Maximal estimates for the Kramers-Fokker-Planck operator with electromagnetic field

TL;DR

This paper establishes maximal estimates for the Kramers-Fokker-Planck operator with electromagnetic fields on R^d, using nilpotent Lie algebra techniques to characterize its domain and resolvent properties.

Contribution

It introduces a nilpotent Lie algebra approach to derive maximal estimates for the KFP operator with electromagnetic fields, extending previous work on the subject.

Findings

Maximal estimate for the KFP operator with electromagnetic field

Optimal domain characterization of the operator

Criterion for resolvent compactness

Abstract

In continuation of a former work by the first author with F. Nier (2009) and of a more recent work by the second author on the torus (2019), we consider the Kramers-Fokker-Planck operator (KFP) with an external electromagnetic field on R d. We show a maximal type estimate on this operator using a nilpotent approach for vector field polynomial operators and induced representations of a nilpotent graded Lie algebra. This estimate leads to an optimal characterization of the domain of the closure of the KFP operator and a criterion for the compactness of the resolvent.