{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZUZ5ONKNVWALJFT3ATPXIJLKLV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0df8d356878896a9d2a8b50f1c34ca6f47e8761a55783d6e373ef493bafcbf6a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-09-27T17:44:30Z","title_canon_sha256":"0bfe82ee28d5b16b0875d272b99ed62a1f3bf34ecf28a39ef5360d5d2fcc536d"},"schema_version":"1.0","source":{"id":"1909.12818","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.12818","created_at":"2026-07-05T01:45:18Z"},{"alias_kind":"arxiv_version","alias_value":"1909.12818v3","created_at":"2026-07-05T01:45:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.12818","created_at":"2026-07-05T01:45:18Z"},{"alias_kind":"pith_short_12","alias_value":"ZUZ5ONKNVWAL","created_at":"2026-07-05T01:45:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZUZ5ONKNVWALJFT3","created_at":"2026-07-05T01:45:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZUZ5ONKN","created_at":"2026-07-05T01:45:18Z"}],"graph_snapshots":[{"event_id":"sha256:844b22a68ca818886229b6b0c01c27bfcc7c290d76e1ea474b7b1cacc6eb2eb3","target":"graph","created_at":"2026-07-05T01:45:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1909.12818/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Oscar Dom\\'inguez, Sergey Tikhonov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-09-27T17:44:30Z","title":"Sobolev embeddings, extrapolations, and related inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.12818","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4a8ed8cfa3889c6ee4a2b8b5f0fd9641d03aeb1d5043f667a9d9b4074797be7c","target":"record","created_at":"2026-07-05T01:45:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0df8d356878896a9d2a8b50f1c34ca6f47e8761a55783d6e373ef493bafcbf6a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-09-27T17:44:30Z","title_canon_sha256":"0bfe82ee28d5b16b0875d272b99ed62a1f3bf34ecf28a39ef5360d5d2fcc536d"},"schema_version":"1.0","source":{"id":"1909.12818","kind":"arxiv","version":3}},"canonical_sha256":"cd33d7354dad80b4967b04df74256a5d67a6d900b1d2968010e5ddb518e23f3b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd33d7354dad80b4967b04df74256a5d67a6d900b1d2968010e5ddb518e23f3b","first_computed_at":"2026-07-05T01:45:18.721391Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:45:18.721391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t1NcMZeioiZKkbPVIgkeWiZ9S11TNKPDK330EMpwPBV/pB/7BaiXuydljbB7CO2hUIdtDRJIPUh2AXa/ut1fCw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:45:18.721812Z","signed_message":"canonical_sha256_bytes"},"source_id":"1909.12818","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a8ed8cfa3889c6ee4a2b8b5f0fd9641d03aeb1d5043f667a9d9b4074797be7c","sha256:844b22a68ca818886229b6b0c01c27bfcc7c290d76e1ea474b7b1cacc6eb2eb3"],"state_sha256":"0aab26161947346f1cae941f7320f9fb4214621b67174d6540286a22fc9eba16"}