Extensions and approximations of Banach-valued Sobolev functions

Abstract

In complete metric measure spaces Z equipped with a doubling measure and supporting a weak Poincaré inequality, we consider a measurable subset Ω satisfying a measure-density condition. We investigate when a given Banach–valued Sobolev function defined on Ω is the restriction of a Banach–valued Sobolev function defined on the whole space Z. We study the problem for Hajłasz– and Newton–Sobolev spaces, respectively. First, we show that Hajłasz–Sobolev extendability holds for real-valued functions if and only if it holds for all 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 socalled metric approximation property.