Traces of Sobolev spaces to irregular subsets of metric measure spaces

Abstract

Given $p\in (1,\infty )$, let $(X,d,\mu )$ 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 $W_p^1(X)$-space to lower codimension-$\theta$ content regular closed subsets $S\subset X$. In particular, if the space $(X,d,\mu )$ is Ahlfors $Q$-regular for some $Q\ge 1$ and $p\in (Q,\infty )$, then we get an intrinsic description of the trace-space of the Sobolev $W_p^1(X)$-space to arbitrary closed nonempty set $S\subset X$.

arXiv commentMany typos in the previous versions have been corrected. Some proofs have been clarified