Calculus of Variations and Geometric Measure Theory

A. Tyulenev

Tracaes of Sobolev spaces to irregular subsets of metric measure spaces

created by tyulenev on 22 Feb 2024


Published Paper

Inserted: 22 feb 2024
Last Updated: 22 feb 2024

Journal: Sbornik:Mathematics
Volume: 214
Number: 9
Pages: 1241-1320
Year: 2023


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 $W_{p}^{1}(\operatorname{X})$-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 $W_{p}^{1}(\operatorname{X})$ to arbitrary closed nonempty sets $S \subset \operatorname{X}$.

Tags: GeoMeG
Keywords: Sobolev spaces, traces, extensions