2022/08/26 by Miguel García‐Bravo, Toni Ikonen, García-Bravo, Miguel +3 · 1 citation
Mathematics · Social Sciences · #30L99 #Complex Variables (math.CV) #Differential Equations and Boundary Problems #Diverse Research Studies Overview #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Primary: 46E35. Secondary: 46E36
paper · pdf · doi:10.48550/arxiv.2208.12594
openalex publication_date 2022/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality, we investigate when a given Banach-valued Sobolev function defined on a subset satisfying a measure-density condition is the restriction of a Banach-valued Sobolev function defined on the whole space. We investigate the problem for Hajłasz- and Newton-Sobolev spaces, respectively. First, we show that Hajłasz-Sobolev extendability is independent of the target Banach spaces. We also show that every c0-valued Newton-Sobolev extension set is a Banach-valued Newton-Sobolev extension set for every Banach space. We also prove that any measurable set satisfying a measure-density condition and a weak Poincaré inequality up to some scale is a Banach-valued Newton-Sobolev extension set for every Banach space. Conversely, we verify a folklore result stating that when n≤ p<∞, every W1,p-extension domain Ω⊂ ℝn supports a weak (1,p)-Poincaré inequality up to some scale. As a related result of independent interest, we prove that in any metric measure space when 1 ≤ p < ∞ and real-valued Lipschitz functions with bounded support are norm-dense in the real-valued W1,p-space, then Banach-valued Lipschitz functions with bounded support are energy-dense in every Banach-valued W1,p-space whenever the Banach space has the so-called metric approximation property.