On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system
Abstract. Let G be the Walsh group. In this paper we prove for f ∈ L1(G) integrable functions the a.e. convergence σnf → f (n → ∞), where σn is the n-th (C, 1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator σ∗f := supn |σnf |. We prove that σ∗ is of type (p, p) for all 1 < p ≤ ∞ and of weak type (1, 1). Moreover, ‖σ∗f‖1 ≤ c‖|f |‖H , where H is the Hardy space on the Walsh group.