Integrals of weighted maximal logarithmic kernels on bounded Vilenkin groups (Q2428649)

Let \(\{m_k\}_{k=0}^\infty\) be a sequence of natural numbers that are greater than \(1\), let \(M_0=1\), \(M_{k+1}=M_km_k\), \(k\geq 1\), and let \(\mathbb Z_{m_k}\) be the \(m_k\)-th cyclic group with discrete topology and measure \(\mu(x)=1/m_k\) for all \(x\in \mathbb Z_{m_k}\). Then \(G_m:=\prod^\infty_{k=0} \mathbb Z_{m_k}\) is a compact Vilenkin group. Let \(\{\psi_n\}_{n=0}^\infty\) be the character group of \(G_m\) with Vilenkin numeration (see Appendix 0.7. in [\textit{F. Schipp} et al., Walsh series. An introduction to dyadic harmonic analysis. Bristol etc.: Adam Hilger (1990; Zbl 0727.42017)]). By definition, \(D_n:=\sum^{n-1}_{k=0}\psi_k\), \(n\in\mathbb N=\{1,2,\dots\}\), \(H_n=\sum^{n-1}_{k=1}1/k\), \(R_n=H_n^{-1}\sum^{n-1}_{k=1}D_k/k\), \(N_n=H_n^{-1}\sum^{n-1}_{k=1}D_k/(n-k)\) for \(n\geq 2\) and \(H_1=1\), \(R_1=N_1=0\). Let \(R^*_{\alpha}(x)=\sup_{n\geq 1}| R_n(x)| /\alpha(n)\), \(N^*_{\alpha}(x)=\sup_{n\geq 1}| N_n(x)| /\alpha(n)\), where \(\alpha:[0,+\infty)\to [1,+\infty)\) is a monotone increasing function. Theorem 1. If \(\{m_k\}_{k=0}^\infty\) is bounded, then \[ 0<C_1\sum^\infty_{n=1}\frac{1}{n\alpha(M_n)}\leq \int_{G_m}R^*_{\alpha}(x)\,dx\leq C_2\sum^\infty_{n=1} \frac{1}{n\alpha(M_n)}. \] Theorem 2. If \(\{m_k\}_{k=0}^\infty\) is bounded, then \[ 0<C_1\sum^\infty_{n=1}\frac{1}{\alpha(M_n)}\leq \int_{G_m}N^*_{\alpha}(x)\,dx\leq C_2\sum^\infty_{n=1} \frac{1}{\alpha(M_n)}. \]