On the algebra of $A^k$-functions

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 int}\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 $A^k$.