{"paper":{"title":"The foundational inequalities of D.L. Burkholder and some of their ramifications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA"],"primary_cat":"math.PR","authors_text":"Rodrigo Ba\\~nuelos","submitted_at":"2010-12-22T00:54:48Z","abstract_excerpt":"This paper present an overview of some of the applications of the martingale inequalities of D.L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling-Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calder\\'on-Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4850","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"}