The lattice structure of negative Sobolev and extrapolation spaces

TL;DR

This paper investigates the lattice structure of negative Sobolev and extrapolation spaces, revealing that their positive cones span vector lattices, which extends the understanding of positivity in infinite-dimensional analysis.

Contribution

It demonstrates that the span of the positive cone in negative Sobolev spaces forms a vector lattice, and generalizes this to extrapolation spaces of positive semigroups on Banach lattices.

Findings

The span of the positive cone in negative Sobolev spaces is a vector lattice.

The span of the cone in extrapolation spaces of positive semigroups is a vector lattice.

Results extend the theory of positivity in infinite-dimensional spaces.

Abstract

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 $(T(t))_{t \in [0,\infty)}$ 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 B\'atkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.