{"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"}