| 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) |
