Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series

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^∞n_i2ⁱ, n_i ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I²) under the condition sup_AV (nA) < ∞, the subsequence of quadratic partial sums Sn□A(f) of two-dimensional Walsh–Fourier series converges 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 {n_A : A ≥ 1} with the condition sup_AV(nA) < ∞ and a function f ∈ φ(L)(I²) for which supA|Sn◻A(x1,x2;f)|=∞ for almost all (x¹, x²) ∈ I².