  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 our previous work), they yield smoothing estimates for classes
of functions of the operators .
  A important particular case is the derivation of global spacetime estimates
for a perturbed operator $H+V$ on the basis of its comparison with the
unperturbed operator $H.$
  A number of applications are given, including smoothing estimates for
fractional Laplacians, Stark Hamiltonians and Schr\"odinger operators with
potentials.
