Nonexistence of Bounded Linear Extension Operators for Limiting Besov Traces on Metric Measure Spaces

Abstract

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