A sharp Leibniz rule for BV functions in metric spaces

TL;DR

This paper establishes a sharp Leibniz rule for BV functions in general metric spaces with doubling measures and Poincaré inequalities, removing boundedness assumptions and constants from previous results.

Contribution

It proves a more general Leibniz rule for BV functions in metric spaces without boundedness assumptions, achieving an optimal form.

Findings

Leibniz rule holds without boundedness assumptions

Variation measures converge weakly* with quasi semicontinuous test functions

Result is essentially optimal in general metric spaces

Abstract

We prove a Leibniz rule for BV functions in a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality. Unlike in previous versions of the rule, we do not assume the functions to be locally essentially bounded and the end result does not involve a constant $C\ge 1$, and so our result seems to be essentially the best possible. In order to obtain the rule in such generality, we first study the weak* convergence of the variation measure of BV functions, with quasi semicontinuous test functions.