Comptes Rendus
Analyse harmonique, Équations aux dérivées partielles
The Caffarelli–Kohn–Nirenberg inequalities for radial functions
Arka Mallick1 ; Hoai-Minh Nguyen2 
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.

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≤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.

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DOI : 10.5802/crmath.503
Classification : 26D10, 26A54
Mots clés : Caffarelli–Kohn–Nirenberg inequality, radial functions, compact embedding
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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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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