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Precisely, let $\\mathbb{B}$ be the unit ball in $\\mathbb{R}^N$ $(N\\geq 2)$, $p>1$, $g=|x|^{\\frac{2p}{N}\\beta}(dx_1^2+\\cdots+dx_N^2)$ be a conical metric on $\\mathbb{B}$, and $\\lambda_p(\\mathbb{B})=\\inf\\left\\{\\int_\\mathbb{B}|\\nabla u|^Ndx: u\\in W_0^{1,N}(\\mathbb{B}),\\,\\int_\\mathbb{B}|u|^pdx=1\\right\\}$. We prove that for any $\\beta\\geq 0$ and $\\alpha<(1+\\frac{p}{N}\\beta)^{N-1+\\frac{N}{p}}\\lambda_p(\\mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $u\\in W_0^{1,N}(\\mathbb"},"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":"1808.05316","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2018-08-16T00:46:09Z","cross_cats_sorted":[],"title_canon_sha256":"61e600fb14e95f60f0efc7d16cbe226e45630639c98d46966a7dae6642086352","abstract_canon_sha256":"7a76adea8ce41c0bcdba97d4dff8f31f6dfd0a1a1c2584c1b27e895da7b8481b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:57.574572Z","signature_b64":"q6cwvZ6AS+X+JCg0kbY6naXFj8mVIZw6leCF7FiLjtqPNOVPwHSyV5PJEZvsUlTfrqQ5IKe7Q2hvjiQef5yfBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f23bd31094da0d1f45cd5db74d6a2b0727582b74d01a05f9bd9d72d24b872463","last_reissued_at":"2026-05-18T00:07:57.573880Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:57.573880Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Trudinger-Moser inequality for conical metric in the unit ball","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaobao Zhu, Yunyan Yang","submitted_at":"2018-08-16T00:46:09Z","abstract_excerpt":"In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $\\mathbb{B}$ be the unit ball in $\\mathbb{R}^N$ $(N\\geq 2)$, $p>1$, $g=|x|^{\\frac{2p}{N}\\beta}(dx_1^2+\\cdots+dx_N^2)$ be a conical metric on $\\mathbb{B}$, and $\\lambda_p(\\mathbb{B})=\\inf\\left\\{\\int_\\mathbb{B}|\\nabla u|^Ndx: u\\in W_0^{1,N}(\\mathbb{B}),\\,\\int_\\mathbb{B}|u|^pdx=1\\right\\}$. We prove that for any $\\beta\\geq 0$ and $\\alpha<(1+\\frac{p}{N}\\beta)^{N-1+\\frac{N}{p}}\\lambda_p(\\mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $u\\in W_0^{1,N}(\\mathbb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05316","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.05316","created_at":"2026-05-18T00:07:57.573981+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.05316v1","created_at":"2026-05-18T00:07:57.573981+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.05316","created_at":"2026-05-18T00:07:57.573981+00:00"},{"alias_kind":"pith_short_12","alias_value":"6I55GEEU3IGR","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"6I55GEEU3IGR6RON","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"6I55GEEU","created_at":"2026-05-18T12:32:08.215937+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4","json":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4.json","graph_json":"https://pith.science/api/pith-number/6I55GEEU3IGR6RONLW3U22RLA4/graph.json","events_json":"https://pith.science/api/pith-number/6I55GEEU3IGR6RONLW3U22RLA4/events.json","paper":"https://pith.science/paper/6I55GEEU"},"agent_actions":{"view_html":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4","download_json":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4.json","view_paper":"https://pith.science/paper/6I55GEEU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.05316&json=true","fetch_graph":"https://pith.science/api/pith-number/6I55GEEU3IGR6RONLW3U22RLA4/graph.json","fetch_events":"https://pith.science/api/pith-number/6I55GEEU3IGR6RONLW3U22RLA4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4/action/storage_attestation","attest_author":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4/action/author_attestation","sign_citation":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4/action/citation_signature","submit_replication":"https://pith.science/pith/6I55GEEU3IGR6RONLW3U22RLA4/action/replication_record"}},"created_at":"2026-05-18T00:07:57.573981+00:00","updated_at":"2026-05-18T00:07:57.573981+00:00"}