Interpolations and Fractional Sobolev Spaces in Carnot Groups

In this paper we present an interpolation approach to the fractional Sobolev spaces in Carnot groups using the K-method. This approach provides us with a different characterization of these Sobolev spaces, moreover, it provides us with the limiting behavior of the fractional Sobolev norms at the end-points. This allows us to deduce results similar to the Bourgain-Brezis-Mironescu and Maz’ya-Shaposhnikova in the case p > 1 and D´avila’s result in the case p = 1. Also, this allows us to deduce the limiting behavior of the fractional perimeter in Carnot groups.