# On BV functions and essentially bounded divergence-measure fields in metric spaces

### Vito Buffa

Smiling International School, Ferrara, Italy### Giovanni E. Comi

Universität Hamburg, Germany### Michele Miranda Jr.

Università degli Studi di Ferrara, Italy

## Abstract

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation (BV) in terms of suitable vector fields on a complete and separable metric measure space $(\mathbb{X},d,\mu)$ equipped with a non-negative Radon measure $\mu$ finite on bounded sets. Then, we extend the concept of divergence-measure vector fields $\mathcal{DM}^p(\mathbb{X})$ for any $p\in[1,\infty]$ and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss–Green formula in terms of the normal trace of a $\mathcal{DM}^\infty(\mathbb{X})$ vector field. This differential machinery is also the natural framework to specialize our analysis for $\mathsf{RCD}(K,\infty)$ spaces, where we exploit the underlying geometry to determine the Leibniz rules for $\mathcal{DM}^\infty(\mathbb{X})$ and ultimately to extend our discussion on the Gauss–Green formulas.

## Cite this article

Vito Buffa, Giovanni E. Comi, Michele Miranda Jr., On BV functions and essentially bounded divergence-measure fields in metric spaces. Rev. Mat. Iberoam. 38 (2022), no. 3, pp. 883–946

DOI 10.4171/RMI/1291