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 $(1,p)$-Poincar\'e inequality. Given $\theta\in(0,p)$ and an Ahlfors--David codimension-$\theta$ regular set $E\subset X$, the trace-space of the Besov space $B^{\theta/p}_{p,1}(X)$ 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 $\operatorname{Ext}:L_p(E,\mathcal{H}_{\theta}\lfloor_E) \to B^{\theta/p}_{p,1}(X)$.