Global subelliptic estimates for geometric Kramers-Fokker-Planck operators on closed manifolds

Abstract: In this article we reconsider the proof of subelliptic estimates for Geometric Kramers-Fokker-Planck operators, a class which includes Bismut's hypoelliptic Laplacian, when the base manifold is closed (no boundary). The method is significantly different from the ones proposed by Bismut-Lebeau in [BiLe] and Lebeau in [Leb1] and [Leb2]. As a new result we are able to prove maximal subelliptic estimates with some control of the constants in the two asymptotic regimes of high (b $\rightarrow$ 0) and low (b $\rightarrow$ +$\infty$) friction. After a dyadic partition in the momentum variable, the analysis is essentially local in the position variable, contrary to the microlocal reduction techniques of the previous works. In particular this method will be easier to adapt on manifolds with boundaries. A byproduct of our analysis is the introduction of a very convenient double exponent Sobolev scale associated with globally defined differential operators. Applications of this convenient parameter dependent functional analysis to accurate spectral problems, in particular for Bismut's hypoelliptic Laplacian with all its specific geometry, is deferred to subsequent works.