2023/04/27 by Jean Van Schaftingen, Van Schaftingen, Jean
Mathematics · #35A23 (Primary) 26D15 #35E05 #42B30 #42B35 #46E35 (Secondary) #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2304.14112
openalex publication_date 2023/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L1 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.