Another Approach to Juhl's Conformally Covariant Differential Operators from $S^n$ to $S^{n-1}$

TL;DR

This paper constructs a new family of conformally covariant differential operators on spheres, extending Juhl's operators from $S^n$ to $S^{n-1}$ by composition and restriction, enhancing understanding of conformal geometry.

Contribution

It introduces a novel family of differential operators on spheres that generalize Juhl's operators through composition and restriction, providing new tools in conformal geometry.

Findings

Constructed a family of differential operators on $S^n$

Operators are conformally covariant under subgroup actions

Recovered Juhl's operators via composition and restriction

Abstract

A family $({\mathbf D}_\lambda)_{\lambda\in \mathbb C}$ of differential operators on the sphere $S^n$ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of $S^n$ which preserve the smaller sphere $S^{n-1}\subset S^n$. The family of conformally covariant differential operators from $S^n$ to $S^{n-1}$ introduced by A. Juhl is obtained by composing these operators on $S^n$ and taking restrictions to $S^{n-1}$.