Abstract
We consider the Sobolev type spaces Hσ(K) with σ≥0, where K is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes of Sobolev spaces HNσ(K),HDσ(K) and Hσ(K), and observe a sequence of critical orders of σ in our comparison theorem. Secondly, we present a general atomic decomposition theorem of Sobolev spaces Hσ(K), where the same critical orders play an important role. At the same time, we provide purely analytic approaches for various Besov type characterizations of Sobolev spaces Hσ(K).
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