The lattice structure of negative Sobolev and extrapolation spaces

It is well-known that the Sobolev spaces $W^{k,p}({\mathbb{R}}^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k\in \{0,1\}$, but not if $k\ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}({\mathbb{R}}^d)$ is a vector lattice in this case. We also prove a related abstract result: if $(T(t){)}_{t\in [0,\mathrm{\infty })}$ is a positive $C_0$-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 Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.