Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems
Shiping Cao, Hua Qiu · arXiv (Cornell University) · 2020
We study the Sobolev spaces $H^σ(K)$ and $H^σ_0(K)$ on p.c.f. self-similar sets in terms of the boundary behavior of functions. First, for $σ\in \mathbb{R}^+$, we make an exact description of the tangents of functions in $H^σ(K)$ at the boundary. Second, we characterize $H_0^σ(K)$ as the space of functions in $H^σ(K)$ with zero tangent of an appropriate order depending on $σ$. Last, we extend $H^σ(K)$ to $σ\in\mathbb{R}$, and obtain various interpolation theorems with $σ\in\mathbb{R}^+$ or $σ\in\mathbb{R}$. We illustrate that there is a countable set of critical orders, that arises naturally in the boundary behavior of functions, such that $H^σ_0(K)$ presents a critical phenomenon if $σ$ is critical. These orders will play a crucial role in our study. They are just the values in $\frac 12+\mathbb{Z}_+$ in the classical case, but are much more complicated in the fractal case.