Traces of Sobolev spaces on piecewise Ahlfors-David regular sets. (English. Russian original) Zbl 1537.46035
Math. Notes 114, No. 3, 351-376 (2023); translation from Mat. Zametki 114, No. 3, 404-434 (2023).
Summary: 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 \(\bigcup_{i=1}^NS^i\), \(N \in \mathbb{N} \), 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.
MSC:
| 46E36 | Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces |
| 28A75 | Length, area, volume, other geometric measure theory |
| 30L99 | Analysis on metric spaces |
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