Let \((h_I)\) denote the standard Haar system on [0, 1], indexed by \(I\in \mathcal {D}\) , the set of dyadic intervals and \(h_I\otimes h_J\) denote the tensor product \((s,t)\mapsto h_I(s) h_J(t)\) , \(I,J\in \mathcal {D}\) . We consider a class of two-parameter function spaces which are completions of the linear span \(\mathcal {V}(\delta ^2)\) of \(h_I\otimes h_J\) , \(I,J\in \mathcal {D}\) . This class contains all the spaces of the form X(Y), where X and Y are either the Lebesgue spaces \(L^p[0,1]\) or the Hardy spaces \(H^p[0,1]\) , \(1\le p < \infty \) . We say that \(D:X(Y)\rightarrow X(Y)\) is a Haar multiplier if \(D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J\) , where \(d_{I,J}\in \mathbb {R}\) , and ask which more elementary operators factor through D. A decisive role is played by the Capon projection \(\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)\) given by \(\mathcal {C} h_I\otimes h_J = h_I\otimes h_J\) if \(|I|\le |J|\) , and \(\mathcal {C} h_I\otimes h_J = 0\) if \(|I| > |J|\) , as our main result highlights: Given any bounded Haar multiplier \(D:X(Y)\rightarrow X(Y)\) , there exist \(\lambda ,\mu \in \mathbb {R}\) such that \(\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}\) i.e., for all \(\eta > 0\) , there exist bounded operators A, B so that AB is the identity operator \({{\,\textrm{Id}\,}}\) , \(\Vert A\Vert \cdot \Vert B\Vert = 1\) and \(\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta \) . Additionally, if \(\mathcal {C}\) is unbounded on X(Y), then \(\lambda = \mu \) and then \({{\,\textrm{Id}\,}}\) either factors through D or \({{\,\textrm{Id}\,}}-D\) .
