We extend the Dezern-Waterman version of the Lebesgue test for convergence of Vilenkin-Fourier series to the case of unbounded Vilenkin groups.

Abstract. We prove that certain means of the quadratical partial sums of the two-dimensional Vilenkin-Fourier series are uniformly bounded operators from the Hardy space H p to the space L p for 0 < p ≤ 1. We also prove that the sequence in the denominator cannot be improved.2010 Mathematics Subject Classification. 42C10.

In [14] we investigated some Vilenkin-Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H 1/(1+α) to the space weak -L 1/(1+α) , (0 < α 1). In this paper we construct a martingale in the space H 1/(1+α) , which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin-Nörlund means with non-increasing coefficients are not bounded from the Hardy space H 1/(1+α) to the space L 1/(1+α) . In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

We give an explicit characterization of ergodicity of rational functions on the set of units of the [Formula: see text]-adic group by means of the coefficients in their nominators and denominators. Although rational functions are not ergodic on the set of p-adic numbers, this was proved by Diao and Silva, we study their ergodicity on the 2-adic unit sphere if they only contain integer coefficients. The first results of this paper provide isometricity tests of polynomials and rational functions on the set of 2-adic units with 2-adic integer coefficients. Then, follow results on ergodicity tests on general isometric functions, then on rational functions on the set of 2-adic units with 2-adic integer coefficients.

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