# zbMATH — the first resource for mathematics

Infinitely generated ideals in $$H^ \infty$$ on a class of infinitely connected domains. (Chinese. English summary) Zbl 0843.46039
The following result is developed on $$C$$-domains, which are Denjoy domains satisfying certain conditions.
Theorem 1. Let $$D$$ be a $$C$$-domain. If $$g, f_1,f_2, f_3,\dots\in H^\infty(D)$$ such that $\sup_{z\in D} (\Sigma |f_i(z)|^2)^{1/2}< \infty\quad\text{and} \quad |g(z)|^2\leq \Sigma |f_i(z)|^2,$ then there exist $$g_1, g_2, g_3,\dots\in H^\infty(D)$$ such that $$g^3= \Sigma f_i g_i$$.
Theorem 2. Let $$D$$ be a $$C$$-domain. For each $$i$$, $$F_i= E_i\cup G_i$$ is a bounded simply connected closed set in the plane, where $$G_i$$ is a connected subset or a union of two connected subsets in $$D$$. Let $$A= D- \cup F_i$$. The statement in Theorem 1 is still true if $$D$$ is replaced by $$A$$.
Theorem 3. Let $$D$$ be an $$L$$-domain obtained by deleting from the punctured disc a disjoint sequence of closed discs $$C_n= \{|z- x_n|\leq r_n\}$$ centered on the positive $$x$$-axis and accumulating only at the origin, i.e. $$D= \{0< |z|< 1\}- \cup C_n$$. Further, assume $$d_n\leq {r_n(x_n- r_n)\over Ax_n+ (A+ 1)r_n}$$ for sufficiently large $$n$$, where $$A$$ is a constant and $$d_n$$ denotes the distance between $$C_n$$ and $$C_{n+ 1}$$. The statement in Theorem 1 is still true for this kind of $$L$$-domain $$D$$.

##### MSC:
 46J20 Ideals, maximal ideals, boundaries 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 30D55 $$H^p$$-classes (MSC2000) 30H05 Spaces of bounded analytic functions of one complex variable
##### Keywords:
Hardy spaces; Carleson measure; Denjoy domains; $$L$$-domain