{"paper":{"title":"The lattice structure of negative Sobolev and extrapolation spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Felix L. Schwenninger, Jochen Gl\\\"uck, Sahiba Arora","submitted_at":"2024-04-02T17:20:12Z","abstract_excerpt":"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.\n  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 ob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.02116","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.02116/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}