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Onofri had established that $J_\\alpha$ is non-negative on $H^1(S^2)$ provided $\\alpha \\geq 1$. In this note, we show that if $J_\\alpha$ is restricted to those $u\\in H^1(S^2)$ that satisfy the Aubin condition: \\int_{S^2}e^u x_j dw=0\\quad\\text"},"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":"0910.0890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-10-05T23:49:39Z","cross_cats_sorted":[],"title_canon_sha256":"4ac2cc8b3d808f6f8599474037e37dd4e725e66ebd023032c3bac89833fa453f","abstract_canon_sha256":"b3d0b0368f75cc93fda6ab8fe736d47925490a86bbc9f709754aa7836bcfd18d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:11:26.454580Z","signature_b64":"rXWpsm/Lo+TSzkgScru4GC3Wxn7Kbxn/F6Do8PWtb1rU2hYJrJdv4/ThmwDNDY794u5m1pDpaGjHXLJ63LFcAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aad34ef02cd5a7cbf1de3362117d6f576e327be800fced87b4db62049e93025d","last_reissued_at":"2026-05-18T02:11:26.453931Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:11:26.453931Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Best Constant in the Moser-Onofri-Aubin Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Nassif Ghoussoub","submitted_at":"2009-10-05T23:49:39Z","abstract_excerpt":"Let $S^2$ be the 2-dimensional unit sphere and let $J_\\alpha $ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$ J_\\alpha(u) = \\frac{\\alpha}{4}\\int_{S^2}|\\nabla u|^2 d\\omega + \\int_{S^2} u d\\omega -\\ln \\int_{S^2} e^{u} d\\omega, $$ where $d\\omega$ denotes Lebesgue measure on $S^2$, normalized so that $\\int_{S^2} d\\omega = 1$. Onofri had established that $J_\\alpha$ is non-negative on $H^1(S^2)$ provided $\\alpha \\geq 1$. 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