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

Let $S_{n}f$ be the $n$th partial sum of the Fourier series of a function $f$ in $L^1(\text{D})$, where $\text{D}$ is the ring of integers of a local field $K$. For $1<p<\mathrm{\infty }$, we characterize all weight functions $w$ so that the partial sum operators $S_n$, $n\ge 0$, are uniformly bounded on the weighted space $L^p(\text{D},w)$ and that $S_{n}f$ converges to $f$ in $L^p(\text{D},w)$. 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 $\text{D}$ 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 $\phi \in L^2(K)$ for which the collection of translates of $\phi$ forms a Schauder basis for its closed linear span. Moreover, we establish sharp bounds for the Hardy-Littlewood maximal operator.