{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:T7G5WKHCX6QZADFBQHZE2CBEAR","short_pith_number":"pith:T7G5WKHC","canonical_record":{"source":{"id":"1312.5038","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-18T04:32:56Z","cross_cats_sorted":["math.AP","math.FA"],"title_canon_sha256":"ef811b29bc07b6294cc1d2d297f74a3048b6c2df00e5693b6fa07536596e3f27","abstract_canon_sha256":"d4d86a48fbe0cec1911c01fbde5df8d122ffb2e0b886fac65997441e36054476"},"schema_version":"1.0"},"canonical_sha256":"9fcddb28e2bfa1900ca181f24d0824047e63f2f497cba7361190eecd5ed07dda","source":{"kind":"arxiv","id":"1312.5038","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.5038","created_at":"2026-05-18T03:04:16Z"},{"alias_kind":"arxiv_version","alias_value":"1312.5038v1","created_at":"2026-05-18T03:04:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5038","created_at":"2026-05-18T03:04:16Z"},{"alias_kind":"pith_short_12","alias_value":"T7G5WKHCX6QZ","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"T7G5WKHCX6QZADFB","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"T7G5WKHC","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:T7G5WKHCX6QZADFBQHZE2CBEAR","target":"record","payload":{"canonical_record":{"source":{"id":"1312.5038","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-18T04:32:56Z","cross_cats_sorted":["math.AP","math.FA"],"title_canon_sha256":"ef811b29bc07b6294cc1d2d297f74a3048b6c2df00e5693b6fa07536596e3f27","abstract_canon_sha256":"d4d86a48fbe0cec1911c01fbde5df8d122ffb2e0b886fac65997441e36054476"},"schema_version":"1.0"},"canonical_sha256":"9fcddb28e2bfa1900ca181f24d0824047e63f2f497cba7361190eecd5ed07dda","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:16.511202Z","signature_b64":"8DcPo3WSLQomU4wxh3gYFljsakEj+FaezLhQ11WzSCiDX1TPChTwgtbWVudD9B6GJ6Ta5xx4rVPvLDpcGrSZDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fcddb28e2bfa1900ca181f24d0824047e63f2f497cba7361190eecd5ed07dda","last_reissued_at":"2026-05-18T03:04:16.510784Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:16.510784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.5038","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":"2026-05-18T03:04:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RH0HZayd/HbGoOujhVGKkUnblxBxQX3Mj8JayhH760A8/R9oHjr/mmFwdb3WB0EpPA8n9/K4dkYtXhxYdgQFAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:56:45.050611Z"},"content_sha256":"698e2ca4acd2e5439f17c741524439a2fd1f8d857727d8b582d8d077e219ffe8","schema_version":"1.0","event_id":"sha256:698e2ca4acd2e5439f17c741524439a2fd1f8d857727d8b582d8d077e219ffe8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:T7G5WKHCX6QZADFBQHZE2CBEAR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp maximal $L^p$-estimates for martingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA"],"primary_cat":"math.PR","authors_text":"Adam Osekowski, Rodrigo Ba\\~nuelos","submitted_at":"2013-12-18T04:32:56Z","abstract_excerpt":"Let $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_p$, $C_p$ and $\\mathfrak{c}_p$ such that $$ \\,\\,\\,\\,\\sup_{t\\geq 0}\\left|\\left|X_t\\right|\\right|_p\\leq C_p\\left|\\left|-\\inf_{t\\geq 0}X_t\\right|\\right|_p,$$ $$ \\,\\,||\\sup_{t\\geq 0}X_t||_p\\leq c_p\\left|\\left|-\\inf_{t\\geq 0}X_t\\right|\\right|_p$$ and $$ ||\\sup_{t\\geq 0}|X_t|\\;||_p\\leq \\mathfrak{c}_p\\left|\\left|-\\inf_{t\\geq 0}X_t\\right|\\right|_p.$$ The estimates are shown to be sharp if $X$ is assumed to be a stopped one-dimensional Brownian motion. The inequalities ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5038","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":"2026-05-18T03:04:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4gGH0PTIRqqCAswMZi7GhLHeV5K0s1vMf7vO6XyNrMbWBli23yaJkpao3S+ihnyK3FzcUIL4lkBGA1V5YJ31Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:56:45.051462Z"},"content_sha256":"7a334bf980c278366d11f85c001b1fb502e25cd87e5aa4ff28d35cc8e96a68e4","schema_version":"1.0","event_id":"sha256:7a334bf980c278366d11f85c001b1fb502e25cd87e5aa4ff28d35cc8e96a68e4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T7G5WKHCX6QZADFBQHZE2CBEAR/bundle.json","state_url":"https://pith.science/pith/T7G5WKHCX6QZADFBQHZE2CBEAR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T7G5WKHCX6QZADFBQHZE2CBEAR/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":"2026-05-31T12:56:45Z","links":{"resolver":"https://pith.science/pith/T7G5WKHCX6QZADFBQHZE2CBEAR","bundle":"https://pith.science/pith/T7G5WKHCX6QZADFBQHZE2CBEAR/bundle.json","state":"https://pith.science/pith/T7G5WKHCX6QZADFBQHZE2CBEAR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T7G5WKHCX6QZADFBQHZE2CBEAR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:T7G5WKHCX6QZADFBQHZE2CBEAR","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":"d4d86a48fbe0cec1911c01fbde5df8d122ffb2e0b886fac65997441e36054476","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-18T04:32:56Z","title_canon_sha256":"ef811b29bc07b6294cc1d2d297f74a3048b6c2df00e5693b6fa07536596e3f27"},"schema_version":"1.0","source":{"id":"1312.5038","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.5038","created_at":"2026-05-18T03:04:16Z"},{"alias_kind":"arxiv_version","alias_value":"1312.5038v1","created_at":"2026-05-18T03:04:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5038","created_at":"2026-05-18T03:04:16Z"},{"alias_kind":"pith_short_12","alias_value":"T7G5WKHCX6QZ","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"T7G5WKHCX6QZADFB","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"T7G5WKHC","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:7a334bf980c278366d11f85c001b1fb502e25cd87e5aa4ff28d35cc8e96a68e4","target":"graph","created_at":"2026-05-18T03:04:16Z","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 $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_p$, $C_p$ and $\\mathfrak{c}_p$ such that $$ \\,\\,\\,\\,\\sup_{t\\geq 0}\\left|\\left|X_t\\right|\\right|_p\\leq C_p\\left|\\left|-\\inf_{t\\geq 0}X_t\\right|\\right|_p,$$ $$ \\,\\,||\\sup_{t\\geq 0}X_t||_p\\leq c_p\\left|\\left|-\\inf_{t\\geq 0}X_t\\right|\\right|_p$$ and $$ ||\\sup_{t\\geq 0}|X_t|\\;||_p\\leq \\mathfrak{c}_p\\left|\\left|-\\inf_{t\\geq 0}X_t\\right|\\right|_p.$$ The estimates are shown to be sharp if $X$ is assumed to be a stopped one-dimensional Brownian motion. The inequalities ar","authors_text":"Adam Osekowski, Rodrigo Ba\\~nuelos","cross_cats":["math.AP","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-18T04:32:56Z","title":"Sharp maximal $L^p$-estimates for martingales"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5038","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":"sha256:698e2ca4acd2e5439f17c741524439a2fd1f8d857727d8b582d8d077e219ffe8","target":"record","created_at":"2026-05-18T03:04:16Z","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":"d4d86a48fbe0cec1911c01fbde5df8d122ffb2e0b886fac65997441e36054476","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-18T04:32:56Z","title_canon_sha256":"ef811b29bc07b6294cc1d2d297f74a3048b6c2df00e5693b6fa07536596e3f27"},"schema_version":"1.0","source":{"id":"1312.5038","kind":"arxiv","version":1}},"canonical_sha256":"9fcddb28e2bfa1900ca181f24d0824047e63f2f497cba7361190eecd5ed07dda","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9fcddb28e2bfa1900ca181f24d0824047e63f2f497cba7361190eecd5ed07dda","first_computed_at":"2026-05-18T03:04:16.510784Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:16.510784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8DcPo3WSLQomU4wxh3gYFljsakEj+FaezLhQ11WzSCiDX1TPChTwgtbWVudD9B6GJ6Ta5xx4rVPvLDpcGrSZDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:16.511202Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.5038","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:698e2ca4acd2e5439f17c741524439a2fd1f8d857727d8b582d8d077e219ffe8","sha256:7a334bf980c278366d11f85c001b1fb502e25cd87e5aa4ff28d35cc8e96a68e4"],"state_sha256":"857767af33afec1dcbcf5e08f5b4656d7003d876b8cb18e45d262ec40394d525"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AmhwIBJduP4W9NiAfnHEIcpJkzeWqsjSA5OYoxsJfINlcYilupmhMD2KRz35UqEdTT3Iu7dsIvmcAwmb763yCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T12:56:45.056207Z","bundle_sha256":"9ab2c2759b32df0ef5eca89e63c09c1f9c2bda93557e3262d00fb156f9028b27"}}