Kavli Affiliate: Zheng Zhu
| First 5 Authors: Miguel García-Bravo, Toni Ikonen, Zheng Zhu, ,
| Summary:
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 $nleq p<infty$, 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.
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