Inserted: 11 feb 2024
Last Updated: 11 feb 2024
We find a surprising link between Maz'ya\textendash Shaposhnikova's well-known asymptotic formula concerning fractional Sobolev seminorms and the generalized Bishop--Gromov inequality. In the setting of abstract metric measure spaces we prove a large family of asymptotic formulas concerning non-local energies. Important examples in which our approach applies are Carnot groups, Riemannian manifolds with Ricci curvature bounded from below and non-collapsed RCD spaces. We also extend the classical Maz'ya\textendash Shaposhnikova's asymptotic formula to a wide class of mollifiers.