On almost everywhere convergence and divergence of Marcinkiewicz-like means of integrable functions with respect to the two-dimensional Walsh system

Abstract

In this paper we generalize the notion of the two dimensional Marcinkiewicz means of Fourier series of two-variable integrable functions as $t^{\alpha}_n f:= \frac{1}{n}\sum_{k=0}^{n-1}S_{\alpha_1(|n|,k), \alpha_2(|n|,k)}f$ ($|n|=\lfloor\log_2 n\rfloor$) and give a kind of necessary and sufficient condition for functions $\alpha :\mathbb{N}^2\to \mathbb{N}^2$ in order to have the almost everywhere relation $t^{\alpha}_n f\to f$ for all \(f\in L^1([0,1)^2)\) with respect to the Walsh-Paley system. The original version of the Marcinkiewicz means is defined by $\alpha_j(|n|, k)=k \, (j=1,2)$ and discussed a lot of authors. See for instance \cite{zhi, Mar, gog2, gatgr, wmar}.