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Archiv der Mathematik

, Volume 103, Issue 2, pp 167–175 | Cite as

Asymmetric anisotropic fractional Sobolev norms

Article

Abstract

Bourgain, Brezis, and Mironescu showed that (with suitable scaling) the fractional Sobolev s-seminorm of a function \({f \in W^{1,p}(\mathbb{R}^n)}\) converges to the Sobolev seminorm of f as \({s\rightarrow1^-}\) . Ludwig introduced the anisotropic fractional Sobolev s-seminorms of f defined by a norm on \({\mathbb{R}^n}\) with unit ball K and showed that they converge to the anisotropic Sobolev seminorm of f defined by the norm whose unit ball is the polar L p moment body of K, as \({s \rightarrow 1^-}\) . The asymmetric anisotropic s-seminorms are shown to converge to the anisotropic Sobolev seminorm of f defined by the Minkowski functional of the polar asymmetric L p moment body of K.

Mathematics Subject Classification

46E35 52A20 

Keywords

Fractional sobolev norm Lp moment body 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWienAustria

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