Weighted norm inequalities for Fourier series on the ring of integers of a local field
Abstract
Let $S_n f$ be the $n$th partial sum of the Fourier series of a function $f$ in $L^1(\mathfrak{D})$, where $\mathfrak{D}$ is the ring of integers of a local field $K$. For $1<p<\infty$, we characterize all weight functions $w$ such that for any function $f\in L^p(\mathfrak{D}, w)$, we have $f\in L^1(\mathfrak{D})$ and the partial sum operators $S_n$, $n\geq 0$, are uniformly bounded on $L^p(\mathfrak{D}, w)$. This includes the case where $K$ is a $p$adic number field or a $p$series field, and in particular, when $\mathfrak{D}$ is the ring of the integers of WalshPaley 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 $\varphi\in L^2(K)$ so that the collection of translates of $\varphi$ forms a Schauder basis for its closed linear span. Moreover, we establish sharp bounds for the HardyLittlewood maximal operator.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.00837
 Bibcode:
 2020arXiv200500837N
 Keywords:

 Mathematics  Functional Analysis;
 Primary: 43A70;
 Secondary: 42B25;
 43A25