On an estimate of Calder-Zygmund operators by dyadic positive operators

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• 作者：Andrei K. Lerner (1)
  
• 刊名：Journal d'Analyse Math¨|matique
• 出版年：2013
• 出版时间：October 2013
• 年：2013
• 卷：121
• 期：1
• 页码：141-161
• 全文大小：
• 作者单位：Andrei K. Lerner (1)
  
  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat Gan, Israel
  
• ISSN：1565-8538

文摘

Given a general dyadic grid D and a sparse family of cubes S = {Q j k D, define a dyadic positive operator A D,S by $${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$ . Given a Banach function space X( ^{n ) and the maximal Calder-Zygmund operator ${T_\natural }$ , we show that $${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$ This result is applied to weighted inequalities. In particular, it implies (i) the woweight conjectureby D. Cruz-Uribe and C. Pez in full generality; (ii) a simplification of the proof of the em class="a-plus-plus">A 2 conjecture (iii) an extension of certain mixed A p A r estimates to general Calder-Zygmund operators; (iv) an extension of sharp A 1 estimates (known for T ) to the maximal Calder-Zygmund operator $\natural $ .}