In this paper we study how the (normalised) Gagliardo semi-norms ${\left[u\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)}$ control translations. In particular, we prove that ${\Vert u(\cdot +y)-u\Vert }_{L^p\left({\mathbb{R}}^n\right)}\le C{\left[u\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)}{\left|y\right|}^s$ for $n\ge 1$, $s\in [0,1]$ 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 $W^{s,p}\left({\mathbb{R}}^n\right)$ 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 discretisations for $W^{s,p}\left({\mathbb{R}}^n\right)$.

在本文中，我们研究了（规范化的）Gagliardo半范数 ${\left[ü\right]}_{{\mathrm{w\; ^}}^{s，p}（{\mathbb{[R}}^{ñ}）}$控制翻译。特别是，我们证明${\Vert ü（\cdot +ÿ）-ü\Vert }_{{\mathrm{大号}}^p（{\mathbb{[R}}^{ñ}）}\le C{\left[ü\right]}_{{\mathrm{w\; ^}}^{s，p}（{\mathbb{[R}}^{ñ}）}{\left|ÿ\right|}^s$ 对于 $ñ\ge \mathrm{1个}$， $s\in [0，\mathrm{1个}]$ 和 $p\in [\mathrm{1个}，+\infty ]$，在哪里 $C$ 仅取决于 $ñ$。然后，我们获得此结果的相应高阶版本：在泰勒展开式中，获得误差项的分数率。我们还介绍了我们两个结果的相关含义。首先，我们获得了几个紧凑嵌入的直接证明。${\mathrm{w\; ^}}^{s，p}（{\mathbb{[R}}^{ñ}）$Fréchet–Kolmogorov定理适用于已知比率。我们还推导了函数与合适的缓和器的卷积收敛的分数速率。第三，我们获得了有限差分离散化的分数收敛速率${\mathrm{w\; ^}}^{s，p}（{\mathbb{[R}}^{ñ}）$。