BV functions and sets of finite perimeter in sub-Riemannian manifolds

Abstract We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms Gp:TpM→[0,∞] are given. When we consider sub-Riemannian manifolds, our definition coincides with the one given in the more general context of metric measure spaces which are doubling and support a Poincaré inequality. We focus on finite perimeter sets, i.e., sets whose characteristic function is BV, in sub-Riemannian manifolds. Under an assumption on the nilpotent approximation, we prove a blowup theorem, generalizing the one obtained for step-2 Carnot groups in [24].

First we study in detail the tensorization properties of weak gradients in metric measure spaces $(X,d,mm)$. Then, we compare potentially different notions of Sobolev space $H^{1,1}(X,d,mm)$ and of weak gradient with exponent 1. Eventually we apply these results to compare the area functional $intsqrt{1+| abla f|_w^2},dmm$ with the perimeter of the subgraph of $f$, in the same spirit as the classical theory.