On the algebra of $A^k$-functions

Backlund, Ulf, and Fällström, Anders. "On the algebra of $A^k$-functions." Mathematica Bohemica 131.1 (2006): 49-61. <http://eudml.org/doc/249919>.

@article{Backlund2006,
abstract = {For a domain $\Omega \subset \{\mathbb \{C\}\}^n$ let $H(\Omega )$ be the holomorphic functions on $\Omega $ and for any $k\in \mathbb \{N\}$ let $A^k(\Omega )=H(\Omega )\cap C^k(\overline\{\Omega \})$. Denote by $\{\mathcal \{A\}\}_D^k(\Omega )$ the set of functions $f\: \Omega \rightarrow [0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k(\Omega )$ such that $\lbrace |f_j|\rbrace $ is a nonincreasing sequence and such that $ f(z)=\lim _\{j\rightarrow \infty \}|f_j(z)|$. By $\{\mathcal \{A\}\}_I^k(\Omega )$ denote the set of functions $f\: \Omega \rightarrow (0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k(\Omega )$ such that $\lbrace |f_j|\rbrace $ is a nondecreasing sequence and such that $ f(z)=\lim _\{j\rightarrow \infty \}|f_j(z)|$. Let $k\in \mathbb \{N\}$ and let $\Omega _1$ and $\Omega _2$ be bounded $A^k$-domains of holomorphy in $\mathbb \{C\}^\{m_1\}$ and $\mathbb \{C\}^\{m_2\}$ respectively. Let $g_1\in \{\mathcal \{A\}\}_D^k(\Omega _1)$, $g_2\in \{\mathcal \{A\}\}_I^k(\Omega _1)$ and $h\in \{\mathcal \{A\}\}_D^k(\Omega _2)\cap \{\mathcal \{A\}\}_I^k(\Omega _2)$. We prove that the domains $\Omega =\left\rbrace (z,w)\in \Omega _1\times \Omega _2\: g_1(z)<h(w)<g_2(z)\right\lbrace $ are $A^k$-domains of holomorphy if $\mathop \{\mathrm \{i\}nt\}\overline\{\Omega \}=\Omega $. We also prove that under certain assumptions they have a Stein neighbourhood basis and are convex with respect to the class of $A^k$-functions. If these domains in addition have $C^1$-boundary, then we prove that the $A^k$-corona problem can be solved. Furthermore we prove two general theorems concerning the projection on $\{\mathbb \{C\}\}^n$ of the spectrum of the algebra $A^k$.},
author = {Backlund, Ulf, Fällström, Anders},
journal = {Mathematica Bohemica},
keywords = {$A^k$-domains of holomorphy; $A^k$-convexity; -domains of holomorphy; -convexity},
language = {eng},
number = {1},
pages = {49-61},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the algebra of $A^k$-functions},
url = {http://eudml.org/doc/249919},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Backlund, Ulf
AU - Fällström, Anders
TI - On the algebra of $A^k$-functions
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 1
SP - 49
EP - 61
AB - For a domain $\Omega \subset {\mathbb {C}}^n$ let $H(\Omega )$ be the holomorphic functions on $\Omega $ and for any $k\in \mathbb {N}$ let $A^k(\Omega )=H(\Omega )\cap C^k(\overline{\Omega })$. Denote by ${\mathcal {A}}_D^k(\Omega )$ the set of functions $f\: \Omega \rightarrow [0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k(\Omega )$ such that $\lbrace |f_j|\rbrace $ is a nonincreasing sequence and such that $ f(z)=\lim _{j\rightarrow \infty }|f_j(z)|$. By ${\mathcal {A}}_I^k(\Omega )$ denote the set of functions $f\: \Omega \rightarrow (0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k(\Omega )$ such that $\lbrace |f_j|\rbrace $ is a nondecreasing sequence and such that $ f(z)=\lim _{j\rightarrow \infty }|f_j(z)|$. Let $k\in \mathbb {N}$ and let $\Omega _1$ and $\Omega _2$ be bounded $A^k$-domains of holomorphy in $\mathbb {C}^{m_1}$ and $\mathbb {C}^{m_2}$ respectively. Let $g_1\in {\mathcal {A}}_D^k(\Omega _1)$, $g_2\in {\mathcal {A}}_I^k(\Omega _1)$ and $h\in {\mathcal {A}}_D^k(\Omega _2)\cap {\mathcal {A}}_I^k(\Omega _2)$. We prove that the domains $\Omega =\left\rbrace (z,w)\in \Omega _1\times \Omega _2\: g_1(z)<h(w)<g_2(z)\right\lbrace $ are $A^k$-domains of holomorphy if $\mathop {\mathrm {i}nt}\overline{\Omega }=\Omega $. We also prove that under certain assumptions they have a Stein neighbourhood basis and are convex with respect to the class of $A^k$-functions. If these domains in addition have $C^1$-boundary, then we prove that the $A^k$-corona problem can be solved. Furthermore we prove two general theorems concerning the projection on ${\mathbb {C}}^n$ of the spectrum of the algebra $A^k$.
LA - eng
KW - $A^k$-domains of holomorphy; $A^k$-convexity; -domains of holomorphy; -convexity
UR - http://eudml.org/doc/249919
ER -