Banach-valued versus real-valued W^{1,p} extension domains for low integrability

Ascertain whether, for exponents p ∈ [1,n), a domain Ω ⊂ R^n is a Banach-valued W^{1,p}-extension domain for every Banach space V if and only if it is a real-valued W^{1,p}-extension domain.

Background

The authors recall their recent work on Banach-valued W^{1,p}-extension domains, where they established that for p ≥ n, a domain is a Banach-valued extension domain if and only if it is a real-valued extension domain. They highlight that the corresponding equivalence for the range p ∈ [1,n) has not been settled.

Resolving this would align the Banach-valued extension theory for low integrability exponents with the established high-integrability case and clarify whether vector-valued extension properties introduce genuinely new geometric constraints at low p.