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 $f \Omega\to[0,\infty)$ with the property that there exists a sequence of functions $f_j\in A^k(\Omega)$ such that $\{|f_j|\}$ is a nonincreasing sequence and such that $ f(z)=\lim_{j\to\infty}|f_j(z)|$. By ${\mathcal A}_I^k(\Omega)$ denote the set of functions $f \Omega\to(0,\infty)$ with the property that there exists a sequence of functions $f_j\in A^k(\Omega)$ such that $\{|f_j|\}$ is a nondecreasing sequence and such that $ f(z)=\lim_{j\to\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 $\Omega=\left\{(z,w)\in\Omega_1\times\Omega_2 g_1(z)<h(w)<g_2(z)\right\}$ are $A^k$-domains of holomorphy if $\Int\overline\Omega=\Omega$. 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 $A^k$.

Keywords: $A^k$-domains of holomorphy, $A^k$-convexity

Classification (MSC2000): 32A38

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