On subsequences of Lebesgue functions of general uniformly bounded ONS (Q2089919)

In this paper, the author proves the following Olevskii-type inequality about the growth of Lebesgue functions of the general uniformly bounded orthonormal system (ONS): Theorem 2. Let \(\{\phi_n(x)\}_{n=1}^\infty\) be an ONS on \([0,1]\) that satisfies the following condition \[ |\phi_n(x)|\leq M, \quad x\in [0,1], \quad n\in\mathbb N, \] where \(M\) is a given positive constant. Let \(\{\nu_k(x)\}_{k=1}^\infty\) be a strictly increasing sequence of positive integers and \(h(x)\) be a positive, differentiable on \((a,\infty)\), function that satisfies the following conditions: \[ C_1 [h(x)]^\beta \leq h^\prime(x)\leq C_2 [h(x)]^\beta, \quad x\in(a,\infty) \] \[ \nu_k=[h(k)], \quad k\in\mathbb N, \] where \(\alpha\), \(\beta\), \(C_1\), \(C_2\) are constants such that \[ 0<\alpha<1, \quad 0\leq \beta<1, \quad C_1>0, \quad C_2>0. \] Then for any number \(A>A_0\) there exists an integer \(m_0=m_0(A)\) such that \(\nu_{m_0}\geq A/2\) and \[ \mu_1\left \{x\in[0,1]: L_{\nu_{m_0}}^\phi\geq C_3\frac{\log_2\nu_{m_0}}{\log_2 \log_2\nu_{m_0}}\right \}\geq \gamma_1>0, \] where \(A_0\), \(\gamma_1\) and \(C_3\) are positive constants.