{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:OH4M3BWII7AD7QLR3MDNUO32EA","short_pith_number":"pith:OH4M3BWI","schema_version":"1.0","canonical_sha256":"71f8cd86c847c03fc171db06da3b7a2021fc089fece9e07a3d812bfb602bccde","source":{"kind":"arxiv","id":"2408.15071","version":1},"attestation_state":"computed","paper":{"title":"Sobolev spaces via chains in metric measure spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Emanuele Caputo, Nicola Cavallucci","submitted_at":"2024-08-27T13:59:12Z","abstract_excerpt":"We define the chain Sobolev space on a possibly non-complete metric measure space in terms of chain upper gradients. In this context, $\\varepsilon$-chains are a finite collection of points with distance at most $\\varepsilon$ between consecutive points. They play the role of discrete versions of curves. Chain upper gradients are defined accordingly and the chain Sobolev space is defined by letting the size parameter $\\varepsilon$ going to zero. In the complete setting, we prove that the chain Sobolev space is equal to the classical notions of Sobolev spaces in terms of relaxation of upper gradi"},"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":"2408.15071","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2024-08-27T13:59:12Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"aa86c8938c67f317eaed7e58f5eb637ac9f100a5ff14ba195b4684e84e72d84d","abstract_canon_sha256":"8d5c7bce9b243b050aec10708ea0902e930757a3783e45f615b29b7ce7898c8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:48:45.921148Z","signature_b64":"rmtgK91X5/nOrcIKKn37JlhzSPQcc2KOvjL3Ci8x/u69KGiodH4eXIXtMs5DGHC7viREfOBjvGfEmnoZmWFsBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71f8cd86c847c03fc171db06da3b7a2021fc089fece9e07a3d812bfb602bccde","last_reissued_at":"2026-07-05T11:48:45.920705Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:48:45.920705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sobolev spaces via chains in metric measure spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Emanuele Caputo, Nicola Cavallucci","submitted_at":"2024-08-27T13:59:12Z","abstract_excerpt":"We define the chain Sobolev space on a possibly non-complete metric measure space in terms of chain upper gradients. In this context, $\\varepsilon$-chains are a finite collection of points with distance at most $\\varepsilon$ between consecutive points. They play the role of discrete versions of curves. Chain upper gradients are defined accordingly and the chain Sobolev space is defined by letting the size parameter $\\varepsilon$ going to zero. In the complete setting, we prove that the chain Sobolev space is equal to the classical notions of Sobolev spaces in terms of relaxation of upper gradi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.15071","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.15071/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":"2408.15071","created_at":"2026-07-05T11:48:45.920761+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.15071v1","created_at":"2026-07-05T11:48:45.920761+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.15071","created_at":"2026-07-05T11:48:45.920761+00:00"},{"alias_kind":"pith_short_12","alias_value":"OH4M3BWII7AD","created_at":"2026-07-05T11:48:45.920761+00:00"},{"alias_kind":"pith_short_16","alias_value":"OH4M3BWII7AD7QLR","created_at":"2026-07-05T11:48:45.920761+00:00"},{"alias_kind":"pith_short_8","alias_value":"OH4M3BWI","created_at":"2026-07-05T11:48:45.920761+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/OH4M3BWII7AD7QLR3MDNUO32EA","json":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA.json","graph_json":"https://pith.science/api/pith-number/OH4M3BWII7AD7QLR3MDNUO32EA/graph.json","events_json":"https://pith.science/api/pith-number/OH4M3BWII7AD7QLR3MDNUO32EA/events.json","paper":"https://pith.science/paper/OH4M3BWI"},"agent_actions":{"view_html":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA","download_json":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA.json","view_paper":"https://pith.science/paper/OH4M3BWI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.15071&json=true","fetch_graph":"https://pith.science/api/pith-number/OH4M3BWII7AD7QLR3MDNUO32EA/graph.json","fetch_events":"https://pith.science/api/pith-number/OH4M3BWII7AD7QLR3MDNUO32EA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA/action/storage_attestation","attest_author":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA/action/author_attestation","sign_citation":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA/action/citation_signature","submit_replication":"https://pith.science/pith/OH4M3BWII7AD7QLR3MDNUO32EA/action/replication_record"}},"created_at":"2026-07-05T11:48:45.920761+00:00","updated_at":"2026-07-05T11:48:45.920761+00:00"}