$\mathbb{R}^n$上某些加权Sobolev空间和$\mathbb{R}^n$的Ahlfors正则紧子集上Besov函数的迹和扩张
Traces and extensions of certain weighted Sobolev spaces on $\mathbb{R}^n$ and Besov functions on Ahlfors regular compact subsets of $\mathbb{R}^n$
摘要：
本文的重点是Ahlfors $Q$ -正则紧集$E\subset\mathbb{R}^n$，使得对于每个$Q-2<α\le 0$，通过积分密度$ω(x)=\text{dist}(x, E)^α$给出的加权测度$μ_α$在包含$E$的球$B$中产生Muckenhoupt $\mathcal{A}_p$ -权。对于这样的集合$E$，我们证明了存在一个有界线性迹算子从$W^{1,p}(B,μ_α)$到$B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$当$0<θ<1-\tfrac{α+n-Q}{p}$，和存在一个有界线性扩展算子从$B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$到$W^{1,p}(B, μ_α)$当$1-\tfrac{α+n-Q}{p}\le θ<1$。我们说明这些结果与$E$的Sierpienski地毯，Sierpienski垫片，和冯科赫雪花。
英文摘要：
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.
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