{"paper":{"title":"Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","math.SP"],"primary_cat":"math.AP","authors_text":"Amaury Hayat, Emmanuel Tr\\'elat, Vincent Boulard","submitted_at":"2026-07-14T21:30:39Z","abstract_excerpt":"Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-\\gamma/t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set.\n  Let $A$ be a nonnegative self-adjoint operator on sections of a finite-rank Euclidean vector bundle over a doubling metric measure space, satisfying ultracontractivity, Davies-Gaffney estimates (equivalently, finite speed of propagation for the wave equation) and a pointwise local Weyl law. If a weighted integrated observability inequa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13279","kind":"arxiv","version":1},"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/2607.13279/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"}