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Paley Type Inequalities for Several Parameter Vilenkin Systems

Authors:

P Simon F Weisz

Institution:

(1) Department of Numerical Analysis, Eötvös Loránd University, Pázmány P. Sétány I/D, 1117 Budapest, Hungary

Abstract:

The aim of this paper is to prove Paley type inequalities for two-parameter Vilenkin system. Our main result is the following estimate:

$(\mathop \Sigma \limits_{n,k}^\infty \left( {p_n q_k } \right)^{^{1 - 2/p} } \left( {P_n Q_k } \right)^{^{2 - 2/p} } \mathop \Sigma \limits_{j = 1}^{p_{_n - 1} } \mathop \Sigma \limits_{l = 1}^{q_{_k - 1} } \left| {\hat f\left( {jP_n ,lQ_k } \right)} \right|^2 )^{1/2} {\text{ }} \leqslant C_p \left\| f \right\|_{H^p }$

for martingales f isin

H ^p(G _p × G _q) (0 < p

1). Here G _p and G _q are Vilenkin groups generated by the sequences p = (p _n) and q = (q _n), respectively, and f^(u, v) (u, v isin

N) is the (u,v)th (two-parameter) Vilenkin-Fourier coefficient of f. The Hardy space H ^p(G _p × G _q) is defined by means of a usual martingal maximal function.We get the inequality (*) from its dual version, especially it follows from a BMO-result in the case p = 1. Furthermore, interpolation leads to an L ^p-variant of (*) for 1 < p

2. We also formulate an analogous statement for another Hardy space. In the so-called unbounded case, i.e. when p or q is not bounded, we shall investigate whether (*) can be improved. Our results hold also in the case of higher dimensions.

Keywords:

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