{"paper":{"title":"The maximal Beurling transform associated with squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Anna Bosch-Cam\\'os, Joan Mateu, Joan Orobitg","submitted_at":"2014-04-08T16:29:35Z","abstract_excerpt":"It is known that the improved Cotlar's inequality $B^{*}f(z) \\le C M(Bf)(z)$, $z\\in\\mathbb C$, holds for the Beurling transform $B$, the maximal Beurling transform $B^{*}f(z)=$ $\\displaystyle\\sup_{\\varepsilon >0}\\left|\\int_{|w|>\\varepsilon}f(z-w) \\frac{1}{w^2} \\,dw\\right|$, $z\\in\\mathbb C$, and the Hardy--Littlewood maximal operator $M$. In this note we consider the maximal Beurling transform associated with squares, namely, $B^{*}_Sf(z)=\\displaystyle\\sup_{\\varepsilon >0}\\left|\\int_{w\\notin Q(0,\\varepsilon)}f(z-w) \\frac{1}{w^2} \\,dw \\right|$, $z\\in\\mathbb C$, $Q(0,\\varepsilon)$ being the squar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2196","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"}