%0 Journal Article
%A Del Teso Méndez, Félix
%A Gómez-Castro, D.
%A Vázquez, Juan Luis
%T Estimates on translations and Taylor expansions in fractional Sobolev spaces
%D 2020
%@ 0362-546X
%U https://hdl.handle.net/20.500.14352/7245
%X In this paper we study how the (normalised) Gagliardo semi-norms [u]Ws,p(Rn) control translations. In particular, we prove that ‖u(⋅+y)−u‖Lp(Rn)≤C[u]Ws,p(Rn)|y|s for n≥1, s∈[0,1] and p∈[1,+∞], 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 Ws,p(Rn) 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 Ws,p(Rn).
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