  In this paper, we prove a new functional inequality of Hardy-Littlewood type
for generalized rearrangements of functions. We then show how this inequality
provides {\em quantitative} stability results of steady states to evolution
systems that essentially preserve the rearrangements and some suitable energy
functional, under minimal regularity assumptions on the perturbations. In
particular, this inequality yields a {\em quantitative} stability result of a
large class of steady state solutions to the Vlasov-Poisson systems, and more
precisely we derive a quantitative control of the $L^1$ norm of the
perturbation by the relative Hamiltonian (the energy functional) and
rearrangements. A general non linear stability result has been obtained in
\cite{LMR} in the gravitational context, however the proof relied in a crucial
way on compactness arguments which by construction provides no quantitative
control of the perturbation. Our functional inequality is also applied to the
context of 2D-Euler system and also provides quantitative stability results of
a large class of steady-states to this system in a natural energy space.
