Injective Ellipticity, Cancelling Operators, and Endpoint Gagliardo-Nirenberg-Sobolev Inequalities for Vector Fields

Although Ornstein’s nonestimate entails the impossibility to control in general all the \(L^1\) -norm of derivatives of a function by the \(L^1\) -norm of a constant coefficient homogeneous vector differential operator, the corresponding endpoint Sobolev inequality has been known to hold in many cases: 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). The class of differential operators for which estimates holds can be characterized by a cancelling condition. The proof of the estimates rely on a duality estimate for \(L^1\) -vector fields lying in the kernel of a cocancelling differential operator, combined with classical linear algebra and harmonic analysis techniques. This characterization unifies classes of known Sobolev inequalities and extends to fractional Sobolev and Hardy inequalities. A similar weaker condition introduced by Raiṫă characterizes the operators for which there is an \(L^\infty \) -estimate on lower-order derivatives.