Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909820388376576 |
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| author | Caamaño, Iván Kline, Josh |
| author_facet | Caamaño, Iván Kline, Josh |
| contents | 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 $\partialΩ:=\overlineΩ\setminusΩ$ with a measure which is codimension $θ$ Ahlfors regular with respect to $μ$, these operators take the form
\[
T:B^α_{p,q}(Ω)\to B^{α-θ/p}_{p,q}(\partialΩ),\quad E:B^α_{p,q}(\partialΩ)\to B^{α+θ/p}_{p,q}(Ω).
\] 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces Caamaño, Iván Kline, Josh Metric Geometry Primary: 46E36, Secondary: 46E35, 30L15 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 $\partialΩ:=\overlineΩ\setminusΩ$ with a measure which is codimension $θ$ Ahlfors regular with respect to $μ$, these operators take the form \[ T:B^α_{p,q}(Ω)\to B^{α-θ/p}_{p,q}(\partialΩ),\quad E:B^α_{p,q}(\partialΩ)\to B^{α+θ/p}_{p,q}(Ω). \] 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. |
| title | Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces |
| topic | Metric Geometry Primary: 46E36, Secondary: 46E35, 30L15 |
| url | https://arxiv.org/abs/2510.01385 |