{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4XSLMY23S6OOSXFBYMFBLF77QA","short_pith_number":"pith:4XSLMY23","canonical_record":{"source":{"id":"1611.04724","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-11-15T07:36:36Z","cross_cats_sorted":[],"title_canon_sha256":"3ded740fbc4cfba6f19d74e36e16354cff8b7fe50cc3fb9d5d9add8c5b90d495","abstract_canon_sha256":"fb801ab3a78aee5f0347443da1b138edcc0e1abf808b665c80d950a645c87e35"},"schema_version":"1.0"},"canonical_sha256":"e5e4b6635b979ce95ca1c30a1597ff80355b9ead93580e352a9e6b785964c13b","source":{"kind":"arxiv","id":"1611.04724","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04724","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04724v2","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04724","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"pith_short_12","alias_value":"4XSLMY23S6OO","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4XSLMY23S6OOSXFB","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4XSLMY23","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4XSLMY23S6OOSXFBYMFBLF77QA","target":"record","payload":{"canonical_record":{"source":{"id":"1611.04724","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-11-15T07:36:36Z","cross_cats_sorted":[],"title_canon_sha256":"3ded740fbc4cfba6f19d74e36e16354cff8b7fe50cc3fb9d5d9add8c5b90d495","abstract_canon_sha256":"fb801ab3a78aee5f0347443da1b138edcc0e1abf808b665c80d950a645c87e35"},"schema_version":"1.0"},"canonical_sha256":"e5e4b6635b979ce95ca1c30a1597ff80355b9ead93580e352a9e6b785964c13b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:50.443020Z","signature_b64":"4SPw/4tMCVs9z9OjMsa4xN099BFIO0BQGHjMUNN7fc4ofZKrSV9vaS5MuzrFeGL2mAg2r9bMkNYe0BfjsVzrDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5e4b6635b979ce95ca1c30a1597ff80355b9ead93580e352a9e6b785964c13b","last_reissued_at":"2026-05-17T23:58:50.442331Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:50.442331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.04724","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:58:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i/LwucXsyxDhGSHWhuwb+7Lessb8/8MF7y4yVoPDM61nsAlUuI4iMqYmC6SxBT6PbS4+ho2RchqbxdI7K16MBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:16:40.124573Z"},"content_sha256":"74ecd4bfe0fe49abdfdca898b3c10aa5656cde7bcb33d8b9410532a56d4f6a08","schema_version":"1.0","event_id":"sha256:74ecd4bfe0fe49abdfdca898b3c10aa5656cde7bcb33d8b9410532a56d4f6a08"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4XSLMY23S6OOSXFBYMFBLF77QA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Boumediene Abdellaoui, Rachid Bentifour","submitted_at":"2016-11-15T07:36:36Z","abstract_excerpt":"Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\\ge 2$ For all $q<p$, there exists a positive constant $C\\equiv C(\\Omega, q, N, s)$ such that\n  $$ \\int_{{\\mathbb R}^N}\\int_{{\\mathbb R}^N} \\, \\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\\,dx\\,dy - \\Lambda_{N,p,s} \\int_{{\\mathbb R}^N} \\frac{|u(x)|^p}{|x|^{p}}\\,dx\\geq C \\int_{\\Omega}\\dint_{\\Omega}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \\in \\mathcal{C}_0^\\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04724","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:58:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NsEorJ4ddgEm2rlXWEdgHBXin5sPb9Zliv6eeZo6xfbMISb9AufzwzGHXX4apIGiR1B2ohqVDhBSzzls78LoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:16:40.125231Z"},"content_sha256":"9225a8d4c64f9e67fb0251460895e8cf86b5dc81f95e622fbe202654ed605e44","schema_version":"1.0","event_id":"sha256:9225a8d4c64f9e67fb0251460895e8cf86b5dc81f95e622fbe202654ed605e44"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/bundle.json","state_url":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4XSLMY23S6OOSXFBYMFBLF77QA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T17:16:40Z","links":{"resolver":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA","bundle":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/bundle.json","state":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4XSLMY23S6OOSXFBYMFBLF77QA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4XSLMY23S6OOSXFBYMFBLF77QA","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":"fb801ab3a78aee5f0347443da1b138edcc0e1abf808b665c80d950a645c87e35","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-11-15T07:36:36Z","title_canon_sha256":"3ded740fbc4cfba6f19d74e36e16354cff8b7fe50cc3fb9d5d9add8c5b90d495"},"schema_version":"1.0","source":{"id":"1611.04724","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04724","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04724v2","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04724","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"pith_short_12","alias_value":"4XSLMY23S6OO","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4XSLMY23S6OOSXFB","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4XSLMY23","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:9225a8d4c64f9e67fb0251460895e8cf86b5dc81f95e622fbe202654ed605e44","target":"graph","created_at":"2026-05-17T23:58:50Z","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":"Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\\ge 2$ For all $q<p$, there exists a positive constant $C\\equiv C(\\Omega, q, N, s)$ such that\n  $$ \\int_{{\\mathbb R}^N}\\int_{{\\mathbb R}^N} \\, \\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\\,dx\\,dy - \\Lambda_{N,p,s} \\int_{{\\mathbb R}^N} \\frac{|u(x)|^p}{|x|^{p}}\\,dx\\geq C \\int_{\\Omega}\\dint_{\\Omega}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \\in \\mathcal{C}_0^\\in","authors_text":"Boumediene Abdellaoui, Rachid Bentifour","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-11-15T07:36:36Z","title":"Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04724","kind":"arxiv","version":2},"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":"sha256:74ecd4bfe0fe49abdfdca898b3c10aa5656cde7bcb33d8b9410532a56d4f6a08","target":"record","created_at":"2026-05-17T23:58:50Z","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":"fb801ab3a78aee5f0347443da1b138edcc0e1abf808b665c80d950a645c87e35","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-11-15T07:36:36Z","title_canon_sha256":"3ded740fbc4cfba6f19d74e36e16354cff8b7fe50cc3fb9d5d9add8c5b90d495"},"schema_version":"1.0","source":{"id":"1611.04724","kind":"arxiv","version":2}},"canonical_sha256":"e5e4b6635b979ce95ca1c30a1597ff80355b9ead93580e352a9e6b785964c13b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5e4b6635b979ce95ca1c30a1597ff80355b9ead93580e352a9e6b785964c13b","first_computed_at":"2026-05-17T23:58:50.442331Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:50.442331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4SPw/4tMCVs9z9OjMsa4xN099BFIO0BQGHjMUNN7fc4ofZKrSV9vaS5MuzrFeGL2mAg2r9bMkNYe0BfjsVzrDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:50.443020Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.04724","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74ecd4bfe0fe49abdfdca898b3c10aa5656cde7bcb33d8b9410532a56d4f6a08","sha256:9225a8d4c64f9e67fb0251460895e8cf86b5dc81f95e622fbe202654ed605e44"],"state_sha256":"d794db656a42a34a5f528e6c957e1c49238c50448774059f19d531d9164308f3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JP35By2j2dXOqSXgfB3XdjHca2VbXKcEwukEZmwk99F2IqCW8kfowPsYh/FsbAhIs/csRRY/oRYEP6wYqdikAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T17:16:40.128696Z","bundle_sha256":"12aa964b993301ebf8db18751fb0af18bdaf7f7bf1b6684993b93bcb677fbe57"}}