Endpoint Sobolev Inequalities for Vector Fields and Cancelling Operators

The injectively elliptic vector differential operators \(A (\mathrm {D})\) from V  to E on \(\mathbb {R}^n\) such that the estimate \(\displaystyle \Vert \mathrm {D}^\ell u\Vert _{L^{n/(n - (k - \ell ))} (\mathbb {R}^n)} \le \Vert A (\mathrm {D}) u\Vert _{L^1 (\mathbb {R}^n)} \) holds can be characterized as the operators satisfying a cancellation condition \(\displaystyle \bigcap _{\xi \in \mathbb {R}^n \setminus \{0\}} A (\xi )[V] = \{0\}\;. \) These estimates unify existing endpoint Sobolev inequalities for the gradient of scalar functions (Gagliardo and Nirenberg), the deformation operator (Korn–Sobolev inequality by M.J. Strauss) and the Hodge complex (Bourgain and Brezis). Their proof is based on the fact that \(A (\mathrm {D}) u\) lies in the kernel of a cocancelling differential operator.