Trace and Extension Theorems for Homogeneous Sobolev and Besov Spaces for Unbounded Uniform Domains in Metric Measure Spaces

Pub Date : 2024-03-06 DOI:10.1134/s0081543823050061

Ryan Gibara, Nageswari Shanmugalingam

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Abstract

In this paper we fix \(1\le p<\infty\) and consider \((\Omega,d,\mu)\) to be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure \(\mu\) supporting a \(p\)-Poincaré inequality such that \(\Omega\) is a uniform domain in its completion \(\overline\Omega\). We realize the trace of functions in the Dirichlet–Sobolev space \(D^{1,p}(\Omega)\) on the boundary \(\partial\Omega\) as functions in the homogeneous Besov space \(H\kern-1pt B^\alpha_{p,p}(\partial\Omega)\) for suitable \(\alpha\); here, \(\partial\Omega\) is equipped with a non-atomic Borel regular measure \(\nu\). We show that if \(\nu\) satisfies a \(\theta\)-codimensional condition with respect to \(\mu\) for some \(0<\theta<p\), then there is a bounded linear trace operator \(T \colon\, D^{1,p}(\Omega)\to H\kern-1pt B^{1-\theta/p}(\partial\Omega)\) and a bounded linear extension operator \(E \colon\, H\kern-1pt B^{1-\theta/p}(\partial\Omega)\to D^{1,p}(\Omega)\) that is a right-inverse of \(T\).

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公度量空间中无边界均匀域的同质索波列夫和贝索夫空间的踪迹和扩展定理

Abstract In this paper we fix\(1\le p<\infty\) and consider \((\Omega,d,\mu)\) to be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure \(\mu\) supporting a \(p\)-Poincaré inequality such that \(\Omega\) is a uniform domain in its completion \(\overline\Omega\).我们将边界 \(\partial\Omega\) 上的 Dirichlet-Sobolev 空间 \(D^{1,p}(\Omega)\) 中的函数的迹作为同质 Besov 空间 \(H\kern-1pt B^\alpha_{p,p}(\partial\Omega)\) 中的函数来实现，对于合适的 \(\alpha\)；这里，\(\partial\Omega\) 配备了一个非原子的波尔正则量度\(\nu\)。我们证明，如果\(\nu\)满足一个关于\(\mu\)的\(\theta\)-codimensional条件，对于某个\(0<\theta<；p），那么存在一个有界线性迹算子（T \colon\, D^{1,p}(\Omega)\to H\kern-1pt B^{1-\theta/p}(\partial\Omega)\) 和一个有界线性扩展算子（E \colon\、H\kern-1pt B^{1-\theta/p}(\partial\Omega)\to D^{1,p}(\Omega)\) 是 \(T\)的右逆。

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