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arXiv · 2410.15317

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

Ryosuke Shimizu
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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.

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Ryosuke Shimizu. 2024-10-20. Characterizations of Sobolev functions via Besov-type energy functionals in fractals. https://arxiv.org/abs/2410.15317

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