Limiting Sobolev estimates for vector fields and cancelling differential operators

Abstract

These notes 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 showed that there is no nontrivial estimate on same-order derivatives. On the other hand endpoint Sobolev estimates were proved 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 such Sobolev estimates holds can be characterized by a cancelling condition. The estimates rely on a duality estimate for $L^1$ vector fields satisfying some conditions on the derivatives, combined with classical algebraic and harmonic analysis techniques. This characterization unifies classes of known inequalities and extends to the case of Hardy inequalities.

22 pages, lecture notes for the Spring School on Analysis 2023 "Function Spaces and Applications XII" held at Paseky nad Jizerou, May 28-June 3, 2023, corrected typos; Function spaces and applications XII (Pazeky nad Jizerou 2023), MatfyzPress, Charles University, Prague, 2023, 135-152