Almost Everywhere Strong Summability of Double Walsh-Fourier Series

Abstract

In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation $(\frac{1}{n}\sum _{m=0}^{n-1}{\left|S_{mm}f-f\right|}^p{)}^{1/p}\to 0$ for every two-dimensional functions belonging to $L\log L$ and $0<p\le 2$. From the theorem of Getsadze \cite{Gets} it follows that the space $L\log L$ can not be enlarged with preserving this strong summability property.