Rank-one theorem and subgraphs of BV functions in Carnot groups

TL;DR

This paper extends the rank-one theorem to BV functions in Carnot groups, including Heisenberg groups, revealing new structural properties of derivatives in these non-Euclidean settings.

Contribution

It proves a rank-one theorem for BV functions in Carnot groups, a significant generalization of classical Euclidean results.

Findings

Established a rank-one theorem for BV functions in Carnot groups.

Linked horizontal derivatives of BV functions to their subgraph properties.

Extended results to include Heisenberg groups for n ≥ 2.

Abstract

We prove a rank-one theorem \`a la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups $\mathbb H^n$ for $n\geq 2$. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph.