Smirnov domain

domain of type $C$, domain of type $S$

A bounded simply-connected domain $G$ with a rectifiable Jordan boundary in the complex plane $\mathbf C$ having the following property: there is a univalent conformal mapping $z=\phi(w)$ from the disc $|w|<1$ onto $G$ such that for $|w|<1$ the harmonic function $\ln|\phi'(w)|$ can be written a the Poisson integral of its non-tangential boundary values $\ln|\phi'(e^{i\theta})|$:

$$\ln|\phi'(re^{i\theta})|=\frac{1}{2\pi}\int\limits_0^{2\pi}\frac{1-r^2}{1+r^2-2r\cos(t-\theta)}\ln|\phi'(e^{it})|dt.$$

These domains were introduced by V.I. Smirnov [1] in 1928 in the course of investigating the completeness of a system of polynomials in the Smirnov class $E_2(G)$. The problem of the existence of non-Smirnov domains with rectifiable Jordan boundaries was solved by M.V. Keldysh and M.A. Lavrent'ev [2], who gave a sophisticated and intricate construction of such domains and of the corresponding mapping functions $\phi$, with the additional property that $|\phi'(e^{i\theta})|=1$ for almost-all $e^{i\theta}$. The basic boundary properties of analytic functions in the disc also hold for functions analytic in a Smirnov domain, and many of these properties hold only in Smirnov domains. Examples of Smirnov domains are Jordan domains whose boundaries are Lyapunov curves or piecewise Lyapunov curves with non-zero angles (cf. Lyapunov surfaces and curves).

References

[1]	V.I. Smirnov, "Sur la théorie des polynomes orthogonaux à une variable complexe" Zh. Leningrad. Fiz.-Mat. Obshch. , 2 : 1 (1928) pp. 155–179
[2]	M.V. Keldysh, M.A. Lavrent'ev, "Sur la répresentation conforme des domaines limités" Ann. Sci. École Normale Sup. , 54 (1937) pp. 1–38
[3]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4]	A.J. Lohwater, "The boundary behavior of analytic functions" Itogi Nauk. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)
[5]	G.Ts. Tumarkin, "A sufficient condition for a domain to belong to class $S$" Vestnik. Leningrad. Univ. , 17 : 13 (1962) pp. 47–55 (In Russian) (English summary)

Comments

The German translation of Privalov's book is the most detailed Western reference on Smirnov classes and domains. A reference in English is [a1].

References

[a1]	P.L. Duren, "Theory of $H^p$ spaces" , Acad. Press (1970)