{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:SHZFBYUYTUPUSG72AZ3XGXYPNM","short_pith_number":"pith:SHZFBYUY","schema_version":"1.0","canonical_sha256":"91f250e2989d1f491bfa0677735f0f6b17b362aacfce15a0db016e1b1bd19a3b","source":{"kind":"arxiv","id":"1011.5465","version":1},"attestation_state":"computed","paper":{"title":"About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Dolbeault (CEREMADE), Maria J. Esteban (CEREMADE)","submitted_at":"2010-11-24T19:49:35Z","abstract_excerpt":"This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (C"},"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":"1011.5465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-11-24T19:49:35Z","cross_cats_sorted":[],"title_canon_sha256":"227c7e1e1fff25dabe3fcdf7deb5fe81c81a844d90c5b15aa7fc55496dd3dd3d","abstract_canon_sha256":"d2f96242776280e756ed192294b7a9452a861b8010d6263b344cf9456c8c8968"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:47.249567Z","signature_b64":"An9pY3Bplp3DKOoUzNtMPwuPv/gyrAUBeKmfmOsdNYofy3LPuamW+vsHbVSnk1VkPr0M9jdE/MfCCfTe59kdBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91f250e2989d1f491bfa0677735f0f6b17b362aacfce15a0db016e1b1bd19a3b","last_reissued_at":"2026-05-18T04:34:47.249156Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:47.249156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Dolbeault (CEREMADE), Maria J. Esteban (CEREMADE)","submitted_at":"2010-11-24T19:49:35Z","abstract_excerpt":"This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (C"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.5465","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":"1011.5465","created_at":"2026-05-18T04:34:47.249218+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.5465v1","created_at":"2026-05-18T04:34:47.249218+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.5465","created_at":"2026-05-18T04:34:47.249218+00:00"},{"alias_kind":"pith_short_12","alias_value":"SHZFBYUYTUPU","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_16","alias_value":"SHZFBYUYTUPUSG72","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_8","alias_value":"SHZFBYUY","created_at":"2026-05-18T12:26:13.927090+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/SHZFBYUYTUPUSG72AZ3XGXYPNM","json":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM.json","graph_json":"https://pith.science/api/pith-number/SHZFBYUYTUPUSG72AZ3XGXYPNM/graph.json","events_json":"https://pith.science/api/pith-number/SHZFBYUYTUPUSG72AZ3XGXYPNM/events.json","paper":"https://pith.science/paper/SHZFBYUY"},"agent_actions":{"view_html":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM","download_json":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM.json","view_paper":"https://pith.science/paper/SHZFBYUY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.5465&json=true","fetch_graph":"https://pith.science/api/pith-number/SHZFBYUYTUPUSG72AZ3XGXYPNM/graph.json","fetch_events":"https://pith.science/api/pith-number/SHZFBYUYTUPUSG72AZ3XGXYPNM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM/action/storage_attestation","attest_author":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM/action/author_attestation","sign_citation":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM/action/citation_signature","submit_replication":"https://pith.science/pith/SHZFBYUYTUPUSG72AZ3XGXYPNM/action/replication_record"}},"created_at":"2026-05-18T04:34:47.249218+00:00","updated_at":"2026-05-18T04:34:47.249218+00:00"}