Abstract

Let \((\operatorname{X},\operatorname{d},\mu)\) be a metric measure space with uniformly locally doubling measure \(\mu\). Given \(p \in (1,\infty)\), assume that \((\operatorname{X},\operatorname{d},\mu)\) supports a weak local \((1,p)\)-Poincaré inequality. We characterize trace spaces of the first-order Sobolev \(W^{1}_{p}(\operatorname{X})\)-spaces to subsets \(S\) of \(\operatorname{X}\) that can be represented as a finite union \(\bigcup_{i=1}^{N}S^{i}\), \(N \in \mathbb{N}\), of Ahlfors–David regular subsets \(S^{i} \subset \operatorname{X}\), \(i \in \{1,\dots,N\}\), of different codimensions. Furthermore, we explicitly compute the corresponding trace norms up to some universal constants.

中文翻译：

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分段阿尔福斯-大卫正则集上索博列夫空间的迹

摘要

令\((\operatorname{X},\operatorname{d},\mu)\)为具有均匀局部加倍测度\(\mu\)的度量测度空间。给定\(p \in (1,\infty)\)，假设\((\operatorname{X},\operatorname{d},\mu)\)支持弱局部\((1,p)\) -庞加莱不等式。我们将一阶 Sobolev 的迹空间\(W^{1}_{p}(\operatorname{X})\)表征为 \(\operatorname{X}\) 的子集\ ( S \ )可以表示为Ahlfors – David 正则子集\ (S ^{i} \subset \operatorname{X}\)、\(i \in \{1,\dots,N\}\)，具有不同的余维。此外，我们显式地计算相应的迹范数直至一些通用常数。