On the algebra of Ak-functions
Ulf Bäcklund, Anders Fällström · 2006
Abstract. For a domain Ω ⊂ n let H(Ω) be the holomorphic functions on Ω and for any k ∈ let Ak(Ω) = H(Ω) ∩ Ck(Ω). Denote by A kD(Ω) the set of functions f: Ω → [0,∞) with the property that there exists a sequence of functions fj ∈ Ak(Ω) such that {|fj |} is a nonincreasing sequence and such that f(z) = lim j→∞ |fj(z)|. By A kI (Ω) denote the set of functions f: Ω → (0,∞) with the property that there exists a sequence of functions fj ∈ Ak(Ω) such that {|fj |} is a nondecreasing sequence and such that f(z) = lim j→∞ |fj (z)|. Let k ∈ and let Ω1 and Ω2 be bounded Ak-domains of holomorphy in m1 and m2 respectively. Let g1 ∈ A kD(Ω1), g2 ∈ A kI (Ω1) and h ∈ A kD(Ω2)∩A kI (Ω2). We prove that the domains Ω = {(z, w) ∈ Ω1 × Ω2: g1(z) < h(w) < g2(z)} are Ak-domains of holomorphy if intΩ = Ω. We also prove that under certain assumptions they have a Stein neighbourhood basis and are convex with respect to the class of Ak-functions. If these domains in addition have C1-boundary, then we prove that the Ak-corona problem can be solved. Furthermore we prove two general theorems concerning the projection on n of the spectrum of the algebra Ak.