论文标题
$ \ mathbb {r}^n $上某些加权sobolev空间的痕迹和扩展，并且在ahlfors上的besov函数$ \ mathbb {r}^n $的常规紧凑型子集
Traces and extensions of certain weighted Sobolev spaces on $\mathbb{R}^n$ and Besov functions on Ahlfors regular compact subsets of $\mathbb{R}^n$
论文作者
论文摘要
本文的重点是ahlfors $ q $ - 惯用紧凑型设置$ e \ subset \ mathbb {r}^n $，这样，对于每个$ q-2 <α\ le 0 $，加权度量$μ_α$，通过整合密度$ω（x）= \ text {x）= \ text {x，x，x，x，e） $ \ Mathcal {a} _p $ - 含有$ e $的球$ b $中的$。对于此类$ e $，我们显示了有界线性跟踪操作员的存在，该操作员的作用于$ w^{1，p，b，μ_α）$到$ b^θ_{p，p，p，p，p}（e，\ mathcal {h}^q \ vert_e）$ n时，有界线性扩展运算符，来自$ b^θ_{p，p}（e，\ nathcal {h}^q \ vert_e）$到$ w^{1，p}（b，μ_α）$时，$ 1- \ tfrac {α+n-q} {α+n-q} {p} {p} {p} {p} {p} {p} {p} {p} {p} {我们用$ e $作为Sierpiński地毯，Sierpiński垫圈和von Koch Snowflake说明了这些结果。
The focus of this paper is on Ahlfors $Q$-regular compact sets $E\subset\mathbb{R}^n$ such that, for each $Q-2<α\le 0$, the weighted measure $μ_α$ given by integrating the density $ω(x)=\text{dist}(x, E)^α$ yields a Muckenhoupt $\mathcal{A}_p$-weight in a ball $B$ containing $E$. For such sets $E$ we show the existence of a bounded linear trace operator acting from $W^{1,p}(B,μ_α)$ to $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ when $0<θ<1-\tfrac{α+n-Q}{p}$, and the existence of a bounded linear extension operator from $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ to $W^{1,p}(B, μ_α)$ when $1-\tfrac{α+n-Q}{p}\le θ<1$. We illustrate these results with $E$ as the Sierpiński carpet, the Sierpiński gasket, and the von Koch snowflake.
