The moduli of holomorphic functions in Lipschitz spaces
Let \(\Lambda^\alpha_A\), \(0<\alpha<1\), denote the class of bounded functions \(f\) on \(\mathbb{R}\) satisfying the Lipschitz condition of order \(\alpha\) and such that the Poisson integral of \(f\) is holomorphic in the upper half-plane. The problem of description of the class \(\{|f|: f\in\Lambda^\alpha_A\}\) is treated. Trivially, the conditions \(\varphi\in \Lambda^\alpha\) and \(\int_\mathbb{R} (\log\varphi) (1+t^2)^{-1} dt>-\infty\) are necessary for \(\varphi\) to belong to this class. However, they are known to be insufficient. The author proves that a set of necessary and sufficient conditions is obtained if we supplement the above two ones by any of the following (the symbol \(\Phi\) denotes the outer function with modulus \(\varphi)\): (a) \(|\Phi|\) is of class \(\Lambda^\alpha\) in the upper half-plane \(\mathbb{C}_+\); (b) for \(z=x +iy\in\mathbb{C}_+\), we have \(\varphi(x)- |\Phi(z) |=O(y^\alpha)\); (c) for \(z=x+i y\in \mathbb{C} +\), we have \(\int\varphi d\mu_z- \exp\int \log\varphi d\mu_z =O(y^\alpha)\), where \(\mu_z\) is the harmonic measure at \(z\) (the Poisson kernel). There is also a simpler criterion applicable if \(0<\alpha <1/2\). Another criterion for any nonintegral \(\alpha\), expressed in quite different terms, was obtained by N. A. Shirokov by quite a different method; see \{\textit{N. A. Shirokov}, Analytic functions smooth up to the boundary (1988; Zbl 0656.30029).
scientific article; zbMATH DE number 1046972