Studia Math. 130 (2) (1998), 135-148 ON (C, 1) SUMMABILITY OF INTEGRABLE FUNCTIONS WITH RESPECT TO THE WALSH-KACZMARZ SYSTEM G. G´ at Abstract. Let G be the Walsh group. In this paper we prove for f ∈ L 1 (G) integrable functions the a.e. convergence σ n f → f (n →∞), where σ n is the n-th (C, 1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator σ * f := sup n |σ n f |. We prove that σ * is of type (p, p) for all 1 <p ≤∞ and of weak type (1, 1). Moreover, kσ * f k 1 ≤ ck|f |k H , where H is the Hardy space on the Walsh group. Introduction And The Main Results This paper is devoted to the problem of a.e. convergence of the (C, 1) means of integrable functions with respect to the Walsh-Kaczmarz system. The Walsh system in the Kaczmarz enumeration was studied by a lot of authors (see [SCH], [SCH2], [SKV], [SKV2], [BAL], [SWS], [WY]). In [SNE] it has been pointed out that the behavior of the Dirichlet kernel of the Walsh-Kaczmarz system is worse than of the kernel of the Walsh- Paley system considered more often. Namely, it is proved [SNE] that for the Dirichlet kernel D n (x) of the Walsh-Kaczmarz system the inequality lim sup n→∞ D n (x) log n ≥ C> 0 holds a.e. This “spreadness” of this system makes easier to construct examples of divergent Fourier series [BAL]. A number of pathological properties is due to this “spreadness” property of the kernel. For example, for Fourier series with respect to the Walsh- Kaczmarz system it is impossible to establish any local test for convergence at a point or on an interval, since the principle of localization does not hold for this system. On the other hand the global behavior of the Fourier series with respect to this system is similar in many aspects to the case of the Walsh-Paley system. Schipp [SCH1] and Wo-Sang Young [WY] proved that the Walsh- Kaczmarz system is a convergence system. Skvorcov proved for continuous functions f , that Fej´ er means converges uniformly to f . In this paper we prove for integrable functions that the Fej´ er means (with respect to the Walsh-Kaczmarz system) converges almost everywhere to the function. Let P denote the set of positive integers, N := P ∪{0} the set of nonnega- tive integers and Z 2 the discrete cyclic group of order 2, respectively. That 1991 Mathematics Subject Classification. primary: 42C10, secondary: 43A75, 40 G05. Research supported by the Hungarian National Foundation for Scientific Research (OTKA) , grant no. F020334 and by the Hungarian “M˝ uvel˝od´ esi ´ esK¨ozoktat´asiMin- iszt´ erium”, grant no. FKFP 0710/1997 Typeset by A M S-T E X 1