no. 3
Multiparameter singular integrals and maximal functions
Annales de l'Institut Fourier, Tome 42 (1992) no. 3, pp. 637-670.

On donne des estimations dans L p pour une classe d’opérateurs, à intégrales singulières et maximales, associés à une famille quelconque de dilatations diagonales à k paramètres sur R n . Cette classe comprend les opérateurs homogènes définis par noyaux à support sur des variétés homogènes. Pour les intégrales singulières, l’annulation qu’on impose sur le noyau est d’un type “minimal”, défini à partir de la famille de dilatations considérées.

We prove L p -boundedness for a class of singular integral operators and maximal operators associated with a general k-parameter family of dilations on R n . This class includes homogeneous operators defined by kernels supported on homogeneous manifolds. For singular integrals, only certain “minimal” cancellation is required of the kernels, depending on the given set of dilations.

@article{AIF_1992__42_3_637_0,
     author = {Ricci, Fulvio and Stein, Elias M.},
     title = {Multiparameter singular integrals and maximal functions},
     journal = {Annales de l'Institut Fourier},
     pages = {637--670},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {3},
     year = {1992},
     doi = {10.5802/aif.1304},
     zbl = {0760.42008},
     mrnumber = {94d:42020},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1304/}
}
TY  - JOUR
AU  - Ricci, Fulvio
AU  - Stein, Elias M.
TI  - Multiparameter singular integrals and maximal functions
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 637
EP  - 670
VL  - 42
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1304/
DO  - 10.5802/aif.1304
LA  - en
ID  - AIF_1992__42_3_637_0
ER  - 
%0 Journal Article
%A Ricci, Fulvio
%A Stein, Elias M.
%T Multiparameter singular integrals and maximal functions
%J Annales de l'Institut Fourier
%D 1992
%P 637-670
%V 42
%N 3
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1304/
%R 10.5802/aif.1304
%G en
%F AIF_1992__42_3_637_0
Ricci, Fulvio; Stein, Elias M. Multiparameter singular integrals and maximal functions. Annales de l'Institut Fourier, Tome 42 (1992) no. 3, pp. 637-670. doi : 10.5802/aif.1304. https://aif.centre-mersenne.org/articles/10.5802/aif.1304/

[1] A. Carbery, Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem, Ann. Inst. Fourier, Grenoble, 38-1 (1988), 157-168. | Numdam | MR | Zbl

[2] A. Carbery, A. Seeger, Hp- and Lp- variants of multiparameter Calderón-Zygmund theory, preprint. | Zbl

[3] H. Carlsson, P. Sjögren, Estimates for maximal functions along hypersurfaces, Ark. För Math., 25 (1987), 1-14. | MR | Zbl

[4] H. Carlsson, P. Sjögren, J.-O. Strömberg, Multiparameter maximal functions along dilation-invariant hypersurfaces, Trans. Amer. Math. Soc., 292 (1985), 335-343. | MR | Zbl

[5] M. Christ, Hilbert transforms along curves. I. Nilpotent groups, Ann. of Math., 122 (1985), 575-596. | MR | Zbl

[6] M. Christ, The strong maximal function on a nilpotent group, to appear in Trans. Amer. Math. Soc. | Zbl

[7] A. Córdoba, R. Fefferman, On the equivalence between the boundedness of certain classes of maximal operators and multiplier operators in Fourier analysis, Proc. Nat. Acad. Sci. USA, 74 (1977), 423-425. | MR | Zbl

[8] J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Inst. Fourier, Grenoble, 36-4 (1986), 185-206. | Numdam | MR | Zbl

[9] J. Duoandikoetxea, J.L. Rubio De Francia, Maximal and singular integral operators via Fourier transform estimates, Inv. Math., 84 (1986), 541-561. | MR | Zbl

[10] R. Fefferman, Calderón-Zygmund theory for product domains. Hp spaces, Proc. Nat. Acad. Sci. USA, 83 (1986), 840-843. | MR | Zbl

[11] R. Fefferman, E.M. Stein, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117-143. | MR | Zbl

[12] J.L. Journé, Calderón-Zygmund operators on products spaces, Rev. Mat. Ibero-Amer., 3 (1985), 55-91. | MR | Zbl

[13] A. Nagel, E.M. Stein, S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci., USA, 75 (1978), 1060-1062. | MR | Zbl

[14] A. Nagel, S. Wainger, L2-boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math., 99 (1977), 761-785. | MR | Zbl

[15] J. Pipher, Journé's covering lemma and its extension to higher dimensions, Duke Math. J., 53 (1986), 683-690. | MR | Zbl

[16] F. Ricci, E.M. Stein, Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds, J. Funct. Anal., 78 (1988), 56-84. | MR | Zbl

[17] E.M. Stein, Oscillatory integrals in Fourier analysis, in Bejing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, 1986. | MR | Zbl

[18] E.M. Stein, S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84 (1978), 1239-1295. | MR | Zbl

[19] R. Strichartz, Singular integrals supported on submanifolds, Studia Math., 74 (1982), 137-151. | MR | Zbl

[20] J.-O. Strömberg, Dissertation (1976, Mittag-Leffler Inst., Djursholm, Sweden.

[21] J.T. Vance, Lp-boundedness of the multiple Hilbert transform along a surface, Pac. J. Math., 108 (1983), 221-241. | MR | Zbl

Cité par Sources :