Traces of Besov spaces to regular subsets of metric measure spaces: the limiting case

Abstract

Let $(X,d,\mu )$ be a metric measure space whose measure $\mu$ is uniformly locally doubling and which supports a local weak $(1,p)$-Poincaré inequality for some $p\in [1,\infty )$. Given $\theta \in (0,p)$ and an Ahlfors--David codimension-$\theta$ regular subset $E\subset X$, we provide a complete intrinsic description of the trace-space of the Besov space $B_{p,1}^{\theta /p}(X)$ to $E$. More precisely, we show that the trace operator is well defined and bounded from $B_{p,1}^{\theta /p}(X)$ to $L_p(E,{\mathcal{H}}_{\theta }{\lfloor }_E)$. We also show that the upper estimate in the Ahlfors--David codimension-$\theta$ regularity condition is necessary for such boundedness under the local weak Poincaré inequality. Conversely, assuming that $E$ is Ahlfors--David codimension-$\theta$ regular, we construct a bounded nonlinear extension operator from $L_p(E,{\mathcal{H}}_{\theta }{\lfloor }_E)$ to $B_{p,1}^{\theta /p}(X)$. Thus the trace-space is identified intrinsically with $L_p(E,{\mathcal{H}}_{\theta }{\lfloor }_E)$. This extends the classical limiting case of the trace theorem obtained by Burenkov and Gol'dman. Finally, we apply the general theory to $K$-regular trees, $K\ge 1$, for which we additionally derive a necessary and sufficient criterion for the existence of traces.