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 [2], and is equivalent to V.I. Smirnov's definition [1] 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.

References

[1]	V.I. Smirnov, "Sur les formules de Cauchy et de Green et quelques problèmes qui s'y rattachent" Izv. Akad. Nauk SSSR. Otdel. Mat. i Estestv. Nauk , 3 (1932) pp. 337–372
[2]	M.V. Keldysh, M.A. Lavrent'ev, "Sur la répresentation conforme des domaines limités par des courbes rectifiables" Ann. Sci. Ecole Norm. Sup. , 54 (1937) pp. 1–38
[3]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[5]	P.L. Duren, "Theory of spaces" , Acad. Press (1970)