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

Vito Buffa, Giovanni E Comi, Michele Miranda Jr.

• 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,μ) equipped with a non-negative Radon measure μ finite on bounded sets. Then, we extend the concept of divergence-measure vector fields DMp(X) for any p∈[1,∞] 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 DM∞(X) vector field. This differential machinery is also the natural framework to specialize our analysis for RCD(K,∞) spaces, where we exploit the underlying geometry to determine the Leibniz rules for DM∞(X) and ultimately to extend our discussion on the Gauss–Green formulas.