The range of Hardy numbers for comb domains (Q2100148)

The author states a relation between comb domains and their Hardy numbers. In case \(D\neq\mathbb C\) is a simply connected domain and \(f\) is a Riemann mapping from \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) onto \(D\), the Hardy number \(h(D)\) of \(D\) is defined by \(h(D)=\sup\{p>0:f\in H^p(\mathbb D)\}\), where \(H^p(\mathbb D)\) is the Hardy space, \(p>0\). Let \(\{x_n\}_{n\in\mathbb Z}\) be a strictly increasing sequence of real numbers, \(x_0=0\), \(\inf_{n\in\mathbb Z}(x_n-x_{n-1})>0\), and let \(\{b_n\}_{n\in\mathbb Z}\) be an associated sequence of positive numbers. A comb domain \(C\) is of the form \[ C=\mathbb C\setminus\bigcup_{n\in\mathbb Z}\{x_n+iy:|y|\geq b_n\}. \] The main result is given in the following theorem. Theorem 1.1. If \(C\) is a comb domain, then \(h(C)\geq1\). Moreover, for any \(p\in[1,+\infty)\), there is a comb domain \(C\) with \(h(C)=p\).