JournalsjemsVol. 10 , No. 4DOI 10.4171/jems/133

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

  • Jean Van Schaftingen

    Université Catholique de Louvain, Belgium
Estimates for L<sup>1</sup>-vector fields under higher-order differential conditions cover

Abstract

We prove that an L1\mathrm{L}^1 vector field whose components satisfy some condition on kk-th order derivatives induce linear functionals on the Sobolev space W1,n(Rn)\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.