Analysis Math., 44 (1) (2018), 73–88 DOI: 10.1007/s10476-018-0107-2 ALMOST EVERYWHERE CONVERGENCE OF SUBSEQUENCE OF QUADRATIC PARTIAL SUMS OF TWO-DIMENSIONAL WALSH–FOURIER SERIES G. G ´ AT 1,∗,† and U. GOGINAVA 2 1 Institute of Mathematics, University of Debrecen, H-4002 Debrecen, Pf. 400, Hungary e-mail: gat.gyorgy@science.unideb.hu 2 Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi, Georgia e-mail: zazagoginava@gmail.com (Received December 3, 2017; accepted January 27, 2018) Abstract. For a non-negative integer n let us denote the dyadic variation of a natural number n by V (n) := ∞  j=0 |nj-nj+1| + n0, where n := ∑ ∞ i=0 ni 2 i , ni ∈{0, 1}. In this paper we prove that for a func- tion f ∈ L log L(I 2 ) under the condition sup A V (nA) < ∞, the subsequence of quadratic partial sums S  n A (f ) of two-dimensional Walsh–Fourier series con- verges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function ϕ : [0, ∞) → [0, ∞) such that ϕ(u)= o(u log u) as u →∞ there exists a sequence {nA : A ≥ 1} with the condition sup A V (nA) < ∞ and a function f ∈ ϕ(L)(I 2 ) for which sup A |S  n A (x 1 ,x 2 ; f )| = ∞ for almost all (x 1 ,x 2 ) ∈ I 2 . 1. Introduction We shall denote the set of all non-negative integers by N, the set of all integers by Z and the set of dyadic rational numbers in the unit interval ∗ Corresponding author. † The first author is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K111651 and by project EFOP-3.6.2-16-2017-00015 supported by the European Union, cofinanced by the European Social Fund. Key words and phrases: double Walsh–Fourier series, quadratic partial sum, almost every- where convergence. Mathematics Subject Classification: 42C10. 0133-3852/$ 20.00 c  2018 Akad´ emiai Kiad´o, Budapest