Traces of Sobolev spaces to irregular subsets of metric measure spaces

Abstract

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