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 (), where and 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 . A decisive role is played by the 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 ,  so that is the identity operator , and . Additionally, if is unbounded on (), then and then either factors through or .