Infinitely generated ideals in \(H^ \infty\) on a class of infinitely connected domains.

*(Chinese. English summary)*Zbl 0843.46039The 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 \(D\).

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 \(D\).

Reviewer: Chow Kwan-nan (Northridge)