Estimates on translations and Taylor expansions in fractional Sobolev spaces

Abstract

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 $\Vert u(\cdot +y)-u{\Vert }_{L^p({\mathbb{R}}^n)}\le C[u{]}_{W^{s,p}({\mathbb{R}}^n)}|y{|}^s$ 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 $W^{s,p}({\mathbb{R}}^n)$ 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 $W^{s,p}({\mathbb{R}}^n))$.

arXiv comment12 pages

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• Gamma-convergence of nonlocal perimeter functionals

Cited by in corpus (1)

• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II