About the general chain rule for functions of bounded variation

TL;DR

This paper provides an alternative proof of the general chain rule for functions of bounded variation, extending its applicability to non-smooth finite-dimensional RCD spaces by leveraging recent theoretical links.

Contribution

It introduces a new proof method for the chain rule in BV functions, adaptable to non-smooth RCD spaces, building on recent advances in differentiation operator closability.

Findings

New proof of the chain rule for BV functions

Extension to non-smooth RCD spaces

Connection between closability and differentiability

Abstract

We give an alternative proof of the general chain rule for functions of bounded variation ([ADM90]), which allows to compute the distributional differential of $\varphi\circ F$, where $\varphi\in \mathrm{LIP}(\mathbb{R}^m)$ and $F\in\mathrm{BV}(\mathbb{R}^n,\mathbb{R}^m)$. In our argument we build on top of recently established links between `closability of certain differentiation operators' and `differentiability of Lipschitz functions in related directions' ([ABM23]): we couple this with the observation that `the map that takes $\varphi$ and returns the distributional differential of $\varphi\circ F$ is closable' to conclude. Unlike previous results in this direction, our proof can directly be adapted to the non-smooth setting of finite dimensional RCD spaces.