Multiple singular integrals and maximal functions along hypersurfaces
Annales de l'Institut Fourier, Tome 36 (1986) no. 4, pp. 185-206.

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.

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.

@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},
     pages = {185--206},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {36},
     number = {4},
     year = {1986},
     doi = {10.5802/aif.1073},
     mrnumber = {88f:42037},
     zbl = {0568.42011},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1073/}
}
TY  - JOUR
AU  - Duoandikoetxea, Javier
TI  - Multiple singular integrals and maximal functions along hypersurfaces
JO  - Annales de l'Institut Fourier
PY  - 1986
SP  - 185
EP  - 206
VL  - 36
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1073/
DO  - 10.5802/aif.1073
LA  - en
ID  - AIF_1986__36_4_185_0
ER  - 
%0 Journal Article
%A Duoandikoetxea, Javier
%T Multiple singular integrals and maximal functions along hypersurfaces
%J Annales de l'Institut Fourier
%D 1986
%P 185-206
%V 36
%N 4
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1073/
%R 10.5802/aif.1073
%G en
%F AIF_1986__36_4_185_0
Duoandikoetxea, Javier. Multiple singular integrals and maximal functions along hypersurfaces. Annales de l'Institut Fourier, Tome 36 (1986) no. 4, pp. 185-206. doi : 10.5802/aif.1073. http://www.numdam.org/articles/10.5802/aif.1073/

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

Cité par Sources :