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Radial limits and star invariant subspaces of bounded mean oscillation. (English) Zbl 0607.30034
Let D denote the unit disk in the complex plane and \(H^ p\) the usual classes of analytic functions on D. If \(\psi\) is an inner function on D, \(0<p<\infty\), \(K_ p\) denotes the subspace of \(H^ p\) defined by: \[ K_ p=\psi \bar H^ p_ 0\cap H^ p, \] where \(\bar H^ p_ 0=\{e^{-i\theta}\overline{f(e^{i\theta})}:\) \(f\in H^ p\}\). A function \(f\in H^ p\) belongs to \(K_ p\) if and only if \[ f(e^{i\theta})=\psi (e^{i\theta})h(e^{i\theta})e^{-i\theta}\quad [a.e.] \] for some \(h\in H^ p\). In the case \(p=2\), \(K_ 2=(\psi H^ 2)^{\bot}\). Finally, let \(K_*=K_ 2\cap BMOA\). Since \(\psi\) is inner, \(\psi\) can be written as \(\psi =\lambda BS_{\sigma}\) where \(\lambda\in {\mathbb{C}}\), \(| \lambda | =1\), B is a Blaschke product whose zeroes \(\{a_ k\}\) satisfy \(\sum (1-| a_ k|)<\infty\), and \(S_{\sigma}\) is a singular inner function for the positive singular measure \(\sigma\). In the paper, the author proves the following: Let \(\xi_ 0\in T\) and suppose \(\psi =BS_{\sigma}\). Then a necessary and sufficient condition that \((*)\quad \lim_{r\to 1^-}f(r\xi_ 0)=L\) exist for all \(f\in K_*\) is that \[ \sum (1-| a_ k|)/| \xi_ 0-a_ k| +\int_{T}d\sigma(\xi)/| \xi_ 0-\xi | <\infty. \] Furthermore, it is shown that if \(1<p<\infty\), (*) holds for all \(f\in K_ p\) if and only if \[ \sum (1-| a_ k|)/| \xi_ 0-a_ k|^ q+\int_{T}d\sigma (\xi)/| \xi_ 0-\xi |^ q<\infty, \] where q is the conjugate exponent of p.
Reviewer: M.Stoll

MSC:
30D55 \(H^p\)-classes (MSC2000)
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