JournalsjemsVol. 15, No. 3pp. 877–921

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

  • Jean Van Schaftingen

    Université Catholique de Louvain, Belgium
Limiting Sobolev inequalities for vector fields and canceling linear differential operators cover
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Abstract

The estimate

Dk1uLn/(n1)A(D)uL1\|{D^{k-1}u}\|_{L^{n/(n-1)}} \le \|{A(D)u}\|_{L^1}

is shown to hold if and only if A(D)A(D) is elliptic and canceling. Here A(D)A(D) is a homogeneous linear differential operator A(D)A(D) of order kk on Rn\mathbb R^n from a vector space VV to a vector space EE. The operator A(D)A(D) is defined to be canceling if

ξRn{0}A(ξ)[V]={0}.\bigcap_{\xi \in \mathbb R^n \setminus \{0\}} A(\xi)[V]=\{0\}.

This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator L(D)L(D) of order kk on Rn\mathbb R^n from a vector space EE to a vector space FF is introduced. It is proved that L(D)L(D) is cocanceling if and only if for every fL1(Rn;E)f \in L^1(\mathbb R^n; E) such that L(D)f=0L(D)f=0, one has fW˙1,n/(n1)(Rn;E)f \in \dot{W}^{-1, n/(n-1)}(\mathbb R^n; E). The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.

Cite this article

Jean Van Schaftingen, Limiting Sobolev inequalities for vector fields and canceling linear differential operators. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 877–921

DOI 10.4171/JEMS/380