Multiple singular integrals and maximal functions along hypersurfaces

Duoandikoetxea, Javier

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 $L^p$, $1<p<\infty$, 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 $L^p$, $1<p<\infty$. 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.

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