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.
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Ma, D. Asymmetric anisotropic fractional Sobolev norms. Arch. Math. 103, 167–175 (2014). https://doi.org/10.1007/s00013-014-0680-y
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DOI: https://doi.org/10.1007/s00013-014-0680-y