{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7KUH3C2EH3Z57LNMZPIHTARURG","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":"1d73ab643394b316116cc03602f76e27d25c03617d8e0aba5cbaf9533c6d3c14","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-04T13:27:17Z","title_canon_sha256":"0413d31dd98bd23236cf05e202aa4d61c42398dc2e781427b38f07a578605a5e"},"schema_version":"1.0","source":{"id":"1502.01190","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01190","created_at":"2026-05-18T00:57:44Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01190v1","created_at":"2026-05-18T00:57:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01190","created_at":"2026-05-18T00:57:44Z"},{"alias_kind":"pith_short_12","alias_value":"7KUH3C2EH3Z5","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7KUH3C2EH3Z57LNM","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7KUH3C2E","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:61cf07db4328bf4dc4bc3080f5a26bae22e6c3411e153f7554d66f52e9afa24f","target":"graph","created_at":"2026-05-18T00:57:44Z","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"},"paper":{"abstract_excerpt":"We establish both sufficient and necessary conditions for the validity of the so-called Hardy-Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.","authors_text":"Antti V. V\\\"ah\\\"akangas, Juha Lehrb\\\"ack","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-04T13:27:17Z","title":"In between the inequalities of Sobolev and Hardy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01190","kind":"arxiv","version":1},"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:b2f410cd7a6bb89826f76692740fb03b7a16e1dd997056e7510cb5fbbfec5724","target":"record","created_at":"2026-05-18T00:57:44Z","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":"1d73ab643394b316116cc03602f76e27d25c03617d8e0aba5cbaf9533c6d3c14","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-04T13:27:17Z","title_canon_sha256":"0413d31dd98bd23236cf05e202aa4d61c42398dc2e781427b38f07a578605a5e"},"schema_version":"1.0","source":{"id":"1502.01190","kind":"arxiv","version":1}},"canonical_sha256":"faa87d8b443ef3dfadaccbd079823489ae299640e0b9c7611f5794fa5102a6fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"faa87d8b443ef3dfadaccbd079823489ae299640e0b9c7611f5794fa5102a6fe","first_computed_at":"2026-05-18T00:57:44.451568Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:57:44.451568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hRvSYD6yudLls577NrjfDBFAlm56yVVt9SFY0YCfHChOWJDXaZiNjQvIfOJFWuFwifNKqwTN5vrs3AnoQ8rODQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:57:44.452196Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.01190","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2f410cd7a6bb89826f76692740fb03b7a16e1dd997056e7510cb5fbbfec5724","sha256:61cf07db4328bf4dc4bc3080f5a26bae22e6c3411e153f7554d66f52e9afa24f"],"state_sha256":"a57c00ccf410f77fb54f23061fee9c9bd5ec632c047dde3f00ab8206ba0cf5f3"}