  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 $D\colon X(Y)\to X(Y)$ 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 plays the
  {\em Capon projection} $\mathcal{C}\colon \mathcal{V}(\delta^2)\to
\mathcal{V}(\delta^2)$ given by $\mathcal{C} h_I\otimes h_J = h_I\otimes h_J$
if $|I|\leq |J|$, and $\mathcal{C} h_I\otimes h_J = 0$ if $|I| > |J|$, as our
main result highlights:
  Given any bounded Haar multiplier $D\colon X(Y)\to X(Y)$, there exist
$\lambda,\mu\in \mathbb{R}$ such that \begin{equation*}
  \text{$\lambda \mathcal{C} + \mu (\mathrm{Id}-\mathcal{C})$ approximately
$1$-projectionally factors through $D$,} \end{equation*} i.e., for all
$\eta>0$, there exist bounded operators $A,B$ so that $AB$ is the identity
operator $\mathrm{Id}$, $\|A\|\cdot\|B\|=1$ and $\|\lambda \mathcal{C} + \mu
(\mathrm{Id}-\mathcal{C}) - ADB\|<\eta$. Additionally, if $\mathcal{C}$ is
unbounded on $X(Y)$, then $\lambda = \mu$ and then $\mathrm{Id}$ either factors
through $D$ or $\mathrm{Id}-D$.
