{"paper":{"title":"Borderline weighted estimates for commutators of singular integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos P\\'erez, Israel P. Rivera-R\\'ios","submitted_at":"2015-07-30T16:28:41Z","abstract_excerpt":"In this paper we establish the following estimate \\[ w\\left(\\left\\{ x\\in\\mathbb{R}^{n}\\,:\\,\\left|[b,T]f(x)\\right| > \\lambda\\right\\} \\right)\\leq \\frac{c_{T}}{\\varepsilon^{2}}\\int_{\\mathbb{R}^{n}}\\Phi\\left(\\|b\\|_{BMO}\\frac{|f(x)|}{\\lambda}\\right)M_{L(\\log L)^{1+\\varepsilon}}w(x)dx \\] where $w\\geq0, \\, 0<\\varepsilon<1$ and $\\Phi(t)=t(t+\\log^+(t))$. This inequality relies upon the following sharp $L^p$ estimate \\[ \\|[b,T]f\\|_{L^{p}(w)}\\leq c_{T}\\left(p'\\right)^{2}p^{2}\\left(\\frac{p-1}{\\delta}\\right)^{\\frac{1}{p'}} \\|b\\|_{BMO} \\, \\|f \\|_{L^{p}(M_{L(\\log L)^{2p-1+\\delta}}w)} \\]where $1<p<\\infty, w\\g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08568","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"}