  We develop a functional analytic approach to the study of the Kramers and
kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of
uniformly elliptic equations. In particular, we identify a function space
analogous to $H^1$ and develop a well-posedness theory for weak solutions in
this space. In the case of a conservative force, we identify the weak solution
as the minimizer of a uniformly convex functional. We prove new functional
inequalities of Poincar\'e and H\"ormander type and combine them with basic
energy estimates (analogous to the Caccioppoli inequality) in an iteration
procedure to obtain the $C^\infty$ regularity of weak solutions. We also use
the Poincar\'e-type inequality to give an elementary proof of the exponential
convergence to equilibrium for solutions of the kinetic Fokker-Planck equation
which mirrors the classic dissipative estimate for the heat equation. Finally,
we prove enhanced dissipation in a weakly collisional limit.
