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The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H p, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H p, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.

In this paper we prove that the maximal operator $${\tilde{\sigma }}^{\kappa ,*}f:=\underset{n\in \mathbb{P}}{sup}\frac{|{\sigma }_n^{\kappa }f|}{{log}^2(n+1)},$$ where ${\sigma }_n^{\kappa }f$ is the $n$-th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_{1/2}\left(G\right)$ to the space $L_{1/2}\left(G\right).$