On the asymptotic behaviour of the fractional Sobolev seminorms: A geometric approach

Abstract

We study the well-known asymptotic formulas for fractional Sobolev functions à la Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova, in a geometric approach. We show that the key to these asymptotic formulas are Rademacher's theorem and volume growth at infinity respectively. Examples fitting our framework includes Euclidean spaces, Riemannian manifolds, Alexandrov spaces, finite dimensional Banach spaces, and some ideal sub-Riemannian manifolds.