{"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"}