is shown to hold if and only if is elliptic and canceling. Here is a homogeneous linear differential operator of order on from a vector space to a vector space . The operator is defined to be canceling if
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 of order on from a vector space to a vector space is introduced. It is proved that is cocanceling if and only if for every such that , one has . 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. 877–921DOI 10.4171/JEMS/380