{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:FLTURINNYOYH3QS5WVCNRMACYL","short_pith_number":"pith:FLTURINN","schema_version":"1.0","canonical_sha256":"2ae748a1adc3b07dc25db544d8b002c2ef0322054c175b3985e352d67591130f","source":{"kind":"arxiv","id":"2211.14185","version":4},"attestation_state":"computed","paper":{"title":"Stability for the Sobolev inequality: existence of a minimizer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Tobias K\\\"onig","submitted_at":"2022-11-25T15:39:37Z","abstract_excerpt":"We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\\mathcal M$ and given by \\[ \\frac{\\|\\nabla f\\|_{L^2(\\mathbb R^d)}^2 - S_d \\|f\\|_{L^\\frac{2d}{d-2}(\\mathbb R^d)}^2}{ \\inf_{h \\in \\mathcal M} \\|\\nabla (f - h)\\|_{L^2(\\mathbb R^d)}^2 } \\geq c_{BE} > 0 \\qquad \\text{ for every } f \\in \\dot{H}^1(\\mathbb R^d),\\] which is due to Bianchi and Egnell, admits a minimizer for every $d \\geq 3$. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality $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":"2211.14185","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-11-25T15:39:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"dfefe0c6aedc5a57eeaab7cc6c8894f8f1afe8b77790b27b07fc16638239496d","abstract_canon_sha256":"ec9fcdc314dab6a030bdad869828e7ccc0237d1ae58daac0f0b788c139b2b696"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:57:51.401368Z","signature_b64":"dMjA+HUr4RnEKAjQpUeURBFovkwyEy20zIWmlKIYxHl++zdiDTXN0L2H6u0bcxtv7oWgSQI2siQ5oDWwhCphDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ae748a1adc3b07dc25db544d8b002c2ef0322054c175b3985e352d67591130f","last_reissued_at":"2026-07-05T06:57:51.400897Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:57:51.400897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stability for the Sobolev inequality: existence of a minimizer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Tobias K\\\"onig","submitted_at":"2022-11-25T15:39:37Z","abstract_excerpt":"We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\\mathcal M$ and given by \\[ \\frac{\\|\\nabla f\\|_{L^2(\\mathbb R^d)}^2 - S_d \\|f\\|_{L^\\frac{2d}{d-2}(\\mathbb R^d)}^2}{ \\inf_{h \\in \\mathcal M} \\|\\nabla (f - h)\\|_{L^2(\\mathbb R^d)}^2 } \\geq c_{BE} > 0 \\qquad \\text{ for every } f \\in \\dot{H}^1(\\mathbb R^d),\\] which is due to Bianchi and Egnell, admits a minimizer for every $d \\geq 3$. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality $c_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.14185","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.14185/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2211.14185","created_at":"2026-07-05T06:57:51.400952+00:00"},{"alias_kind":"arxiv_version","alias_value":"2211.14185v4","created_at":"2026-07-05T06:57:51.400952+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.14185","created_at":"2026-07-05T06:57:51.400952+00:00"},{"alias_kind":"pith_short_12","alias_value":"FLTURINNYOYH","created_at":"2026-07-05T06:57:51.400952+00:00"},{"alias_kind":"pith_short_16","alias_value":"FLTURINNYOYH3QS5","created_at":"2026-07-05T06:57:51.400952+00:00"},{"alias_kind":"pith_short_8","alias_value":"FLTURINN","created_at":"2026-07-05T06:57:51.400952+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.25692","citing_title":"The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates","ref_index":70,"is_internal_anchor":false},{"citing_arxiv_id":"2604.16791","citing_title":"Log-Sobolev and Beckner inequalities and stability of Poincar\\'e inequality with weighted Gaussian measures","ref_index":46,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL","json":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL.json","graph_json":"https://pith.science/api/pith-number/FLTURINNYOYH3QS5WVCNRMACYL/graph.json","events_json":"https://pith.science/api/pith-number/FLTURINNYOYH3QS5WVCNRMACYL/events.json","paper":"https://pith.science/paper/FLTURINN"},"agent_actions":{"view_html":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL","download_json":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL.json","view_paper":"https://pith.science/paper/FLTURINN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2211.14185&json=true","fetch_graph":"https://pith.science/api/pith-number/FLTURINNYOYH3QS5WVCNRMACYL/graph.json","fetch_events":"https://pith.science/api/pith-number/FLTURINNYOYH3QS5WVCNRMACYL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL/action/storage_attestation","attest_author":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL/action/author_attestation","sign_citation":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL/action/citation_signature","submit_replication":"https://pith.science/pith/FLTURINNYOYH3QS5WVCNRMACYL/action/replication_record"}},"created_at":"2026-07-05T06:57:51.400952+00:00","updated_at":"2026-07-05T06:57:51.400952+00:00"}