Estimates for $L^1$-vector fields under higher-order differential conditions

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.