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

TL;DR

This paper extends spectral multiplier theorems for the Kohn Laplacian on spheres to the boundary cases, establishing sharp results and endpoint estimates using an abstract multivariate multiplier framework.

Contribution

It proves a sharp spectral multiplier theorem for the Kohn Laplacian on spheres in the cases j=0 and j=n-1, including endpoint estimates, complementing previous results.

Findings

Established sharp spectral multiplier theorems for boundary cases j=0 and j=n-1.

Included weak type (1,1) endpoint estimates.

Demonstrated the sharpness of the theorems.

Abstract

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.