On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity

TL;DR

This paper investigates the asymptotic behavior of fractional Sobolev seminorms in metric measure spaces, establishing new proofs, linking asymptotic formulas to volume ratios, and proving novel rigidity results under Ricci curvature bounds.

Contribution

It provides new proofs of known theorems in a broader setting, introduces new spaces satisfying asymptotic formulas, and establishes new rigidity results for metric measure spaces with curvature bounds.

Findings

New proofs of Maz'ya-Shaposhnikova and Ludwig's theorems in metric measure spaces

Identification of new spaces satisfying asymptotic formulas

Rigidity results for spaces with synthetic Ricci curvature bounds

Abstract

We study the asymptotic behaviour of suitably defined seminorms in general metric measure spaces. As a particular case we provide new and shorter proofs of the Maz'ya-Shaposhnikova's theorem on the asymptotic behaviour of the fractional Sobolev $s$-seminorm, in the setting of metric measure spaces and with general mollifiers, as well as of the Ludwig's result on finite dimensional Banach spaces. Our result also provides new spaces satisfying an asymptotic formula and it also builds a link between the asymptotic formula for functions and the asymptotic volume ratio of a metric measure space. In addition, we prove two related rigidity results for metric measure spaces with synthetic Ricci curvature bound which are new even in the smooth setting.