# Estimates for L<sup>1</sup>-vector fields under higher-order differential conditions

### Jean Van Schaftingen

Université Catholique de Louvain, Belgium

## Abstract

We prove that an $\mathrm{L}^1$ vector field whose components satisfy some condition on $k$-th order derivatives induce linear functionals on the Sobolev space $\mathrm{W}^{1,n}(\R^n)$. Two proofs are provided, relying on the two distinct methods developed by Bourgain and Brezis (J.\ Eur.\ Math.\ Soc.\ (JEMS), to appear) and by the author (C.\ R.\ Math.\ Acad.\ Sci.\ Paris, 2004) to prove the same result for divergence-free vector fields and partial extensions to higher-order conditions.

## Cite this article

Jean Van Schaftingen, Estimates for L<sup>1</sup>-vector fields under higher-order differential conditions. J. Eur. Math. Soc. 10 (2008), no. 4, pp. 867–882

DOI 10.4171/JEMS/133