no. 3
Multiplier conditions for Boundedness into Hardy spaces
[Sur une condition de multiplicateur de continuité a l’espace de Hardy]
Grafakos, Loukas1 ; Nakamura, Shohei2 ; Nguyen, Hanh Van3 ; Sawano, Yoshihiro4
1 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
2 Department of Mathematics Osaka University Toyonaka 560-0043, Osaka (Japan)
3 Department of Mathematics University of Alabama Tuscaloosa, AL 35487 (USA)
4 Department of Mathematics Chuo University Kasuga, Bunkyo-ku, 112-8551, Tokyo (Japan)
Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1047-1064.

Dans ce travail, on donne des conditions utilisables et explicites pour que des multiplicateurs linéaires et multilinéaires de type Coifman-Meyer, des sommes de produits d’opérateurs de Calderon-Zygmund, et aussi des opérateurs de type intermédiaire, soient bornés de produits d’espaces de Lebesgue ou de Hardy dans un espace de Hardy. Ces conditions affirment que les symboles des multiplicateurs

σ(ξ 1 ,...,ξ m )

et leurs dérivées s’annulent sur l’hyperplan ξ 1 +⋯+ξ m =0.

In the present work we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman–Meyer type, finite sum of products of Calderón–Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers

σ(ξ 1 ,...,ξ m )

and their derivatives vanish on the hyperplane ξ 1 +⋯+ξ m =0.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3416
Classification : 10X99, 14A12, 11L05
Keywords: multiplier, multilinear, Hardy spaces
Mot clés : multiplicateur, multilineaire, espaces de Hardy
Grafakos, Loukas 1 ; Nakamura, Shohei 2 ; Nguyen, Hanh Van 3 ; Sawano, Yoshihiro 4

1 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
2 Department of Mathematics Osaka University Toyonaka 560-0043, Osaka (Japan)
3 Department of Mathematics University of Alabama Tuscaloosa, AL 35487 (USA)
4 Department of Mathematics Chuo University Kasuga, Bunkyo-ku, 112-8551, Tokyo (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Multiplier conditions for {Boundedness} into {Hardy} spaces},
     journal = {Annales de l'Institut Fourier},
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Grafakos, Loukas; Nakamura, Shohei; Nguyen, Hanh Van; Sawano, Yoshihiro. Multiplier conditions for Boundedness into Hardy spaces. Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1047-1064. doi : 10.5802/aif.3416. https://aif.centre-mersenne.org/articles/10.5802/aif.3416/

[1] Bényi, ’A’rpád Bilinear singular integral operators, smooth atoms and molecules, J. Fourier Anal. Appl., Volume 9 (2003) no. 3, pp. 301-319 | DOI | MR | Zbl

[2] Bernicot, Frédéric; Grafakos, Loukas; Song, Liang; Yan, Lixin The bilinear Bochner-Riesz problem, J. Anal. Math., Volume 127 (2015), pp. 179-217 | DOI | MR | Zbl

[3] Coifman, Ronald R.; Grafakos, Loukas Hardy space estimates for multilinear operators, I, Rev. Mat. Iberoam., Volume 8 (1992) no. 1, pp. 45-67 | DOI | Zbl

[4] Coifman, Ronald R.; Lions, Pierre-Louis; Meyer, Yves; Semmes, Stephen Compensated compactness and Hardy spaces, J. Math. Pures Appl., Volume 72 (1993) no. 3, pp. 247-286 | MR

[5] Coifman, Ronald R.; Meyer, Yves Commutateurs d’intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier, Volume 28 (1978) no. 3, pp. 177-202 | DOI | Zbl

[6] Dobyinski, Sylvia Ondelettes, renormalisations du produit et applications à certains opérateurs bilinéaires, Ph. D. Thesis, Université de Paris 9 (France) (1992)

[7] García-Cuerva, José; Rubio de Francia, José L. Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116, North-Holland, 1985 | Zbl

[8] Grafakos, Loukas Hardy space estimates for multilinear operators, II, Rev. Mat. Iberoam., Volume 8 (1992) no. 1, pp. 69-92 | DOI | MR | Zbl

[9] Grafakos, Loukas; Kalton, Nigel J. The Marcinkiewicz multiplier condition for bilinear operators, Stud. Math., Volume 146 (2001) no. 2, pp. 115-156 | DOI | MR | Zbl

[10] Grafakos, Loukas; Kalton, Nigel J. Multilinear Calderón–Zygmund operators on Hardy spaces, Collect. Math., Volume 52 (2001) no. 2, pp. 169-179 | Zbl

[11] Grafakos, Loukas; Li, Xinwei Bilinear operators on homogeneous groups, J. Oper. Theory, Volume 44 (2000) no. 1, pp. 63-90 | MR | Zbl

[12] Grafakos, Loukas; Nakamura, Shohei; Van Nguyen, Hanh; Sawano, Yoshihiro Conditions for Boundedness into Hardy Spaces, Math. Nachr., Volume 292 (2019), pp. 2383-2410 | DOI | MR | Zbl

[13] Grafakos, Loukas; Torres, Rodolfo H. Discrete decompositions for bilinear operators and almost diagonal conditions, Trans. Am. Math. Soc., Volume 354 (2002) no. 3, pp. 1153-1176 | DOI | MR | Zbl

[14] Grafakos, Loukas; Torres, Rodolfo H. Multilinear Calderón–Zygmund theory, Adv. Math., Volume 165 (2002) no. 1, pp. 124-164 | DOI | Zbl

[15] Jones, Peter W.; Journé, Jean-Lin On weak convergence in H 1 (ℝ d ), Proc. Am. Math. Soc., Volume 120 (1994) no. 1, pp. 137-138 | Zbl

[16] Stein, Elias M. Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993

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