Spectral multipliers for the Kohn Laplacian on forms on the sphere in ${\mathbb{C}}^n$

Abstract

The unit sphere $\mathbb{S}$ in ${\mathbb{C}}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian ${\square }_b$. We prove a Hörmander spectral multiplier theorem for ${\square }_b$ with critical index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\mathbb{S}$.

28 pages