Notes on limits of Sobolev spaces and the continuity of interpolation scales

We extend lemmas by Bourgain-Brezis-Mironescu ( 2001), and Maz'ya-Shaposhnikova ( 2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.