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

TL;DR

This paper establishes a spectral multiplier theorem for the Kohn Laplacian on the sphere in complex space, using representation theory and analysis of differential forms, advancing understanding of harmonic analysis on CR manifolds.

Contribution

It proves a Hörmander spectral multiplier theorem for the Kohn Laplacian with a critical index, employing representation theory and form space analysis.

Findings

Proved spectral multiplier theorem for the Kohn Laplacian on the sphere.

Identified critical index as half the topological dimension.

Utilized representation theory and differential form analysis.

Abstract

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a H\"ormander spectral multiplier theorem for $\Box_b$ 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 $\mathbb{S}$.