Multipliers on bi-parameter Haar system Hardy spaces

Abstract Let $$(h_I)$$ ( h I ) denote the standard Haar system on [0, 1], indexed by $$I\in \mathcal {D}$$ I ∈ D , the set of dyadic intervals and $$h_I\otimes h_J$$ h I ⊗ h J denote the tensor product $$(s,t)\mapsto h_I(s) h_J(t)$$ ( s , t ) ↦ h I ( s ) h J ( t ) , $$I,J\in \mathcal {D}$$ I , J ∈ D . We consider a class of two-parameter function spaces which are completions of the linear span $$\mathcal {V}(\delta ^2)$$ V ( δ 2 ) of $$h_I\otimes h_J$$ h I ⊗ h J , $$I,J\in \mathcal {D}$$ I , J ∈ D . This class contains all the spaces of the form X(Y), where X and Y are either the Lebesgue spaces $$L^p[0,1]$$ L p [ 0 , 1 ] or the Hardy spaces $$H^p[0,1]$$ H p [ 0 , 1 ] , $$1\le p < \infty $$ 1 ≤ p < ∞ . We say that $$D:X(Y)\rightarrow X(Y)$$ D : X ( Y ) → X ( Y ) is a Haar multiplier if $$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ D ( h I ⊗ h J ) = d I , J h I ⊗ h J , where $$d_{I,J}\in \mathbb {R}$$ d I , J ∈ R

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