Peak functions on convex domains

Kolář, Martin

Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books, (2000), p. 103-112 / Harvested from

Résumé

Let $\Omega \subset \mathbb {C}^{n}$ 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 \lbrace p\rbrace $. If $\Omega $ is strongly pseudoconvex, then $C^{\infty }$ peak functions exist. On the other hand, J. E. Fornaess constructed an example in $\mathbb {C}^2$ 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 $\mathbb {C}^2$ [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 $\mathbb {C}^{n}$. The construct!

Détail

MR MR1758085 | Zbl 0976.32018

EUDML-ID : urn:eudml:doc:221240
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