  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.
