Analytic functions smooth up to the boundary
The principal theme of the book is the inner-outer factorization in classes of functions analytic in the open unit disc \({\mathbb{D}}\) and ``smoothly'' extendable to its boundary \({\mathbb{T}}\). It is a detailed account of the impressive progress achieved by the author in the investigation of the multiplicative structure of the above classes. In what follows \(I_ f\) will denote the inner part of a function \(f\in H^ 1({\mathbb{D}})\) (the Hardy class). A set \(X\subset H^ 1({\mathbb{D}})\) is said to possess the F-property (F-p.) if \(I^{-1}\cdot f\in X\) whenever \(f\in X\) and I is an inner function dividing \(I_ f\). It is known that many natural classes X possess the F-p. (e.g. \(\{f\in H^ 1:\) \(f^{(n)}\in H^ p\}\) (n\(\geq 0\), \(p\in (1,+\infty))\), \(\{f\in H^ 1:\) \(f\in Lip_{\alpha}({\mathbb{D}})\}\) \((0<\alpha <1))\). One of the main results by the author presented in the first chapter states that for every continuity modulus \(\omega\) and \(n\in {\mathbb{Z}}_+\) the class \[ \{f\in H^ 1:\quad \forall z,\zeta \in {\mathbb{D}},\quad | f^{(n)}(z)- f^{(n)}(\zeta)| \leq C_ f\omega (| z-\zeta |)\} \] possesses the F-p.. Classes with ``varying boundary smoothness'' are also considered. Some new classes not possessing the F-p. are exhibited. The second chapter treats the problem of description of moduli \(| f| /{\mathbb{T}}\) for f's belonging to a ``smooth'' class \(X\subset H^ 1\). General results obtained by the author imply quite a few previously known and new facts and, in particular, the famous (though never published) Carleson-Jacobs theorem (f outer, \(| f| /{\mathbb{T}}\in C^{\alpha}({\mathbb{T}})\Rightarrow f| {\mathbb{D}}\in C^{\alpha /2}({\mathbb{D}}\cup {\mathbb{T}})\) for an arbitrary positive \(\alpha\) with the usual Zygmund-like interpretation of \(C^{\alpha}\) for an integer \(\alpha)\). The third chapter is devoted to the description of zero sets. It contains the following remarkable result. Let \(\omega\) be an arbitrary continuity modulus, E a closed set in \({\mathbb{D}}\cup {\mathbb{T}}\) satisfying \[ \int_{{\mathbb{T}}}\log \omega (dist(z,E))| dz| >-\infty \] and the Blaschke condition \(\sum_{z\in E}(1-| z|)<+\infty\). Let I be an inner function with spec \(I\subset E\). Then there exists a function \(f\in H^ 1\cap Lip_{\omega}({\mathbb{D}})\) such that \(f^{-1}(\{0\})=E\), \(I_ f=I\). This theorem is relatively simple for ``good'' \(\omega\) 's, but the point is that no extra conditions are imposed on \(\omega\). The third chapter contains among other things a rather complete characterization of zeros of \(H^{\infty}\)-functions with rare Taylor coefficients. This characterization involves parameters occuring in the inner-outer factorization. The book is concluded by the fourth chapter yielding a description of closed ideals for a wide class of algebras of ``smooth'' analtic functions.
scientific article; zbMATH DE number 193001