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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
Citations:
Zbl 0571.58027
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