Polar Coordinates in Carnot groups II

TL;DR

This paper establishes a characterization of polarizable Carnot groups by showing that the existence of a horizontal polar coordinate system implies the group is polarizable, linking geometric structure to analytical properties.

Contribution

It proves the converse that a Carnot group with a suitable horizontal polar coordinate system must be polarizable, completing the characterization.

Findings

Horizontal polar coordinates imply polarizability.

Polarizable groups admit fundamental solutions for all p.

Characterization of Carnot groups via coordinate systems.

Abstract

A Carnot group is polarizable if it carries a homogeneous norm whose powers are fundamental solutions for the $p$-sub-Laplacian operators for all $1<p \le \infty$. Such groups also support a system of horizontal polar coordinates. We prove that the converse statement is true: if a Carnot group supports a horizontal polar coordinate system with suitable properties, then it is polarizable.