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Joint Mathematical seminar of Saint Petersburg State University and Peking University
June 3, 2021 15:00–16:00, St. Petersburg, online

Nevanlinna factorization in classes of analytic functions smooth up to the boundary

N. A. Shirokov

Abstract: The definition of all objects used in this announcement will be given in the talk. Let an analytical function $f$ belong to the Hardy class $H^p$ in the unit disc $\mathbb{D}$. Then f may be represented as the product, $f=IO_f$, where $I$ is the so-called inner function, it means that $|I(z)|<1$ for $z$ belonging $\mathbb{D}$, and $|I(z)|=1$ for almost every $z$ on the unit circle, and the so-called outer function $O_f$ is defined by values of $|f(z)|$ on the unit circle. One of those two factors may be absent. The cited statement is the classical result, the factors $I$ and $O_f$ are in a sense independent for the functions $f$ from $H^p$. Let us consider a class $X$ which is contained in $H^1$ and consists of functions $f$ continuous in the closed disc $\overline{\mathbb{D}}$. Then $f=IO_f$ ,the inner function $I$ is in general discontinuous in $\overline{\mathbb{D}}$ what implies that the outer function $O_f$ is to compensate the points of discontinuity of the inner function $I$. The talk is devoted to the concrete way of this compensation and to the specific properties of outer functions belonging to the analytical Holder classes and to the classes of functions of variable smoothness. The consequences about the half-smoothness of an analytical function in comparison with the smoothness of its modulus on the boundary will be given too.