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Title: Peak functions on convex domains (English)
Author: Kolář, Martin
Language: English
Journal: Proceedings of the 19th Winter School "Geometry and Physics"
Issue: 1999
Pages: 103-112
Category: math
Summary: Let $\Omega\subset\Bbb C^{n}$ be a domain with smooth boundary and $p\in\partial\Omega$. A holomorphic function $f$ on $\Omega$ is called a $C^k$ ($k=0,1,2,\dots$) peak function at $p$ if $f\in C^{k}(\overline\Omega)$, $f(p)=1$, and $|f(q)|<1$ for all $q\in\overline\Omega\setminus\{p\}$. If $\Omega$ is strongly pseudoconvex, then $C^{\infty}$ peak functions exist. On the other hand, J. E. Fornaess constructed an example in $\Bbb C^2$ to show that this result fails, even for $C^1$ functions, on a weakly pseudoconvex domain [Math. Ann. 227, 173-175 (1977; Zbl 0346.32026)]. Subsequently, E. Bedford and J. E. Fornaess showed that there is always a continuous peak function on a pseudoconvex domain of finite type in $\Bbb C^2$ [Ann. Math. (2) 107, 555-568 (1978; Zbl 0392.32004)]. In the present paper, the author constructs a continuous and a Hölder continuous peak function at a point of finite type on a convex domain in $\Bbb C^{n}$. The construct! (English)
MSC: 32F18
MSC: 32T40
idZBL: Zbl 0976.32018
idMR: MR1758085
Date available: 2009-07-13T21:43:27Z
Last updated: 2012-09-18
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