Weighted estimates for maximal multilinear commutators

<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>T</jats:italic> be a Calderón–Zygmund operator with regular kernel <jats:italic>K</jats:italic> and <jats:italic>T</jats:italic> *<jats:sub><jats:bold><jats:italic>b</jats:italic></jats:bold></jats:sub> be the maximal multilinear commutator defined by <jats:disp-formula> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" position="anchor" xlink:href="urn:x-wiley:0025584X:media:MANA200310370:mana1037001"><jats:alt-text>equation image</jats:alt-text></jats:graphic> </jats:disp-formula>.</jats:p><jats:p> In this paper, the following weighted estimates for <jats:italic>T</jats:italic> *<jats:sub><jats:bold><jats:italic>b</jats:italic></jats:bold></jats:sub> are discussed. Precisely, for 0 < <jats:italic>p</jats:italic> < ∞, <jats:italic>ω</jats:italic> ∈ <jats:italic>A</jats:italic> <jats:sub>∞</jats:sub> and <jats:italic>b</jats:italic> <jats:sub><jats:italic>j</jats:italic> </jats:sub> ∈ <jats:italic>Osc</jats:italic> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="equation/tex2gif-inf-5.gif" xlink:title="urn:x-wiley:0025584X:media:MANA200310370:tex2gif-inf-5" />, <jats:italic>r<jats:sub>j</jats:sub></jats:italic> ≥ 1, <jats:italic>j</jats:italic> = 1, … , <jats:italic>m</jats:italic> , there exists a positive constant <jats:italic>C</jats:italic> such that <jats:disp-formula> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" position="anchor" xlink:href="urn:x-wiley:0025584X:media:MANA200310370:mana1037002"><jats:alt-text>equation image</jats:alt-text></jats:graphic> </jats:disp-formula>.</jats:p><jats:p>For <jats:italic>p</jats:italic> = 1 and <jats:italic>ω</jats:italic> ∈ <jats:italic>A</jats:italic> <jats:sub>1</jats:sub>, the weighted weak <jats:italic>L</jats:italic> (log <jats:italic>L</jats:italic> )<jats:sup>1/<jats:italic>r</jats:italic> </jats:sup>‐type estimates are also established.</jats:p><jats:p>Our theorems are parallel to the ones of the multilinear commutators of Calderón–Zygmund operators obtained in [18] and extend the main result in [14] essentially. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)</jats:p>