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.
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Funding
This work was supported by the Russian Science Foundation under grant no. 23-71-30001, https://rscf.ru/en/project/23-71-30001/, at Lomonosov Moscow State University.
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Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 404–434 https://doi.org/10.4213/mzm14097.
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Tyulenev, A.I. Traces of Sobolev Spaces on Piecewise Ahlfors–David Regular Sets. Math Notes 114, 351–376 (2023). https://doi.org/10.1134/S0001434623090079
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DOI: https://doi.org/10.1134/S0001434623090079