论文标题
扩展和痕量结果，用于加倍度量测量空间及其双曲线填充物
Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
论文作者
论文摘要
在本文中，我们研究了BESOV函数空间之间的连接在紧凑的度量空间$ z $上，配备了双倍度量，以及在均匀域上的newton-sobolev功能空间$ x_ \ varepsilon $。该统一域是作为$ z $的（Gromov）双曲线填充的均匀化。为此，我们以Bonk和Kleiner的作品以及Bourdon和Pajot的作品的风格建造了一个双曲线填充物。然后，对于每个参数$β> 0 $，我们构建了$ z $上的lift $μ_β$ to $ z $ to $ x_ \ varepsilon $，并表明$μ_β$正在翻倍，并支持$ 1 $-POINCARé的不平等。然后，我们证明，对于每个$θ$，$ 0 <θ<1 $和$ p \ ge 1 $都有$β= p（1-θ）\logα$的选择，使得besov space $ b^θ_{p，p}（z）$是Newton-sobolev space $ n^{1，p}的痕迹， $ \ varepsilon = \logα$。最后，我们在$ x_ \ varepsilon $上利用潜在理论的工具，以获得$ b^θ_{p，p}（z）$中的功能的良好属性，例如它们的quasicontinuitionuity and quasievery where $ l^q $ lebesgue的存在与$ q =s_νp/$s_νp/s $ s $ s $在$ z $上。将此应用于紧凑的欧几里得空间子集，在$ \ mathbb {r}^n $中的Netrusov结果上有所改善。
In this paper we study connections between Besov spaces of functions on a compact metric space $Z$, equipped with a doubling measure, and the Newton--Sobolev space of functions on a uniform domain $X_\varepsilon$. This uniform domain is obtained as a uniformization of a (Gromov) hyperbolic filling of $Z$. To do so, we construct a family of hyperbolic fillings in the style of the work of Bonk and Kleiner and the work of Bourdon and Pajot. Then for each parameter $β>0$ we construct a lift $μ_β$ of the doubling measure $ν$ on $Z$ to $X_\varepsilon$, and show that $μ_β$ is doubling and supports a $1$-Poincaré inequality. We then show that for each $θ$ with $0<θ<1$ and $p\ge 1$ there is a choice of $β=p(1-θ)\logα$ such that the Besov space $B^θ_{p,p}(Z)$ is the trace space of the Newton--Sobolev space $N^{1,p}(X_\varepsilon,μ_β)$ when $\varepsilon=\logα$. Finally, we exploit the tools of potential theory on $X_\varepsilon$ to obtain fine properties of functions in $B^θ_{p,p}(Z)$, such as their quasicontinuity and quasieverywhere existence of $L^q$-Lebesgue points with $q=s_νp/(s_ν-pθ)$, where $s_ν$ is a doubling dimension associated with the measure $ν$ on $Z$. Applying this to compact subsets of Euclidean spaces improves upon a result of Netrusov in $\mathbb{R}^n$.
