Published Paper

Inserted: 21 jun 2023
Last Updated: 28 aug 2024

Journal: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
Pages: 39
Year: 2023

Abstract:

In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume.

Surprisingly enough, when $M$ is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on $M$.

As a corollary, we can find the asymptotics of the fractional s-perimeter on (essentially) every complete manifold, generalizing both the existing results for $\mathbb{R}^n$ and for the Gaussian space. In doing so, from many sets $E ⊂ M$ we are able to produce a bounded harmonic function associated to $E$, which in general can be non-constant.

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• Asymptotics_v4.pdf