The lattice structure of negative Sobolev and extrapolation spaces
Sahiba Arora, Jochen Glück, Felix L. Schwenninger
TL;DR
This work investigates when the natural positive cone in negative-order Sobolev and extrapolation spaces induces a lattice structure. By developing abstract lattice-transfer theorems for ordered Banach spaces and applying them to Sobolev spaces $W^{-k,p}$ and to extrapolation spaces $X_{-1}$ of positive semigroups, the authors show that the span of the positive cone is a vector lattice (and a Banach lattice under suitable norms) even when the full space is not. The results unify and extend known phenomena from concrete Sobolev spaces to abstract extrapolation settings, yielding concrete lattice properties for negative Sobolev spaces on $\\mathbb{R}^d$ and on domains, as well as for extrapolation spaces arising in positive infinite-dimensional systems. They also identify conditions under which these spans are KB-spaces and discuss open problems related to removing order-continuity assumptions. Collectively, the paper strengthens the toolbox for analyzing order structure in negative-order function spaces and their role in perturbation theory and positive system theory.
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 Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.