On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

Let G be the Walsh group. For ${\displaystyle f\in L^1\left(G\right)}$ we prove the a. e. convergence σf → f(n → ∞), where ${\displaystyle {\sigma }_n}$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator ${\displaystyle \sigma \cdot f≔{\supset }_n\left|{\sigma }_{n}f\right|.}$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, ${\displaystyle \parallel \sigma \cdot f{\parallel }_1\le c\parallel \left|f\right|{\parallel }_H}$, where H is the Hardy space on the Walsh group.