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Abstract. The main aim of this paper is to prove that the maximal operator σ * 0 := sup n |σ n,n | of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space H 1/2 to the space weak-L 1/2 .Let N + denote the set of positive integers, N := N + ∪ {0}. Let m := (m 0 , m 1 , . . .) be a sequence of positive integers not less than 2. Denote by Z m k := {0, 1, . . . , m k − 1} the additive group of integers modulo m k . Define the group G m as the complete direct product of the groups Z m j , with the product of the discrete topologies of Z m j 's. The direct product µ of the measuresis the Haar measure on G m with µ(G m ) = 1. If the sequence m is bounded, then G m is called a bounded Vilenkin group, otherwise it is an unbounded Vilenkin group.

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