The Fefferman-Phong uncertainty principle for representations of Lie groups and applications

Abstract

The Fefferman-Phong uncertainty principle is a deep result concerning the order of magnitude of the bottom of the spectrum of second order pseudodifferential operators with nonnegative (Weyl) symbol. We show that a similar uncertainty principle holds for discrete series representations of connected Lie groups, where the concentration of the matrix coefficients is measured in terms of weighted $L^p$ norms, with weights in the local Muckenhoupt class $A_{\infty,{\rm loc}}$ associated with a subRiemannian left-invariant metric and a relatively invariant measure. The proof relies on a certain connection with lower bounds for left-invariant subLaplacians. As a consequence of this result (in the case of the Schrödinger representation of the reduced Heisenberg group), we provide an explicit formula for the order of magnitude of the bottom of the spectrum and also of the essential spectrum of semiclassical anti-Wick operators with nonnegative symbols of arbitrary order, hence providing the analog, for the anti-Wick quantization, of the above mentioned result by Fefferman and Phong. We consider symbols in standard symbol classes appearing in semiclassical analysis, and also in the Muckenhoupt classes (hence possibly non-smooth).