{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:ZUZ5ONKNVWALJFT3ATPXIJLKLV","short_pith_number":"pith:ZUZ5ONKN","schema_version":"1.0","canonical_sha256":"cd33d7354dad80b4967b04df74256a5d67a6d900b1d2968010e5ddb518e23f3b","source":{"kind":"arxiv","id":"1909.12818","version":3},"attestation_state":"computed","paper":{"title":"Sobolev embeddings, extrapolations, and related inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Oscar Dom\\'inguez, Sergey Tikhonov","submitted_at":"2019-09-27T17:44:30Z","abstract_excerpt":"In this paper we propose a unified approach, based on limiting interpolation, to investigate the embeddings for the Sobolev space $(\\dot{W}^k_p(\\mathcal{X}))_0, \\, \\mathcal{X} \\in \\{\\mathbb{R}^d, \\mathbb{T}^d, \\Omega\\}$, in the subcritical case ($k < d/p$), critical case ($k = d/p$) and supercritical case ($k > d/p$). We characterize the Sobolev embeddings in terms of pointwise inequalities involving rearrangements and moduli of smoothness/derivatives of functions and via extrapolation theorems for corresponding smooth function spaces. Applications include Ulyanov-Kolyada type inequalities for"},"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":"1909.12818","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-09-27T17:44:30Z","cross_cats_sorted":[],"title_canon_sha256":"0bfe82ee28d5b16b0875d272b99ed62a1f3bf34ecf28a39ef5360d5d2fcc536d","abstract_canon_sha256":"0df8d356878896a9d2a8b50f1c34ca6f47e8761a55783d6e373ef493bafcbf6a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:45:18.721812Z","signature_b64":"t1NcMZeioiZKkbPVIgkeWiZ9S11TNKPDK330EMpwPBV/pB/7BaiXuydljbB7CO2hUIdtDRJIPUh2AXa/ut1fCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd33d7354dad80b4967b04df74256a5d67a6d900b1d2968010e5ddb518e23f3b","last_reissued_at":"2026-07-05T01:45:18.721391Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:45:18.721391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sobolev embeddings, extrapolations, and related inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Oscar Dom\\'inguez, Sergey Tikhonov","submitted_at":"2019-09-27T17:44:30Z","abstract_excerpt":"In this paper we propose a unified approach, based on limiting interpolation, to investigate the embeddings for the Sobolev space $(\\dot{W}^k_p(\\mathcal{X}))_0, \\, \\mathcal{X} \\in \\{\\mathbb{R}^d, \\mathbb{T}^d, \\Omega\\}$, in the subcritical case ($k < d/p$), critical case ($k = d/p$) and supercritical case ($k > d/p$). We characterize the Sobolev embeddings in terms of pointwise inequalities involving rearrangements and moduli of smoothness/derivatives of functions and via extrapolation theorems for corresponding smooth function spaces. Applications include Ulyanov-Kolyada type inequalities for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.12818","kind":"arxiv","version":3},"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/1909.12818/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":"1909.12818","created_at":"2026-07-05T01:45:18.721446+00:00"},{"alias_kind":"arxiv_version","alias_value":"1909.12818v3","created_at":"2026-07-05T01:45:18.721446+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.12818","created_at":"2026-07-05T01:45:18.721446+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZUZ5ONKNVWAL","created_at":"2026-07-05T01:45:18.721446+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZUZ5ONKNVWALJFT3","created_at":"2026-07-05T01:45:18.721446+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZUZ5ONKN","created_at":"2026-07-05T01:45:18.721446+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.16110","citing_title":"Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces","ref_index":33,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV","json":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV.json","graph_json":"https://pith.science/api/pith-number/ZUZ5ONKNVWALJFT3ATPXIJLKLV/graph.json","events_json":"https://pith.science/api/pith-number/ZUZ5ONKNVWALJFT3ATPXIJLKLV/events.json","paper":"https://pith.science/paper/ZUZ5ONKN"},"agent_actions":{"view_html":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV","download_json":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV.json","view_paper":"https://pith.science/paper/ZUZ5ONKN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1909.12818&json=true","fetch_graph":"https://pith.science/api/pith-number/ZUZ5ONKNVWALJFT3ATPXIJLKLV/graph.json","fetch_events":"https://pith.science/api/pith-number/ZUZ5ONKNVWALJFT3ATPXIJLKLV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV/action/storage_attestation","attest_author":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV/action/author_attestation","sign_citation":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV/action/citation_signature","submit_replication":"https://pith.science/pith/ZUZ5ONKNVWALJFT3ATPXIJLKLV/action/replication_record"}},"created_at":"2026-07-05T01:45:18.721446+00:00","updated_at":"2026-07-05T01:45:18.721446+00:00"}