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 (Ambrosio and Maso, 1990), which allows to compute the distributional differential of $\phi \circ F$, where $\phi \in \mathrm{LIP}\left(R^m\right)$ and $F\in \mathrm{BV}(R^n,R^m)$. 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” (Alberti et al., 2023): 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.