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

Let In I be the lower integer part of the binary logarithm of the positive integer n and alpha : N-2 -> N-2. In this paper we generalize the notion of the two dimensional Marcinkiewicz means of Fourier series of two-variable integrable functions as t(n)(alpha) f := 1/n Sigma(n-1)(k=0) S-alpha(vertical bar n vertical bar,S- k) f and give a kind of necessary and sufficient condition for functions in order to have the almost everywhere relation t(n)(alpha) f -> f for all f is an element of L-1([0, 1)(2)) with respect to the Walsh-Paley system. The original version of the Marcinkiewicz means are defined by alpha (vertical bar n vertical bar, k) = (k, k) and discussed by a lot of authors. See for instance [13,8,6,3,11]. (C) 2011 Elsevier Inc. All rights reserved.