The Fejér means of the double Walsh-Fourier series of a martingale \(f\) are defined by \[ \sigma_{n,m}f:= {1\over nm} \sum^n_{k=1} \sum^m_{\ell= 1} s_{k,\ell} f, \] where the \(s_{k,\ell}f\) are the rectangular partial sums of the double series in question. The maximal Fejér operator is defined by \(\sigma_* f:= \sup\{|\sigma_{n,m}f|:n,m\in\mathbb{N}\}\).
Iterating the corresponding one-dimensional result gives Theorem 4.1, which says that if \(f\in L_p[0,1)^2\) for some \(1< p\leq\infty\), then \[ \|\sigma_* f\|_p\leq C_p\| f\|_p, \] where \(C_p\) is a constant depending only on \(p\).
The main result of the present paper is Theorem 4.2, which says that if \(f\in H_p[0,1)^2\), the two-parameter martingale Hardy space, for some \(1/2< p\leq\infty\), then \[ \|\sigma_* f\|_p\leq C_p\| f\|_{H_p}. \] Pointwise convergence of the Fejér means follows from these theorems by applying the density argument of J. Marcinkiewicz and A. Zygmund [Fundam. Math. 32, 122–132 (1939; Zbl 0022.01804)]. In particular, it follows that if \(f\in L(\log L)[0, 1)^2\), then the Fejér means of the two-dimensional Walsh-Fourier series of \(f\) converge almost everywhere.