Journal of the European Mathematical Society


Full-Text PDF (191 KB) | Table of Contents | JEMS summary
Volume 10, Issue 4, 2008, pp. 867–882
DOI: 10.4171/JEMS/133

Estimates for L1-vector fields under higher-order differential conditions

Jean Van Schaftingen[1]

(1) Département de Mathématiques, Université Catholique de Louvain, Chemin du Cylcotron, 2, B-1348, LOUVAIN-LA-NEUVE, BELGIUM

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.

Keywords: Critical Sobolev spaces, compensation, Sobolev inequality, Korn–Sobolev inequality Schrödinger equations, L2-critical NLS, pseudo-conformal blow-up

Van Schaftingen Jean: Estimates for L1-vector fields under higher-order differential conditions. J. Eur. Math. Soc. 10 (2008), 867-882. doi: 10.4171/JEMS/133