{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:E7Q5APWUOCPHQWAJAPTMP2XDGI","short_pith_number":"pith:E7Q5APWU","schema_version":"1.0","canonical_sha256":"27e1d03ed4709e78580903e6c7eae3320a04748e41622c3d9e8f7914a13478db","source":{"kind":"arxiv","id":"1507.08568","version":2},"attestation_state":"computed","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 \\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