# The $$L^p$$ L p CR Hartogs separate analyticity theorem for convex domains

The $$L^p$$ L p CR Hartogs separate analyticity theorem for convex domains In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in $$\mathbf{C}^{\mathbf{n}}$$ C n with real analytic boundary. Given such a domain D, and given an $$L^p$$ L p function f on $$\partial D$$ ∂ 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 $$H^p(D)$$ 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 $$L^1$$ L 1 functions. As a corollary, the Szegő kernel is shown to be the strong operator limit of $$(\pi _1 \pi _2)^n$$ ( π 1 π 2 ) n , where $$\pi _i$$ π i is the projection onto $$z_i$$ z i holomorphically extendible $$L^2(\partial D)$$ L 2 ( ∂ D ) functions (in $$\mathbf{C}^{\mathbf{2}}$$ C 2 , with a slightly more complicated formula in $$\mathbf{C}^{\mathbf{n}}$$ C n ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

# The $$L^p$$ L p CR Hartogs separate analyticity theorem for convex domains

, Volume 288 (2) – May 13, 2017
14 pages

/lp/springer_journal/the-l-p-l-p-cr-hartogs-separate-analyticity-theorem-for-convex-domains-K4lvYvDQLI
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1894-z
Publisher site
See Article on Publisher Site

### Abstract

In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in $$\mathbf{C}^{\mathbf{n}}$$ C n with real analytic boundary. Given such a domain D, and given an $$L^p$$ L p function f on $$\partial D$$ ∂ 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 $$H^p(D)$$ 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 $$L^1$$ L 1 functions. As a corollary, the Szegő kernel is shown to be the strong operator limit of $$(\pi _1 \pi _2)^n$$ ( π 1 π 2 ) n , where $$\pi _i$$ π i is the projection onto $$z_i$$ z i holomorphically extendible $$L^2(\partial D)$$ L 2 ( ∂ D ) functions (in $$\mathbf{C}^{\mathbf{2}}$$ C 2 , with a slightly more complicated formula in $$\mathbf{C}^{\mathbf{n}}$$ C n ).

### Journal

Mathematische ZeitschriftSpringer Journals

Published: May 13, 2017

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