A maximal inequality for $H^{1}$-functions on a generalized Walsh-Paley group

Abstract:

Let $G = \prod _{i = 0}^\infty Z({p_i})$ be the countable product of discrete cyclic groups of order ${p_i}$. We assume that ${\sup _{i \geqslant 0}}{p_i} < \infty$. We consider Walsh-Fourier series on G and define ${H^1}$-functions on G by the Coifman-Weiss atoms. Let ${K_n}(x)$ be the nth (C, 1)-kernel. We prove that ${\smallint _G {{{\sup }_{n \geqslant 1}}|({K_n} \ast f)(x)|d\mu \leqslant C\left \| f \right \|} _{{H^1}}}$. Here $d\mu$ is the normalized Haar measure, ${\left \| \right \|_{{H^1}}}$ is the ${H^1}$-norm and C is a constant independent of f.