We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = Sigma(i=1)(m) (XiXi)-X-T + 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 epsilon > 0 we show that an inequality of the form parallel touparallel to(delta,delta) < C(parallel touparallel to(0,epsilon) + parallel to(K +iy)uparallel to0,0) holds for suitable delta and C which are independent of y is an element of 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 greater than or equal to \y(tau) - c, tau is an element of (0,1], c is an element of R}.