High order regularity for subelliptic operators on Lie groups of polynomial growth

Let G be a Lie group of polynomial volume growth, with Lie algebra g. Consider a second-order, right-invariant, subelliptic differential operator H on G, and the associated semigroup St = e-tH. We identify an ideal n' of g such that H satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of n'. The regularity is expressed as L2 estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in Lp, 1 < p < 8, of some associated Riesz transform operators. Finally, we show that n' is the largest ideal of g for which the regularity results hold.Various algebraic characterizations of n' are given. In particular, n' = s A n where n is the nilradical of g and s is tha largest semisimple ideal of g.Additional features of this article include an exposition of the structure theory for G in Section 2, and a concept of twisted multiplications on Lie groups which includes semidirect products in the Appendix.