Injective Ellipticity, Cancelling Operators, and Endpoint Gagliardo-Nirenberg-Sobolev Inequalities for Vector Fields
(2024) Geometric and Analytic Aspects of Functional Variational Principles — ISBN: [9783031676000], p. 259-317, published
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- Abstract
- Although Ornstein’s nonestimate entails the impossibility to control in general all the -norm of derivatives of a function by the -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 -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 -estimate on lower-order derivatives.
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Citations
Van Schaftingen, J. (2024). Injective Ellipticity, Cancelling Operators, and Endpoint Gagliardo-Nirenberg-Sobolev Inequalities for Vector Fields. In Andrea Cianchi, Vladimir G. Maz’ya, Tobias Weth (ed.), Geometric and Analytic Aspects of Functional Variational Principles (p. p. 259-317). Springer. https://doi.org/10.1007/978-3-031-67601-7_5