On positive Rockland operators

Let G be a homogeneous Lie group with a left Haar measure dg and L the action of G as left translations on ${\displaystyle L_p(G;dg)}$. Further, let H = dL(C) denote a homogeneous operator associated with L. If H is positive and hypoelliptic on ${\displaystyle L_2}$ we prove that it is closed on each of the ${\displaystyle L_p}$-spaces, p ∈ 〈 1,∞〉, and that it generates a semigroup S with a smooth kernel K which, with its derivatives, satisfies Gaussian bounds. The semigroup is holomorphic in the open right half-plane on all the ${\displaystyle L_p}$-spaces, p ∈ [1,∞]. Further extensions of these results to nonhomogeneous operators and general representations are also given.