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

In the setting of a non-complete doubling metric measure space $(Ω,d,μ)$, we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces B^αp,q. Equipping the boundary ∂Ω:=ΩΩ with a measure which is codimension $θ$ Ahlfors regular with respect to $μ$, these operators take the form \[ T:B^αp,q(Ω)→ Bα-θ/pp,q(∂Ω), E:B^αp,q(∂Ω)→ Bα+θ/pp,q(Ω). \] The trace operators are first constructed under the additional assumption that $Ω$ 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 $Ω$ 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.