Keldysh-Lavrent'ev example

An example of a simply-connected domain $\Delta$ in the complex $z$- plane bounded by a rectifiable Jordan curve but not belonging to the class $S$ of Smirnov domains (cf. Smirnov domain).

Let $z = f ( w )$ be a function realizing a conformal mapping of the unit disc $E = \{ {w } : {| w | < 1 } \}$ onto a simply-connected domain $D$ bounded by a rectifiable Jordan curve. It is known that $f ( w)$ is continuous and one-to-one in the closed disc $\overline{E}\;$ and that the logarithm of the modulus of the derivative $\mathop{\rm ln} | f ^ { \prime } ( w) |$ can be represented in $E$ by the Poisson–Stieltjes integral

$$\tag{* } \mathop{\rm ln} | f ^ { \prime } ( \rho e ^ {i \phi } ) | = \int\limits \frac{1 - \rho ^ {2} }{1 + \rho ^ {2} - 2 \rho \cos ( \phi - \theta ) } \ d \mu ( \theta ) ,$$

where $\mu$ is the normalized Borel measure on $\partial E$, $\int d \mu ( \theta ) = 1$. The class $S$ consists of those closed domains $D$ for which the measure $\mu$ in the representation (*) is absolutely continuous with respect to the Lebesgue measure on $\partial E$ and the integral (*) becomes the Poisson–Lebesgue integral (cf. Poisson integral) of the boundary values $\mathop{\rm ln} | f ^ { \prime } ( e ^ {i \theta } ) |$, which exist almost-everywhere on $E$.

M.V. Keldysh and M.A. Lavrent'ev [1] constructed for any $h$, $0 < h < 1$, a simply-connected domain $\Delta$ bounded by a rectifiable Jordan curve $\Gamma$, located in the disc $| z | < h$, $0 \in \Delta$, such that under a conformal mapping of $\Delta$ onto $E$,

$$z = 0 \leftrightarrow w = 0 ,$$

and the image of any arc $\Gamma$ on the circle $\partial E = \{ {w } : {| w | = 1 } \}$ is an arc of the same length. This domain $\Delta$ does not belong to the class $S$, since $\mathop{\rm ln} | f ^ { \prime } ( e ^ {i \theta } ) | = 0$ almost-everywhere on $\partial E$.

A complete solution of the problem of characterizing domains of class $S$( domains of Smirnov type) has so far (1989) not been obtained (see [2], [3]).

References

[1]	M.V. Keldysh, M.A. Lavrent'ev, "Sur la répresentation conforme des domains limités par des courbes rectifiables" Ann. Ecole Norm. Sup. , 54 (1937) pp. 1–38
[2]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[3]	A.J. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)

Comments

The construction of Keldysh and Lavrent'ev is extraordinarily complicated. For a more accessible treatment, see [a1] and [a2].

References

[a1]	P.L. Duren, H.S. Shapiro, A.L. Shields, "Singular measures and domains not of Smirnov type" Duke Math. J. , 33 (1966) pp. 247–254
[a2]	G. Piranian, "Two monotonic, singular, uniformly almost smooth functions" Duke Math. J. , 33 (1966) pp. 254–262