Multiple singular integrals and maximal functions along hypersurfaces
Annales de l'Institut Fourier, Volume 36 (1986) no. 4, p. 185-206
Maximal functions written as convolution with a multiparametric family of positive measures, and singular integrals whose kernel is decomposed as a multiple series of measures, are shown to be bounded in L p , 1<p<. The proofs are based on the decomposition of the operators according to the size of the Fourier transform of the measures, assuming some regularity at zero and decay at infinity of these Fourier transforms. Applications are given to homogeneous singular integrals in product spaces with size conditions on the kernel and maximal functions and multiple Hilbert transforms along different types of surfaces.
On prouve que certaines fonctions maximales écrites comme convolution avec une suite double (ou multiple) de mesures, et certains opérateurs invariants par translation dont le noyau est décomposé en séries doubles (ou multiples) de mesures, sont bornés dans L p , 1<p<, à partir de certaines conditions de régularité et décroissance de la transformée de Fourier de ces mesures. On donne ensuite quelques applications aux intégrales singulières homogènes dans un espace produit et aux fonctions maximales et aux transformées de Hilbert sur une hypersurface.
@article{AIF_1986__36_4_185_0,
     author = {Duoandikoetxea, Javier},
     title = {Multiple singular integrals and maximal functions along hypersurfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {36},
     number = {4},
     year = {1986},
     pages = {185-206},
     doi = {10.5802/aif.1073},
     mrnumber = {88f:42037},
     zbl = {0568.42011},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1986__36_4_185_0}
}
Multiple singular integrals and maximal functions along hypersurfaces. Annales de l'Institut Fourier, Volume 36 (1986) no. 4, pp. 185-206. doi : 10.5802/aif.1073. https://aif.centre-mersenne.org/item/AIF_1986__36_4_185_0/

[1] A. P. Calderon, A. Zygmund, On singular integrals, Amer. J. Math., 78 (1956), 289-309. | MR 18,894a | Zbl 0072.11501

[2] H. Carlsson, P. Sjögren, Estimates for maximal functions along hypersurfaces, Univ. of Göteborg, preprint, 1984.

[3] H. Carlsson, P. Sjögren, J. O. Stromberg, Multiparameter maximal functions along dilation invariant hypersurfaces, Univ. of Göteborg, preprint, 1984.

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

[5] R. Fefferman, Singular integrals on product domains, Bull. Amer. Math. Soc., 4 (1981), 195-201. | MR 83i:42014 | Zbl 0466.42007

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

[7] D. S. Kurtz, Littlewood-Paley and multiplier theorems on weighted Lp spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254. | MR 80f:42013 | Zbl 0436.42012

[8] 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

[9] J. L. Rubio De Francia, Factorization theory and Ap weights, Amer. J. Math., 106 (1984), 533-547. | MR 86a:47028a | Zbl 0558.42012

[10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton N.J., 1970. | MR 44 #7280 | Zbl 0207.13501

[11] 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

[12] R. S. Strichartz, Singular integrals supported in manifolds, Studia Math., 74 (1982), 137-151. | MR 85c:42019 | Zbl 0501.43007

[13] J. T. Vance, Lp-boundedness of the multiple Hilbert transform along a surface, Pacific J. Math., 108 (1983), 221-241. | MR 85h:44010 | Zbl 0462.44001