Geodesic
Analyse harmonique, Équations aux dérivées partielles
The Caffarelli–Kohn–Nirenberg inequalities for radial functions
Mallick, Arka1 ; Nguyen, Hoai-Minh2 
1 Department of Mathematics, IISc, Bengaluru, India
2 Laboratoire Jacques Louis Lions, Sorbonne Université, Paris, France
Comptes Rendus. Mathématique, Tome 361 (2023) no. G7, pp. 1175-1189

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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
Keywords: Caffarelli–Kohn–Nirenberg inequality, radial functions, compact embedding

Mallick, Arka 1 ; Nguyen, Hoai-Minh 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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Mallick, Arka; Nguyen, Hoai-Minh. The Caffarelli–Kohn–Nirenberg inequalities for radial functions. Comptes Rendus. Mathématique, Tome 361 (2023) no. G7, pp. 1175-1189. doi: 10.5802/crmath.503

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