  We prove quantitative decay estimates of macroscopic quantities generated by
the solutions to linear transport equations driven by a general family of
Hamiltonians. The associated particle trajectories are all trapped in a compact
region of phase-space and feature a non-degenerate elliptic stagnation point.
The analysis covers a large class of Hamiltonians generated by the radially
symmetric compactly supported equilibria of the gravitational Vlasov-Poisson
system. Working in radial symmetry, our analysis features both the
1+2-dimensional case and the harder 1+1-dimensional case, where all the
particles have the same value of the modulus of angular momentum. The latter
case is also of importance in both the plasma physics case and two dimensional
incompressible fluid flows.
