Multipliers on bi-parameter Haar system Hardy spaces

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