On the $L^2 \to L^{\infty}$ norms of spectral multipliers of ``quasi-homogeneous'' operators on homogeneous groups

We study the $L^2 \to L^{\infty}$ 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 $L^2 \to L^{\infty}$ 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.