Multipliers on bi-parameter Haar system Hardy spaces

Let $$ denote the standard Haar system on [0, 1], indexed by $$ , the set of dyadic intervals and $$ denote the tensor product $$ , $$ . We consider a class of two-parameter function spaces which are completions of the linear span $$ of $$ , $$ . This class contains all the spaces of the form X(Y), where X and Y are either the Lebesgue spaces $$ or the Hardy spaces $$ , $$ . We say that $$ is a Haar multiplier if $$ , where $$ , and ask which more elementary operators factor through D. A decisive role is played by the Capon projection $$ given by $$ if $$ , and $$ if $$ , as our main result highlights: Given any bounded Haar multiplier $$ , there exist $$ such that i.e., for all $$ , there exist bounded operators A, B so that AB is the identity operator $$ , $$ and $$ . Additionally, if $$ is unbounded on X(Y), then $$ and then $$ either factors through D or $$ .