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Multiparameter singular integrals and maximal functions

Fulvio RicciElias M. Stein — 1992

Annales de l'institut Fourier

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 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.

On the product theory of singular integrals.

Alexander NagelElias M. Stein — 2004

Revista Matemática Iberoamericana

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...

Corrigenda: .

Alexander NagelElias M. Stein — 2005

Revista Matemática Iberoamericana

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.

An algebra of pseudo-differential operators and quantum mechanics in phase space

A. GrossmannGuy LoupiasElias M. Stein — 1968

Annales de l'institut Fourier

Nous étudions une algèbre 𝒫 de fonctions infiniment différentiables définies sur l’espace de phase et satisfaisant des conditions de croissance à l’infini. Le produit dans 𝒫 est la transformée de Fourier symplectique de la convolution gauche. On montre que 𝒫 est une généralisation naturelle de l’algèbre des opérateurs pseudodifférentiels.

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