Asymmetric anisotropic fractional Sobolev norms

Abstract

Bourgain, Brezis & 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 $s\rightarrow1^-$. 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 $K$.