Vector \(A_2\) weights and a Hardy-Littlewood maximal function (Q2701692)

Let \(\mathcal M\) be the space of all nonnegative Hermitian operators on a Hilbert space \(\mathcal H\). A measurable locally intergable function \(w: {\mathbb R}^n\to{\mathcal M}\) is said to satisfy the \(A_2\)-condition if there is \(C<\infty\) such that \(\|w_Q^{1/2}(w^{-1})_Q^{1/2}\|\leq C\) for every cube \(Q\subset{\mathbb R}^n\) (the subscript \(Q\) means taking the average over \(Q\)). Next, the maximal function associated with \(w\) is an operator from \({\mathcal H}\)-valued to scalar valued functions defined by NEWLINE\[NEWLINE M_w f(x)=\sup_{x\in Q} |Q|^{-1} \int_Q \|w^{1/2}(x)w^{-1/2}(y)f(y)\|dy. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEIt is proved that if \({\mathcal H}\) is finite-dimensional and \(w\) satisfies the \(A_2\)-condition, then there is \(\delta >0\) such that \(M_w\) is bounded from \(L^p ({\mathbb R}^n, {\mathcal H})\) to \(L^p ({\mathbb R}^n)\) whenever \(|p-2|<\delta\). Conversely, if \(M_w\) is bounded for \(p=2\), then \(w\in A_2\). NEWLINENEWLINENEWLINEA forthcoming paper of the second author is mentioned in which a similar result is proved for \(1<p<\infty\). NEWLINENEWLINENEWLINEThe question of introducing a proper analog of the maximal function in the context of operator-valued weights was first raised in [\textit{A. Volberg}, J. Am. Math. Soc. 10, No. 2, 445-466 (1997; Zbl 0877.42003)] and [\textit{S. Treil} and \textit{A. Volberg}, J. Funct. Anal. 143, No. 2, 269-308 (1997; Zbl 0876.42027)].