Spectral identities and smoothing estimates for evolution operators

Abstract

Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting. Spectral identities for the time evolution of such operators are derived, enabling results concerning “best constants” for smoothing estimates. When combined with suitable “comparison principles” (analogous to those established in [39]), they yield smoothing estimates for classes of functions of the operators.

An important contribution of this study involves the comparison of the spectral density of an operator $H$ and its perturbation $H+V.$ It entails a derivation of global spacetime estimates for $H+V$ on the basis of analogous estimates for $H.$

A number of applications are given, including smoothing estimates for fractional Laplacians, Stark Hamiltonians and Schrödinger operators with potentials.