Regularity properties of commutators and BMO-Triebel-Lizorkin spaces
Annales de l'Institut Fourier, Volume 45 (1995) no. 3, p. 795-807
In this paper we consider the regularity problem for the commutators ([b,R k ]) 1kn where b is a locally integrable function and (R j ) 1jn are the Riesz transforms in the n-dimensional euclidean space n . More precisely, we prove that these commutators ([b,R k ]) 1kn are bounded from L p into the Besov space B ˙ p s,p for 1<p<+ and 0<s<1 if and only if b is in the BMO-Triebel-Lizorkin space F ˙ s,p . The reduction of our result to the case p=2 gives in particular that the commutators ([b,R k ]) 1kn are bounded form L 2 into the Sobolev space H ˙ s if and only if b is in the BMO-Sobolev space F ˙ s,2 .
Nous nous intéressons à la régularité des commutateurs ([b,R k ]) 1kn b est une fonction localement intégrable et (R j ) 1jn désignent les transformées de Riesz. Nous montrons que si 1<p<+ et 0<s<1, alors les commutateurs ([b,R k ]) 1kn sont continus de L p ( n ) dans l’espace de Besov B ˙ p s,p si et seulement si b appartient à l’espace BMO-Triebel-Lizorkin F ˙ s,p . En particulier, si p=2, les commutateurs ([b,R k ]) 1kn sont continus de L 2 ( n ) dans l’espace de Sobolev H ˙ s si et seulement si b appartient à l’espace BMO-Sobolev F ˙ s,2 .
@article{AIF_1995__45_3_795_0,
     author = {Youssfi, Abdellah},
     title = {Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     pages = {795-807},
     doi = {10.5802/aif.1474},
     zbl = {0827.46030},
     mrnumber = {96k:47089},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1995__45_3_795_0}
}
Youssfi, Abdellah. Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces. Annales de l'Institut Fourier, Volume 45 (1995) no. 3, pp. 795-807. doi : 10.5802/aif.1474. http://www.numdam.org/item/AIF_1995__45_3_795_0/

[1] G. Bourdaud, Analyse fonctionnelle dans l'espace euclidien, Pub. Math. Paris VII, 23 (1987). | Zbl 0627.46048

[2] G. Bourdaud, Réalisations des espaces de Besov homogènes, Arkiv for Math., 26 (1988), 41-54. | MR 90d:46046 | Zbl 0661.46026

[3] A.P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci., 53 (1965), 1092-1099. | MR 31 #1575 | Zbl 0151.16901

[4] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math Pures et Appl., 72 (1993), 247-286. | MR 95d:46033 | Zbl 0864.42009

[5] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. | MR 54 #843 | Zbl 0326.32011

[6] M. Frazier, and B. Jawerth, A discrete transform and applications to distribution spaces, J. Funct. Anal., 93 (1990), 34-170. | MR 92a:46042 | Zbl 0716.46031

[7] Y. Meyer, Ondelettes et opérateurs. Tome II, Hermann, 1990. | Zbl 0694.41037

[8] M.A.M. Murray, Commutateurs with fractional differentiation and BMO-Sobolev spaces, Indiana Univ. Math. J., 34 (1985), 205-215. | MR 86c:47042 | Zbl 0537.46035

[9] J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Series I Durham, North Carolina, 1976. | MR 57 #1108 | Zbl 0356.46038

[10] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971. | MR 46 #4102 | Zbl 0232.42007

[11] R.S. Strichartz, Bounded mean oscillations and Sobolev spaces, Indiana Univ. Math. J., 29 (1980), 539-558. | MR 82f:46040 | Zbl 0437.46028

[12] H. Triebel, Theory of function spaces (Leipzig 1983). | Zbl 0546.46028

[13] H. Triebel, Theory of function spaces II, Basel-Boston-Berlin, Birkhäuser, 1992. | MR 93f:46029 | Zbl 0763.46025

[14] A. Youssfi, Localisation des espaces de Lizorkin-Triebel homogènes, Math. Na-chr., 147 (1990), 107-121. | MR 92j:46059 | Zbl 0737.46026

[15] A. Youssfi, Commutators on Besov spaces and factorization of the paraproduct, Bull. Sc. Math., 119 (1995), 157-186. | MR 96e:46050 | Zbl 0827.46031