We investigate convergence and divergence of specific subsequences of partial sums with respect to the Walsh system on martingale Hardy spaces. By using these results we obtain a relationship of the ratio of convergence of the partial sums of the Walsh series and the modulus of continuity of the martingale. These conditions are in a sense necessary and sufficient.
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.
We prove inclusion relations between generalizing Waterman's and generalized Wiener's classes for functions of two variable.
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