Self-improving behaviour of inner functions as multipliers
A bounded analytic function \(\theta\) on the unit disk \(\mathbb D=\{z\in\mathbb{C}:| z| <1\}\) is said to be inner if \(\lim_{r\to1^{-}}| \theta(r\zeta)| =1\) for \(m\)-almost all \(\zeta\in\mathbb T\), where \(\mathbb T=\partial\mathbb{D}\) is the unit circle and \(m\) is the normalized arclength measure on \(\mathbb T\). Let \(X\) and \(Y\) be two classes of analytic functions on \(\mathbb{D}\) with \(X\subset Y\). Then \(\theta\) is called \((X,Y)\)-improving if every function \(f\in X\) satisfying \(f\theta\in Y\) must actually satisfy \(f\theta\in X\), that is, the implication \[ (f\in X,f\theta\in Y)\Longrightarrow f\theta\in X \] holds. Let \(\mathcal{A}\) stand for the disk algebra, i.e., the set of analytic functions on \(\mathbb D\) that admit a continuous extension to \(\mathbb D\cup\mathbb T\) and let the Lipschitz space consisting of those \(f\) which satisfy \[ | f(z_1)-f(z_2)| \leq C| z_1-z_2| ^{\alpha},\qquad 0<\alpha\leq 1,\quad \text{\(z_1\), \(z_2\in\mathbb D\)}, \] be denoted by \(A^{\alpha}\). Further, BMOA (respectively, VMOA) is the analytic subspace of \(\text{BMO}(\mathbb T)\) \(\bigl(\text{respectively, VMO(}\mathbb T\))
scientific article; zbMATH DE number 5083456
scientific article