Internal-external factorization in Lumer's Hardy spaces
Assume M is a simply connected analytic manifold in \({\mathbb{C}}^ n\). The Lumer-Nevanlinna class L-N consists of those f holomorphic on M for which there is \(u=Re F\), F holomorphic on M, and \(\log^+| f(Q)| \leq u(Q)\) all \(Q\in M\). Similarly the Lumer-Hardy class \((LH)^ p\) consists of f holomorhic on M for which there exists \(u=Re F\), F holomorphic on M and \(| f(Q)|^ p\leq u(Q)\) all \(Q\in M\). The author gives a natural definition of internal and external functions in \((LH)^ p\). He then shows two interesting results. Theorem. Each function f in \((LH)^ p\) admits internal-external factorization (i.e. \(f=I\times E\), where I is internal and E is external). A natural question arises. Is the external E in this factorization also in \((LH)^ p?\) A negative answer is given by Walter Rudin and his counterexample is included in the paper. However, the author shows that the following is valid: Proposition. For \(f\in(LH)^ p\) and \(p>\epsilon>0\), there exists an internal \(I_{\epsilon}\) and an external \(E_{\epsilon}\) such that \(f=I_{\epsilon}\times E_{\epsilon}\) and \(E_{\epsilon}\in(LH)^{p-\epsilon}.\)
scientific article; zbMATH DE number 3857474
scientific article