Calculus and fine properties of functions of bounded variation on RCD spaces

TL;DR

This paper extends calculus rules for functions of bounded variation to RCD spaces, defining a distributional differential in infinite dimensions and establishing fine properties and calculus rules in finite dimensions.

Contribution

It introduces a generalized calculus framework for BV functions on RCD spaces, including an analogue of the distributional differential and the Vol'pert chain rule.

Findings

Defined a distributional differential in infinite-dimensional RCD spaces

Proved fine properties of BV functions in finite-dimensional RCD spaces

Established calculus rules such as the Vol'pert chain rule for vector-valued functions

Abstract

We generalize the classical calculus rules satisfied by functions of bounded variation to the framework of RCD spaces. In the infinite dimensional setting we are able to define an analogue of the distributional differential and on finite dimensional spaces we prove fine properties and suitable calculus rules, such as the Vol'pert chain rule for vector valued functions.