Estimates on translations and Taylor expansions in fractional Sobolev spaces - Teso, FD; Gomez Castro, D; Vazquez, JL (Elsevier, 2020)
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 discretizations for Ws,p(Rn).