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

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 Raiţă characterizes the operators for which there is an $L^{\infty }$-estimate on lower-order derivatives.

arXiv comment48 pages, lecture notes for the CIME summer school "Geometric and analytic aspects of functional variational principles'', June 27 - July 1, 2022, final corrections

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Cited by in corpus (2)

• Elliptic Pre-Complexes, Hodge-like Decompositions and Overdetermined Boundary-Value Problems
• Trace conjunction inequalities