Limiting behaviour of intrinsic semi-norms in fractional order Sobolev spaces

TL;DR

This paper investigates the limiting behavior of intrinsic semi-norms in fractional Sobolev spaces as the fractional order approaches 0 or 1, providing explicit constants and semi-norms, with special Fourier-based results for the case p=2.

Contribution

It extends and refines existing results on the limits of fractional Sobolev semi-norms, explicitly characterizing the limits with constants and semi-norms, including Fourier analysis for p=2.

Findings

Limits of semi-norms are explicitly characterized as fractional order approaches 0 or 1.

Explicit constants and semi-norms are provided for the limits.

Fourier transform methods are used for the case p=2 in Euclidean space.

Abstract

We collect and extend results on the limit of $\sigma^{1-k}(1-\sigma)^k |v|_{l+\sigma,p,\Omega}^p$ as $\sigma$ tends to $0^+$ or $1^-$, where $\Omega$ is $\mathbb{R}^n$ or a smooth bounded domain, $k$ is 0 or 1, $l$ is a nonnegative integer, $p\in[1,\infty)$, and $|.|_{l+\sigma,p,\Omega}$ is the intrinsic semi-norm of order $l+\sigma$ in the Sobolev space $W^{l+\sigma,p}(\Omega)$. In general, the above limit is equal to $c[v]^p$, where $c$ and $[.]$ are, respectively, a constant and a semi-norm that we explicitly provide. The particular case $p=2$ for $\Omega=\mathbb{R}^n$ is also examined and the results are then proved by using the Fourier transform.