BCAM Course | Limiting Sobolev estimates for vector fields and cancelling differential operators

Jean Van Schaftingen will present Sobolev–Gagliardo–Nirenberg endpoint estimates for classes of homogeneous vector differential operators.

There are available grants for this course: https://www.bcamath.org/events/bcamcourse-20250303-lmsevfcdo/en/

The 10-hour course will take place at the Basque Center for Applied Mathematics (BCAM), Bilbao, Spian. 

From Monday March 3rd, to Friday March 7th, 2025 from 11:00-13:00. 

Abstract

I will present Sobolev–Gagliardo–Nirenberg endpoint estimates for classes of homogeneous vector differential operators. Away of the endpoint cases, the classical Calderón–Zygmund estimates show that the ellipticity is necessary and sufficient to control all the derivatives of the vector field. In the endpoint case, Ornstein has showed that there is no nontrivial estimate on same-order derivatives and the ellipticity is necessary for endpoint Sobolev estimates. Such endpoint estimates were proved first for the deformation operator (Korn–Sobolev inequality by M.J. Strauss) and for the Hodge complex (Bourgain and Brezis). The class of operators for which estimates holds can be characterized by a cancelling condition. The estimates rely on a duality estimate for L1 vector fields satisfying some conditions on the derivatives, combined with classical algebraic and harmonic analysis techniques. This characterisation unifies classes of known inequalities and extends to the case of Hardy inequalities. The estimates obtained in such a way are stronger than those obtained thanks to real Hardy spaces and Sobolev embeddings.