On permutations in the space BMO\(([0,1]\times [0,1])\) (Q1358607)

Let \(A= A_0\cup\{[0, 1]\}\) where \(A_0=\{[k2^{-n},(k+ 1)2^{-n}):0\leq k<2^n,\;k,n\in N\cup\{0\}\}\), and \(A^{(2)}= A\times A\). Let \(\chi^{(2)}= \{h_B: B\in A^{(2)}\}\) be the Haar double system on \(Q=[0, 1]\times[0,1]\). Every bijection \({\mathcal P}:A^{(2)}\to A^{(2)}\) generates on the linear span \({\mathcal L}^{(2)}\) of \(\chi^{(2)}\) the operator \(R^{(2)}_{\mathcal P}f= \sum_{B\in A^{(2)}}\widehat f_Bh_{{\mathcal P}(B)}\), where \(\widehat f_B= \int_Q f(t)h_B(t)dt\), \(B\in A^{(2)}\). In the article, the operator \(R^{(2)}_{\mathcal P}\) is considered on the dyadic space \(\text{BMO}(Q)\) in which \({\mathcal L}^{(2)}\) is dense and the quantities \(|R^{(2)}_{\mathcal P}|_{\text{BMO}(Q)\to\text{BMO}(Q)}\) are investigated for the measure preserving bijection \(\mathcal P\).