Estimates on translations and Taylor expansions in fractional Sobolev spaces. (November 2020)

• Record Type: Journal Article
• Authors: del Teso, Félix; Gómez-Castro, David; Vázquez, Juan Luis
• Is Part Of: Nonlinear analysis. Volume 200(2020)
• Publication Date: 2020-11
• Subjects: 46E35 -- 26D10; Fractional sobolev spaces -- Gagliardo norms -- Translation estimates -- Taylor expansions -- Discretisation error -- Interpolation -- Convolution; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111995
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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 discretisations for W s, p ( R n ) .