Traces of Sobolev spaces to irregular subsets of metric measure spaces

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 $(1,p)$-Poincaré inequality. For each $\theta \in [0,p)$ we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower $\theta$-codimensional content regular closed sets $S \subset \operatorname{X}$. 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 obtain an intrinsic description of the trace-space of the Sobolev space $W^{1}_{p}(\operatorname{X})$ to arbitrary closed nonempty sets $S \subset \operatorname{X}$. Bibliography: 43 titles.