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

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