Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces

Youssfi, Abdellah

doi:10.5802/aif.1474

Regularity properties of commutators and

B M O

-Triebel-Lizorkin spaces

Youssfi, Abdellah

Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 795-807.

Résumé
Abstract

Nous nous intéressons à la régularité des commutateurs $([b, R_{k}])_{1 \leq k \leq n}$ où $b$ est une fonction localement intégrable et $(R_{j})_{1 \leq j \leq n}$ désignent les transformées de Riesz. Nous montrons que si $1 < p < + \infty$ et $0 < s < 1$ , alors les commutateurs $([b, R_{k}])_{1 \leq k \leq n}$ sont continus de $L^{p} (ℝ^{n})$ dans l’espace de Besov ${\dot{B}}_{p}^{s, p}$ si et seulement si $b$ appartient à l’espace $B M O$ -Triebel-Lizorkin ${\dot{F}}_{\infty}^{s, p}$ . En particulier, si $p = 2$ , les commutateurs $([b, R_{k}])_{1 \leq k \leq n}$ sont continus de $L^{2} (ℝ^{n})$ dans l’espace de Sobolev ${\dot{H}}^{s}$ si et seulement si $b$ appartient à l’espace $B M O$ -Sobolev ${\dot{F}}_{\infty}^{s, 2}$ .

In this paper we consider the regularity problem for the commutators $([b, R_{k}])_{1 \leq k \leq n}$ where $b$ is a locally integrable function and $(R_{j})_{1 \leq j \leq n}$ are the Riesz transforms in the $n$ -dimensional euclidean space $ℝ^{n}$ . More precisely, we prove that these commutators $([b, R_{k}])_{1 \leq k \leq n}$ are bounded from $L^{p}$ into the Besov space ${\dot{B}}_{p}^{s, p}$ for $1 < p < + \infty$ and $0 < s < 1$ if and only if $b$ is in the $B M O$ -Triebel-Lizorkin space ${\dot{F}}_{\infty}^{s, p}$ . The reduction of our result to the case $p = 2$ gives in particular that the commutators $([b, R_{k}])_{1 \leq k \leq n}$ are bounded form $L^{2}$ into the Sobolev space ${\dot{H}}^{s}$ if and only if $b$ is in the $B M O$ -Sobolev space ${\dot{F}}_{\infty}^{s, 2}$ .

MR Zbl

DOI : 10.5802/aif.1474

@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},
     pages = {795--807},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     doi = {10.5802/aif.1474},
     mrnumber = {96k:47089},
     zbl = {0827.46030},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1474/}
}

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Youssfi, Abdellah. Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces. Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 795-807. doi : 10.5802/aif.1474. http://archive.numdam.org/articles/10.5802/aif.1474/

Bibliographie
Cité par

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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