Multiparameter singular integrals and maximal functions

Ricci, Fulvio; Stein, Elias M.

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 ${\mathbf{R}}^n$. 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 ${\mathbf{R}}^n$. 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.

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