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

Extended Rearrangement inequalities and applications to some quantitative stability results