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