Uniform norm comparability for Banach-valued M^{1,p} and W^{1,p} spaces

Determine whether, for exponents p ∈ [1,∞), and any Banach space V, the equivalence M^{1,p}(Z; V) = W^{1,p}(Z; V) with comparable norms can be established with constants independent of V whenever the real-valued spaces satisfy M^{1,p}(Z) = W^{1,p}(Z).

Background

For p = ∞, the paper shows that if M^{1,∞}(Z) = W^{1,∞}(Z), then the Banach-valued versions coincide with uniform (Banach-space-independent) norm comparability, leveraging a reduction via linear embeddings/projections. The authors note that it is unclear whether the same uniformity extends to finite p.

A resolution would clarify the robustness of Banach-valued Sobolev theory across integrability scales and determine whether vector-valued norms introduce unavoidable dependence on the target Banach space when p < ∞.