Asymmetric anisotropic fractional Sobolev norms

TL;DR

This paper extends the understanding of fractional Sobolev norms by analyzing their anisotropic and asymmetric variants, demonstrating their convergence to classical anisotropic Sobolev seminorms as the fractional parameter approaches one.

Contribution

It introduces and studies asymmetric anisotropic fractional Sobolev norms, establishing their convergence to anisotropic Sobolev seminorms defined by the Minkowski functional of the polar asymmetric $L_p$ moment body.

Findings

Asymmetric anisotropic fractional Sobolev norms converge to classical anisotropic Sobolev seminorms.

Extension of previous isotropic and symmetric anisotropic results to asymmetric cases.

Provides a new geometric interpretation involving Minkowski functionals and polar asymmetric $L_p$ moment bodies.

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 $K$.