Traces of Sobolev spaces to irregular subsets of metric measure spaces

Given $p\in (1,\mathrm{\infty })$, let $(\mathrm{X},\mathrm{d},\mu )$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincaré inequality. For each $\theta \in [0,p)$, we characterize the trace space of the Sobolev $W_p^1(\mathrm{X})$-space to lower codimension-$\theta$ content regular closed subsets $S\subset \mathrm{X}$. In particular, if the space $(\mathrm{X},\mathrm{d},\mu )$ is Ahlfors $Q$-regular for some $Q\ge 1$ and $p\in (Q,\mathrm{\infty })$, then we get an intrinsic description of the trace-space of the Sobolev $W_p^1(\mathrm{X})$-space to arbitrary closed nonempty set $S\subset \mathrm{X}$.