Abstract. The Burkholder--Gundy inequality is shown for vector valued martingales. Using this we extend a classical result due to Marcinkievicz and Zygmund to the two-parameter Walsh-system. We extend some results of Sunouchi from one dimension to two dimensions and, with the help of martingale theory, we prove that the Sunouchi operators and the supremum operator of the strong $(C,\alpha,\beta,q)$ means are bounded operators from $L_p$ to $L_p$ $(1< p< \infty )$. As a consequence it is obtained that every function $f\in L_p$ $(1< p< \infty )$ is strong $(C,\alpha,\beta,q)$ summable.

AMS Subject Classification (1991): 42C10, 43A75, 40F05, 60G42

Received September 21, 1994. (Registered under 5665/2009.)