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 $$\mathrm{\parallel }{D}^{\mathrm{\ell }}u{\mathrm{\parallel }}_{L^{n\mathrm{/}(n-\mathrm{\ell })}({\mathbb{R}}^n)}\le \mathrm{\parallel }A(\mathrm{D})u{\mathrm{\parallel }}_{L^1({\mathbb{R}}^n)}$$ holds can be characterized as the operators satisfying a cancellation condition $$\bigcap _{\xi \in {\mathbb{R}}^n\setminus \{0\}}A(\xi )[V]=\{0\}\mathrm{.}$$ 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.