Traces of Sobolev spaces to irregular subsets of metric measure spaces

TL;DR

This paper characterizes the trace spaces of Sobolev functions on irregular subsets of metric measure spaces with certain regularity and Poincaré inequalities, extending classical results to more general settings.

Contribution

It provides an intrinsic description of Sobolev trace spaces on lower codimension subsets in metric measure spaces with doubling measures and Poincaré inequalities, including Ahlfors regular spaces.

Findings

Characterization of Sobolev trace spaces on content regular subsets.

Extension of trace space descriptions to Ahlfors regular spaces.

Explicit description for Sobolev spaces on arbitrary closed sets.

Abstract

Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincar\'e inequality. For each $\theta \in [0,p)$, we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower codimension-$\theta$ content regular closed subsets $S \subset \operatorname{X}$. In particular, if the space $(\operatorname{X},\operatorname{d},\mu)$ 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 $S \subset \operatorname{X}$.