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We assume that the Gaussian Sobolev embedding $H^1(\\Omega,\\gammaN)\\hookrightarrow L^2(\\Omega,\\gammaN)$ is compact; a sufficient condition is the existence of a bounded Gaussian Sobolev extension operator from $\\Omega$ to $\\R^N$. Denote by \\[ 0=\\mu_0(\\Omega)<\\mu_1(\\Omega)\\leq\\mu_2(\\Omega)\\leq\\cdots \\] the Neumann eigenvalues of the positive Ornstein--Uhlenbeck operator $-\\Delta+x\\cdot\\nabla$ in $\\Omega$. We prove the sharp recip"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.28328","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2026-07-30T14:58:34Z","cross_cats_sorted":[],"title_canon_sha256":"424a416e6a1960c55c8de79b8056b315d9e76ddc5906c802ea84f7256be07180","abstract_canon_sha256":"984d762d460e2b6313d853d7721393ed2575f1da30738c94e1b4007ffe3309ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11802fbfa79bb49d0f5ff09dee4fdb41fa27b30437493feda6d3f454f6b37aeb","last_reissued_at":"2026-07-31T01:37:14.994796Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-31T01:37:14.994796Z"},"graph_snapshot":{"paper":{"title":"A sharp Gaussian harmonic-mean inequality for Neumann eigenvalues of the Ornstein-Uhlenbeck operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Francesco Chiacchio","submitted_at":"2026-07-30T14:58:34Z","abstract_excerpt":"Let $N\\geq2$ and let $\\Omega\\subset\\R^N$ be a connected Lipschitz domain, possibly unbounded, symmetric with respect to the origin, and such that $0<\\gammaN(\\Omega)<1$. We assume that the Gaussian Sobolev embedding $H^1(\\Omega,\\gammaN)\\hookrightarrow L^2(\\Omega,\\gammaN)$ is compact; a sufficient condition is the existence of a bounded Gaussian Sobolev extension operator from $\\Omega$ to $\\R^N$. Denote by \\[ 0=\\mu_0(\\Omega)<\\mu_1(\\Omega)\\leq\\mu_2(\\Omega)\\leq\\cdots \\] the Neumann eigenvalues of the positive Ornstein--Uhlenbeck operator $-\\Delta+x\\cdot\\nabla$ in $\\Omega$. 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