Characterizations of Sobolev functions via Besov-type energy functionals in fractals

TL;DR

This paper extends Sobolev function characterizations via Besov-type energy functionals to fractal metric spaces, providing new insights into function spaces on complex geometries like the Vicsek set and Sierpiński gaskets.

Contribution

It introduces novel characterizations of Sobolev functions on fractals using Besov-type energies, linking energy forms with Besov spaces under specific geometric conditions.

Findings

Equivalent norms on Korevaar--Schoen--Sobolev spaces

Identification of energy form domains with Besov spaces

Extension of Sobolev characterizations to fractals

Abstract

In the spirit of the ground-breaking result of Bourgain--Brezis--Mironescu, we establish some characterizations of Sobolev functions in metric measure spaces including fractals like the Vicsek set, the Sierpi\'{n}ski gasket and the Sierpi\'{n}ski carpet. As corollaries of our characterizations, we present equivalent norms on the Korevaar--Schoen--Sobolev space, and show that the domain of a $p$-energy form is identified with a Besov-type function space under a suitable $(p,p)$-Poincar\'e inequality, capacity upper bound and the volume doubling property.