论文标题
傅立叶部分总和运算符在本地田地的加权空间上的均匀界限
Uniform boundedness of the Fourier partial sum operators on the weighted spaces of local fields
论文作者
论文摘要
令$ s_n f $为$ n $ th $ f $ in $ l^1（\ d）$的傅立叶系列的部分总和，其中$ \ d $是本地字段$ k $的整数。对于$ 1 <p <\ infty $，我们表征所有权重函数$ w $，以便部分总和运算符$ s_n $，$ n \ geq 0 $在加权空间$ l^p（\ d，w）$上均匀地界限，该$ s_n f $ to $ s_n f $收敛到$ f $ in $ l^p（\ l^p（\ d，w，w，w）$。这包括$ k $是$ p $ - adik数字字段或正式的laurent系列$ \ mathbb {f} _q（（x））$上的有限字段$ \ mathbb {f} _q $，尤其是$ \ d $是$ \ d $是walsh-paley或dyadic group $ 2^oper，作为一个应用程序，在本地字段$ k $的积极特征中，我们在l^2（k）$中的函数$φ\中提供了必要的条件，为此，$φ$的集合构成了其封闭线性跨度的schauder基础。此外，我们为Hardy-Littlewood最大运算符建立了尖锐的界限。
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 $2^ω$. As an application, in a local field $K$ of positive characteristic, we provide a necessary and sufficient condition on a function $φ\in L^2(K)$ for which the collection of translates of $φ$ forms a Schauder basis for its closed linear span. Moreover, we establish sharp bounds for the Hardy-Littlewood maximal operator.
