Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces

Abstract

In the setting of a non-complete doubling metric measure space $(\Omega,d,\mu)$, we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces $B^\alpha_{p,q}$. Equipping the boundary $\partial\Omega:=\overline\Omega\setminus\Omega$ with a measure which is codimension $\theta$ Ahlfors regular with respect to $\mu$, these operators take the form \[ T:B^\alpha_{p,q}(\Omega)\to B^{\alpha-\theta/p}_{p,q}(\partial\Omega),\quad E:B^\alpha_{p,q}(\partial\Omega)\to B^{\alpha+\theta/p}_{p,q}(\Omega). \] The trace operators are first constructed under the additional assumption that $\Omega$ is a uniform domain in its completion. We then use such results along with the technique of hyperbolic filling to remove this assumption in the case that $\Omega$ is bounded. This extends to the doubling setting some earlier results of Marcos and Saksman-Soto proven under the assumption that the ambient measure is Ahlfors regular.