Let G be a homogeneous Lie group with a left Haar measure dg and L the action of G as left translations on Lp(G; dg). Further, let H = dL(C) denote a homogeneous operator associated with L. If H is positive and hypoelliptic on L2 we prove that it is closed on each of the Lp-spaces, p e (1, 8), 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 Lp-spaces, p e [1, 8]. Further extensions of these results to nonhomogeneous operators and general representations are also given.
|Publication status||Published - 1994|