Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929634256355328 |
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| author | Van Schaftingen, Jean |
| author_facet | Van Schaftingen, Jean |
| contents | Although Ornstein's nonestimate entails the impossibility to control in general all the $L^1$-norm of derivatives of a function by the $L^1$-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 $L^1$-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 Raiţă characterizes the operators for which there is an $L^\infty$-estimate on lower-order derivatives. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_01201 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields Van Schaftingen, Jean Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A23 (Primary) 26D15, 35E05, 42B30, 42B35, 46E35 (Secondary) Although Ornstein's nonestimate entails the impossibility to control in general all the $L^1$-norm of derivatives of a function by the $L^1$-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 $L^1$-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 Raiţă characterizes the operators for which there is an $L^\infty$-estimate on lower-order derivatives. |
| title | Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A23 (Primary) 26D15, 35E05, 42B30, 42B35, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2302.01201 |