[Paper Review] Fine properties of functions with bounded variation in Carnot-Carathéodory spaces

Sebastiano Don, Davide Vittone|arXiv (Cornell University)|Aug 29, 2018

Stochastic processes and financial applications3 references3 citations

TL;DR

This paper establishes fine properties of functions with bounded variation (BV) in Carnot-Carathéodory (CC) spaces, proving almost everywhere approximate X-differentiability and characterizing the structure of their distributional derivatives. Under the property $\mathcal{R}$, it shows that almost all discontinuities are of jump type and derives a representation formula for the jump part of the derivative, extending classical BV theory to sub-Riemannian settings.

ABSTRACT

We study properties of functions with bounded variation in Carnot-Ca\-ra\-théo\-do\-ry spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property $\mathcal R$, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative.

Motivation & Objective

• To establish fine differentiability and decomposition properties of BV functions in equiregular Carnot-Carathéodory spaces.
• To define and analyze the notion of approximate X-jumps for BV functions in sub-Riemannian geometry.
• To prove that under property $\mathcal{R}$, the singular part of the derivative is concentrated on jump-type discontinuities.
• To derive a representation formula for the jump part of the distributional derivative in CC spaces.
• To extend classical BV results—such as approximate differentiability and jump set structure—to the non-Euclidean setting of CC spaces.

Proposed method

• Uses a new inequality (Lemma 3.12) generalizing a classical $L^1$-type estimate for BV functions in $\mathbb{R}^n$ to CC spaces, resolving an open problem from [5].
• Introduces the concept of approximate $X$-differentiability and defines the approximate $X$-gradient via the density of the absolutely continuous part of the derivative with respect to Lebesgue measure.
• Defines $X$-jump points and the associated triple $(u^+, u^-, \nu_u)$ using intrinsic $C^1_X$-regular hypersurfaces and blow-up techniques in the nilpotent tangent cone.
• Applies a blow-up procedure via dilations $\delta_r$ and diffeomorphisms $F_p$ to analyze the asymptotic behavior of $u$ near Lebesgue points.
• Uses the homogeneous dimension $Q$ and the $\mathscr{H}^{Q-1}$-measure to characterize the size and structure of the approximate discontinuity set $\mathcal{S}_u$.
• Proves Borel measurability of the approximate gradient and jump triple by approximating characteristic functions and using weak convergence of measures.

Experimental results

Research questions

• RQ1 Can the classical result of almost everywhere approximate differentiability for BV functions be extended to functions in Carnot-Carathéodory spaces?
• RQ2 What is the structure of the distributional derivative $D_X u$ for $BV_X$ functions in equiregular CC spaces?
• RQ3 Under what conditions are the approximate discontinuities of $u$ of jump type in the sub-Riemannian setting?
• RQ4 Can a representation formula for the jump part of $D_X u$ be derived in CC spaces?
• RQ5 How do the notions of approximate limit and approximate jump generalize in the context of $X$-differentiability?

Key findings

• The paper proves that every $u \in BV_X(\Omega; \mathbb{R}^k)$ is approximately $X$-differentiable $\mathscr{L}^n$-almost everywhere in $\Omega$, with the approximate $X$-gradient equal to the density of $D_X^a u$.
• The set $\mathcal{S}_u$ of points without an approximate limit is contained in a countable union of sets of finite $\mathscr{H}^{Q-1}$-measure.
• Under the property $\mathcal{R}$, the singular part of $D_X u$ is entirely concentrated on the jump set $\mathcal{J}_u$, meaning $D_X^s u = D_X^j u$.
• The jump part $D_X^j u$ admits a representation formula involving the jump triple $(u^+, u^-, \nu_u)$, with $\mathcal{J}_u$ being $\mathscr{H}^{Q-1}$-rectifiable.
• The approximate gradient $D_X^{ap} u$ and the jump triple are Borel measurable functions on $\Omega \setminus \mathcal{S}_u$.
• The paper establishes that the approximate limit and jump triple are uniquely defined almost everywhere, ensuring consistency of the blow-up and trace definitions.