arXiv 1406.06722014-06math.FAmath.AP

索伯列夫空间$L^p$近似刻画

$L^p$-Taylor approximations characterize the Sobolev space $W^{1,p}$

Daniel E. Spector

备注：: 7 pages. Preprint of an article to appear in Comptes Rendus - the exposition of the two articles is substantially different and the full article will not be available as an arxiv paper. The title and abstract displaying on arxiv have been changed to that of the article in its more polished form

摘要：

本文介绍了Calderón和Zygmund的$L^p$ -可微性-一个\emph{$L^p$-Taylor approximation}概念的一个变体。我们的第一个结果是索伯列夫空间$W^{1,p}(\mathbb{R}^N)$中的函数具有一阶$L^p$ -Taylor逼近.这与Calderón和Zygmund关于Sobolev函数的$L^p$ -可微性的结果类似。事实上，我们在这里宣布的主要结果是，一阶$L^p$ -Taylor逼近刻画了索伯列夫空间$W^{1,p}(\mathbb{R}^N)$，因此蕴涵了$L^p$ -可微性.我们的方法建立了索伯列夫空间的一些特征之间的联系，由于斯旺森使用Calderón-Zygmund类与其他由于Bourgain，Brezis和Mironescu使用非局部泛函与作者和Mengesha使用非局部梯度的其他人。索伯列夫空间的任何两个特征是相关的并不奇怪，然而，我们的分析的一个结果是一个简单的条件，用于确定是否有界变差函数是在一个索伯列夫空间。

英文摘要：

In this note, we introduce a variant of Calderón and Zygmund's notion of $L^p$-differentiability - an \emph{$L^p$-Taylor approximation}. Our first result is that functions in the Sobolev space $W^{1,p}(\mathbb{R}^N)$ possess a first order $L^p$-Taylor approximation. This is in analogy with Calderón and Zygmund's result concerning the $L^p$-differentiability of Sobolev functions. In fact, the main result we announce here is that the first order $L^p$-Taylor approximation characterizes the Sobolev space $W^{1,p}(\mathbb{R}^N)$, and therefore implies $L^p$-differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calderón-Zygmund classes with others due to Bourgain, Brezis, and Mironescu using nonlocal functionals with still others of the author and Mengesha using nonlocal gradients. That any two characterizations of Sobolev spaces are related is not surprising, however, one consequence of our analysis is a simple condition for determining whether a function of bounded variation is in a Sobolev space.

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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