2014/10/22 by Dan Ma
Mathematics · #math.AP #math.FA #msc:46E35 #msc:52A20
paper · pdf · doi:10.1007/s00013-014-0680-y
published as Archiv der Mathematik, 2014, Volume 103, Issue 2, pp 167-175 · 9 pages. arXiv admin note: text overlap with arXiv:1304.0703 by other authors
arxiv created 2014/10/22 · arxiv updated 2016/04/01
Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev s-seminorm of a function f∈ W1,p(ℝn) converges to the Sobolev seminorm of f as s→1-. Ludwig introduced the anisotropic fractional Sobolev s-seminorms of f defined by a norm on ℝn 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 Lp moment body of K, as s→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 Lp moment body of K.