  We prove that the noncommutative Lorentz norm (associated to a semifinite von
Neumann algebra) of a propagator of the form $\varphi(|\mathscr{L}|)$ can be
estimated if the Borel function $\varphi$ is bounded by a positive
monotonically decreasing vanishing at infinity continuous function $\psi$. As a
consequence we obtain the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm
estimates for the solutions of heat, wave and Schr\"odinger type equations (new
in this setting) on a locally compact separable unimodular group $G$ by using a
non-local integro-differential operator in time and any positive left invariant
operator (maybe unbounded and either with discrete or continuous spectrum) on
$G$. We also provide asymptotic estimates (large-time behavior) for the
solutions which in some cases can be claimed to be sharp. Illustrative examples
are given for several groups.
