{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:ST32FQ7WDFBXNUGMPJHPL35RXD","short_pith_number":"pith:ST32FQ7W","canonical_record":{"source":{"id":"1102.3905","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-18T20:38:59Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"ad40830789c4fe845d75801043c692adc1f40b7a82d988ef8fe08fa174fb491e","abstract_canon_sha256":"f215e5675068cb36363a50cb7b1d3a26ca0b8c4e9a05ce598e9196ea4db954d2"},"schema_version":"1.0"},"canonical_sha256":"94f7a2c3f6194376d0cc7a4ef5efb1b8e55418e2e78ff3eaeed15d83bc3e8347","source":{"kind":"arxiv","id":"1102.3905","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.3905","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"arxiv_version","alias_value":"1102.3905v1","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.3905","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"pith_short_12","alias_value":"ST32FQ7WDFBX","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"ST32FQ7WDFBXNUGM","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"ST32FQ7W","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:ST32FQ7WDFBXNUGMPJHPL35RXD","target":"record","payload":{"canonical_record":{"source":{"id":"1102.3905","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-18T20:38:59Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"ad40830789c4fe845d75801043c692adc1f40b7a82d988ef8fe08fa174fb491e","abstract_canon_sha256":"f215e5675068cb36363a50cb7b1d3a26ca0b8c4e9a05ce598e9196ea4db954d2"},"schema_version":"1.0"},"canonical_sha256":"94f7a2c3f6194376d0cc7a4ef5efb1b8e55418e2e78ff3eaeed15d83bc3e8347","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:20.693130Z","signature_b64":"CjPbELSUzWNLLWRiUg2tvvSlaF+3k6QjwO9B5lCBB/N6XVEftcr31GIbTGDg7ochzKI4z9/iygJN0x0G9ct2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94f7a2c3f6194376d0cc7a4ef5efb1b8e55418e2e78ff3eaeed15d83bc3e8347","last_reissued_at":"2026-05-18T04:28:20.692598Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:20.692598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.3905","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":"2026-05-18T04:28:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VpuOrFqOREyhLOrCeKTNN0t3gR+xboLUU3IkqoLzUnxx76ePSmIqXFSIcok/t/F4TEa1EfgZ7QdeVSV/e6F3Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T13:45:20.753103Z"},"content_sha256":"20aade518f775eed1c14c8393fcbe3aa5b4934ab5f315aece322ed55a06081c0","schema_version":"1.0","event_id":"sha256:20aade518f775eed1c14c8393fcbe3aa5b4934ab5f315aece322ed55a06081c0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:ST32FQ7WDFBXNUGMPJHPL35RXD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Perturbation of Burkholder's martingale transform and Monge--Amp\\`ere equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Alexander Volberg, Nicholas Boros, Prabhu Janakiraman","submitted_at":"2011-02-18T20:38:59Z","abstract_excerpt":"Let $\\{d_k\\}_{k \\geq 0}$ be a complex martingale difference in $L^p[0,1],$ where $1<p<\\infty,$ and $\\{\\e_k\\}_{k \\geq 0}$ a sequence in $\\{\\pm 1\\}.$ We obtain the following generalization of Burkholder's famous result. If $\\tau \\in [-\\frac 12, \\frac 12]$ and $n \\in \\Z_+$ then\n  $$|\\sum_{k=0}^n{(\\{c} \\e_k \\tau) d_k}|_{L^p([0,1], \\C^2)} \\leq ((p^*-1)^2 + \\tau^2)^{\\frac 12}|\\sum_{k=0}^n{d_k}|_{L^p([0,1], \\C)},$$\n where $((p^*-1)^2 + \\tau^2)^{\\frac 12}$ is sharp and $p^*-1 = \\max\\{p-1, \\frac 1{p-1}\\}.$ For $2\\leq p<\\infty$ the result is also true with sharp constant for $\\tau \\in \\R.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.3905","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":"2026-05-18T04:28:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RgaMzdEmagRtpK4fpyoLdDCIFvPb2N7/++Yr4AMDsE2dPMH40Lhm90pwuOG4D9OqWP292OfG2M6Nx1u7QKypCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T13:45:20.753795Z"},"content_sha256":"af6d14e5566986754014ecf17cdd7cc1d4ea5082ee628369dd35cafb4db17d82","schema_version":"1.0","event_id":"sha256:af6d14e5566986754014ecf17cdd7cc1d4ea5082ee628369dd35cafb4db17d82"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/bundle.json","state_url":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/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":"2026-05-26T13:45:20Z","links":{"resolver":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD","bundle":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/bundle.json","state":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ST32FQ7WDFBXNUGMPJHPL35RXD","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":"f215e5675068cb36363a50cb7b1d3a26ca0b8c4e9a05ce598e9196ea4db954d2","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-18T20:38:59Z","title_canon_sha256":"ad40830789c4fe845d75801043c692adc1f40b7a82d988ef8fe08fa174fb491e"},"schema_version":"1.0","source":{"id":"1102.3905","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.3905","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"arxiv_version","alias_value":"1102.3905v1","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.3905","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"pith_short_12","alias_value":"ST32FQ7WDFBX","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"ST32FQ7WDFBXNUGM","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"ST32FQ7W","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:af6d14e5566986754014ecf17cdd7cc1d4ea5082ee628369dd35cafb4db17d82","target":"graph","created_at":"2026-05-18T04:28:20Z","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":"Let $\\{d_k\\}_{k \\geq 0}$ be a complex martingale difference in $L^p[0,1],$ where $1<p<\\infty,$ and $\\{\\e_k\\}_{k \\geq 0}$ a sequence in $\\{\\pm 1\\}.$ We obtain the following generalization of Burkholder's famous result. If $\\tau \\in [-\\frac 12, \\frac 12]$ and $n \\in \\Z_+$ then\n  $$|\\sum_{k=0}^n{(\\{c} \\e_k \\tau) d_k}|_{L^p([0,1], \\C^2)} \\leq ((p^*-1)^2 + \\tau^2)^{\\frac 12}|\\sum_{k=0}^n{d_k}|_{L^p([0,1], \\C)},$$\n where $((p^*-1)^2 + \\tau^2)^{\\frac 12}$ is sharp and $p^*-1 = \\max\\{p-1, \\frac 1{p-1}\\}.$ For $2\\leq p<\\infty$ the result is also true with sharp constant for $\\tau \\in \\R.$","authors_text":"Alexander Volberg, Nicholas Boros, Prabhu Janakiraman","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-18T20:38:59Z","title":"Perturbation of Burkholder's martingale transform and Monge--Amp\\`ere equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.3905","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":"sha256:20aade518f775eed1c14c8393fcbe3aa5b4934ab5f315aece322ed55a06081c0","target":"record","created_at":"2026-05-18T04:28:20Z","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":"f215e5675068cb36363a50cb7b1d3a26ca0b8c4e9a05ce598e9196ea4db954d2","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-18T20:38:59Z","title_canon_sha256":"ad40830789c4fe845d75801043c692adc1f40b7a82d988ef8fe08fa174fb491e"},"schema_version":"1.0","source":{"id":"1102.3905","kind":"arxiv","version":1}},"canonical_sha256":"94f7a2c3f6194376d0cc7a4ef5efb1b8e55418e2e78ff3eaeed15d83bc3e8347","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"94f7a2c3f6194376d0cc7a4ef5efb1b8e55418e2e78ff3eaeed15d83bc3e8347","first_computed_at":"2026-05-18T04:28:20.692598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:20.692598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CjPbELSUzWNLLWRiUg2tvvSlaF+3k6QjwO9B5lCBB/N6XVEftcr31GIbTGDg7ochzKI4z9/iygJN0x0G9ct2CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:20.693130Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.3905","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20aade518f775eed1c14c8393fcbe3aa5b4934ab5f315aece322ed55a06081c0","sha256:af6d14e5566986754014ecf17cdd7cc1d4ea5082ee628369dd35cafb4db17d82"],"state_sha256":"a315da5a39f2cc3d09678331b5d52267da0cea3252e0a052a87ee12c393e5320"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wh/Bw6+CBOdYTpECIXs0XJbFFDNsOeZ407DVXcZMKTFM1MnmQY7aLDb8dBV5gyMvWRvwHyXmTncp0Qi8RtcTCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T13:45:20.757357Z","bundle_sha256":"e1f83b91e0a74680a61bf47f6e29b53063ce73534f5cc222059de8ef8b73102c"}}