On rearrangements of the Haar system in \(L^p\) (Q5959462)

Let us set \(I \equiv I_k^n = [k\cdot 2^{ - n},(k+1) \cdot 2^{ - n})\) and \(A_0 = \{ I_k^n :k =0,\dots,2^n - 1, n \in N \}\), where \(N = \{ 0,1,\dots\}\). The author numbers orthonormal Haar system \(\{ h(I), I \in A \}\) by the elements of the set \(A = A_0 \cup\{ \emptyset \}\) as follows: \(h_I ( t) = 2^{n / 2}\) for \(t \in I \equiv I_{2k}^{n + 1} \), \(h_I ( t) = 2^{ - n / 2}\) for \(t \in I \equiv I_{2k + 1}^{n + 1} \), \(h_I ( t) = 0\) for \(t\in [0, 1)\backslash I \equiv I_k^n \). The bijection \(\pi:A \to A\) is said to be measure-preserving if for each \(I \in A\) we have \(|\pi ( I)|= |I |\), where \(|I |\) is Lebesgue measure of the set \(I\). Let us set \[ \|\pi \|= \sup\limits_{I \in A_0 } \left( {|\cup _{J \subset I} \pi (J)|\over |I|}\right)^{1/2}. \] Each bijection \(\pi :A \to A\) for \(f \in L^p[ {0, 1})\), \(1< p < \infty \), generates the operator \[ R_\pi f=\sum\limits_{I \in A} \hat {f}_I h_{\pi ( I)},\quad \hat {f}_I = \int_0^1 f( t) h_I ( t) dt . \] Theorem 1. Suppose that \(\pi :A \to A\) is a nonidentical measure-preserving bejection and \(\|\pi\|= 1\). Then \[ \|R_\pi \|_{L^p[0,1)\to L^p[0,1)} \geq \begin{cases} \left( {3^{p/(p-1)} + 1\over 2( 2^{p / (p-1)} + 1)} \right) ^{(p-1)/p},& 1 < p< 2, \\ \left( {3^p + 1\over 2(2^p+1)} \right)^{1/p},& 2 < p< \infty. \end{cases} \] Theorem 2. Suppose that \(\pi :A \to A\) is a measure-preserving bejection and \(\|\pi\|\neq 1\). Then \[ \|R_\pi \|_{L^p[0,1)\to L^p[0,1)} \geq \begin{cases} \left( {2^{1/(p-1)}+1\over 3} \right) ^{(p-1)/p},& 1 < p < 2, \\ \left( {2^{p-1}+1\over 3} \right)^{1/p},& 2 < p < \infty. \end{cases} . \]