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

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 (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 D M-p (X) for any p is an element of [1, infinity] 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 D M-infinity (X) vector field. This differential machinery is also the natural framework to specialize our analysis for RCD(K, infinity) spaces, where we exploit the underlying geometry to determine the Leibniz rules for D M-infinity (X) and ultimately to extend our discussion on the Gauss-Green formulas.