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00925njm 22002777a 4500 dc Del Teso Méndez, Félix author Gómez-Castro, D. author Vázquez, Juan Luis author 2020-11 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). 0362-546X 10.1016/j.na.2020.111995 https://hdl.handle.net/20.500.14352/7245 https://doi.org/10.1016/j.na.2020.111995 Estimates on translations and Taylor expansions in fractional Sobolev spaces