Approximation in higher-order Sobolev spaces and Hodge systems

Abstract

Let $d\ge 2$ be an integer, $1\le l\le d-1$ and $\phi$ 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 $d\phi =d\psi$. 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 $d-\kappa \le l\le d-1$, 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.