Weighted norm inequalities for Vilenkin-Fourier series

Abstract:

Let ${S_n}f$ be the $n$th partial sum of the Vilenkin-Fourier series of $f \in {L^1}$. For $1 < p < \infty$, we characterize all weight functions $w$ such that if $f \in {L^p}(w)$, ${S_n}f$ converges to $f$ in ${L^p}(w)$. We also determine all weight functions $w$ such that $\{ {S_n}\}$ is uniformly of weak type $(1,1)$ with respect to $w$.