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 $(1,p)$-Poincar\'e 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 $B^{\theta/p}_{p,1}(X)$ to $E$. More precisely, we show that the trace operator is well defined and bounded from $B^{\theta/p}_{p,1}(X)$ 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 Poincar\'e 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 $B^{\theta/p}_{p,1}(X)$. 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.