# Smirnov class

The set $E _ {p} ( G)$ of all functions $f( z)$ holomorphic in a simply-connected domain $G \subset \mathbf C$ with rectifiable Jordan boundary $\Gamma$, such that for every function in it there is a sequence of closed rectifiable Jordan curves $\Gamma _ {n} ( f ) \subset G$, $n = 1, 2 \dots$ with the following properties:

1) $\Gamma _ {n} ( f )$ tends to $\Gamma$ as $n \rightarrow \infty$ in the sense that if $G _ {n} ( f )$ is the bounded domain with boundary $\Gamma _ {n} ( f )$, then

$$G _ {1} ( f ) \subset \dots \subset G _ {n} ( f ) \subset G \ \ \textrm{ and } \ \cup _ {n= 1 } ^ \infty G _ {n} ( f ) = G;$$

2)

$$\sup _ { n } \left \{ \int\limits _ {\Gamma _ {n} ( f ) } | f( z) | ^ {p} | dz | \right \} \langle \infty \ ( p\rangle 0 \ \textrm{ fixed } ).$$

This definition was proposed by M.V. Keldysh and M.A. Lavrent'ev , and is equivalent to V.I. Smirnov's definition  in which curves $\gamma ( \rho )$ are used instead of $\Gamma _ {n} ( f )$. These curves are the images of the circles $| w | = \rho < 1$ under some univalent conformal mapping $z= \phi ( w)$ from the disc $| w | < 1$ onto the domain $G$, and the supremum is taken over all $\rho \in ( 0, 1)$.

The classes $E _ {p} ( G)$ are the best known and most thoroughly studied generalization of the Hardy classes $H _ {p}$, and are connected with them by the following relation: $f \in E _ {p} ( G)$ if and only if

$$f( \phi ( w))( \phi ^ \prime ( w)) ^ {1/p} \in H _ {p} .$$

The properties of the classes $E _ {p} ( G)$ are closest to those of $H _ {p}$ in the case when $G$ is a Smirnov domain. They have been generalized to domains $G$ with boundaries of finite Hausdorff length. See also Boundary properties of analytic functions.

How to Cite This Entry:
Smirnov class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smirnov_class&oldid=48738
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article