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

@article{CRMATH_2023__361_G7_1175_0,
     author = {Arka Mallick and Hoai-Minh Nguyen},
     title = {The {Caffarelli{\textendash}Kohn{\textendash}Nirenberg} inequalities for radial functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1175--1189},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.503},
     language = {en},
}

TY  - JOUR
AU  - Arka Mallick
AU  - Hoai-Minh Nguyen
TI  - The Caffarelli–Kohn–Nirenberg inequalities for radial functions
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1175
EP  - 1189
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.503
LA  - en
ID  - CRMATH_2023__361_G7_1175_0
ER  -

%0 Journal Article
%A Arka Mallick
%A Hoai-Minh Nguyen
%T The Caffarelli–Kohn–Nirenberg inequalities for radial functions
%J Comptes Rendus. Mathématique
%D 2023
%P 1175-1189
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.503
%G en
%F CRMATH_2023__361_G7_1175_0

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/

Bibliographie
Cité par

[1] Boumediene Abdellaoui; Rachid Bentifour Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications, J. Funct. Anal., Volume 272 (2017) no. 10, pp. 3998-4029 | DOI | MR | Zbl

[2] Jacopo Bellazzini; Rupert L. Frank; Nicola Visciglia Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems, Math. Ann., Volume 360 (2014) no. 3-4, pp. 653-673 | DOI | MR | Zbl

[3] Jacopo Bellazzini; Marco Ghimenti; Carlo Mercuri; Vitaly Moroz; Jean Van Schaftingen Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces, Trans. Am. Math. Soc., Volume 370 (2018) no. 11, pp. 8285-8310 | DOI | MR | Zbl

[4] Jacopo Bellazzini; Marco Ghimenti; Tohru Ozawa Sharp lower bounds for Coulomb energy, Math. Res. Lett., Volume 23 (2016) no. 3, pp. 621-632 | DOI | MR | Zbl

[5] Henri Berestycki; Pierre-Louis Lions Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., Volume 82 (1983) no. 4, pp. 313-345 | DOI | MR | Zbl

[6] Jean Bourgain; Haïm Brezis; Petru Mironescu Another look at Sobolev spaces, Optimal control and partial differential equations, IOS Press, 2001, pp. 439-455 | Zbl

[7] Jean Bourgain; Hoai-Minh Nguyen A new characterization of Sobolev spaces, C. R. Math. Acad. Sci. Paris, Volume 343 (2006) no. 2, pp. 75-80 | DOI | Numdam | MR | Zbl

[8] Haïm Brezis How to recognize constant functions. A connections with Sobolev spaces, Usp. Mat. Nauk, Volume 57 (2002), pp. 59-74 | MR

[9] Haïm Brezis; Hoai-Minh Nguyen Non-local functionals related to the total variation and connections with image processing, Ann. PDE, Volume 4 (2018) no. 1, 9, 77 pages | MR | Zbl

[10] Haïm Brezis; Jean Van Schaftingen; Po-Lam Yung A surprising formula for Sobolev norms, Proc. Natl. Acad. Sci. USA, Volume 118 (2021) no. 8, e2025254118 | MR

[11] Luis A. Caffarelli; Robert V. Kohn; Louis Nirenberg Partial regularity of suitable weak solutions of the Navier-Stokes equations, Pure Appl. Math., Volume 35 (1982) no. 6, pp. 771-831 | DOI | MR | Zbl

[12] Luis A. Caffarelli; Robert V. Kohn; Louis Nirenberg First order interpolation inequalities with weights, Compos. Math., Volume 53 (1984) no. 3, pp. 259-275 | Numdam | MR | Zbl

[13] Florin Catrina; Zhi-Qiang Wang On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Pure Appl. Math., Volume 54 (2001) no. 2, pp. 229-258 | DOI | MR | Zbl

[14] Kai-Seng Chou; Chiu-Wing Chu On the best constant for a weighted Sobolev-Hardy inequality, J. Lond. Math. Soc., Volume 48 (1993) no. 1, pp. 137-151 | DOI | MR | Zbl

[15] Pablo L. De Nápoli; Irene Drelichman; Ricardo G. Durán Improved Caffarelli–Kohn–Nirenberg and trace inequalities for radial functions, Commun. Pure Appl. Anal., Volume 11 (2012) no. 5, pp. 1629-1642 | MR | Zbl

[16] Veronica Felli; Matthias Schneider Perturbation results of critical elliptic equations of Caffarelli–Kohn–Nirenberg type, J. Differ. Equations, Volume 191 (2003) no. 1, pp. 121-142 | DOI | MR | Zbl

[17] Rupert L. Frank; Robert Seiringer Non-linear ground state representations and sharp hardy inequalities, J. Funct. Anal., Volume 255 (2008) no. 12, pp. 3407-3430 | DOI | MR | Zbl

[18] Elliott H. Lieb Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies Appl. Math., Volume 57 (1977), pp. 93-105 | DOI | Zbl

[19] Pierre-Louis Lions Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., Volume 49 (1982) no. 3, pp. 315-334 | DOI | Zbl

[20] Arka Mallick; Hoai-Minh Nguyen Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg interpolation inequalities associated with Coulomb–Sobolev spaces, J. Funct. Anal., Volume 283 (2022) no. 10, 109662, 33 pages | MR | Zbl

[21] Vladimir Gilelevich Mazýa; Tatyana O. Shaposhnikova On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002) no. 2, pp. 230-238 | MR

[22] Hoai-Minh Nguyen Some new characterizations of Sobolev spaces, J. Funct. Anal., Volume 237 (2006) no. 2, pp. 689-720 | DOI | MR | Zbl

[23] Hoai-Minh Nguyen Some inequalities related to Sobolev norms, Calc. Var. Partial Differ. Equ., Volume 41 (2011) no. 3-4, pp. 483-509 | DOI | MR | Zbl

[24] Hoai-Minh Nguyen; Marco Squassina Fractional Caffarelli–Kohn–Nirenberg inequalities, J. Funct. Anal., Volume 274 (2018) no. 9, pp. 2661-2672 | DOI | MR | Zbl

[25] Hoai-Minh Nguyen; Marco Squassina On Hardy and Caffarelli-Kohn-Nirenberg inequalities, J. Anal. Math., Volume 139 (2019) no. 2, pp. 773-797 | DOI | MR | Zbl

[26] Boris Rubin One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions, Mat. Zametki, Volume 34 (1983) no. 4, pp. 521-533 | MR | Zbl

[27] Winfried Sickel; Leszek Skrzypczak Radial subspaces of Besov and Lizorkin–Triebel classes: extended Strauss lemma and compactness of embeddings, J. Fourier Anal. Appl., Volume 6 (2000) no. 6, pp. 639-662 | DOI | MR | Zbl

[28] Walter A. Strauss Existence of solitary waves in higher dimensions, Commun. Math. Phys., Volume 55 (1977) no. 2, pp. 149-162 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique