{"paper":{"title":"A note on sharp one-sided bounds for the Hilbert transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Micha{\\l} Strzelecki","submitted_at":"2014-11-17T17:04:04Z","abstract_excerpt":"Let $\\mathcal{H}^{\\mathbb{T}}$ denote the Hilbert transform on the circle. The paper contains the proofs of the sharp estimates \\begin{equation*} \\frac{1}{2\\pi}|\\{ \\xi\\in\\mathbb{T} : \\mathcal{H}^{\\mathbb{T}}f(\\xi) \\geq 1 \\}| \\leq \\frac{4}{\\pi}\\arctan\\left(\\exp\\left(\\frac{\\pi}{2}\\|f\\|_1\\right)\\right) -1, \\quad f\\in L^{1}(\\mathbb{T}), \\end{equation*} and \\begin{equation*} \\frac{1}{2\\pi}|\\{ \\xi\\in\\mathbb{T} : \\mathcal{H}^{\\mathbb{T}}f(\\xi) \\geq 1 \\}| \\leq \\frac{\\|f\\|_2^2}{1+\\|f\\|_2^2},\\quad f\\in L^{2}(\\mathbb{T}). \\end{equation*} Related estimates for orthogonal martingales satisfying a subordina"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.4551","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"}