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

Iván Caamaño, Josh Kline

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 $\partialΩ:=\overlineΩ\setminusΩ$ 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.

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

Abstract

In the setting of a non-complete doubling metric measure space , we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces . Equipping the boundary with a measure which is codimension Ahlfors regular with respect to , these operators take the form The trace operators are first constructed under the additional assumption that 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 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.

Paper Structure

This paper contains 12 sections, 23 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Doubling measures.
  4. Uniform domains and Harnack chains
  5. Besov spaces.
  6. Proof of Trace Theorems \ref{['thm:Trace']} and \ref{['thm:Intro Inhom Trace']}
  7. Existence of traces and $L^p$ bounds.
  8. Energy bounds.
  9. Proof of Extension Theorems \ref{['thm:Extension']} and \ref{['thm:InhomExtension']}
  10. Hyperbolic fillings and the proof of Theorem \ref{['thm:Intro gen trace']}
  11. Uniformized hyperbolic fillings
  12. Trace theorem

Key Result

Theorem 1.1

Let $(\Omega,d,\mu)$ be a locally compact, non-complete metric measure space, with $\mu$ a doubling measure, such that $\Omega$ is a uniform domain in its completion $\overline\Omega$. Suppose also that $\partial\Omega:=\overline\Omega\setminus\Omega$, the boundary of $\Omega$, is equipped with a Bo such that for all $u\in HB^\alpha_{p,q}(\Omega,\mu)$, we have for $\nu$-a.e. $z\in\partial\Omega$,

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.4
  • Theorem 1.6
  • Corollary 1.8
  • Theorem 1.10
  • Theorem 1.11
  • Lemma 2.1: HKST, Proposition 4.1.15
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 34 more