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

Let |n||n| be the lower integer part of the binary logarithm of the positive integer nn and α:N2→N2α:N2→N2. In this paper we generalize the notion of the two dimensional Marcinkiewicz means of Fourier series of two-variable integrable functions as tnαf≔1n∑k=0n−1Sα(|n|,k)f and give a kind of necessary and sufficient condition for functions in order to have the almost everywhere relation tnαf→f for all f∈L1([0,1)2)f∈L1([0,1)2) with respect to the Walsh–Paley system. The original version of the Marcinkiewicz means are defined by α(|n|,k)=(k,k)α(|n|,k)=(k,k) and discussed by a lot of authors. See for instance [13], [8], [6], [3] and [11].