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

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.

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.

@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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {42},
     number = {3},
     year = {1992},
     pages = {637-670},
     doi = {10.5802/aif.1304},
     zbl = {0760.42008},
     mrnumber = {94d:42020},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1992__42_3_637_0}
}
Ricci, Fulvio; Stein, Elias M. Multiparameter singular integrals and maximal functions. Annales de l'Institut Fourier, Volume 42 (1992) no. 3, pp. 637-670. doi : 10.5802/aif.1304. http://www.numdam.org/item/AIF_1992__42_3_637_0/

[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 89h:42026 | Zbl 0607.42009

[2] A. Carbery, A. Seeger, Hp- and Lp- variants of multiparameter Calderón-Zygmund theory, preprint. | Zbl 0770.42010

[3] H. Carlsson, P. Sjögren, Estimates for maximal functions along hypersurfaces, Ark. För Math., 25 (1987), 1-14. | MR 89a:42027 | Zbl 0629.42010

[4] H. Carlsson, P. Sjögren, J.-O. Strömberg, Multiparameter maximal functions along dilation-invariant hypersurfaces, Trans. Amer. Math. Soc., 292 (1985), 335-343. | MR 86k:42031 | Zbl 0578.42018

[5] M. Christ, Hilbert transforms along curves. I. Nilpotent groups, Ann. of Math., 122 (1985), 575-596. | MR 87f:42039a | Zbl 0593.43011

[6] M. Christ, The strong maximal function on a nilpotent group, to appear in Trans. Amer. Math. Soc. | Zbl 0765.43002

[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 55 #6096 | Zbl 0342.42003

[8] J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Inst. Fourier, Grenoble, 36-4 (1986), 185-206. | Numdam | MR 88f:42037 | Zbl 0568.42011

[9] J. Duoandikoetxea, J.L. Rubio De Francia, Maximal and singular integral operators via Fourier transform estimates, Inv. Math., 84 (1986), 541-561. | MR 87f:42046 | Zbl 0568.42012

[10] R. Fefferman, Calderón-Zygmund theory for product domains. Hp spaces, Proc. Nat. Acad. Sci. USA, 83 (1986), 840-843. | MR 87h:42032 | Zbl 0602.42023

[11] R. Fefferman, E.M. Stein, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117-143. | MR 84d:42023 | Zbl 0517.42024

[12] J.L. Journé, Calderón-Zygmund operators on products spaces, Rev. Mat. Ibero-Amer., 3 (1985), 55-91. | MR 88d:42028 | Zbl 0634.42015

[13] A. Nagel, E.M. Stein, S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci., USA, 75 (1978), 1060-1062. | MR 57 #6349 | Zbl 0391.42015

[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 56 #9192 | Zbl 0374.44003

[15] J. Pipher, Journé's covering lemma and its extension to higher dimensions, Duke Math. J., 53 (1986), 683-690. | MR 88a:42019 | Zbl 0645.42018

[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 89g:42030 | Zbl 0645.42019

[17] E.M. Stein, Oscillatory integrals in Fourier analysis, in Bejing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, 1986. | MR 88g:42022 | Zbl 0618.42006

[18] E.M. Stein, S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84 (1978), 1239-1295. | MR 80k:42023 | Zbl 0393.42010

[19] R. Strichartz, Singular integrals supported on submanifolds, Studia Math., 74 (1982), 137-151. | MR 85c:42019 | Zbl 0501.43007

[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 85h:44010 | Zbl 0462.44001