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On some intrinsic differentiability properties for Absolutely continuous functions between Carnot groups and the Area formula

Andrea Pinamonti, Francesco Serra Cassano, Kilian Zambanini

TL;DR

This work develops a robust intrinsic regularity theory for maps between Carnot groups, centering on $Q$-absolutely continuous ($AC^Q$) maps and their relation to bounded $Q$-variation ($BV^Q$). It extends Stein’s sharp Lorentz-space gradient condition to the Carnot setting by leveraging $L^{Q,1}$ bounds on horizontal sections, yielding continuity, a.e. Pansu differentiability, and an area formula for $AC^Q$ maps. The authors also connect Orlicz–Sobolev spaces with $AC^Q$ regularity, provide a detailed area-change formula, and compare two foundational approaches to metric-space–valued Sobolev maps (embedding vs. sections). Collectively, these results advance the understanding of Sobolev-type regularity and geometric measure properties for maps in sub-Riemannian geometry, with implications for sub-Laplacian problems and geometric analysis on Carnot groups.

Abstract

We discuss Q-absolutely continuous functions between Carnot groups, following Maly's definition for maps of several variables. Such maps enjoy nice regularity properties, like continuity, Pansu differentiability a.e., weak differentiability and an Area formula. Furthermore, we extend Stein's result concerning the sharp condition for continuity and differentiability a.e. of a Sobolev map in terms of the integrability of the weak gradient: more precisely, we prove that a Sobolev map between Carnot groups with horizontal gradient of its sections uniformly bounded in L(Q,1) admits a representative which is Q-absolutely continuous.

On some intrinsic differentiability properties for Absolutely continuous functions between Carnot groups and the Area formula

TL;DR

This work develops a robust intrinsic regularity theory for maps between Carnot groups, centering on -absolutely continuous () maps and their relation to bounded -variation (). It extends Stein’s sharp Lorentz-space gradient condition to the Carnot setting by leveraging bounds on horizontal sections, yielding continuity, a.e. Pansu differentiability, and an area formula for maps. The authors also connect Orlicz–Sobolev spaces with regularity, provide a detailed area-change formula, and compare two foundational approaches to metric-space–valued Sobolev maps (embedding vs. sections). Collectively, these results advance the understanding of Sobolev-type regularity and geometric measure properties for maps in sub-Riemannian geometry, with implications for sub-Laplacian problems and geometric analysis on Carnot groups.

Abstract

We discuss Q-absolutely continuous functions between Carnot groups, following Maly's definition for maps of several variables. Such maps enjoy nice regularity properties, like continuity, Pansu differentiability a.e., weak differentiability and an Area formula. Furthermore, we extend Stein's result concerning the sharp condition for continuity and differentiability a.e. of a Sobolev map in terms of the integrability of the weak gradient: more precisely, we prove that a Sobolev map between Carnot groups with horizontal gradient of its sections uniformly bounded in L(Q,1) admits a representative which is Q-absolutely continuous.
Paper Structure (16 sections, 35 theorems, 183 equations)

This paper contains 16 sections, 35 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Carnot groups
  4. Differential Calculus on Carnot Groups
  5. Lorentz space
  6. Sobolev spaces on Carnot groups.
  7. Sobolev functions from a Carnot group to $\mathbb{R}$
  8. Sobolev functions between Carnot groups.
  9. Absolutely continuous functions between Carnot groups and some properties
  10. Sobolev maps between Carnot groups with horizontal gradient of sections in $L^{Q,1}$
  11. Area formula and change of variables for absolutely continuous functions between Carnot groups
  12. Area formula between Carnot groups
  13. Appendix: Orlicz-Sobolev spaces on Carnot groups
  14. Appendix: Differentiation for maps between metric spaces and metric space-valued Sobolev maps
  15. Differentiation for maps between metric spaces:
  16. ...and 1 more sections

Key Result

Proposition 2.1

The group product has the form where $\mathcal{Q}=(\mathcal{Q}_1,\dots,\mathcal{Q}_n):\mathbb R^n\times\mathbb R^n\to\mathbb R^n$ and each $\mathcal{Q}_i$ is a homogeneous polynomial of degree $d(i)$ i.e. Moreover $\mathcal{Q}$ is anti-symmetric, which means and also

Theorems & Definitions (77)

  • Definition 1.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Pansu pansu
  • Theorem 2.7: Stepanov's theorem for Carnot groups
  • Definition 2.8
  • Lemma 2.9
  • ...and 67 more