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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 discuss the properties of the maximal operator of two-dimensional Walsh-Kaczmarz-Fejér means, where the set of indices is inside a cone-like set. We prove that the maximal operator σk,*L is bounded from Hp to Lp for 1/2 < p and is of weak type (1, 1). As a corollary, we get Simon's theorem [13] on the convergence almost everywhere of cone-restricted two-dimensional Fejér means of integrable functions. In the endpoint case p = 1/2, we show that the maximal operator σk,*L is not bounded from the Hardy space H1/2 to the space L1/2. [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]

In 2009, Goginava showed that , where is the th Marcinkiewicz-Fejér mean of the Walsh-Kaczmarz-Fourier series, is not bounded from the Hardy space to the space . The main aim of this paper is to show that the maximal operator is bounded from the Hardy space to the space . Moreover, it is proved that the order of deviant behavior of the th Walsh-Kacmarz-Marcinkiewicz-Fejér mean is exactly . [ABSTRACT FROM AUTHOR]