About the general chain rule for functions of bounded variation

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 $\phi \circ F$, where $\phi \in \mathrm{LIP}({\mathbb{R}}^m)$ 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 $\phi$ and returns the distributional differential of $\phi \circ F$ 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.