Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields

TL;DR

This paper characterizes differential operators that satisfy endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields, unifying known results and extending to fractional and Hardy inequalities through a cancelling condition.

Contribution

It introduces a unifying cancelling condition that characterizes when endpoint Sobolev inequalities hold for various differential operators, extending previous results.

Findings

Characterization of operators satisfying endpoint inequalities via cancelling condition

Unification of known Sobolev inequalities under a common framework

Extension of inequalities to fractional Sobolev and Hardy cases

Abstract

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 Rai\c{t}\u{a} characterizes the operators for which there is an $L^\infty$-estimate on lower-order derivatives.