Endpoint Sobolev Inequalities for Vector Fields and Cancelling Operators
(2024) ISBN: [978-3-031-48579-4], 10 pages, published
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- Abstract
- The injectively elliptic vector differential operators A(D) from V to E on Rn such that the estimate ∥Dℓu∥Ln/(n−ℓ)(Rn)≤∥A(D)u∥L1(Rn) holds can be characterized as the operators satisfying a cancellation condition ⋂ξ∈Rn∖{0}A(ξ)[V]={0}. These estimates unify existing endpoint Sobolev inequalities for 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). Their proof is based on the fact that A(D)u lies in the kernel of a cocancelling differential operator.
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Citations
Van Schaftingen, J. (2024). Endpoint Sobolev Inequalities for Vector Fields and Cancelling Operators. Birkhäuser. https://doi.org/10.1007/978-3-031-48579-4_5