Peak functions on convex domains

Kolář, Martin. "Peak functions on convex domains." Proceedings of the 19th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 2000. 103-112. <http://eudml.org/doc/221240>.

@inProceedings{Kolář2000,
abstract = {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!},
author = {Kolář, Martin},
booktitle = {Proceedings of the 19th Winter School "Geometry and Physics"},
keywords = {Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)},
location = {Palermo},
pages = {103-112},
publisher = {Circolo Matematico di Palermo},
title = {Peak functions on convex domains},
url = {http://eudml.org/doc/221240},
year = {2000},
}

TY - CLSWK
AU - Kolář, Martin
TI - Peak functions on convex domains
T2 - Proceedings of the 19th Winter School "Geometry and Physics"
PY - 2000
CY - Palermo
PB - Circolo Matematico di Palermo
SP - 103
EP - 112
AB - 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!
KW - Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)
UR - http://eudml.org/doc/221240
ER -