Endpoint Sobolev inequalities for vector fields and cancelling operators

Abstract: The injectively elliptic vector differential operators $A (\mathrm{D})$ from $V$ to $E$ on $\mathbb{R}^n$ such that the estimate [ \Vert D^\ell u\Vert_{L^{n/(n - \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 [ \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.