Absolute values of BMOA functions
Let \(D\) denote the unit disk and let \(T=\partial D\). The space BMOA consists of all functions \(f\in H^1(D)\) satisfying \[ \|f\|_*= \sup_{z\in D} \int_T|f(\zeta)- f(z)|{1-|z|^2\over|\zeta- z|^2} dm(\zeta)< \infty, \] wehre \(dm(\zeta)\) denotes the element of linear measure on \(T\). A function \(f\) analytic in \(D\) is called a Bloch function if \(\sup_{z\in D}|f'(z)|(1-|z|^2)< \infty\). It is known that each function in BMOA is a Bloch function. The author is concerned with the following problem: if \(\phi\geq 0\) and \(\phi\in L^1(T)\), when does there exist a function \(f\in\text{BMOA}\) such that \(\phi=|f|\). An equivalent formulation of this question is to ask when the outer function \[ {\mathcal O}_\phi(z)= \exp\Biggl\{ \int_T{\zeta+ z\over\zeta- z}\log\phi(\zeta) dm(\zeta)\Biggr\} \] is in BMOA. Clearly, a necessary condition is that \(\phi\in \text{BMO}\), that is, \[ \sup_{z\in D} \int_T|\phi(\zeta)- \phi(z)|{1-|z|^2\over|\zeta- z|^2} dm(\zeta)< \infty. \] The author proves that if \(\phi\geq 0\), \(\phi\in \text{BMO}\), and \(\int_T\log\phi(\zeta) dm(\zeta)> -\infty\), then the condition \({\mathcal O}_\phi\in\text{BMOA}\) is equivalent to the condition that for some \(M>0\), \[ \sup_{\phi(z)\geq M} \Biggl\{(\phi(z))^2 \int_T \log^-\phi(\zeta) {1-|z|^2\over|\zeta- z|^2} dm(\zeta)\Biggr\}< \infty, \] where \(\log^-\phi(\zeta)= \max\{-\log\phi(\zeta), 0\}\). It is further shown that if \(\phi\geq 0\), \(g\in \text{BMO}\), \[ \int_T \log\phi(\zeta) dm(\zeta)>- \infty, \] and, for some \(M>0\), \[ \sup_{\phi(z)\geq M} \Biggl\{\phi(z) \int_T \log^-\phi(\zeta) {1-|z|^2\over|\zeta- z|^2} dm(\zeta)\Biggr\}< \infty, \] then \({\mathcal O}_\phi\) is a Bloch function. However, it is also shown that there exists a function \(\phi\) with \(\phi\geq 0\), \(\phi\in \text{BMO}\), and \(\int_T\log \phi(\zeta) dm(\zeta)> -\infty\) such that \({\mathcal O}_\phi\not\in \text{BMOA}\). And, finally, it is shown that if \(f\in \text{BMOA}\) is an outer function with \(|f|^2\in \text{BMO}\), then \(f^2\) is a Bloch function.
scientific article; zbMATH DE number 1419785