Schrödinger equations on locally symmetric spaces

Abstract

We prove dispersive and Strichartz estimates for Schrödinger equations on a class of locally symmetric spaces Γ\X, where X = G/K is a symmetric space and Γ is a torsion free discrete subgroup of G. We deal with the cases when either X has rank one or G is complex. We present Strichartz estimates applications to the well-posedness and scattering for nonlinear Schrödinger equations.1991 Mathematics Subject Classification. 35Q55, 43A85, 22E30, 35P25, 47J35, 58D25.

Wave and Klein–Gordon Equations on Certain Locally Symmetric Spaces

“…Remark 1.1. The Schrödinger equation is studied in [16] under slightly different assumptions, our well-posedness results hold also in that setting.…”

“…In the recent paper [16], the Schrödinger equation was considered on certain locally symmetric spaces. In the present paper, we study the wave and Klein-Gordon equations in the same spirit.…”

Oscillating multipliers on symmetric spaces and locally symmetric spaces

“…In this section we recall some basic facts about symmetric spaces and locally symmetric spaces we will use for the proof of our results. For details see [3,16,10,20].…”

“…Consider a cut-off function ζ ∈ C ∞ (K\G/K), as in (10), and set κ 0 j = ζκ j , T 0 j = * κ 0 j and m 0 j = H(κ 0 j ).…”

“…For the proof of the proposition above we shall make use, as in [3,20,10,8], of Kunze and Stein phenomenon. We shall give the proof of the L p -boundedness only for T ∞ α,β .…”