{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:ACLW5CILYAMOYAGZP7AVR5JGKE","short_pith_number":"pith:ACLW5CIL","schema_version":"1.0","canonical_sha256":"00976e890bc018ec00d97fc158f52651182f2f4c3ea802cd278957612df01b00","source":{"kind":"arxiv","id":"1907.06380","version":1},"attestation_state":"computed","paper":{"title":"Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Carlo Sbordone, Karl-Mikael Perfekt, Luigi D'Onofrio, Luigi Greco, Roberta Schiattarella","submitted_at":"2019-07-15T09:15:59Z","abstract_excerpt":"Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\\mathcal{B}_\\ast$, the biduality result that $\\mathcal{B}_0^\\ast = \\mathcal{B}_\\ast$ and $\\mathcal{B}_\\ast^\\ast = \\mathcal{B}$, and a formula for the distance from an element $f \\in \\mathcal{B}$ to $"},"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":"1907.06380","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-07-15T09:15:59Z","cross_cats_sorted":[],"title_canon_sha256":"07fadaea5c371e88d9429e49a909b1e3c5420a612debf3b8ec9ca02aa9342e5d","abstract_canon_sha256":"74491e81fddecad9abe0a0c0f7652fd1fb3c09bd0bb138abf80690ca92be009c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:37.776751Z","signature_b64":"qlZ5IHM+tUpUJD8gQDkHoOq3NYO42Wvg4Jz+0tGnzxY70EpQNBuq9qntuJWB1ijUmfWoflgqhLcE3HqdMLLnAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00976e890bc018ec00d97fc158f52651182f2f4c3ea802cd278957612df01b00","last_reissued_at":"2026-05-17T23:40:37.776164Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:37.776164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Carlo Sbordone, Karl-Mikael Perfekt, Luigi D'Onofrio, Luigi Greco, Roberta Schiattarella","submitted_at":"2019-07-15T09:15:59Z","abstract_excerpt":"Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\\mathcal{B}_\\ast$, the biduality result that $\\mathcal{B}_0^\\ast = \\mathcal{B}_\\ast$ and $\\mathcal{B}_\\ast^\\ast = \\mathcal{B}$, and a formula for the distance from an element $f \\in \\mathcal{B}$ to $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.06380","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":"1907.06380","created_at":"2026-05-17T23:40:37.776248+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.06380v1","created_at":"2026-05-17T23:40:37.776248+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.06380","created_at":"2026-05-17T23:40:37.776248+00:00"},{"alias_kind":"pith_short_12","alias_value":"ACLW5CILYAMO","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"ACLW5CILYAMOYAGZ","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"ACLW5CIL","created_at":"2026-05-18T12:33:12.712433+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/ACLW5CILYAMOYAGZP7AVR5JGKE","json":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE.json","graph_json":"https://pith.science/api/pith-number/ACLW5CILYAMOYAGZP7AVR5JGKE/graph.json","events_json":"https://pith.science/api/pith-number/ACLW5CILYAMOYAGZP7AVR5JGKE/events.json","paper":"https://pith.science/paper/ACLW5CIL"},"agent_actions":{"view_html":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE","download_json":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE.json","view_paper":"https://pith.science/paper/ACLW5CIL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.06380&json=true","fetch_graph":"https://pith.science/api/pith-number/ACLW5CILYAMOYAGZP7AVR5JGKE/graph.json","fetch_events":"https://pith.science/api/pith-number/ACLW5CILYAMOYAGZP7AVR5JGKE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE/action/storage_attestation","attest_author":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE/action/author_attestation","sign_citation":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE/action/citation_signature","submit_replication":"https://pith.science/pith/ACLW5CILYAMOYAGZP7AVR5JGKE/action/replication_record"}},"created_at":"2026-05-17T23:40:37.776248+00:00","updated_at":"2026-05-17T23:40:37.776248+00:00"}