Hardy and Paley inequalities for fully-odd Vilenkin systems (Q1307379)

Choose a nonatomic probability space \(\Omega\). Let \((\psi_n)^\infty_{n=-\infty}\) for be a Vilenkin system on \(\Omega\). Denote the probability measure on \(\Omega\) by \(d\omega\). Given a function \(f\) in \(L^1(d\omega)\) and an integer \(n\), let \[ \widehat f (n)=\int_\Omega f(\omega)\overline{\psi_n(\omega)}d\omega. \] We also regard \(\widehat f\) as a function on the set \(\Gamma=\{\psi_n\}^\infty_{n=-\infty}\) and use the notation \(\widehat f(\psi_n)\) rather than \(\widehat f(n)\). One of the main results is: Theorem 1. Let \(f\in L^1(d\omega)\) and \(\widehat f(\psi_n)=0\) for all \(n<0\). Then \[ \sum_{n>0}\frac{| \widehat f(\psi_n)| }{n}<\infty. \] Moreover, if \((\lambda_j)\) is a Paley sequence, then \[ \sum^\infty_{j=1} | \widehat f(\psi_{\lambda_j})| ^2<\infty. \]