no. 7
Partial differential equations/Functional analysis
Brézis–Gallouet–Wainger-type inequality with critical fractional Sobolev space and BMO
[Inégalité de type Brézis–Gallouet–Wainger pour un espace de Sobolev fractionnaire critique et BMO]

Présenté par : Haïm Brézis

Dao, Nguyen-Anh1 ; Nguyen, Quoc-Hung2
1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
2 Scuola Normale Superiore, Centro Ennio de Giorgi, Piazza dei Cavalieri 3, I-56100 Pisa, Italy
Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 747-756.

Dans cette Note, nous montrons l'inégalité de type Brézis–Gallouet–Wainger faisant intervenir la norme BMO, la norme fractionnaire de Sobolev et la norme logarithmique de C˙η, pour η∈(0,1).

In this paper, we prove the Brézis–Gallouet–Wainger-type inequality involving the BMO norm, the fractional Sobolev norm, and the logarithmic norm of C˙η, for η∈(0,1).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.05.009
Dao, Nguyen-Anh 1 ; Nguyen, Quoc-Hung 2

1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
2 Scuola Normale Superiore, Centro Ennio de Giorgi, Piazza dei Cavalieri 3, I-56100 Pisa, Italy 
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Dao, Nguyen-Anh; Nguyen, Quoc-Hung. Brézis–Gallouet–Wainger-type inequality with critical fractional Sobolev space and BMO. Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 747-756. doi : 10.1016/j.crma.2018.05.009. http://www.numdam.org/articles/10.1016/j.crma.2018.05.009/

[1] Beale, J.T.; Kato, T.; Majda, A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., Volume 94 (1984), pp. 61-66

[2] Brézis, H.; Gallouet, T. Nonlinear Schrodinger evolution equations, Nonlinear Anal., Volume 4 (1980), pp. 677-681

[3] Brézis, H.; Wainger, S. A note on limiting cases of Sobolev embeddings and convolution inequalities, Commun. Partial Differ. Equ., Volume 5 (1980), pp. 773-789

[4] Engler, H. An alternative proof of the Brézis–Wainger inequality, Commun. Partial Differ. Equ., Volume 14 (1989), pp. 541-544

[5] Kozono, H.; Taniuchi, Y. Limiting case of the Sobolev inequality in BMO with application to the Euler equations, Commun. Math. Phys., Volume 214 (2000), pp. 191-200

[6] Kozono, H.; Wadade, H. Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO, Math. Z., Volume 295 (2008), pp. 935-950

[7] Kozono, H.; Ogawa, T.; Taniuchi, Y. The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., Volume 242 (2002), pp. 251-278

[8] Di Nezza, E.; Palatucci, G.; Valdinoci, E. Hitchhiker's guide to the fractional Sobolev spaces, Bull. Soc. Math. Fr., Volume 136 (2012), pp. 521-573

[9] Ogawa, T.; Taniuchi, Y. On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain, J. Differ. Equ., Volume 190 (2003), pp. 39-63

[10] Ozawa, T. On critical cases of Sobolev's inequalities, J. Funct. Anal., Volume 127 (1995), pp. 259-269

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