Journal of the European Mathematical Society

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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) Université Catholique de Louvain, 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