Approximations of Sobolev Norms in Carnot Groups

This paper deals with a notion of Sobolev space W 1, p introduced by Bourgain, Brezis and Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by Ponce to obtain a Poincaré-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressed in terms of the intrinsic distance is equivalent to the L p -norm of the intrinsic gradient, and provide a Poincaré-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest.