# zbMATH — the first resource for mathematics

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)
##### Keywords:
inner function; Blaschke product
Full Text: