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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 $\\tau \\in \\R.$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1102.3905","created_at":"2026-05-18T04:28:20.692657+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.3905v1","created_at":"2026-05-18T04:28:20.692657+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.3905","created_at":"2026-05-18T04:28:20.692657+00:00"},{"alias_kind":"pith_short_12","alias_value":"ST32FQ7WDFBX","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"ST32FQ7WDFBXNUGM","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"ST32FQ7W","created_at":"2026-05-18T12:26:41.206345+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD","json":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD.json","graph_json":"https://pith.science/api/pith-number/ST32FQ7WDFBXNUGMPJHPL35RXD/graph.json","events_json":"https://pith.science/api/pith-number/ST32FQ7WDFBXNUGMPJHPL35RXD/events.json","paper":"https://pith.science/paper/ST32FQ7W"},"agent_actions":{"view_html":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD","download_json":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD.json","view_paper":"https://pith.science/paper/ST32FQ7W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.3905&json=true","fetch_graph":"https://pith.science/api/pith-number/ST32FQ7WDFBXNUGMPJHPL35RXD/graph.json","fetch_events":"https://pith.science/api/pith-number/ST32FQ7WDFBXNUGMPJHPL35RXD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/action/storage_attestation","attest_author":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/action/author_attestation","sign_citation":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/action/citation_signature","submit_replication":"https://pith.science/pith/ST32FQ7WDFBXNUGMPJHPL35RXD/action/replication_record"}},"created_at":"2026-05-18T04:28:20.692657+00:00","updated_at":"2026-05-18T04:28:20.692657+00:00"}