On the l² l^∞ norms of spectral multipliers of "quasi-homogeneous" operators on homogeneous groups

We study the L² → L^∞ norms of spectral projectors and spectral multipliers of left-invariant elliptic and subelliptic second-order differential operators on homogeneous Lie groups. We obtain a precise description of the L² → L^∞norms of spectral multipliers for some class of operators which we call quasi-homogeneous. As an application we prove a stronger version of Alexopoulos' spectral multiplier theorem for this class of operators.