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Commutation properties and Lipschitz estimates for the Bergman and Szegö projections. (English) Zbl 0866.32012

It is well known that the Bergman and Szegö projections satisfy Lipschitz estimates in pseudo-convex domains of finite type of \(\mathbb{C}^2\), both isotropic and anisotropic (see the works of Nagel, Rosay, Stein and Wainger, Mc Neal, Fefferman and Kohn). We give accurate estimates in this context: Lipschitz estimates in the direction \(T\), where \(T\) is any \(C^\infty\) vector field which is tangential and transverse to the complex tangential space at the boundary, or Lipschitz estimates in the complex tangential direction \(L\). The proofs rely on commutation properties between \(L\) or \(T\) and the Bergman or Szegö type in which these commutation properties may be written. Such precised estimates had been given, in strictly pseudo-convex domains, respectively by Ahern and Schneider for the first ones, by Cohn for the second ones.
Reviewer: A.Bonami

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32T99 Pseudoconvex domains
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References:

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