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

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We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by M⊗(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 fg ᵖ,∞(w₁ᵖ/ᵖ¹ w₂ᵖ/ᵖ²) ≤ C f ₗᵖ¹,∞₍w₁ ₎ g ₗᵖ²,∞₍w₂ ₎. Unfortunately, we shall prove that this inequality does not hold, in general, and we shall have to consider a weaker version of it.