{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:6JDKEEE4IRF5FS3NCGXAXMXL3S","short_pith_number":"pith:6JDKEEE4","canonical_record":{"source":{"id":"1310.0182","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-01T08:26:05Z","cross_cats_sorted":[],"title_canon_sha256":"952da93b09ac228623bcceb1307ffb708fa0ec02a20f2c846621b7f1c4d1255b","abstract_canon_sha256":"57ceaf367629561bba729b8b615c25885b4a95a7861d57b11c97b72e7243d144"},"schema_version":"1.0"},"canonical_sha256":"f246a2109c444bd2cb6d11ae0bb2ebdc8b0a70b44231ba1a0a47d2e866aa1d1a","source":{"kind":"arxiv","id":"1310.0182","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.0182","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"arxiv_version","alias_value":"1310.0182v1","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.0182","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"pith_short_12","alias_value":"6JDKEEE4IRF5","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6JDKEEE4IRF5FS3N","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6JDKEEE4","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:6JDKEEE4IRF5FS3NCGXAXMXL3S","target":"record","payload":{"canonical_record":{"source":{"id":"1310.0182","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-01T08:26:05Z","cross_cats_sorted":[],"title_canon_sha256":"952da93b09ac228623bcceb1307ffb708fa0ec02a20f2c846621b7f1c4d1255b","abstract_canon_sha256":"57ceaf367629561bba729b8b615c25885b4a95a7861d57b11c97b72e7243d144"},"schema_version":"1.0"},"canonical_sha256":"f246a2109c444bd2cb6d11ae0bb2ebdc8b0a70b44231ba1a0a47d2e866aa1d1a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:45.261814Z","signature_b64":"p++SQgQ09VY1hJr5ZBk2x0o4bFQU4CgUO7kyW0l1LUNx7SkDgWNFdtPa9YPz5h2FcbunBE7l8tLm5yp9PqwFAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f246a2109c444bd2cb6d11ae0bb2ebdc8b0a70b44231ba1a0a47d2e866aa1d1a","last_reissued_at":"2026-05-18T03:11:45.260985Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:45.260985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.0182","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":"2026-05-18T03:11:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ze3qb2eC3jomsQYzOffoPejC8g+9IZ7QinXf2Q5rWS80ROYzR0FR1Sx8Rd5e702UnRPx1ciGJNjz2QFTj0RtDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T09:38:03.813723Z"},"content_sha256":"215fd45632448d4bf3f3b1183a7e7a97a69f4feb4d8993d7ab929286da47cd28","schema_version":"1.0","event_id":"sha256:215fd45632448d4bf3f3b1183a7e7a97a69f4feb4d8993d7ab929286da47cd28"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:6JDKEEE4IRF5FS3NCGXAXMXL3S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Probabilistic Approach to Fractional Integrals and the Hardy-Littlewood-Sobolev Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Applebaum, Rodrigo Banuelos","submitted_at":"2013-10-01T08:26:05Z","abstract_excerpt":"We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic representation of the classical fractional integral operators on $\\R^n$ as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy and Doob's inequalities for martingales."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0182","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":"2026-05-18T03:11:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ONwHU8y500mOCTnlY02+RVeW6jT/Y+NbKOqCEPHR2C/ZeRC5HB/zY0IqkyZGpnkHTcN+qDcT6GthHSHFMFLhDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T09:38:03.814070Z"},"content_sha256":"5840d0b80a9f52910bf12a4a186b568b4c35f942de4f258396a78038cb4415cf","schema_version":"1.0","event_id":"sha256:5840d0b80a9f52910bf12a4a186b568b4c35f942de4f258396a78038cb4415cf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6JDKEEE4IRF5FS3NCGXAXMXL3S/bundle.json","state_url":"https://pith.science/pith/6JDKEEE4IRF5FS3NCGXAXMXL3S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6JDKEEE4IRF5FS3NCGXAXMXL3S/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":"2026-06-11T09:38:03Z","links":{"resolver":"https://pith.science/pith/6JDKEEE4IRF5FS3NCGXAXMXL3S","bundle":"https://pith.science/pith/6JDKEEE4IRF5FS3NCGXAXMXL3S/bundle.json","state":"https://pith.science/pith/6JDKEEE4IRF5FS3NCGXAXMXL3S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6JDKEEE4IRF5FS3NCGXAXMXL3S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6JDKEEE4IRF5FS3NCGXAXMXL3S","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":"57ceaf367629561bba729b8b615c25885b4a95a7861d57b11c97b72e7243d144","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-01T08:26:05Z","title_canon_sha256":"952da93b09ac228623bcceb1307ffb708fa0ec02a20f2c846621b7f1c4d1255b"},"schema_version":"1.0","source":{"id":"1310.0182","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.0182","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"arxiv_version","alias_value":"1310.0182v1","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.0182","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"pith_short_12","alias_value":"6JDKEEE4IRF5","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6JDKEEE4IRF5FS3N","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6JDKEEE4","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:5840d0b80a9f52910bf12a4a186b568b4c35f942de4f258396a78038cb4415cf","target":"graph","created_at":"2026-05-18T03:11:45Z","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 give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic representation of the classical fractional integral operators on $\\R^n$ as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy and Doob's inequalities for martingales.","authors_text":"David Applebaum, Rodrigo Banuelos","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-01T08:26:05Z","title":"Probabilistic Approach to Fractional Integrals and the Hardy-Littlewood-Sobolev Inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0182","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":"sha256:215fd45632448d4bf3f3b1183a7e7a97a69f4feb4d8993d7ab929286da47cd28","target":"record","created_at":"2026-05-18T03:11:45Z","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":"57ceaf367629561bba729b8b615c25885b4a95a7861d57b11c97b72e7243d144","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-01T08:26:05Z","title_canon_sha256":"952da93b09ac228623bcceb1307ffb708fa0ec02a20f2c846621b7f1c4d1255b"},"schema_version":"1.0","source":{"id":"1310.0182","kind":"arxiv","version":1}},"canonical_sha256":"f246a2109c444bd2cb6d11ae0bb2ebdc8b0a70b44231ba1a0a47d2e866aa1d1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f246a2109c444bd2cb6d11ae0bb2ebdc8b0a70b44231ba1a0a47d2e866aa1d1a","first_computed_at":"2026-05-18T03:11:45.260985Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:11:45.260985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p++SQgQ09VY1hJr5ZBk2x0o4bFQU4CgUO7kyW0l1LUNx7SkDgWNFdtPa9YPz5h2FcbunBE7l8tLm5yp9PqwFAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:11:45.261814Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.0182","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:215fd45632448d4bf3f3b1183a7e7a97a69f4feb4d8993d7ab929286da47cd28","sha256:5840d0b80a9f52910bf12a4a186b568b4c35f942de4f258396a78038cb4415cf"],"state_sha256":"c482a599925c6603319112826f61cb163ee7fbb75c3224ff8f94e96bd52dda75"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xa9rUUoiyKOOxtVKw1npnwWfx0Qimzv+xPASKl99FHWAKY+lBgk7W2aE/T/wC6Il5bCEZtLdD29IuFGFxeCBAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T09:38:03.816000Z","bundle_sha256":"c21bae1fb25db27753bd85d1007c66eb9e0e3d5b677445a3a6accae376cedcf4"}}