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

We consider the Sobolev type spaces $H^{\sigma }(K)$ with $\sigma \ge 0$, where $K$ is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes of Sobolev spaces $H_N^{\sigma }(K),H_D^{\sigma }(K)$ and $H^{\sigma }(K)$, {and observe} a sequence of critical orders of $\sigma$ in our comparison theorem. Secondly, We present a general atomic decomposition theorem of Sobolev spaces $H^{\sigma }(K)$, 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 $H^{\sigma }(K)$.