Lp spectral theory and heat dynamics of locally symmetric spaces

In this paper we first derive several results concerning the Lp spectrum of locally symmetric spaces with rank one. In particular, we show that there is an open subset of C consisting of eigenvalues of the Lp Laplacian if p <2 and that corresponding eigenfunctions are given by certain Eisenstein series. On the other hand, if p >2 there is at most a discrete set of real eigenvalues of the Lp Laplacian. These results are used in the second part of this paper in order to show that the dynamics of the Lp heat semigroups for p <2 is very different from the dynamics of the Lp heat semigroups if p 2. © 2009 Elsevier Inc. All rights reserved