In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d=2, we describe all Haar functions.

Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $({\sum }_{k=1}^{\infty }{\sum }_{j=1}^{\infty }{\left|f̂(k,j)\right|}^p{\left(kj\right)}^{p-2}{)}^{1/p}\le C_p\parallel f{\parallel }_{H_{**}^p}$ (1/2 < p≤2) where f belongs to the Hardy space $H_{**}^p\left(G_m\times G_s\right)$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.