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 φ∘F, where φ∈LIP(Rm) and F∈BV(Rn,Rm). 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 φ and returns the distributional differential of φ∘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.
