Approximation in higherorder Sobolev spaces and Hodge systems
Abstract
Let $d\geq 2$ be an integer, $1\leq l\leq d1$ 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 TriebelLizorkin spaces, provided that $d\kappa\leq l\leq d1$, 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1709.01762
 Bibcode:
 2017arXiv170901762B
 Keywords:

 Mathematics  Classical Analysis and ODEs
 EPrint:
 doi:10.1016/j.jfa.2018.08.003