Embedding theorems for star-invariant subspaces generated by smooth inner functions
Let \(\mathbb D\) denote the unit disc in the complex plane, \(\mathbb T\) its boundary, and \(m\) the normalized arclength measure on \(\mathbb T\). Let \(H^p\) \((0<p\leq \infty) \) stand for the usual Hardy space of holomorphic functions on \(\mathbb D\). One also treat \(H^p\) as a subspace of \(L^p(\mathbb T,m)\). Suppose \(\theta\) is an inner function (i.e., \(\theta\) is a bounded holomorphic function with \(| \theta| =1\) a.e. on \(\mathbb T\). For \(1\leq p\leq \infty\), set \(K_\theta^p :=H^p\cap \theta \bar z \bar{H^p}\): a star-invariant subspace of \(H^p\). For \(0<\alpha<1\), one says that an inner function is \(\alpha\)-regular if \(\theta' \in N\): the Nevanlinna class in \(\mathbb D\) and if there is \(C>0\) such that \(\int_{\mathbb T} | \theta'(\xi)| ^\alpha| \xi-z| ^{-1}dm(\xi)\leq\) \(C/(1-| z|)^\alpha\). The author treats embedding theorems of the form \(K_\theta^p \subset L^q(\mu)\), where \(\mu\) is a measure on \(\mathbb D\). In particular, assuming that \(\theta\) is \(\alpha\)-regular, he characterizes the \(\mu\)'s such that every \(f\in K_\theta^1\) satisfies \(\big(\int_{\mathbb D}| f| ^qd\mu\big)^{1/q}\leq\) \(C\int_{\mathbb T}| f| | \theta'| ^\alpha| dz| \).
scientific article; zbMATH DE number 1211645
scientific article