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

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 Calder\'on-Zygmund operators, the bilinear Hardy-Littlewood maximal operator introduced by Calder\'on or in the study of pseudodifferential operators. To this end, we need to study H\"older'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.