On the weighted maximal operators of Marcinkiewicz type Ces\`aro means of two-dimensional Walsh-Fourier series

Abstract

In this paper we investigate the behaviour of the weighted maximal operators of  Marcinkiewicz type (C,$\alpha$)-means $\tilde{\sigma}^{\alpha,*}_pf:=\sup_{n\in\mathbb{P}}\frac{|\sigma^{\alpha}_n f|}{n^{2/p-(2+\alpha)}}$ in the Hardy space $H_p(G^2)$  ($0<\alpha<1$ and $p<2/(2+\alpha)$). It is showed that the maximal operators $\tilde{\sigma}^{\alpha,*}_pf$ are bounded from the dyadic Hardy space $H_p(G^2)$ to the Lebesgue space $L^p(G^2)$, and that this is in a sense sharp. It was also proved a strong convergence theorem for the Marcinkiewicz type $(C,\alpha)$ means of Walsh-Fourier series in $H_p(G^2)$.