proceedings of the american mathematical society Volume 120. Number 3, March 1994 SPECTRALMULTIPLIERS ON LIE GROUPS OF POLYNOMIAL GROWTH G. ALEXOPOULOS (Communicated by J. Marshall Ash) Abstract. Let L be a left invariant sub-Laplacian on a connected Lie group G of polynomial volume growth, and let {Ex, A > 0} be the spectral resolution of L and m a bounded Borel measurable function on [0, oo). In this article we give a sufficient condition on m for the operator m{L) = /0°° m{k) dE^ to extend to an operator bounded on LP{G), 1 < p < co , and also from L1{G) to weak-L'(G). Let G be a connected Lie group, and let us fix a left invariant Haar measure dg on G. If A is a Borel measurable subset of G, then we write \A\ = dg-xneas\xxe(A). We assume that G has polynomial volume growth, i.e., if U is a compact neighborhood of the identity element e of G, then there is a constant c > 0 such that \Un\ < cnc, n e N. Then G becomes unimodular. Furthermore, there is an integer D > 0, such that (cf. Guivarc'h [7]) \Un\~nD (n-oo). Notice that every connected nilpotent Lie group has polynomial volume growth. Let Xi, ... , Xn be left invariant vector fields on G that satisfy Hormander's condition, i.e., they generate together with their successive Lie brackets [Xjt, [Xj2,[..., Xik]...], at every point of G, the tangent space of G. To those vector fields is associated, in a canonical way, the control distance d(-, •) (cf. [22, 23], which are our basic references for results concerning the Harmonic analysis on 67). This distance is left invariant and compatible with the topol- ogy on 67. We put |jc| = d(e, x) and Br(x) = {y € 67: d(x, y) < r} , x e G, r > 0. Then, we know that there is d e N, not depending on x (cf. [15, 22]), such that (1) |2K*)|~a* (r-0), \Br(x)\^r° (r-oo). We consider the sub-Laplacian L = -(X2 + --- + X2). L is a left-invariant hypoelliptic second-order differential operator (cf. [9]). It is also positive and selfadjoint (having as domain of definition the set {/ e Received by the editors April 24, 1992 and, in revised form, July 6, 1992. 1991Mathematics Subject Classification. Primary 22E25, 22E30, 43A80. Key words and phrases. Lie group, volume growth, multiplier, sub-Laplacian, wave equation. ©1994 American Mathematical Society 973 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use