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We consider the Sobolev type spaces $H^\sigma(K)$ with $\sigma\geq 0$, where$K$ is a post-critically finite self-similar set with the natural boundary.Firstly, we compare different classes of Sobolev spaces$H^\sigma_N(K),H^\sigma_D(K)$ and $H^\sigma(K)$, {and observe} a sequence ofcritical orders of $\sigma$ in our comparison theorem. Secondly, We present ageneral atomic decomposition theorem of Sobolev spaces $H^\sigma(K)$, where thesame critical orders play an important role. At the same time, we providepurely analytic approaches for various Besov type characterizations of Sobolevspaces $H^\sigma(K)$.

Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions