Traces of Sobolev spaces to piecewise Ahlfors--David regular sets

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 $(1,p)$-Poincar\'e 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 $\cup_{i=1}^{N}S^{i}$, $N \in \mathbb{N}$, of Ahlfors--David regular subsets $S^{i} \subset \operatorname{X}$, $i \in \{1,...,N\}$, of different codimensions. Furthermore, we explicitly compute the corresponding trace norms up to some universal constants.