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arXiv 2005.00837math.FA
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Fourier部分和算子在局部域加权空间上的一致有界性

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

Md Nurul Molla, Biswaranjan Behera

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arXiv admin note: text overlap with arXiv:2009.12772
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摘要：

设$S_n f$是函数$f$在$L^1(\D)$中的傅立叶级数的$n$次部分和，其中$\D$是局部域$K$的整数环.对于$1<p<\infty$，我们刻画了所有权函数$w$，使得部分和算子$S_n$，$n\geq 0$在加权空间$L^p(\D, w)$上一致有界，并且$S_n f$在$L^p(\D,w)$中收敛于$f$.这包括$K$是$p$ -进数域或有限域$\mathbb{F}_q$上的形式劳伦级数$\mathbb{F}_q((X))$的域的情况，并且特别地，当$\D$是Walsh-Paley或并元群$2^ω$时。作为应用，在正特征的局部域$K$中，给出了函数$φ\in L^2(K)$的平移集合构成其闭线性跨度的Schauder基的一个充要条件.此外，我们建立了Hardy-Littlewood极大算子的精确界。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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