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For $\\alpha \\in [0,\\lambda_1(\\Omega))$, we define $\\|u\\|_{2,\\alpha}^2 = \\|\\Delta u\\|_2^2 - \\alpha \\|u\\|_2^2$, for $u \\in H_0^2(\\Omega)$. In this paper, we will prove the following inequality \\[ \\sup_{u\\in H_0^2(\\Omega),\\, \\|u\\|_{2,\\alpha} \\leq 1} \\int_{\\Omega} e^{32 \\pi^2 u(x)^2} dx < \\infty. \\] This strengthens a recent result of L"},"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":"1701.08249","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-01-28T04:55:09Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"946074c644a6b51b46c3bd8e14fa63b109709157aeeb463b02c3088616c6c7c7","abstract_canon_sha256":"65e381cfb61a66f267028fa7f635de6fc7df00634d898f3353535a2796bd8884"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:54.778975Z","signature_b64":"tC0wnHD/qGIerMnZJ7qvW722CZOuqP+EZFWoaGIZBu7fQB02jE7kqM6UTD+0THlD/DgCU6EKc+4t0KjyJnP3BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70910e5444a439c5fc51ddee3015cede20a1483fd5e491ebf4b90088e51404b1","last_reissued_at":"2026-05-18T00:51:54.778270Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:54.778270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp Adams inequality in dimension four and its extremal functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Van Hoang Nguyen","submitted_at":"2017-01-28T04:55:09Z","abstract_excerpt":"Let $\\Omega$ be a smooth oriented bounded domain in $\\mathbb R^4$, $H_0^2(\\Omega)$ be the Sobolev space, and $\\lambda_1(\\Omega)= \\inf \\{\\|\\Delta u\\|_2^2 : u\\in H_0^2(\\Omega), \\|u\\|_2 =1\\}$ be the first eigenvalue of the bi-Laplacian operator $\\Delta^2$ on $\\Omega$. 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