Normal limits and star-invariant subspaces of bounded mean oscillation in multiply connected domains (Q2716494)

\textit{W. S. Cohn} [Am. J. Math. 108, 719-749 (1986; Zbl 0607.30034)] proved the following theorem: Let \(\varphi\) be an inner function in the unit disk \(U\) with zero set \(\{a_k\}\), Blaschke product \(B\) and singular inner part \(I_\sigma\) such that \(\varphi= BI_\sigma\). Let further be \(H^2\) the usual Hardy class, \(K_2= H^2\ominus\varphi H^2\) the star invariant subspace generated by \(\varphi\) and \(K_*= K_2\cap \text{BMOA}\). Then a necessary and sufficient condition for the existence of \(\lim_{r\to 1} f(ru_0)\) for all \(f\in K_*\), \(|u_0|= 1\), is NEWLINE\[NEWLINE\sum {1-|a_k|\over |u_0- a_k|}+ \int_{\partial U} {d\sigma(u)\over|u_0- u|}< \infty.NEWLINE\]NEWLINE In the present paper the author proves an analoguous necessary and sufficient condition for the existence of boundary values in the case where \(U\) is replaced by a plane domain bounded by \(n+1\) analytic Jordan curves and the definitions used above are extended appropriately.