On rearrangements of the Haar system in the space BMO (Q1900722)

Let \(A\) be the set of the dyadic intervals of \([0,1)\) including the empty interval \(\emptyset\) and consider the Haar system \(\chi : = \{h_I : I \in A\}\). Every bijection \(\Pi : A \to A\) generates an operator \(R_\Pi\) defined by \[ R_\Pi f : = \sum_{I \in A} \widehat f_I h_{\Pi (I)}, \quad \widehat f_I : = \int^1_0 f(t)h_I (t)dt. \] Denote by \(H\) the dyadic Hardy space, and by BMO the dyadic space dual to \(H\). The author proves that if \(\Pi : A \to A\) is a bijection and \(\Pi (\emptyset) = \emptyset\), then \[ |R_\Pi |_H \sim |R_{\Pi^{-1}} |_{BMO}. \] This improves and generalizes an earlier result by \textit{F. Schipp} [Anal. Math. 16, No. 2, 135-141 (1990; Zbl 0736.42019)].