Traces of Sobolev spaces to piecewise Ahlfors--David regular sets

Abstract

Let $(X,d,\mu )$ be a metric measure space with uniformly locally doubling measure $\mu$. Given $p\in (1,\infty )$, assume that $(X,d,\mu )$ supports a weak local $(1,p)$-Poincaré inequality. We characterize trace spaces of the first-order Sobolev $W_p^1(X)$-spaces to subsets $S$ of $X$ that can be represented as a finite union ${\cup }_{i=1}^N{S}^i$, $N\in \mathbb{N}$, of Ahlfors--David regular subsets $S^i\subset X$, $i\in \{1,...,N\}$, of different codimensions. Furthermore, we explicitly compute the corresponding trace norms up to some universal constants.