On Lebesgue functions of uniformly bounded orthonormal systems (Q5951096)

Let \(\{\varphi_n(x): n= 1,2,\dots\}\) be an ONS on \([0,1]\) such that \[ |\varphi_n(x)|\leq M\quad\text{for all }n\text{ and }x, \] and let \[ L_m(x):= \int^1_0 \Biggl|\sum^m_{k=1}\varphi_k(x) \varphi_k(y)\Biggr|dy,\qquad m= 1,2,\dots, \] be the \(m\)th Lebesgue function of the system \(\{\varphi_n(x)\}\). \textit{A. M. Olevskij} [Izv. Akad. Nauk SSSR, Ser. Mat. 30, 387-432 (1966; Zbl 0161.25902)] proved that \[ \mu\Biggl\{x\in [0,1]: \max_{1\leq m\leq n} L_m(x)\geq C\log n\Biggr\}\geq \gamma, \] where \(\mu\) is the Lebesgue measure, \(C\) and \(\gamma\) are positive constants depending only on \(M\). The present author proves the following complement to this inequality: There exist positive constants \(p_0\), \(C_1\) and \(\gamma_1\) depending only on \(M\) such that for each \(n> p_0\) there exists an integer \(m_n< n\) such that \[ \mu\Biggl\{x\in [0,1]: L_{m_n}(x)\geq C_1 {\log n\over\log\log\log n}\Biggr\}\geq \gamma_1. \] The method of proof goes back to the one elaborated by A. M. Olevskij.