Uniform boundedness of the Fourier partial sum operators on the weighted spaces of local fields

Abstract

Let $S_n f$ be the $n$th partial sum of the Fourier series of a function $f$ in $L^1(\D)$, where $\D$ is the ring of integers of a local field $K$. For $1<p<\infty$, we characterize all weight functions $w$ so that the partial sum operators $S_n$, $n\geq 0$, are uniformly bounded on the weighted space $L^p(\D, w)$ and that $S_n f$ converges to $f$ in $L^p(\D,w)$. This includes the case where $K$ is a $p$-adic number field or a field of formal Laurent series $\mathbb{F}_q((X))$ over a finite field $\mathbb{F}_q$, and in particular, when $\D$ is the Walsh-Paley or dyadic group $2^\omega$. As an application, in a local field $K$ of positive characteristic, we provide a necessary and sufficient condition on a function $\varphi\in L^2(K)$ for which the collection of translates of $\varphi$ forms a Schauder basis for its closed linear span. Moreover, we establish sharp bounds for the Hardy-Littlewood maximal operator.