Multipliers on bi-parameter Haar system Hardy spaces

Let \((h_I)\) denote the standard Haar system on [0, 1], indexed by \(I\in \mathcal {D}\), the set of dyadic intervals and \(h_I\otimes h_J\) denote the tensor product \((s,t)\mapsto h_I(s) h_J(t)\), \(I,J\in \mathcal {D}\). We consider a class of two-parameter function spaces which are completions of the linear span \(\mathcal {V}(\delta ^2)\) of \(h_I\otimes h_J\), \(I,J\in \mathcal {D}\). 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 \(H^p[0,1]\), \(1\le p < \infty \). We say that \(D:X(Y)\rightarrow X(Y)\) is a Haar multiplier if \(D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J\), where \(d_{I,J}\in \mathbb {R}\), and ask which more elementary operators factor through D. A decisive role is played by the Capon projection \(\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)\) given by \(\mathcal {C} h_I\otimes h_J = h_I\otimes h_J\) if \(|I|\le |J|\), and \(\mathcal {C} h_I\otimes h_J = 0\) if \(|I| > |J|\), as our main result highlights: Given any bounded Haar multiplier \(D:X(Y)\rightarrow X(Y)\), there exist \(\lambda ,\mu \in \mathbb {R}\) such that

$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$

i.e., for all \(\eta > 0\), there exist bounded operators A, B so that AB is the identity operator \({{\,\textrm{Id}\,}}\), \(\Vert A\Vert \cdot \Vert B\Vert = 1\) and \(\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta \). Additionally, if \(\mathcal {C}\) is unbounded on X(Y), then \(\lambda = \mu \) and then \({{\,\textrm{Id}\,}}\) either factors through D or \({{\,\textrm{Id}\,}}-D\).