On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system
Let G be the Walsh group. For $f ∈ L^1(G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_n$ is the nth (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, $∥σ*f∥_1 ≤ c∥|f|∥_H$, where H is the Hardy space on the Walsh group.