Operator-valued dyadic harmonic analysis beyond doubling measures (Q2814407)

A sequence \(\Phi=\{\phi_Q\}_{Q\in\mathcal{D}}\) is said to be a generalized Haar system in \(\mathbb{R}^d\) adapted to a locally finite Borel measure \(\mu\) and a dyadic lattice \(\mathcal{D}\), if the following conditions hold:NEWLINENEWLINE(a) For every \(Q\in\mathcal{D}\), \(\phi_Q\) is supported in \(Q\);NEWLINENEWLINE(b) If \(Q'\), \(Q\in\mathcal{D}\) and \(Q'\) is a proper subset of \(Q\), then \(\phi_Q\) is a constant on \(Q'\);NEWLINENEWLINE(c) For every \(Q\in\mathcal{D}\), \(\int_{\mathbb{R}^d}\phi_Q\,d\mu=0\);NEWLINENEWLINE(d) For every \(Q\in\mathcal{D}\), either \(\|\phi_Q\|_{L^2(\mu)}=1\) or \(\phi_Q\equiv0\) and \(\mu(Q)=0\).NEWLINENEWLINEThis class is a generalization of the classical Haar wavelet system in \(\mathbb{R}^d\). If the condition (c) is not imposed, then the Haar system is said to be noncancellative.NEWLINENEWLINEGiven two nonnecessarily cancellative generalized Haar systems \(\Phi=\{\phi_Q\}_{Q\in\mathcal{D}}\) and \(\Psi=\{\psi_Q\}_{Q\in\mathcal{D}}\), the Haar shift operator of complexity \((r,s)\in \mathbb{N}\times\mathbb{N}\) is defined as NEWLINE\[NEWLINE\amalg_{r,s}f(x)=\sum_{Q\in\mathcal{D}}\sum_{R,S\in \mathcal{D}_r(Q)} a^Q_{R,S}\langle f,\phi_R\rangle \psi_S(x)NEWLINE\]NEWLINE with \(\sup_{Q,R,S}|a^Q_{R,S}|<\infty\), where \(\mathcal{D}_k(Q)\) with \(k\in\mathbb{N}\) is the family of \(k\)-dyadic descendants of \(Q\). It is well known that many classical operators in dyadic harmonic analysis are special cases of Haar shift operators, such as the Haar multipliers, dyadic paraproducts and so on. In this paper, the authors provide a sufficient condition for the weak type (1,1) boundedness of the Haar shift operator in the operator-valued setting. More precisely, it is proved that, if the Haar shift operator satisfies the restricted local vector-valued \(L_2\) estimate NEWLINE\[NEWLINE\int_{\mathbb{R}^d}\|\amalg_{r,s}^{Q_0}(1_{Q_0})(x)\|_{\mathcal{M}}^2\,d\mu(x)\leq C\mu(Q_0)NEWLINE\]NEWLINE uniformly in \(Q_0\in\mathcal{D}\), where \(\mathcal{M}\) is a von Neumann algebra and \(\amalg_{r,s}^{Q_0}\) is defined via replacing the summation of \(Q\) over all \(\mathcal{D}\) in the definition of \(\amalg_{r,s}\) by the summation over all dyadic subcubes of \(Q_0\), and NEWLINE\[NEWLINE\sup\{\|\phi_R\|_{L_\infty(\mu)}\|\psi_S\|_{L_1(\mu)}:\;R\in \mathcal{D}_r(Q),\;S\in \mathcal{D}_s(Q),\;Q\in \mathcal{D}\}<\infty,NEWLINE\]NEWLINE then the Haar shift operator is of weak type (1,1).