On the Linearity of Extension Operators for Traces of Besov Spaces on Metric Measure Spaces: The Limiting Case

Abstract

Let $p\in [1,\infty )$, and let $X=(X,d,\mu )$ be a metric measure space such that $\mu$ is uniformly locally doubling and $X$ supports a local weak $(1,p)$-Poincaré inequality. Given $\theta \in (0,p)$ and an Ahlfors--David codimension-$\theta$ regular set $E\subset X$, the trace-space of the Besov space $B_{p,1}^{\theta /p}(X)$ to $E$ can be identified with $L_p(E,{\mathcal{H}}_{\theta }{\lfloor }_E)$. If there exists a measurable set $A\subset E$ with $0<{\mathcal{H}}_{\theta }(A)<\infty$ such that ${\mathcal{H}}_{\theta }{\lfloor }_A$ is nonatomic, we prove that there is no bounded linear extension operator $\mathrm{Ext}:L_p(E,{\mathcal{H}}_{\theta }{\lfloor }_E)\to B_{p,1}^{\theta /p}(X)$.