The Caffarelli–Kohn–Nirenberg inequalities for radial functions

Arka Mallick; Hoai-Minh Nguyen

doi:10.5802/crmath.503

Analyse harmonique, Équations aux dérivées partielles

The Caffarelli–Kohn–Nirenberg inequalities for radial functions

Arka Mallick ¹ ; Hoai-Minh Nguyen ²

¹ Department of Mathematics, IISc, Bengaluru, India
² Laboratoire Jacques Louis Lions, Sorbonne Université, Paris, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1175-1189.

Résumé

We establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order $0 < s \leq 1$ . 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.

Reçu le : 2023-01-16
Révisé le : 2023-04-04
Accepté le : 2023-04-04
Publié le : 2023-10-24

DOI : 10.5802/crmath.503

Classification : 26D10, 26A54
Mots clés : Caffarelli–Kohn–Nirenberg inequality, radial functions, compact embedding

Affiliations des auteurs :

Arka Mallick ¹ ; Hoai-Minh Nguyen ²

¹ Department of Mathematics, IISc, Bengaluru, India
² Laboratoire Jacques Louis Lions, Sorbonne Université, Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Comptes Rendus. Math\'ematique},
     pages = {1175--1189},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2023},
     doi = {10.5802/crmath.503},
     language = {en},
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Arka Mallick; Hoai-Minh Nguyen. The Caffarelli–Kohn–Nirenberg inequalities for radial functions. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1175-1189. doi : 10.5802/crmath.503. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.503/

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