{"paper":{"title":"The Hilbert transform and orthogonal martingales in Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.PR"],"primary_cat":"math.FA","authors_text":"Adam Os\\k{e}kowski, Ivan Yaroslavtsev","submitted_at":"2018-05-10T13:08:10Z","abstract_excerpt":"Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \\mathbb E \\Psi(N_t) \\leq C_{\\Phi,\\Psi,X} \\mathbb E \\Phi(M_t),\\;\\;\\; t\\geq 0, $$ where $\\Phi, \\Psi:X \\to \\mathbb R_+$ are convex continuous functions and the least admissible constant $C_{\\Phi,\\Psi,X}$ coincides with the $\\Phi,\\Psi$-norm of the periodic Hilbert transform. As a corollary, it is shown that the $\\Phi,\\Psi$-norms of the periodic Hilbert transform, the Hilbert transform on the real l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03948","kind":"arxiv","version":2},"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"}