  We study hypoelliptic operators with polynomially bounded coefficients that
are of the form K = sum_{i=1}^m X_i^T X_i + X_0 + f, where the X_j denote first
order differential operators, f is a function with at most polynomial growth,
and X_i^T denotes the formal adjoint of X_i in L^2. For any e > 0 we show that
an inequality of the form |u|_{delta,delta} <= C(|u|_{0,eps} + |(K+iy)u|_{0,0})
holds for suitable delta and C which are independent of y in R, in weighted
Sobolev spaces (the first index is the derivative, and the second the growth).
We apply this result to the Fokker-Planck operator for an anharmonic chain of
oscillators coupled to two heat baths. Using a method of Herau and Nier [HN02],
we conclude that its spectrum lies in a cusp {x+iy|x >= |y|^tau-c, tau in
(0,1], c in R}.
