I have just uploaded a note on the issue of “enlarging the functional space of decay estimates on semigroups”:
This is part of long-term project which started with my paper hal-00076709 (arxiv-0605688) on how to obtain relaxation estimates in larger spaces than the usual “symmetrization” space for PDEs arising in statistical physics. A paper with M.P. Gualdani and S. Mischler developing a full abstract method will soon be available.
Here is the abstract:
This note briefly presents a new method for enlarging the functional space of a “spectral-gap-like” estimate of exponential decay on a semigroup. A particular case of the method was first devised in hal-00076709 for the spatially homogeneous Boltzmann equation, and a variant was used in Ref. hal-00124876 in the same context for inelastic collisions. We present a generalized abstract version of it, a short proof of the algebraic core of the method, and a new application to the Fokker-Planck equation. More details and other applications shall be found in the work in preparation Ref.  (another application to quantum kinetic theory can be found in the work in preparation Ref. ).