The atomic decomposition of weak Hardy spaces consisting of Vilenkin martingales is formulated. Some sufficient conditions for a sublinear operator T to be bounded from the weak Hardy space wHp to the weak wLp space are given. As applications a weak version of the Hardy-Littlewood inequality is obtained and it is shown that the maximal operator of the Cesàro means of a Vilenkin-Fourier series is bounded from wHp to wLp and is of weak type (1, 1). This yields that the Cesàro means of a function f ∈ L1 converge a.e. to the function in question, provided that the Vilenkin system is bounded.
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