Extensions and approximations of Banach-valued Sobolev functions

In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincar\'e 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 Haj{\l}asz- and Newton-Sobolev spaces, respectively. First, we show that Haj{\l}asz-Sobolev extendability is independent of the target Banach spaces. We also show that every $c_0$-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 Poincar\'e 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 $n\leq p<\infty$, every $W^{1,p}$-extension domain $\Omega \subset \mathbb{R}^n$ supports a weak $(1,p)$-Poincar\'e inequality up to some scale. As a related result of independent interest, we prove that in any metric measure space when $1 \leq p < \infty$ and real-valued Lipschitz functions with bounded support are norm-dense in the real-valued $W^{1,p}$-space, then Banach-valued Lipschitz functions with bounded support are energy-dense in every Banach-valued $W^{1,p}$-space whenever the Banach space has the so-called metric approximation property.