Univalent functions, Hardy spaces and spaces of Dirichlet type
This is an outstandingly fine study of the relationships between several classical classes of analytic functions. Let \(H^p\), \(0 < p \leq \infty\) be the Hardy space of functions \(f(z)\) that are analytic in the unit disk \(\Delta\) and that are such that the supremum of \(M_p(r,f) = ((1/2\pi)\int_{-\pi}^{\pi} | f(re^{it})| ^p \,dt)^{1/p}\) is finite (\(M_\infty(r,f) = \sup_{| z| =r}| f(z)| \)). Let \(A_\alpha^p\), \(\alpha > -1\), be the standard weighted Bergman space of those \(f(z)\), analytic in \(\Delta\) such that \(\int_\Delta (1-| z| )^\alpha| f(z)| ^p \,dA\) is finite. Let \(\mathcal D^p_{p-1}\) be the Dirichlet type space of those \(f\) whose derivative belongs to \(A^p_{p-1}\). And finally let \(\mathcal U\) be the class of \(f(z)\) that are analytic and univalent in \(\Delta\). There are many classical results about these classes, particularly those of Hardy and Littlewood. For example it is known that \(H^2 = \mathcal D_1^2\) and that \(H^p \subset \mathcal D_{p-1}^p\) for \(2 \leq p < \infty\). It is easily seen that \(H^p \supset \mathcal D_{p-1}^p\) for \(0 < p \leq 2\). It is also known that \(H^p \subset A^{2p}\) for all \(p\) and an easy argument about power series with Hadamard gaps is given in this paper to show \(H^p \neq \mathcal D_{p-1}^p\) for all \(p \neq 2\). The three main results of this paper are that \(\mathcal U \cap \mathcal D^p_{p-1} = \mathcal U \cap H^p\) for all \(p\); that if \(f \in \mathcal U\) then the conditions (i) \(f \in A^p\) and \(\int_0^1 \int_0^r M_\infty^p(\rho,f) \,d\rho \,dr < \infty\) are equivalent; and that if \(1/2 \leq p < \infty\) then there exists an \(f \in \mathcal U\) which is in \(A^{2p}\) but not in \(H^p\). The proofs are detailed and lengthy but appear to be quite accessible. Several portions of the proofs that I checked were correct and well written. I believe that this paper is a valuable contribution to the study of these classes and is worth careful study.
scientific article; zbMATH DE number 2140251