Israel Journal of Mathematics· 2026Q1
The lattice structure of negative Sobolev and extrapolation spaces
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- Q1SCImago
- 2026year
Short summary
The span of the positive cone in negative Sobolev spaces W^{-k,p}(ℝ^d) is proven to be a vector lattice, a property not generally held for positive indices k ≥ 2.
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Key points
- The span of the positive cone in W^{-k,p}(ℝ^d) is a vector lattice for negative indices.
- A partial result is shown for Sobolev spaces on bounded domains.
- An abstract theorem proves vector lattice properties for extrapolation spaces of positive C0-semigroups.
- This work complements existing results in perturbation theory and infinite-dimensional positive systems.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract It is well-known that the Sobolev spaces W k,p (ℝ d ) 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 indices − k and show that the span of the positive cone in W − k,p (ℝ d ) is a vector lattice in this case. On bounded domains Ω ⊆ ℝ d we obtain a partial result in this direction. We also prove a related abstract theorem: if ( T ( t )) t ∈[0,∞) 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.
The authors' abstract, as published at the source. Israel Journal of Mathematics, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics