{"paper":{"title":"The full range of uniform bounds for the bilinear Hilbert transform","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Gennady Uraltsev, Micha{\\l} Warchalski","submitted_at":"2022-05-19T20:51:05Z","abstract_excerpt":"We prove uniform uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $\\mathrm{BHT}_{\\beta} [f_1, f_2] (x) := \\mathrm{p.v.} \\int_{\\mathbb{R}} f_1 (x - t) f_2 (x + \\beta t) \\frac{\\mathrm{d} t}{t}$. We show that the operator $\\mathrm{BHT}_{\\beta}$ maps $L^{p_{1}}\\times L^{p_{2}}$ into $L^{p}$ as long as $p_1 \\in (1, \\infty)$, $p_2 \\in (1, \\infty)$, and $p > \\frac{2}{3}$ with a bound independent of $\\beta\\in(0,1]$. This is the full open range of exponents where the modulation invariant class of bilinear operators containing $\\mathrm{BHT}_{\\beta}$ can be bounded uniformly. This is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.09851","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2205.09851/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}