SPECTRAL MULTIPLIERS ON LIE-GROUPS OF POLYNOMIAL-GROWTH

Let L be a left invariant sub-Laplacian on a connected Lie group G of polynomial volume growth, and let {E(lambda), lambda greater-than-or-e qual-to 0} be the spectral resolution of L and m a bounded Borel measu rable function on [0, infinity) . In this article we give a sufficient condition on m for the operator m(L) = integral-infinity/0 m(lambda) dE(lambda) to extend to an operator bounded on L(p)(G), 1 < p < infini ty, and also from L1(G) to weak-L1(G).