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A construction of peak functions on some finite type domains. (English) Zbl 0809.32005

A peak function is a holomorphic function \(f\) in \(\Omega \subset \mathbb{C}^ n\) which peaks at a boundary point \(p\) (i.e. \(f(p) = 1\) and \(| f | < 1\) in any other point of \(\overline \Omega)\), and is continuous on \(\overline \Omega\). In this article peak functions are constructed for arbitrary boundary points of pseudoconvex finite type domains in \(\mathbb{C}^ 2\), and of certain diagonisable finite type domains in \(\mathbb{C}^ n\). These results are not so new, but the method used is.
First a local peak function is constructed using the Bergman kernel. If \(q\) is taken close to the point \(p\), the Bergman kernel \(K(z,q)\) will be big near \(q\), and hence near \(p\). This idea is used in the so-called Bishop \(1/4 - 3/4\) technique. The local peak function is then used to construct a global one considering a \(\overline \partial\)-problem. Finally, even Hölder continuous peak functions are obtained.
The method involves the delicate use of estimates on the Bergman kernel. An overview of known results is given, both for estimates on and off the diagonal, together with some new ones.
Reviewer: J.Cnops (Gent)

MSC:

32E35 Global boundary behavior of holomorphic functions of several complex variables
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
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