  These notes present Sobolev-Gagliardo-Nirenberg endpoint estimates for
classes of homogeneous vector differential operators. Away of the endpoint
cases, the classical Calder\'on-Zygmund estimates show that the ellipticity is
necessary and sufficient to control all the derivatives of the vector field. In
the endpoint case, Ornstein showed that there is no nontrivial estimate on
same-order derivatives. On the other hand endpoint Sobolev estimates were
proved for the deformation operator (Korn-Sobolev inequality by M.J. Strauss)
and for the Hodge complex (Bourgain and Brezis). The class of operators for
which such Sobolev estimates holds can be characterized by a cancelling
condition. The estimates rely on a duality estimate for $L^1$ vector fields
satisfying some conditions on the derivatives, combined with classical
algebraic and harmonic analysis techniques. This characterization unifies
classes of known inequalities and extends to the case of Hardy inequalities.
