双度量测度空间及其双曲填充的延拓和迹结果
Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
摘要：
本文研究了紧致度量空间$Z$上的Besov函数空间与一致整环$X_\varepsilon$上的Newton-Sobolev函数空间之间的联系.这个统一的域是作为一个（Gromov）双曲填充$Z$的均匀化。要做到这一点，我们构建了一个家庭的双曲填充的风格的工作邦克和克莱纳和工作的波登和Pajot。然后对每个参数$β>0$，构造了$Z$上的加倍测度$ν$到$X_\varepsilon$的提升$μ_β$，证明了$μ_β$是加倍的，并且支持$1$-Poincaré不等式.然后，我们表明，对于每个$θ$与$0<θ<1$和$p\ge 1$有一个选择的$β=p(1-θ)\logα$，使Besov空间$B^θ_{p,p}(Z)$是牛顿- Sobolev空间$N^{1,p}(X_\varepsilon,μ_β)$的迹空间时，$\varepsilon=\logα$。最后，我们利用$X_\varepsilon$上的势理论工具得到了$B^θ_{p,p}(Z)$中函数的精细性质，如它们的拟连续性和具有$q=s_νp/(s_ν-pθ)$的$L^q$-Lebesgue点的拟处处存在性，其中$s_ν$是$Z$上与测度$ν$相关联的一个加倍维数.应用这一点的紧致子集的欧几里德空间改进后的结果Netrusov在$\mathbb{R}^n$。
英文摘要：
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 $\mathbb{R}^n$.
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