Global Hypoellipticity and Compactness of Resolvent for Fokker-Planck Operator

In this paper we study the Fokker-Planck operator with potential V(x), and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates. As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of H\'erau-Nier and Helffer-Nier, by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.Comment: Lemma 4.6 is correcte

Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff

In this article we provide global subelliptic estimates for the linearized inhomogeneous Boltzmann equation without angular cutoff, and show that some global gain in the spatial direction is available although the corresponding operator is not elliptic in this direction. The proof is based on a multiplier method and the so-called Wick quantization, together with a careful analysis of the symbolic properties of the Weyl symbol of the Boltzmann collision operator

Global Gevrey hypoellipticity for the twisted Laplacian on forms

We study in this paper the global hypoellipticity property in the Gevrey category for the generalized twisted Laplacian on forms. Different from the 0-form case, where the twisted Laplacian is a scalar operator, this is a system of differential operators when acting on forms, each component operator being elliptic locally and degenerate globally. We obtain here the global hypoellipticity in anisotropic Gevrey space