Smirnov class

The set of all functions holomorphic in a simply-connected domain with rectifiable Jordan boundary , such that for every function in it there is a sequence of closed rectifiable Jordan curves , with the following properties:

1) tends to as in the sense that if is the bounded domain with boundary , then

2)

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 are used instead of . These curves are the images of the circles under some univalent conformal mapping from the disc onto the domain , and the supremum is taken over all .

The classes are the best known and most thoroughly studied generalization of the Hardy classes , and are connected with them by the following relation: if and only if

The properties of the classes are closest to those of in the case when is a Smirnov domain. They have been generalized to domains 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)