### Singular and maximal Radon transforms: Analysis and geometry.

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We study sets in the boundary of a domain in ${C}^{n}$, on which a holomorphic function has maximum modulus. In particular we show that in a real analytic strictly pseudoconvex boundary, maximum modulus sets of maximum dimension are real analytic. Maximum modulus sets are related to , which are sets along which appropriate collections of holomorphic and antiholomorphic functions agree.

We establish Lp-boundedness for a class of product singular integral operators on spaces M = M1 x M2 x . . . x Mn. Each factor space Mi is a smooth manifold on which the basic geometry is given by a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on Mi 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. This in...

We wish to acknowledge and correct an error in a proof in our paper , which appeared in Revista Matemática Iberoamericana, volume 20, number 2, 2004, pages 531-561.

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