{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:5Q7RUQFKQ4T3JE4ZKWBKKYD6KA","short_pith_number":"pith:5Q7RUQFK","schema_version":"1.0","canonical_sha256":"ec3f1a40aa8727b493995582a5607e501f6bbc52fabade5b2bdccd5abbde33ca","source":{"kind":"arxiv","id":"1601.04873","version":1},"attestation_state":"computed","paper":{"title":"Four proofs of cocompacness for Sobolev embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Cyril Tintarev","submitted_at":"2016-01-19T11:06:05Z","abstract_excerpt":"Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some non-compact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the domain space) have subsequences that can be represented as a sum of a well-structured \"bubble decomposition\" (or defect of compactness) plus a remainder vanishing in the target space. This note is an exposition of different proofs of cocompactness for Sobolev-type embeddings, which employ methods of classical PDE, potential theory, and harmonic analysis."},"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":"1601.04873","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-19T11:06:05Z","cross_cats_sorted":[],"title_canon_sha256":"a8defb4d716c2c49297e7d7ddef4fc03ddb740a343bf7f9c96c3c07c60d5ed20","abstract_canon_sha256":"d60860bfff41ba9bdcb9d3c01eca9a35f304bf3e3ca65aef2aa605c71ca2f4b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:31.650665Z","signature_b64":"ja37AFmA1D1pDKvbdcA648f+dEQ+SY4MEy7mAWplhytuQSI0aVKknN69PXJ4DaY93iMDOeO3ivh/aCy3G8ciDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec3f1a40aa8727b493995582a5607e501f6bbc52fabade5b2bdccd5abbde33ca","last_reissued_at":"2026-05-18T01:22:31.650228Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:31.650228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Four proofs of cocompacness for Sobolev embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Cyril Tintarev","submitted_at":"2016-01-19T11:06:05Z","abstract_excerpt":"Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some non-compact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the domain space) have subsequences that can be represented as a sum of a well-structured \"bubble decomposition\" (or defect of compactness) plus a remainder vanishing in the target space. This note is an exposition of different proofs of cocompactness for Sobolev-type embeddings, which employ methods of classical PDE, potential theory, and harmonic analysis."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04873","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":"1601.04873","created_at":"2026-05-18T01:22:31.650291+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.04873v1","created_at":"2026-05-18T01:22:31.650291+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04873","created_at":"2026-05-18T01:22:31.650291+00:00"},{"alias_kind":"pith_short_12","alias_value":"5Q7RUQFKQ4T3","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"5Q7RUQFKQ4T3JE4Z","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"5Q7RUQFK","created_at":"2026-05-18T12:30:01.593930+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/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA","json":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA.json","graph_json":"https://pith.science/api/pith-number/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/graph.json","events_json":"https://pith.science/api/pith-number/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/events.json","paper":"https://pith.science/paper/5Q7RUQFK"},"agent_actions":{"view_html":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA","download_json":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA.json","view_paper":"https://pith.science/paper/5Q7RUQFK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.04873&json=true","fetch_graph":"https://pith.science/api/pith-number/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/graph.json","fetch_events":"https://pith.science/api/pith-number/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/action/storage_attestation","attest_author":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/action/author_attestation","sign_citation":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/action/citation_signature","submit_replication":"https://pith.science/pith/5Q7RUQFKQ4T3JE4ZKWBKKYD6KA/action/replication_record"}},"created_at":"2026-05-18T01:22:31.650291+00:00","updated_at":"2026-05-18T01:22:31.650291+00:00"}