Spectral multipliers on Damek--Ricci spaces

Let S be a Damek-Ricci space, and Δ be a distinguished Laplacean on S which is left invariant and selfadjoint in L²(ρ). We prove that S is a Calderón-Zygmund space with respect to the right Haar measure ρ and the left invariant distance. We give sufficient conditions of Hörmander type on a multiplier m so that the operator m(Δ) is bounded on L^p(ρ) when 1 < p < ∞, and of weak type (1, 1). © 2007 Heldermann Verlag