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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"},"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":"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"},"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"},"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. 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