Estimates on translations and Taylor expansions in fractional Sobolev spaces
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In this paper we study how the (normalised) Gagliardo semi-norms [u]_W^s,p (R^n) control translations. In particular, we prove that u(· + y) - u _L^p (R^n)< C [ u ] _W^s,p (R^n) |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 W^s,p(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 (R^n)).
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