Almost everywhere convergence of a subsequence of logarithmic means of Fourier series on the group of 2-adic integers (Q2918678)

The author investigates almost everywhere convergence of a special subsequence of the logarithmic means of integrable functions, NEWLINE\[NEWLINE t_{m_{n}}f:=\frac{1}{\log m_{n}}\sum_{k=1}^{m_{n}-1}\frac{S_{k}f}{ m_{n}-k}\rightarrow f NEWLINE\]NEWLINE for every \(f\in L_{1}\left( I\right) \), where \(I\) is the group of 2-adic integers. It is proved that under the condition NEWLINE\[NEWLINE \sum_{n=1}^{\infty }\frac{\log ^{2}\left( m_{n}-2^{\left[ \log m_{n} \right] }+1\right) }{\log m_{n}}<\infty NEWLINE\]NEWLINE the operator \(t^{\ast }f:=\sup\limits_{n\geq 1}\left| t_{m_{n}}f\right| \) is of weak type (1,1). In particular, for \(f\in L_{1}\left( I\right) \) NEWLINE\[NEWLINE t_{m_{n}}f\rightarrow f \,\,\text{a.e. as } n\rightarrow \infty . NEWLINE\]NEWLINE An analogue of this statement on the Walsh-Paley system was proved by \textit{U. Goginava} [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 21, No. 2, 169--175 (2005; Zbl 1093.42018)] .