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Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 401–414 | Cite as

The \(L^p\) CR Hartogs separate analyticity theorem for convex domains

  • Mark G. Lawrence
Article
  • 68 Downloads

Abstract

In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in \(\mathbf{C}^{\mathbf{n}}\) with real analytic boundary. Given such a domain D, and given an \(L^p\) function f on \(\partial D\) which has holomorphic extensions on the slices of D by complex lines parallel to the coordinate axes, f must be CR—i.e. f has a holomorphic extension to D which is in the Hardy space \(H^p(D)\). This is the first general result of CR Hartogs type which is not for the ball, or some other domain with symmetries, and holds for \(L^1\) functions. As a corollary, the Szegő kernel is shown to be the strong operator limit of \((\pi _1 \pi _2)^n\), where \(\pi _i\) is the projection onto \(z_i\) holomorphically extendible \(L^2(\partial D)\) functions (in \(\mathbf{C}^{\mathbf{2}}\), with a slightly more complicated formula in \(\mathbf{C}^{\mathbf{n}}\)).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.AstanaKazakhstan

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