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

2025/10/01 by Iván Caamaño, Josh Kline, Caamaño, Iván +1
Mathematics · #Advanced Harmonic Analysis Research #Mathematical Approximation and Integration #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2510.01385

Abstract

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^α_p,q. Equipping the boundary ∂Ω:=Ω∖Ω with a measure which is codimension θ Ahlfors regular with respect to μ, these operators take the form T:B^α_p,q(Ω)→ B^α-θ/p_p,q(∂Ω), E:B^α_p,q(∂Ω)→ B^α+θ/p_p,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.

Citations

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Related

• Trace and extension theorems for homogeneous Sobolev and Besov spaces for unbounded uniform domains in metric measure spaces
• Extension and trace theorems for noncompact doubling spaces
• A trace theorem for Besov functions in spaces of homogeneous type
• Besov spaces via hyperbolic fillings
• Traces of Besov, Triebel-Lizorkin and Sobolev spaces on metric spaces