{"paper":{"title":"The $L(\\log L)^{\\epsilon}$ endpoint estimate for maximal singular integral operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos P\\'erez, Tuomas Hyt\\\"onen","submitted_at":"2015-03-13T10:20:24Z","abstract_excerpt":"We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \\|T^*f\\|_{L^{1,\\infty}(w)} \\lesssim \\frac{1}{\\epsilon}\n  \\int_{\\mathbb{R}^n} |f(x)| M_{L(\\log L)^{\\epsilon}} (w)(x)dx$, for $w\\geq 0, 0<\\epsilon \\leq 1.$ This follows from the sharp $L^p$ estimate $ \\|T^*f \\|_{ L^{p}(w) } \\lesssim p' (\\frac{1}{\\delta})^{1/p'} \\|f \\|_{L^{p}(M_{ L(\\log L)^{p-1+\\delta}} (w))}$, for $1<p<\\infty, w\\geq 0, 0<\\delta \\leq 1. $ As as a consequence we deduce that $\n  \\|T^*f\\|_{L^{1,\\infty}(w)} \\lesssim [w]_{A_{1}} \\log(e+ [w]_{A_{\\infty}}) \\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.04008","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"}