{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:UCDPAQTZKN6VPRYCNZW2TF6XZJ","short_pith_number":"pith:UCDPAQTZ","schema_version":"1.0","canonical_sha256":"a086f04279537d57c7026e6da997d7ca7fdbb8b3a470e12c426630eeae9c8a22","source":{"kind":"arxiv","id":"1504.00433","version":1},"attestation_state":"computed","paper":{"title":"Existence of extremal functions for a family of Caffarelli-Kohn-Nirenberg inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-02T02:49:35Z","abstract_excerpt":"Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\\it (Compositio Mathematica,1984):} $$\\Big(\\int_\\Omega \\frac{|u|^r}{|x|^s}dx\\Big)^{\\frac{1}{r}}\\leq C(p,q,r,\\mu,\\sigma,s)\\Big(\\int_\\Omega \\frac{|\\nabla u|^p}{|x|^\\mu}dx\\Big)^{\\frac{a}{p}}\\Big(\\int_\\Omega \\frac{|u|^q}{|x|^\\sigma}dx\\Big)^{\\frac{1-a}{q}},$$ where $\\Omega \\subset \\R^N (N\\geq 2)$ is an open set; $p, q, r, \\mu, \\sigma, s, a$ are some parameters satisfying some balanced conditions. When $\\Omega$ is a cone in $\\R^N$ (for example, $\\Omega=\\R^N)$, we prove the sharp constant $C(p,q,r,\\mu,\\sigma,s)$ can be achieve"},"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":"1504.00433","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-02T02:49:35Z","cross_cats_sorted":[],"title_canon_sha256":"3ed162aa9256edc09057d927f26dcba15e65d4052d92536c141fd1564bd8ba17","abstract_canon_sha256":"1f5015473dcc395d1455ac8baff181b89143edc6453ce0f93af3194f9a2e9209"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:43.324519Z","signature_b64":"dwJW/ErrDljsy1mZEn+2+FiFmxDuuh6J4AG6TcuzrvKGWHTT78SPJhyayi1PINzxvP+ik+7fvsL1QUXX7n2ACA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a086f04279537d57c7026e6da997d7ca7fdbb8b3a470e12c426630eeae9c8a22","last_reissued_at":"2026-05-18T02:19:43.324089Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:43.324089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence of extremal functions for a family of Caffarelli-Kohn-Nirenberg inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-02T02:49:35Z","abstract_excerpt":"Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\\it (Compositio Mathematica,1984):} $$\\Big(\\int_\\Omega \\frac{|u|^r}{|x|^s}dx\\Big)^{\\frac{1}{r}}\\leq C(p,q,r,\\mu,\\sigma,s)\\Big(\\int_\\Omega \\frac{|\\nabla u|^p}{|x|^\\mu}dx\\Big)^{\\frac{a}{p}}\\Big(\\int_\\Omega \\frac{|u|^q}{|x|^\\sigma}dx\\Big)^{\\frac{1-a}{q}},$$ where $\\Omega \\subset \\R^N (N\\geq 2)$ is an open set; $p, q, r, \\mu, \\sigma, s, a$ are some parameters satisfying some balanced conditions. When $\\Omega$ is a cone in $\\R^N$ (for example, $\\Omega=\\R^N)$, we prove the sharp constant $C(p,q,r,\\mu,\\sigma,s)$ can be achieve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00433","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":"1504.00433","created_at":"2026-05-18T02:19:43.324144+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.00433v1","created_at":"2026-05-18T02:19:43.324144+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00433","created_at":"2026-05-18T02:19:43.324144+00:00"},{"alias_kind":"pith_short_12","alias_value":"UCDPAQTZKN6V","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UCDPAQTZKN6VPRYC","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UCDPAQTZ","created_at":"2026-05-18T12:29:44.643036+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/UCDPAQTZKN6VPRYCNZW2TF6XZJ","json":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ.json","graph_json":"https://pith.science/api/pith-number/UCDPAQTZKN6VPRYCNZW2TF6XZJ/graph.json","events_json":"https://pith.science/api/pith-number/UCDPAQTZKN6VPRYCNZW2TF6XZJ/events.json","paper":"https://pith.science/paper/UCDPAQTZ"},"agent_actions":{"view_html":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ","download_json":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ.json","view_paper":"https://pith.science/paper/UCDPAQTZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.00433&json=true","fetch_graph":"https://pith.science/api/pith-number/UCDPAQTZKN6VPRYCNZW2TF6XZJ/graph.json","fetch_events":"https://pith.science/api/pith-number/UCDPAQTZKN6VPRYCNZW2TF6XZJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ/action/storage_attestation","attest_author":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ/action/author_attestation","sign_citation":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ/action/citation_signature","submit_replication":"https://pith.science/pith/UCDPAQTZKN6VPRYCNZW2TF6XZJ/action/replication_record"}},"created_at":"2026-05-18T02:19:43.324144+00:00","updated_at":"2026-05-18T02:19:43.324144+00:00"}