Global Lp estimates for degenerate Ornstein-Uhlenbeck operators

We consider a class of degenerate Ornstein-Uhlenbeck operators in R-N, of the kind A Sigma(p0)(i,j=1) a(ij)partial derivative(2)(xixj) + Sigma(N)(i,j=1) b(ij)x(i)partial derivative(xj) where (a(ij)), (b(ij)) are constant matrices, (a(ij)) is symmetric positive definite on R-p0 (p(0) <= N), and (b(ij)) is such that A is hypoelliptic. For this class of operators we prove global L-p estimates (1 < p < infinity) of the kind: parallel to partial derivative(2)(xixj)u parallel to(Lp(RN)) <= c {parallel to Au parallel to(Lp(RN)) + parallel to u parallel to(Lp(RN)) for i, j = 1, 2,..., p(0) and corresponding weak type (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue L-p spaces, for complete Hormander's operators Sigma X-i(2) + X-0, proved in absence of a structure of homogeneous group. We obtain the previous estimates as a byproduct of the following one, which is of interest in its own: parallel to partial derivative(2)(xixj)u parallel to(Lp(S)) <= c {parallel to Lu parallel to(Lp(S)) for any u is an element of C-0(infinity) (S), where S is the strip R-N x [-1, 1] and L is the Kolmogorov-Fokker-Planck operator A - partial derivative(t). To get this estimate we use in a crucial way the left invariance of L with respect to a Lie group structure in RN+1 and some results on singular integrals on nonhomogeneous spaces recently proved in Bramanti (Revista Matematica Iberoamericana, 2009, in press).