Let $\Omega \subset {ℂ}^n$ be a domain with smooth boundary and $p\in \partial \Omega$. A holomorphic function $f$ on $\Omega$ is called a $C^k$ ($k=0,1,2,\cdots$) peak function at $p$ if $f\in C^k\left(\overline{\Omega }\right)$, $f\left(p\right)=1$, and $\left|f\right(q\left)\right|<1$ for all $q\in \overline{\Omega }\setminus \left\{p\right\}$. If $\Omega$ is strongly pseudoconvex, then $C^{\infty }$ peak functions exist. On the other hand, J. E. Fornaess constructed an example in ${ℂ}^2$ 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 ${ℂ}^2$ [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 ${ℂ}^n$. The construct!

EUDML-ID : urn:eudml:doc:221240
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@article{701653,
     title = {Peak functions on convex domains},
     booktitle = {Proceedings of the 19th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {2000},
     pages = {103-112},
     mrnumber = {MR1758085},
     zbl = {0976.32018},
     url = {http://dml.mathdoc.fr/item/701653}
}

Kolář, Martin. Peak functions on convex domains, dans Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books,  (2000), pp. 103-112. http://gdmltest.u-ga.fr/item/701653/