Paley type inequalities for several parameter Vilenkin systems (Q700381)

In this paper the authors prove Paley type inequalities for two-parameter Vilenkin systems. Their main result is the following estimate: \[ \biggl(\sum_{n,k=0}^{\infty}(p_nq_k)^{1-2/p}(P_nQ_k)^{2-2/p}) \sum_{j=1}^{p_n-1}\sum_{l=1}^{q_k-1}|\hat f(jP_n,lQ_k)|^2\biggr)^{1/2} \leq C_p\|f\|_{H^p} \] for any martingale \(f\) in the Hardy space \(H^p(G_{\mathbf p}\times G_{\mathbf q})\) (\(0<p\leq 1\)). In this paper \(G_{\mathbf p}\), and \(G_{\mathbf q}\) denote the Vilenkin groups generated by sequences \(\mathbf p =(p_n)\), and \(\mathbf q =(q_n)\). The complex numbers \(\hat f(u,v)\) for \(u,v\) natural numbers are the Fourier coefficients of the function \(f\) with respect to the two-parameter Vilenkin system of the group \(G_{\mathbf p}\times G_{\mathbf q}\). The Hardy space \(H^p(G_{\mathbf p}\times G_{\mathbf q})\) is defined by means of a usual maximal function. The authors get this inequality from its dual version, especially from a BMO result in the case \(p=1\). By interpolation they also prove an \(L^p\) variant of this inequality for \(1<p \leq 2\). Besides, they investigate whether this inequality can be improved in the case of unbounded Vilenkin systems. These inequalities are of great importance to prove approximation theorems of functions in Hardy spaces by means of partial sums of the Fourier series.