Almost everywhere summability of two-dimensional Walsh-Fourier series (Q2154922)

The weighted variation of a natural number \(n\) corresponding to a sequence of positive numbers \(\alpha=(\alpha_n)\) is defined as follows \[ V_{\alpha}(n)=\sum_{i=0}^{\infty}|\epsilon_i(n)-\epsilon_{i+1}(n)|2^{\alpha_n i}, \] where \((\epsilon_i(n))\) is the sequence of binary coefficients of \(n\). For a sequence \(\alpha=(\alpha_n)\) and \(C>0\) by \(N(\alpha,C)\) denote the set \( \{n\in \mathbb{N}: V_{\alpha}(n)/n^{\alpha_n}\leq C\}.\) By \(\sigma_{n,m}^{\alpha,\beta}(f)\) denote Cesáro two-dimensional means of the Walsh-Fourier series of a function \(f\in L([0,1]^2)\) corresponding to parameters \(\alpha\) and \(\beta\). In the paper, it is proved the following result concerning the convergence of subsequences of the Cesáro means with variable parameters. Theorem. Let \((\alpha_n)\) and \((\beta_n)\) be sequences of positive numbers with \(\lim_{n\to\infty}\alpha_n=\lim_{n\to\infty}\beta_n=0\). Then for every function \(f\) from the class \(L\log^+ L([0,1]^2)\) and every positive number \(C\), \[ \lim_{n,m\in N(\alpha,C);\; n,m\to \infty}\sigma_{n,m}^{\alpha_n,\beta_n}(f)(x,y)=f(x,y) \] for almost every \((x,y)\in[0,1]^2\). This theorem is deduced from the result on boundedness of the corresponding maximal operators on dyadic Hardy spaces. The obtained results generalize theorems of \textit{F. Weisz} [Acta Math. Hung. 161, No. 1, 292--312 (2020; Zbl 1463.42070)] in which there is considered the quantity \(P_{\alpha}(n)=\sum_{i=0}^{\infty}\epsilon_i(n)2^{\alpha_n i}\) instead of the weighted variation \(V_{\alpha}(n)\).