论文标题
Arzelà和Ascoli将如何证明$ l^1（g）$的PEGO定理（如果他们生活在$ 21^{st} $ Century中）？
How Arzelà and Ascoli would have proved Pego theorem for $L^1(G)$ (if they lived in the $21^{st}$ century)?
论文作者
论文摘要
在论文中，我们努力回答一个问题：``如果Arzelà和Ascoli寿命足够长的时间来看Pego Theorem怎么办？”。 Giulio ascoli和CesareArzelà分别于1896年和1912年去世，因此他们无法欣赏$ l^2（\ Mathbb {r}^n）$在1985年提供的紧凑型家庭的特征。我们的文章旨在用作Arzelà-Ascoli定理和PEGO定理之间的桥梁（对于$ l^1（g）$而不是$ l^2（g）$，$ G $，是本地紧凑的Abelian Group）。从某种意义上说，前者是后者的``理由d'être''。
In the paper we make an effort to answer the question ``What if Arzelà and Ascoli lived long enough to see Pego theorem?''. Giulio Ascoli and Cesare Arzelà died in 1896 and 1912, respectively, so they could not appreciate the characterization of compact families in $L^2(\mathbb{R}^N)$ provided by Robert L. Pego in 1985. Unlike the Italian mathematicians, Pego employed various tools from harmonic analysis in his work (for instance the Fourier transform or the Hausdorff-Young inequality). Our article is meant to serve as a bridge between Arzelà-Ascoli theorem and Pego theorem (for $L^1(G)$ rather than $L^2(G)$, $G$ being a locally compact abelian group). In a sense, the former is the ``raison d'être'' of the latter, as we shall painstakingly demonstrate.
