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

TL;DR

This paper characterizes the weights for which Fourier partial sums are uniformly bounded and converge in weighted $L^p$ spaces on local fields, including $p$-adic and Laurent series fields, with applications to bases and maximal operators.

Contribution

It provides a complete characterization of weight functions ensuring boundedness and convergence of Fourier partial sums on local fields, extending classical harmonic analysis results.

Findings

Characterization of weights for uniform boundedness of Fourier partial sums

Necessary and sufficient conditions for Schauder bases in local fields

Sharp bounds for the Hardy-Littlewood maximal operator

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.