Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series

The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh- Fourier series is of type (H^#,L₁) provided that the supremum in the maximal operator is taken over some special indices. The set of Walsh polynomials is dense in H^#, so by the well-known density argument we have that t_2n,2m f (x¹, x²) → f (x¹, x²) a. e. as m, n → ∞ for all f ∈ H^#(⊃ L log⁺ L). We also prove the sharpness of this result. Namely, For all measurable function δ: [0,+∞) → [0,+∞), lim_t→∞ δ(t) = 0 we have a function f such as f ∈ Llog⁺ Lδ(L) and the two-dimensional Nörlund logarithmic means do not converge to f a.e. (in the Pringsheim sense) on I².