{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:6EMDSJNXXFDFJPHSGQP6FJRT7Q","short_pith_number":"pith:6EMDSJNX","canonical_record":{"source":{"id":"1208.6447","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-31T10:08:52Z","cross_cats_sorted":[],"title_canon_sha256":"d38f293967f1949c99d48d7006bc5ddcb6a31a94d0b44e412226f5122e1dfe0a","abstract_canon_sha256":"95d52fd50cb88611cded7a6c4adf19c08d1557e58900ef337d4cc0060b17e2fe"},"schema_version":"1.0"},"canonical_sha256":"f1183925b7b94654bcf2341fe2a633fc2fb688d976928d380d914b94c7839735","source":{"kind":"arxiv","id":"1208.6447","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.6447","created_at":"2026-05-18T03:27:32Z"},{"alias_kind":"arxiv_version","alias_value":"1208.6447v1","created_at":"2026-05-18T03:27:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.6447","created_at":"2026-05-18T03:27:32Z"},{"alias_kind":"pith_short_12","alias_value":"6EMDSJNXXFDF","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"6EMDSJNXXFDFJPHS","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"6EMDSJNX","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:6EMDSJNXXFDFJPHSGQP6FJRT7Q","target":"record","payload":{"canonical_record":{"source":{"id":"1208.6447","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-31T10:08:52Z","cross_cats_sorted":[],"title_canon_sha256":"d38f293967f1949c99d48d7006bc5ddcb6a31a94d0b44e412226f5122e1dfe0a","abstract_canon_sha256":"95d52fd50cb88611cded7a6c4adf19c08d1557e58900ef337d4cc0060b17e2fe"},"schema_version":"1.0"},"canonical_sha256":"f1183925b7b94654bcf2341fe2a633fc2fb688d976928d380d914b94c7839735","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:32.693046Z","signature_b64":"VrzFYd+i2Tz7wigvyv03LhVGClbEorLRZxa5POQMbm1wnL67Zd3G7Yxz2MrKVh4RYvCVOFjKczzjp0I3AvOsCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1183925b7b94654bcf2341fe2a633fc2fb688d976928d380d914b94c7839735","last_reissued_at":"2026-05-18T03:27:32.692595Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:32.692595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.6447","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:27:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eTK2h2yoOEKG7M91E/6UYd/W1UT5djc2WbKIQdAuuw1aVl8uA/6474FZwEESiiSCdvstyc37NCrieRK/ulrLDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:47:34.723714Z"},"content_sha256":"d57cc6070ee0a3d2f950dd72de0d9bce3efc6873b9f7872591c55e791a1247f6","schema_version":"1.0","event_id":"sha256:d57cc6070ee0a3d2f950dd72de0d9bce3efc6873b9f7872591c55e791a1247f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:6EMDSJNXXFDFJPHSGQP6FJRT7Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlocal Hardy type inequalities with optimal constants and remainder terms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Van Schaftingen, Vitaly Moroz","submitted_at":"2012-08-31T10:08:52Z","abstract_excerpt":"Using a groundstate transformation, we give a new proof of the optimal Stein-Weiss inequality of Herbst [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N,\\alpha, 0}\\int_{\\R^N} \\abs{\\varphi}^2,] and of its combinations with the Hardy inequality by Beckner [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha + s}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha + s}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N, \\alpha, 1} \\int_{\\R^N} \\abs{\\nabla \\varphi}^2,] and with t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.6447","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":""},"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":"2026-05-18T03:27:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2fVE/MsVwi0UeC7psoVSrtSYcYuH2j8JYIbP6l1a8jLfTCNQxaYPqaHpbCG0wrVFIQtuN1oVKYgucc3cDnLEAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:47:34.724409Z"},"content_sha256":"6686e6004a1d4cc4a5ef538a4ada8641c97e1e1023e1c60493ddef3327e6f9cc","schema_version":"1.0","event_id":"sha256:6686e6004a1d4cc4a5ef538a4ada8641c97e1e1023e1c60493ddef3327e6f9cc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6EMDSJNXXFDFJPHSGQP6FJRT7Q/bundle.json","state_url":"https://pith.science/pith/6EMDSJNXXFDFJPHSGQP6FJRT7Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6EMDSJNXXFDFJPHSGQP6FJRT7Q/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":"2026-05-25T15:47:34Z","links":{"resolver":"https://pith.science/pith/6EMDSJNXXFDFJPHSGQP6FJRT7Q","bundle":"https://pith.science/pith/6EMDSJNXXFDFJPHSGQP6FJRT7Q/bundle.json","state":"https://pith.science/pith/6EMDSJNXXFDFJPHSGQP6FJRT7Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6EMDSJNXXFDFJPHSGQP6FJRT7Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:6EMDSJNXXFDFJPHSGQP6FJRT7Q","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":"95d52fd50cb88611cded7a6c4adf19c08d1557e58900ef337d4cc0060b17e2fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-31T10:08:52Z","title_canon_sha256":"d38f293967f1949c99d48d7006bc5ddcb6a31a94d0b44e412226f5122e1dfe0a"},"schema_version":"1.0","source":{"id":"1208.6447","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.6447","created_at":"2026-05-18T03:27:32Z"},{"alias_kind":"arxiv_version","alias_value":"1208.6447v1","created_at":"2026-05-18T03:27:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.6447","created_at":"2026-05-18T03:27:32Z"},{"alias_kind":"pith_short_12","alias_value":"6EMDSJNXXFDF","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"6EMDSJNXXFDFJPHS","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"6EMDSJNX","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:6686e6004a1d4cc4a5ef538a4ada8641c97e1e1023e1c60493ddef3327e6f9cc","target":"graph","created_at":"2026-05-18T03:27:32Z","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":"Using a groundstate transformation, we give a new proof of the optimal Stein-Weiss inequality of Herbst [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N,\\alpha, 0}\\int_{\\R^N} \\abs{\\varphi}^2,] and of its combinations with the Hardy inequality by Beckner [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha + s}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha + s}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N, \\alpha, 1} \\int_{\\R^N} \\abs{\\nabla \\varphi}^2,] and with t","authors_text":"Jean Van Schaftingen, Vitaly Moroz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-31T10:08:52Z","title":"Nonlocal Hardy type inequalities with optimal constants and remainder terms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.6447","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:d57cc6070ee0a3d2f950dd72de0d9bce3efc6873b9f7872591c55e791a1247f6","target":"record","created_at":"2026-05-18T03:27:32Z","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":"95d52fd50cb88611cded7a6c4adf19c08d1557e58900ef337d4cc0060b17e2fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-31T10:08:52Z","title_canon_sha256":"d38f293967f1949c99d48d7006bc5ddcb6a31a94d0b44e412226f5122e1dfe0a"},"schema_version":"1.0","source":{"id":"1208.6447","kind":"arxiv","version":1}},"canonical_sha256":"f1183925b7b94654bcf2341fe2a633fc2fb688d976928d380d914b94c7839735","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1183925b7b94654bcf2341fe2a633fc2fb688d976928d380d914b94c7839735","first_computed_at":"2026-05-18T03:27:32.692595Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:27:32.692595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VrzFYd+i2Tz7wigvyv03LhVGClbEorLRZxa5POQMbm1wnL67Zd3G7Yxz2MrKVh4RYvCVOFjKczzjp0I3AvOsCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:27:32.693046Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.6447","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d57cc6070ee0a3d2f950dd72de0d9bce3efc6873b9f7872591c55e791a1247f6","sha256:6686e6004a1d4cc4a5ef538a4ada8641c97e1e1023e1c60493ddef3327e6f9cc"],"state_sha256":"1e7b8a7fd2b64ad56b4e93b40b365672a0376e8ab6a09fe698b7c4e16a0db1a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mm0pLYkwuT/+C/XJCb7Zg6PrUVhiN94VonKgZl0ti1ikHbQa/fCm7B+8SZ1J74pmwvzEGRXgE2Cnd6LONOv0Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T15:47:34.729222Z","bundle_sha256":"b8fcd0a5900430d1bdaa681607f5ea8c46f849be8191d14230b2858c045ac4a3"}}