{"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 $\\Omega$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08249","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"}