Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Abstract

The estimate ∥Dk-1 u∥ Ln=(n-1)∥A.D/u∥ L 1 is shown to hold if and only if A.D/ is elliptic and canceling. Here A.D/ is a homogeneous linear differential operator A.D/ of order k on Rn from a vector space V to a vector space E. The operator A.D/ is defined to be canceling if \ 2ℝnf0g A &epsiTV U D (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/ of order k on Rn from a vector space E to a vector space F is introduced. It is proved that L.D/ is cocanceling if and only if for every f 2 L1. ℝnIE/ such that L.D/f D 0, one has f 2 PW -1;n=.n-1/.RnIE/. The results extend to fractional and Lorentz spaces and can be strengthened using some tools due to J. Bourgain and H. Brezis. © European Mathematical Society 2013.