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In the present paper we give a necessary and sufficient condition on monotone weights, which guaranties the Nörlund means of Walsh-Fourier series converge in L1 norm and CW norm. We also discuss the almost everywhere summability by Nörlund means and we prove that our condition is sufficient for almost everywhere convergence of Nörlund means for all integrable functions. [ABSTRACT FROM AUTHOR]

Goginava proved that the maximal operator σ α , * {\sigma^{\alpha,*}} ( 0 < α < 1 {0<\alpha<1}) of two-dimensional Marcinkiewicz type (C , α) {(C,\alpha)} means is bounded from the two-dimensional dyadic martingale Hardy space H p ⁢ (G 2) {H_{p}(G^{2})} to the space L p ⁢ (G 2) {L^{p}(G^{2})} for p > 2 2 + α {p>\frac{2}{2+\alpha}}. Moreover, he showed that assumption p > 2 2 + α {p>\frac{2}{2+\alpha}} is essential for the boundedness of the maximal operator σ α , * {\sigma^{\alpha,*}}. It was shown that at the point p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} the maximal operator σ α , * {\sigma^{\alpha,*}} is bounded from the dyadic Hardy space H 2 / (2 + α) ⁢ (G 2) {H_{2/(2+\alpha)}(G^{2})} to the space weak- L 2 / (2 + α) ⁢ (G 2) {L^{2/(2+\alpha)}(G^{2})}. The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α , * {\sigma^{\alpha,*}} means ( 0 < α < 1 {0<\alpha<1}) in the endpoint case p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}}. In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / (2 + α) ⁢ (G 2) {H_{2/(2+\alpha)}(G^{2})} to L 2 / (2 + α) ⁢ (G 2) {L^{2/(2+\alpha)}(G^{2})}. Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / (2 + α) ⁢ (G 2) {H_{2/(2+\alpha)}(G^{2})}. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H, 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, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means. [ABSTRACT FROM AUTHOR]

The main aim of this article is to investigate the (Hp, Lp)-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of Walsh–Kaczmarz–Fourier series. Moreover, we show that the behaviour of Nörlund logarithmic means is worse than the behaviour of Riesz logarithmic means in our special sense. [ABSTRACT FROM AUTHOR]