Multipliers on bi-parameter Haar system Hardy spaces

Let ( h I ) denote the standard Haar system on [0, 1], indexed by I ∈ D , the set of dyadic intervals and h I ⊗ h J denote the tensor product ( s , t ) ↦ h I ( s ) h J ( t ) , I , J ∈ D . We consider a class of two-parameter function spaces which are completions of the linear span V ( δ 2 ) of h I ⊗ h J , I , J ∈ 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 ≤ p < ∞ . We say that D : X ( Y ) → X ( Y ) is a Haar multiplier if D ( h I ⊗ h J ) = d I , J h I ⊗ h J , where d I , J ∈ R , and ask which more elementary operators factor through D. A decisive role is played by the Capon projection C : V ( δ 2 ) → V ( δ 2 ) given by C h I ⊗ h J = h I ⊗ h J if | I | ≤ | J | , and C h I ⊗ h J = 0 if | I | > | J | , as our main result highlights: Given any bounded Haar multiplier D : X ( Y ) → X ( Y ) , there exist λ , μ ∈ R such that λ C + μ ( Id - C ) approximately 1-projectionally factors through D , i.e., for all η > 0 , there exist bounded operators A, B so that AB is the identity operator Id , ‖ A ‖ · ‖ B ‖ = 1 and ‖ λ C + μ ( Id - C ) - A D B ‖ < η . Additionally, if C is unbounded on X( Y), then λ = μ and then Id either factors through D or Id - D .