Endpoint Sobolev inequalities for vector fields and cancelling operators
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916529882267648 |
|---|---|
| author | Van Schaftingen, Jean |
| author_facet | Van Schaftingen, Jean |
| contents | The injectively elliptic vector differential operators $A (\mathrm{D})$ from $V$ to $E$ on $\mathbb{R}^n$ such that the estimate \[
\Vert D^\ell u\Vert_{L^{n/(n - \ell)} (\mathbb{R}^n)}
\le \Vert A (\mathrm{D}) u\Vert_{L^1 (\mathbb{R}^n)} \] holds can be characterized as the operators satisfying a cancellation condition \[
\bigcap_{ξ\in \mathbb{R}^n \setminus \{0\}} A (ξ)[V] = \{0\}\;. \] 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 $A (\mathrm{D}) u$ lies in the kernel of a cocancelling differential operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_00840 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Endpoint Sobolev inequalities for vector fields and cancelling operators Van Schaftingen, Jean Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A23 (Primary) 26D15, 35E05, 42B30, 42B35, 46E35 (Secondary) The injectively elliptic vector differential operators $A (\mathrm{D})$ from $V$ to $E$ on $\mathbb{R}^n$ such that the estimate \[ \Vert D^\ell u\Vert_{L^{n/(n - \ell)} (\mathbb{R}^n)} \le \Vert A (\mathrm{D}) u\Vert_{L^1 (\mathbb{R}^n)} \] holds can be characterized as the operators satisfying a cancellation condition \[ \bigcap_{ξ\in \mathbb{R}^n \setminus \{0\}} A (ξ)[V] = \{0\}\;. \] 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 $A (\mathrm{D}) u$ lies in the kernel of a cocancelling differential operator. |
| title | Endpoint Sobolev inequalities for vector fields and cancelling operators |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A23 (Primary) 26D15, 35E05, 42B30, 42B35, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2305.00840 |