Abstract. The purpose of this work is to show a Paley type inequality for two-parameter Vilenkin system. For the one-dimensional analogue see Simon, Weisz [7]. Hence we will verify the estimation $$(*)\ \ \ C_p\|f\|_{H^p} \geq\left(\sum_{n,k=0}^\infty(m_nm_k)^{1 - 2 / p}(M_nM_k)^{2 - 2 / p} \sum_{j=1}^{m_n-1}\sum_{l = 1 \atop \alpha \leq jM_n / lM_k \leq \beta}^{m_k-1} |{\hat f}(jM_n,lM_k)|^2\right)^{1/2}$$ for martingales $f\in H^p(G_m^2)$ $(2/5\leq p\leq1)$. Here $0<\alpha\leq 1\leq\beta$ and $\hat f(u,v)$ ($u, v\in{\msbm N}$) is the $(u,v)$-th (two-parameter) Vilenkin--Fourier coefficient of $f$. The Hardy space $H^p(G_m^2)$ is defined by means of a diagonal maximal function. By usual interpolation argument it follows an $L^p$-variant of (*) for $1< p\leq2$. As the dual inequality of (*) we formulate a $BMO$-result as well as some analogous statements for other Hardy- and ${\cal BMO}$-spaces. The non-improving of the assumption $2/5\leq p$ as well as the diagonal case of (*) will be investigated.

AMS Subject Classification (1991): 42C10, 60G42

Received August 4, 1998. (Registered under 2728/2009.)