ON SPECTRAL AND MULTIPLIERS FOR HEISENBERG AND RELATED GROUPS

Assume that L is a sub-Laplacian on a Heisenberg group H(n), and let m is-an-element-of L(infinitive) (R+). We prove that the operator m(L) is bounded on L(p)(H(n)) for 1<p<infinitive and of weak type (1,1), provided the multiplier m satisfies a Hormander condition with degree of smoothness greater than (1/2)dim H(n). This result is sharp with respect to the critical degree (1/2)dim H(n). Analogous results are also proved for direct products of Heisenberg and Euclidean groups.