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The Szegö projection need not preserve global analyticity. (English) Zbl 0851.32024
Let $$\Omega$$ be a pseudoconvex domain in $$\mathbb{C}^n$$ with real analytic boundary $$\partial \Omega$$ and $$H^2 (\partial \Omega)$$ be the closed subspace of $$L^2(\partial \Omega)$$, consisting of those functions that extend holomorphically to $$\Omega$$. The Szegö projection $$S$$ is the orthogonal projection from $$L^2(\partial \Omega)$$ onto $$H^2(\partial \Omega)$$ with respect to the Hilbert space structure induced by the surface measure on $$\partial \Omega$$. It is known that if $$f \in C^\infty$$ on $$\partial \Omega$$, then also $$Sf \in C^\infty$$ on $$\partial \Omega$$ [J. J. Kohn, Proc. Symp. Pure Math. 43, 207-217 (1985; Zbl 0571.58027)].
The main result of the author is: There exists a bounded pseudoconvex domain $$\Omega$$ in $$\mathbb{C}^2$$ with real analytic boundary and a real analytic function $$f$$ on $$\partial \Omega$$ such that $$Sf$$ is not real analytic.

##### MSC:
 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32T99 Pseudoconvex domains
Szegö projection
Zbl 0571.58027
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