Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

We study the Sobolev spaces $H^{\sigma }(K)$ and $H_0^{\sigma }(K)$ on p.c.f. self-similar sets in terms of the boundary behavior of functions. First, for $\sigma \in {\mathbb{R}}^{+}$, we make an exact description of the tangents of functions in $H^{\sigma }(K)$ at the boundary. Second, we characterize $H_0^{\sigma }(K)$ as the space of functions in $H^{\sigma }(K)$ with zero tangent of an appropriate order depending on $\sigma$. Last, we extend $H^{\sigma }(K)$ to $\sigma \in \mathbb{R}$, and obtain various interpolation theorems with $\sigma \in {\mathbb{R}}^{+}$ or $\sigma \in \mathbb{R}$. We illustrate that there is a countable set of critical orders, that arises naturally in the boundary behavior of functions, such that $H_0^{\sigma }(K)$ presents a critical phenomenon if $\sigma$ is critical. These orders will play a crucial role in our study. They are just the values in $\frac{1}{2}+{\mathbb{Z}}_{+}$ in the classical case, but are much more complicated in the fractal case.