positive energy "bound states" cannot exist. This is indeed the case for a large class of Schrodinger operators associated to potentials which satisfy certain 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 been growing interest in extending these classical results to more general potentials in various Lebesgue spaces.
