Maximal operators and characterization of Hardy spaces (Q2300137)

The main result (Theorem 2.2): Let \( G \) be a bounded Vilenkin group, and \( p>0. \) Then there exists a positive constant \( C>0 \) depending only on the sequence \( (m_n)_n \) such that \[ \Vert f\Vert_{H^p(G)}^p\leq C\Vert \sup_n\vert \sigma_n(f)\vert \Vert_p^p\] for every function \( f\in L^p \) for which \( \sup_n\vert \sigma_n(f)\vert \in L^p. \)\par Notations here are standard, \( H^p(G) \) is the Hardy space for \( G \) (see, for example, \textit{Gy. Gát} [J. Approx. Theory 124, No. 1, 25--43 (2003; Zbl 1032.43003)]).