论文标题
估计分数Sobolev空间中的翻译和泰勒扩展
Estimates on translations and Taylor expansions in fractional Sobolev spaces
论文作者
论文摘要
在本文中，我们研究了（归一化的）gagliardo semi-Norms $ [u] _ {w^{s，p}（\ Mathbb {r}^n）} $控制翻译。特别是，我们证明了$ \ | u（\ cdot + y） - u \ | _ {l^p（\ mathbb {r}^n）} \ le c [u c [u] _ {w^{s，p}（\ mathbb {r}^n）} | y | y | $ c $仅取决于$ n $。然后，我们获得了此结果的相应高阶版本：我们在泰勒扩展中获得了误差项的分数。我们还提出了两个结果的相关含义。首先，我们获得了$ w^{s，p}的几种紧凑型嵌入（\ mathbb {r}^n）$的直接证明，其中fréchet-kolmogorov Theorem被以已知速率应用。我们还得出了与合适的mollifier的函数卷积的收敛速率。第三，我们获得了$ w^{s，p}（\ mathbb {r}^n））$的有限差异化离散的分数。
In this paper we study how the (normalised) Gagliardo semi-norms $[u]_{W^{s,p} (\mathbb{R}^n)}$ control translations. In particular, we prove that $\| u(\cdot + y) - u \|_{L^p (\mathbb{R}^n)} \le C [ u ] _{W^{s,p} (\mathbb{R}^n)} |y|^s$ for $n\geq1$, $s \in [0,1]$ and $p \in [1,+\infty]$, where $C$ depends only on $n$. We then obtain a corresponding higher-order version of this result: we get fractional rates of the error term in the Taylor expansion. We also present relevant implications of our two results. First, we obtain a direct proof of several compact embedding of $W^{s,p}(\mathbb{R}^n)$ where the Fréchet-Kolmogorov Theorem is applied with known rates. We also derive fractional rates of convergence of the convolution of a function with suitable mollifiers. Thirdly, we obtain fractional rates of convergence of finite-difference discretizations for $W^{s,p} (\mathbb{R}^n))$.
