The Caffarelli–Kohn–Nirenberg inequalities for radial functions
We establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order $0 < s \le 1$. In particular, we show that the range of the parameters for radial functions is strictly larger than the one without symmetric as...
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Académie des sciences
2023-10-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.503/ |
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| author | Mallick, Arka Nguyen, Hoai-Minh |
| author_facet | Mallick, Arka Nguyen, Hoai-Minh |
| author_sort | Mallick, Arka |
| collection | DOAJ |
| description | We establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order $0 < s \le 1$. In particular, we show that the range of the parameters for radial functions is strictly larger than the one without symmetric assumption. Previous known results reveal only some special ranges of parameters even in the case $s=1$. The known proofs used the Riesz potential and inequalities for fractional integrations. Our proof is new, elementary, and is based on one-dimensional case. Applications on compact embeddings are also mentioned. |
| format | Article |
| id | doaj-art-e3d1d837a109497c89945149c1d9db85 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-e3d1d837a109497c89945149c1d9db852025-02-07T11:09:56ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-10-01361G71175118910.5802/crmath.50310.5802/crmath.503The Caffarelli–Kohn–Nirenberg inequalities for radial functionsMallick, Arka0Nguyen, Hoai-Minh1Department of Mathematics, IISc, Bengaluru, IndiaLaboratoire Jacques Louis Lions, Sorbonne Université, Paris, FranceWe establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order $0 < s \le 1$. In particular, we show that the range of the parameters for radial functions is strictly larger than the one without symmetric assumption. Previous known results reveal only some special ranges of parameters even in the case $s=1$. The known proofs used the Riesz potential and inequalities for fractional integrations. Our proof is new, elementary, and is based on one-dimensional case. Applications on compact embeddings are also mentioned.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.503/Caffarelli–Kohn–Nirenberg inequalityradial functionscompact embedding |
| spellingShingle | Mallick, Arka Nguyen, Hoai-Minh The Caffarelli–Kohn–Nirenberg inequalities for radial functions Comptes Rendus. Mathématique Caffarelli–Kohn–Nirenberg inequality radial functions compact embedding |
| title | The Caffarelli–Kohn–Nirenberg inequalities for radial functions |
| title_full | The Caffarelli–Kohn–Nirenberg inequalities for radial functions |
| title_fullStr | The Caffarelli–Kohn–Nirenberg inequalities for radial functions |
| title_full_unstemmed | The Caffarelli–Kohn–Nirenberg inequalities for radial functions |
| title_short | The Caffarelli–Kohn–Nirenberg inequalities for radial functions |
| title_sort | caffarelli kohn nirenberg inequalities for radial functions |
| topic | Caffarelli–Kohn–Nirenberg inequality radial functions compact embedding |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.503/ |
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