Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres

Abstract: Let $\Box_b$ be the Kohn Laplacian acting on $(0,j)$-forms on the unit sphere in $\mathbb{C}^n$. In a 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.