论文标题
P.C.F.上的Sobolev空间自相似集：边界行为和插值定理
Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems
论文作者
论文摘要
我们在P.C.F.上研究Sobolev空间$ H^σ（k）$和$ h^σ_0（k）$自相似的集合函数的边界行为。首先，对于$σ\ in \ mathbb {r}^+$，我们对边界的$ h^σ（k）$中的函数切线进行了确切的描述。其次，我们将$ h_0^σ（k）$描述为$ h^σ（k）$的功能空间，根据$σ$，零切的零切。最后，我们将$ h^σ（k）$扩展到\ Mathbb {r} $中的$σ\，并以$σ\ in \ Mathbb {r}^+$或$σ\ in \ Mathbb {r} $获得各种插值定理。我们说明，有一组可计数的临界顺序，自然而然地在函数的边界行为中产生，因此，如果$ h^σ_0（k）$呈现出一个关键现象，如果$σ$至关重要。这些订单将在我们的研究中发挥至关重要的作用。它们只是$ \ frac 12+ \ mathbb {z} _+$中的值，但在分形情况下更为复杂。
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.
