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arXiv · 1801.08302

Weighted boundedness of the 2-fold product of Hardy-Littlewood maximal operators

María J. Carro·Eduard Roure
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Abstract

We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by $M^{\otimes}(f,g):= MfMg$. This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón-Zygmund operators, the bilinear Hardy-Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures $$ \Vert fg \Vert_{L^{p,\infty}\left(w_1^{p/p_1} w_2^{p/p_2}\right)} \le C \Vert f \Vert_{L^{p_1,\infty}(w_1 )} \Vert g \Vert_{L^{p_2,\infty}(w_2 )}. $$ Unfortunately, we shall prove that this inequality does not hold, in general, and we shall have to consider a weaker version of it.

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María J. Carro, Eduard Roure. 2018-01-25. Weighted boundedness of the 2-fold product of Hardy-Littlewood maximal operators. https://doi.org/10.1002/mana.201700271

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