{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ZNKNDEBJKNCOL75ZG4C4RM7FAB","short_pith_number":"pith:ZNKNDEBJ","canonical_record":{"source":{"id":"1512.01671","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-12-05T14:33:50Z","cross_cats_sorted":[],"title_canon_sha256":"782c8a10e2d9be2dd79735418bcc57f4cd30345edbed62d740bbb829f209621b","abstract_canon_sha256":"c380a0b62a2ac036e3356bd71f6ebc94b8b35dd7eaef7d7b61ecc1913e28c7e1"},"schema_version":"1.0"},"canonical_sha256":"cb54d190295344e5ffb93705c8b3e5007b06d1b2ee46d4d515cf60fa2c92d3c6","source":{"kind":"arxiv","id":"1512.01671","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01671","created_at":"2026-05-18T01:09:42Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01671v2","created_at":"2026-05-18T01:09:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01671","created_at":"2026-05-18T01:09:42Z"},{"alias_kind":"pith_short_12","alias_value":"ZNKNDEBJKNCO","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZNKNDEBJKNCOL75Z","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZNKNDEBJ","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ZNKNDEBJKNCOL75ZG4C4RM7FAB","target":"record","payload":{"canonical_record":{"source":{"id":"1512.01671","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-12-05T14:33:50Z","cross_cats_sorted":[],"title_canon_sha256":"782c8a10e2d9be2dd79735418bcc57f4cd30345edbed62d740bbb829f209621b","abstract_canon_sha256":"c380a0b62a2ac036e3356bd71f6ebc94b8b35dd7eaef7d7b61ecc1913e28c7e1"},"schema_version":"1.0"},"canonical_sha256":"cb54d190295344e5ffb93705c8b3e5007b06d1b2ee46d4d515cf60fa2c92d3c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:42.005309Z","signature_b64":"luweMxSP03sWTpXKNSSSBhsCpekC+9xQAyclXdROa/Sc3IgzVR279kbOGI4Qaj2qZLcaXG89Ci2WdCwbKIoxBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb54d190295344e5ffb93705c8b3e5007b06d1b2ee46d4d515cf60fa2c92d3c6","last_reissued_at":"2026-05-18T01:09:42.004717Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:42.004717Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.01671","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":"2026-05-18T01:09:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hmhLtQI6HKuVia+7TVjrgajUhwuEo1K03/I5Idapivum46DoC7JfgzhzBSN8cNa6yoDmBLMuZ1nvAXkKzBxACw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:04:49.946276Z"},"content_sha256":"1065e807a9c0c9a945bead1efff9b64002510a216135f49cf2b87ebb8e195ff7","schema_version":"1.0","event_id":"sha256:1065e807a9c0c9a945bead1efff9b64002510a216135f49cf2b87ebb8e195ff7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ZNKNDEBJKNCOL75ZG4C4RM7FAB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The fractional Laplacian in power-weighted $L^p$ spaces: integration-by-parts formulas and self-adjointness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Matteo Muratori","submitted_at":"2015-12-05T14:33:50Z","abstract_excerpt":"We consider the fractional Laplacian operator $(-\\Delta)^s$ (let $ s \\in (0,1) $) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $ L^2(\\mathbb{R}^d) $ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space $ \\dot{H}^s(\\mathbb{R}^d) $. More precisely, we focus on functions belonging to some weighted $ L^2 $ space whose fractional Laplacian belongs to another weighted $ L^2 $ space: we prove and disprove the validity of the integration-by-parts formula depending on the be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01671","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":"2026-05-18T01:09:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wSHUUcTtI0E1F/Qa5SP8giM6jyPuJqfqcvom+M2I3u6g9I6G1MRQF1iTlZ/3VwA5qDiS1SwPDMuaWXZmUGWjBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:04:49.946641Z"},"content_sha256":"e19d27fc27a408abc460bc23c97bb618a05f50100ce8021f5db3e82130914733","schema_version":"1.0","event_id":"sha256:e19d27fc27a408abc460bc23c97bb618a05f50100ce8021f5db3e82130914733"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZNKNDEBJKNCOL75ZG4C4RM7FAB/bundle.json","state_url":"https://pith.science/pith/ZNKNDEBJKNCOL75ZG4C4RM7FAB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZNKNDEBJKNCOL75ZG4C4RM7FAB/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":"2026-05-30T13:04:49Z","links":{"resolver":"https://pith.science/pith/ZNKNDEBJKNCOL75ZG4C4RM7FAB","bundle":"https://pith.science/pith/ZNKNDEBJKNCOL75ZG4C4RM7FAB/bundle.json","state":"https://pith.science/pith/ZNKNDEBJKNCOL75ZG4C4RM7FAB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZNKNDEBJKNCOL75ZG4C4RM7FAB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ZNKNDEBJKNCOL75ZG4C4RM7FAB","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":"c380a0b62a2ac036e3356bd71f6ebc94b8b35dd7eaef7d7b61ecc1913e28c7e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-12-05T14:33:50Z","title_canon_sha256":"782c8a10e2d9be2dd79735418bcc57f4cd30345edbed62d740bbb829f209621b"},"schema_version":"1.0","source":{"id":"1512.01671","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01671","created_at":"2026-05-18T01:09:42Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01671v2","created_at":"2026-05-18T01:09:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01671","created_at":"2026-05-18T01:09:42Z"},{"alias_kind":"pith_short_12","alias_value":"ZNKNDEBJKNCO","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZNKNDEBJKNCOL75Z","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZNKNDEBJ","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:e19d27fc27a408abc460bc23c97bb618a05f50100ce8021f5db3e82130914733","target":"graph","created_at":"2026-05-18T01:09:42Z","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 consider the fractional Laplacian operator $(-\\Delta)^s$ (let $ s \\in (0,1) $) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $ L^2(\\mathbb{R}^d) $ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space $ \\dot{H}^s(\\mathbb{R}^d) $. More precisely, we focus on functions belonging to some weighted $ L^2 $ space whose fractional Laplacian belongs to another weighted $ L^2 $ space: we prove and disprove the validity of the integration-by-parts formula depending on the be","authors_text":"Matteo Muratori","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-12-05T14:33:50Z","title":"The fractional Laplacian in power-weighted $L^p$ spaces: integration-by-parts formulas and self-adjointness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01671","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":"sha256:1065e807a9c0c9a945bead1efff9b64002510a216135f49cf2b87ebb8e195ff7","target":"record","created_at":"2026-05-18T01:09:42Z","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":"c380a0b62a2ac036e3356bd71f6ebc94b8b35dd7eaef7d7b61ecc1913e28c7e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-12-05T14:33:50Z","title_canon_sha256":"782c8a10e2d9be2dd79735418bcc57f4cd30345edbed62d740bbb829f209621b"},"schema_version":"1.0","source":{"id":"1512.01671","kind":"arxiv","version":2}},"canonical_sha256":"cb54d190295344e5ffb93705c8b3e5007b06d1b2ee46d4d515cf60fa2c92d3c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb54d190295344e5ffb93705c8b3e5007b06d1b2ee46d4d515cf60fa2c92d3c6","first_computed_at":"2026-05-18T01:09:42.004717Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:42.004717Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"luweMxSP03sWTpXKNSSSBhsCpekC+9xQAyclXdROa/Sc3IgzVR279kbOGI4Qaj2qZLcaXG89Ci2WdCwbKIoxBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:42.005309Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.01671","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1065e807a9c0c9a945bead1efff9b64002510a216135f49cf2b87ebb8e195ff7","sha256:e19d27fc27a408abc460bc23c97bb618a05f50100ce8021f5db3e82130914733"],"state_sha256":"61ddde1d3394649c0476aa5c6dc74ce37b2b4445b8457ff8b283be5db3806cba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZZ85FMsXfU+GshgHrgjV7W9j3xyBO2YXzkA+rsmBeZSi5BKOUUonz878RqVmn9XiLitjixc5ow15ffxxYdtVBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:04:49.948906Z","bundle_sha256":"1163194978111eb74c293b19664c78255fd3d7ff979e1b3561ae012f3d21a65d"}}