Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) ${\displaystyle {\left({\sum }_{k=1}^{\infty }{\sum }_{j=1}^{\infty }{|f̂(k,j)|}^p{\left(kj\right)}^{p-2}\right)}^{\frac{1}{p}}\le C_p\parallel f{\parallel }_{H^p\_\{\ast \}}}$ (1/2 < p≤2) where f belongs to the Hardy space ${\displaystyle H_{\ast }^p(G_m\times G_s)}$ 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.