ON (C, 1) SUMMABILITY OF INTEGRABLE FUNCTIONS WITH RESPECT TO THE WALSH-KACZMARZ SYSTEM

Let G be the Walsh group. For. f is an element of L-1(G) we prove the a.e. convergence a(n)f --> f (n --> infinity), where an is the nth (C, 1) mean of f with respect to the Walsh-Kacamarz system. Define the ma ximal operator sigmaf := sup(n) /sigma(n)f/ We prove that sigma* is o f type (p,p) for all 1 < p < oo and of weak type (1, 1). Moreover, //s igmaf//(1) less than or equal to c///f///(H), where H is the Hardy sp ace on the Walsh group.