Anisotropic Fractional Sobolev Norms

TL;DR

This paper investigates the limiting behavior of anisotropic fractional Sobolev norms as the fractional parameter approaches 0 and 1, revealing convergence to anisotropic Sobolev seminorms related to polar $L_p$ moment bodies.

Contribution

It extends known results on fractional Sobolev norms to the anisotropic setting, establishing convergence to anisotropic Sobolev seminorms involving polar $L_p$ moment bodies.

Findings

Anisotropic $s$-seminorms converge to anisotropic Sobolev seminorms as $s o 1^-$.

The limiting behavior as $s o 0^+$ is characterized.

Results extend previous isotropic cases to anisotropic norms.

Abstract

Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\rn)$ converges to the Sobolev seminorm of $f$ as $s\to 1^-$. The anisotropic $s$-seminorms of $f$ defined by a norm on $\rn$ with unit ball $K$ are shown to converge to the anisotropic Sobolev seminorm of $f$ defined by the norm with unit ball $\,\ompd K$, the polar $L_p$ moment body of $K$. The limiting behavior for $s\to 0^+$ is also determined (extending results by Maz$'$ya & Shaposhnikova).