On the L-2 -> L-infinity norms of spectral multipliers of "quasi-homogeneous" operators on homogeneous groups

We study the L-2 --> L-infinity norms of spectral projectors and spectral m ultipliers of left-invariant elliptic and subelliptic second-order differen tial operators on homogeneous Lie groups. We obtain a precise description o f the L-2 --> L-infinity norms of spectral multipliers for some class of op erators which we call quasi-homogeneous. As an application we prove a stron ger version of Alexopoulos' spectral multiplier theorem for this class of o perators.