Estimates on translations and Taylor expansions in fractional Sobolev spaces

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 Fr\'echet-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 $W^{s,p} (\mathbb{R}^n))$.