Maximal estimates for the Fokker–Planck operator with magnetic field

Zeinab Karaki

doi:10.4171/jst/369

Abstract

We consider the Fokker–Planck operator with a strong external magnetic field. We show a maximal type estimate on this operator using a nilpotent approach on vector field polynomial operators and including the notion of representation of a Lie algebra. This estimate makes it possible to give an optimal characterization of the domain of the closure of the considered operator.

Highlights

In this direction many authors have shown maximal estimates to deduce the compactness of the resolvent of the Fokker-Planck operator and resolvent estimates to address the issue of return to the equilibrium
In the proof of the main theorem, we find a class of representations of a Lie algebra G in the space S(Rk) (k ≥ 1) that have the following form: Definition 2.5
Combined with a result of RothschildStein [17] in the particular case where the order m of the operator P is even and the Lie algebra is stratified of type 1 or 2, the Helffer-Nourrigat Theorem takes the following form

Summary

Introduction and main results

We continue the study of the model case of the Fokker-Planck operator with an external magnetic field Be, started in [12], and we establish a maximal-type estimate for this model, giving a characterization of the domain of its closed extension. Fokker-Planck operator; magnetic field; Lie algebra; irreductible representation; maximal hypoellipticity. C1 > 0, there exists some C > 0 such that for all Be with Be Lipsch(Td) ≤ C1, and for all u ∈ S(Td × Rd), the operator K satisfies the following maximal estimate:.

Review of maximal hypoellipticity in the nilpotent approach

Conclusion