Approximation in higher-order Sobolev spaces and Hodge systems

TL;DR

This paper extends approximation results for differential forms in higher-order Sobolev and Triebel-Lizorkin spaces, showing forms with fractional Sobolev coefficients can be approximated by bounded forms while preserving differential properties.

Contribution

It generalizes Bourgain and Brezis's result to fractional Sobolev spaces within Triebel-Lizorkin spaces, addressing the critical case where classical embeddings fail.

Findings

Positive extension of approximation results to fractional Sobolev spaces

Approximation by bounded forms in Triebel-Lizorkin spaces

Applicable to differential forms with coefficients in critical Sobolev spaces

Abstract

Let $d\geq 2$ be an integer, $1\leq l\leq d-1$ and $\varphi$ be a differential $l$-form on ${\mathbb R}^d$ with $\dot{W}^{1,d}$ coefficients. It was proved by Bourgain and Brezis (\cite[Theorem 5]{MR2293957}) that there exists a differential $l$-form $\psi$ on ${\mathbb R}^d$ with coefficients in $L^{\infty}\cap \dot{W}^{1,d}$ such that $d\varphi=d\psi$. Bourgain and Brezis also asked whether this result can be extended to differential forms with coefficients in the fractional Sobolev space $\dot{W}^{s,p}$ with $sp=d$. We give a positive answer to this question, in the more general context of Triebel-Lizorkin spaces, provided that $d-\kappa\leq l\leq d-1$, where $\kappa$ is the largest positive integer such that $\kappa<\min(p,d)$. The proof relies on an approximation result for functions in $\dot{W}^{s,p}$ by functions in $\dot{W}^{s,p}\cap L^{\infty}$, even though $\dot{W}^{s,p}$ does not embed into $L^{\infty}$ in this critical case.