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

Let b be the Kohn Laplacian acting on (0; j)-forms on the unit sphere in Cn. In a recent paper of Casarino, Cowling, Sikora and the author, a
spectral multiplier theorem of Mihlin{Hormander type for b is proved in the case 0 < j < n &#x100000; 1. Here we prove an analogous theorem in the exceptional
cases j = 0 and j = n &#x100000; 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.