  Let $\Box_b$ be the Kohn Laplacian acting on $(0,j)$-forms on the unit sphere
in $\mathbb{C}^n$. In a recent paper of Casarino, Cowling, Sikora and the
author, a spectral multiplier theorem of Mihlin--H\"ormander type for $\Box_b$
is proved in the case $0<j<n-1$. Here we prove an analogous theorem in the
exceptional cases $j=0$ and $j=n-1$, including a weak type $(1,1)$ endpoint
estimate. We also show that both theorems are sharp. The proof hinges on an
abstract multivariate multiplier theorem for systems of commuting operators.
