preprint

Inserted: 13 jul 2023
Last Updated: 13 jul 2023

Year: 2023

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.