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Namely, we prove the following result: for any $0 \\leq \\lambda < \\left(\\frac{n-1}n\\right)^n$, then we have $$ \\sup_{\\substack{u\\in C_0^\\infty(\\mathbb H^n) \\int_{\\mathbb H^n} |\\nabla_g u|_g^n d\\text{Vol}_g -\\lambda \\int_{\\mathbb H^n} |u|^n d\\text{ Vol}_g \\leq 1}} \\int_{\\mathbb H^n} \\Phi_n(\\alpha_n |u|^{\\frac{n}{n-1}}) d\\text{ Vol}_g < \\infty, $$ where $\\alpha_n = n \\omega_{n-1}^{\\frac1{n-1}}$, $\\omega_{n-1}$ denotes the surface area of the unit sphere in $\\mathbb R^n$ and $\\Phi_n"},"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":"1709.09608","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-27T16:31:04Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"b6663ecfafeb2e12bdc8d7125ba8af6209de8cbaf312aa2d88b7a04d30605198","abstract_canon_sha256":"b248b6f854f031ae88267e0d7ef9843f1e77a9c7044b9a9676796dd33bb31ceb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:28.508387Z","signature_b64":"TA+dcKVMX7zMFwePpROfO5Tb3P38WPZSsQKElTR8GFjNe9Agjh4SXRQ8qbzL17FCQHOHOx5OF3KEVZD293QbDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1c254425d85780ba14c0a9774eae68513a667e51d5e972639c03a3d19d4c7fb","last_reissued_at":"2026-05-18T00:29:28.507761Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:28.507761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved Moser-Trudinger type inequalities in the hyperbolic space $\\mathbb H^n$","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":"2017-09-27T16:31:04Z","abstract_excerpt":"We establish an improved version of the Moser-Trudinger inequality in the hyperbolic space $\\mathbb H^n$, $n\\geq 2$. 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