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

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 Sierpiński gasket and the Sierpiński 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 $(p,p)$-Poincaré inequality, capacity upper bound and the volume doubling property.