{"paper":{"title":"Infinitesimal Hilbertianity of locally CAT($\\kappa$)-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Elefterios Soultanis, Enrico Pasqualetto, Nicola Gigli, Simone Di Marino","submitted_at":"2018-12-05T16:21:26Z","abstract_excerpt":"We show that, given a metric space $(Y,d)$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $\\mu$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,\\mu)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}(Y,d,\\mu)$ is a Hilbert space.\n  The result is obtained by constructing an isometric embedding of the `abstract and analytical' space of derivations into the `concrete and geometrical' bundle whose fibre at $x\\in Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $x"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.02086","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":""},"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"}