F. Móricz, F. Schipp and W. R. Wade [Trans. Am. Math. Soc. 329, No. 1, 131-140 (1992; Zbl 0795.42016)] proved that the double \((C,1)\) means \(\sigma_nf\) of the two-dimensional Walsh-Fourier series of \(f\) converge a.e. to \(f\) as \(n\to\infty\), whenever \(f\in L\log^+L([0, 1)^2)\).
The author shows that this convergence does not hold for all \(f\in L_1([0, 1)^2)\). More exactly, it is verified that for all mesurable functions \(\delta:[0,\infty)\to [0,\infty)\), \(\lim_{t\to\infty} \delta(t)= 0\), there is a function \(f\in L\log^+ L\delta(L)\) such that \(\sigma_n f\) does not converge to \(f\) a.e.