The lattice structure of negative Sobolev and extrapolation spaces

It is well-known that the Sobolev spaces Wk,p( Rᵈ) are vector lattices with respect to the pointwise almost everywhere order if k ∈ \0,1\, but not if k ≥ 2. In this note, we consider negative k and show that the span of the positive cone in Wk,p( Rᵈ) is a vector lattice in this case. We also prove a related abstract result: if (T(t))t ∈ [0,∞) is a positive C₀-semigroup on a Banach lattice X with order continuous norm, then the span of the cone X-1,+ in the extrapolation space X₋₁ is a vector lattice. This complements results obtained by B\'atkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.