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Polar Coordinates in Carnot groups II

Jeremy T. Tyson

TL;DR

This work establishes the converse of a known result for polarizable Carnot groups: if a Carnot group admits a coherent horizontal polar coordinate system (with respect to the Folland homogeneous norm N_u) then the group is polarizable. The authors prove the equivalence of three conditions for a Carnot group of homogeneous dimension Q>2: L_∞ applied to N_u vanishes outside the origin, the existence of a coherent family of singular p-Laplacian solutions for all p∈(1,∞], and the presence of horizontal polar coordinates relative to N_u. The proof of (iii)⇒(i) proceeds by constructing a polar-flow, deriving a divergence-form equation, and exploiting the 2-harmonicity of N_u^{2−Q} to deduce L_∞ N_u = 0, thereby confirming polarizability. The paper also situates these results within the broader framework of rectifiable polar coordinates in metric measure spaces and discusses how horizontal polar coordinates in polarizable groups can be reconciled with rectifiable polar-coordinate theory, including explicit implications for the Heisenberg group and related H-type structures.

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.

Polar Coordinates in Carnot groups II

TL;DR

This work establishes the converse of a known result for polarizable Carnot groups: if a Carnot group admits a coherent horizontal polar coordinate system (with respect to the Folland homogeneous norm N_u) then the group is polarizable. The authors prove the equivalence of three conditions for a Carnot group of homogeneous dimension Q>2: L_∞ applied to N_u vanishes outside the origin, the existence of a coherent family of singular p-Laplacian solutions for all p∈(1,∞], and the presence of horizontal polar coordinates relative to N_u. The proof of (iii)⇒(i) proceeds by constructing a polar-flow, deriving a divergence-form equation, and exploiting the 2-harmonicity of N_u^{2−Q} to deduce L_∞ N_u = 0, thereby confirming polarizability. The paper also situates these results within the broader framework of rectifiable polar coordinates in metric measure spaces and discusses how horizontal polar coordinates in polarizable groups can be reconciled with rectifiable polar-coordinate theory, including explicit implications for the Heisenberg group and related H-type structures.

Abstract

A Carnot group is polarizable if it carries a homogeneous norm whose powers are fundamental solutions for the -sub-Laplacian operators for all . 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.
Paper Structure (10 sections, 5 theorems, 59 equations)

This paper contains 10 sections, 5 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Background, notation and definitions
  4. Nonlinear potential theory in Carnot groups
  5. Polar coordinates in Carnot groups
  6. Polarizable groups
  7. Proof of Theorem \ref{['th:main']}
  8. Concluding remarks
  9. Rectifiable polar coordinates in metric measure spaces
  10. Rectifiable vs. horizontal polar coordinates in polarizable Carnot groups

Key Result

Theorem 1.1

Let $\mathbb{G}$ be a Carnot group of homogeneous dimension $Q>2$. Let ${\mathbf N}_u$ denote Folland's homogeneous quasi-norm. Then the following are equivalent.

Theorems & Definitions (13)

  • Theorem 1.1
  • Example 2.1: Folland, fol:explicit
  • Definition 3.1
  • Theorem 3.2: Polarizable Carnot groups admit horizontal polar coordinates
  • Definition 3.3: Singular solutions to $p$-Laplacians
  • Definition 3.4: Horizontal polar coordinates
  • Remark 3.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 3 more