{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:6IQUCOGKU6XW3R5GDSH4FHD3MK","short_pith_number":"pith:6IQUCOGK","canonical_record":{"source":{"id":"1412.4157","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-12-12T22:56:41Z","cross_cats_sorted":[],"title_canon_sha256":"287e69342eba8477f418563bb04166e9c57e650558fb05590cc192099508a587","abstract_canon_sha256":"a7e74f25cdecf98a5298bb709a5aa52f877893cb737f0087320f6c1fe3066943"},"schema_version":"1.0"},"canonical_sha256":"f2214138caa7af6dc7a61c8fc29c7b62b547597ea6578e31c2400a849497983a","source":{"kind":"arxiv","id":"1412.4157","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.4157","created_at":"2026-05-18T02:31:20Z"},{"alias_kind":"arxiv_version","alias_value":"1412.4157v1","created_at":"2026-05-18T02:31:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4157","created_at":"2026-05-18T02:31:20Z"},{"alias_kind":"pith_short_12","alias_value":"6IQUCOGKU6XW","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6IQUCOGKU6XW3R5G","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6IQUCOGK","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:6IQUCOGKU6XW3R5GDSH4FHD3MK","target":"record","payload":{"canonical_record":{"source":{"id":"1412.4157","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-12-12T22:56:41Z","cross_cats_sorted":[],"title_canon_sha256":"287e69342eba8477f418563bb04166e9c57e650558fb05590cc192099508a587","abstract_canon_sha256":"a7e74f25cdecf98a5298bb709a5aa52f877893cb737f0087320f6c1fe3066943"},"schema_version":"1.0"},"canonical_sha256":"f2214138caa7af6dc7a61c8fc29c7b62b547597ea6578e31c2400a849497983a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:20.062447Z","signature_b64":"kn9k2ktfFB9izWCVsP0Tw1mf0qerq8gIQ5+rtEmyFcSIPm219lteuh/PTC7SPv3tJYt8LhjS1uDaKGOxec7EBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2214138caa7af6dc7a61c8fc29c7b62b547597ea6578e31c2400a849497983a","last_reissued_at":"2026-05-18T02:31:20.061884Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:20.061884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.4157","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":"2026-05-18T02:31:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zk6IMeCNTqRF2G9j6FJ3iQOvYgEryOAKsxkZfy/wtObZmQA4F0+AIWmZDARQbhJA5HrLpbft7GcSfzvwnNtfAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:16:45.825877Z"},"content_sha256":"7ae7ccb28afb18c6eaaadfc4cdc6177b0f62a747d7eb4a65485442638d07d4d4","schema_version":"1.0","event_id":"sha256:7ae7ccb28afb18c6eaaadfc4cdc6177b0f62a747d7eb4a65485442638d07d4d4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:6IQUCOGKU6XW3R5GDSH4FHD3MK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two weight norm inequalities for fractional integral operators and commutators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"David Cruz-Uribe","submitted_at":"2014-12-12T22:56:41Z","abstract_excerpt":"In these lecture notes we describe some recent work on two weight norm inequalities for fractional integral operators, also known as Riesz potentials, and for commutators of fractional integrals. These notes are based on three lectures delivered at the 6th International Course of Mathematical Analysis in Andalucia, held in Antequera, Spain, September 8-12, 2014. They are, however, greatly expanded to include both new results and many details that I did not present in my lectures due to time constraints."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4157","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":"2026-05-18T02:31:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ncz4ZIR80qJJCFM+d8oZ8CMrN+wapsZfVTDEmjjlkh200tzhIPY2wQ9oxT+qrz6xfCDN26lyEX+aQQF0XsMHDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:16:45.826551Z"},"content_sha256":"bc13fe92cc873c3c5b2cfb58d0dcc8274f0e402d9e954b29a0feb16b841598e0","schema_version":"1.0","event_id":"sha256:bc13fe92cc873c3c5b2cfb58d0dcc8274f0e402d9e954b29a0feb16b841598e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6IQUCOGKU6XW3R5GDSH4FHD3MK/bundle.json","state_url":"https://pith.science/pith/6IQUCOGKU6XW3R5GDSH4FHD3MK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6IQUCOGKU6XW3R5GDSH4FHD3MK/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":"2026-05-26T00:16:45Z","links":{"resolver":"https://pith.science/pith/6IQUCOGKU6XW3R5GDSH4FHD3MK","bundle":"https://pith.science/pith/6IQUCOGKU6XW3R5GDSH4FHD3MK/bundle.json","state":"https://pith.science/pith/6IQUCOGKU6XW3R5GDSH4FHD3MK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6IQUCOGKU6XW3R5GDSH4FHD3MK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6IQUCOGKU6XW3R5GDSH4FHD3MK","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":"a7e74f25cdecf98a5298bb709a5aa52f877893cb737f0087320f6c1fe3066943","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-12-12T22:56:41Z","title_canon_sha256":"287e69342eba8477f418563bb04166e9c57e650558fb05590cc192099508a587"},"schema_version":"1.0","source":{"id":"1412.4157","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.4157","created_at":"2026-05-18T02:31:20Z"},{"alias_kind":"arxiv_version","alias_value":"1412.4157v1","created_at":"2026-05-18T02:31:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4157","created_at":"2026-05-18T02:31:20Z"},{"alias_kind":"pith_short_12","alias_value":"6IQUCOGKU6XW","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6IQUCOGKU6XW3R5G","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6IQUCOGK","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:bc13fe92cc873c3c5b2cfb58d0dcc8274f0e402d9e954b29a0feb16b841598e0","target":"graph","created_at":"2026-05-18T02:31:20Z","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":"In these lecture notes we describe some recent work on two weight norm inequalities for fractional integral operators, also known as Riesz potentials, and for commutators of fractional integrals. These notes are based on three lectures delivered at the 6th International Course of Mathematical Analysis in Andalucia, held in Antequera, Spain, September 8-12, 2014. They are, however, greatly expanded to include both new results and many details that I did not present in my lectures due to time constraints.","authors_text":"David Cruz-Uribe","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-12-12T22:56:41Z","title":"Two weight norm inequalities for fractional integral operators and commutators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4157","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":"sha256:7ae7ccb28afb18c6eaaadfc4cdc6177b0f62a747d7eb4a65485442638d07d4d4","target":"record","created_at":"2026-05-18T02:31:20Z","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":"a7e74f25cdecf98a5298bb709a5aa52f877893cb737f0087320f6c1fe3066943","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-12-12T22:56:41Z","title_canon_sha256":"287e69342eba8477f418563bb04166e9c57e650558fb05590cc192099508a587"},"schema_version":"1.0","source":{"id":"1412.4157","kind":"arxiv","version":1}},"canonical_sha256":"f2214138caa7af6dc7a61c8fc29c7b62b547597ea6578e31c2400a849497983a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f2214138caa7af6dc7a61c8fc29c7b62b547597ea6578e31c2400a849497983a","first_computed_at":"2026-05-18T02:31:20.061884Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:31:20.061884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kn9k2ktfFB9izWCVsP0Tw1mf0qerq8gIQ5+rtEmyFcSIPm219lteuh/PTC7SPv3tJYt8LhjS1uDaKGOxec7EBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:31:20.062447Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.4157","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ae7ccb28afb18c6eaaadfc4cdc6177b0f62a747d7eb4a65485442638d07d4d4","sha256:bc13fe92cc873c3c5b2cfb58d0dcc8274f0e402d9e954b29a0feb16b841598e0"],"state_sha256":"6ce891ab1cd01d8fc807c1ecb3e62d5eb7c573bf11f3cbaf02ed552236b5bf7b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z3MQylVjyNf1fBkgc9NjnjnWZ+i8umB8bxoRwAGIVU74LKq4AeL/Slwk+8irY82+OWqwAgYBI/+t/j6rKdG1Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T00:16:45.830121Z","bundle_sha256":"4c3dbbe6665a264eb57c847d7f855f669ab764804d8c0b61d05a8c66d780fc3f"}}