A Remark on the Kernel of the CR Paneitz operator

TL;DR

This paper investigates conditions under which the kernel of the CR Paneitz operator in three-dimensional CR structures includes additional functions beyond the CR pluriharmonic functions, providing insights into the operator's kernel structure.

Contribution

It offers new criteria for the kernel of the CR Paneitz operator to contain a supplementary space, advancing understanding of CR geometric analysis.

Findings

Identifies conditions for the kernel to include extra functions

Provides criteria for the kernel structure of the CR Paneitz operator

Enhances understanding of CR pluriharmonic functions and their relation to the Paneitz operator

Abstract

For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kernel of the CR Paneitz operator contains a supplementary space to the CR pluriharmonic functions.