JournalsrmiVol. 20, No. 2pp. 531–561

On the product theory of singular integrals

  • Alexander Nagel

    University of Wisconsin, Madison, USA
  • Elias M. Stein

    Princeton University, United States
On the product theory of singular integrals cover
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We establish LpL^p-boundedness for a class of product singular integral operators on spaces M~=M1×M2××Mn\widetilde{M} = M_1 \times M_2\times \cdots \times M_n. Each factor space MiM_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 MiM_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 M~\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. Iberoam. 20 (2004), no. 2, pp. 531–561

DOI 10.4171/RMI/400