Open Access
June, 2004 On the product theory of singular integrals
Alexander Nagel, Elias M. Stein
Rev. Mat. Iberoamericana 20(2): 531-561 (June, 2004).


We establish $L^p$-boundedness for a class of product singular integral operators on spaces $\widetilde{M} = M_1 \times M_2\times \cdots \times M_n$. Each factor space $M_i$ is a smooth manifold on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on $M_i$ are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood-Paley theory on $\widetilde M$. This in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor.


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Alexander Nagel. Elias M. Stein. "On the product theory of singular integrals." Rev. Mat. Iberoamericana 20 (2) 531 - 561, June, 2004.


Published: June, 2004
First available in Project Euclid: 17 June 2004

zbMATH: 1057.42016
MathSciNet: MR2073131

Primary: 42B20 , 42B25

Keywords: control metrics , Littlewood-Paley theory , NIS operators , product singular integrals

Rights: Copyright © 2004 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.20 • No. 2 • June, 2004
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