Multipliers on bi-parameter Haar system Hardy spaces

Abstract

AbstractLet $$(h_I)$$ ( 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 , and ask which more elementary operators factor through D. A decisive role is played by the Capon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ C : V ( δ 2 ) → V ( δ 2 ) given by $$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ C h I ⊗ h J = h I ⊗ h J if $$|I|\le |J|$$ | I | ≤ | J | , and $$\mathcal {C} h_I\otimes h_J = 0$$ C h I ⊗ h J = 0 if $$|I| > |J|$$ | I | > | J | , as our main result highlights: Given any bounded Haar multiplier $$D:X(Y)\rightarrow X(Y)$$ D : X ( Y ) → X ( Y ) , there exist $$\lambda ,\mu \in \mathbb {R}$$ λ , μ ∈ R such that $$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$ λ C + μ ( Id - C ) approximately 1-projectionally factors through D , i.e., for all $$\eta > 0$$ η > 0 , there exist bounded operators A, B so that AB is the identity operator $${{\,\textrm{Id}\,}}$$ Id , $$\Vert A\Vert \cdot \Vert B\Vert = 1$$ ‖ A ‖ · ‖ B ‖ = 1 and $$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$ ‖ λ C + μ ( Id - C ) - A D B ‖ < η . Additionally, if $$\mathcal {C}$$ C is unbounded on X(Y), then $$\lambda = \mu $$ λ = μ and then $${{\,\textrm{Id}\,}}$$ Id either factors through D or $${{\,\textrm{Id}\,}}-D$$ Id - D .