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An example of a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551301.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551302.png" />-plane bounded by a rectifiable Jordan curve but not belonging to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551303.png" /> of Smirnov domains (cf. [[Smirnov domain|Smirnov domain]]).
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551304.png" /> be a function realizing a [[Conformal mapping|conformal mapping]] of the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551305.png" /> onto a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551306.png" /> bounded by a rectifiable Jordan curve. It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551307.png" /> is continuous and one-to-one in the closed disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551308.png" /> and that the logarithm of the modulus of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k0551309.png" /> can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513010.png" /> by the Poisson–Stieltjes integral
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
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|Smirnov domain]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513012.png" /> is the normalized Borel measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513014.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513015.png" /> consists of those closed domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513016.png" /> for which the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513017.png" /> in the representation (*) is absolutely continuous with respect to the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513018.png" /> and the integral (*) becomes the Poisson–Lebesgue integral (cf. [[Poisson integral|Poisson integral]]) of the boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513019.png" />, which exist almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513020.png" />.
+
Let  $  z = f ( w ) $
 +
be a function realizing a [[Conformal mapping|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
  
M.V. Keldysh and M.A. Lavrent'ev [[#References|[1]]] constructed for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513022.png" />, a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513023.png" /> bounded by a rectifiable Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513024.png" />, located in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513026.png" />, such that under a conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513027.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513028.png" />,
+
$$ \tag{* }
 +
\mathop{\rm ln}  | f ^ { \prime } ( \rho e ^ {i \phi } ) |  = \int\limits
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513029.png" /></td> </tr></table>
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\frac{1 - \rho  ^ {2} }{1 + \rho  ^ {2} - 2 \rho  \cos  ( \phi - \theta ) }
 +
\
 +
d \mu ( \theta ) ,
 +
$$
  
and the image of any arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513030.png" /> on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513031.png" /> is an arc of the same length. This domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513032.png" /> does not belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513033.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513034.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513035.png" />.
+
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|Poisson integral]]) of the boundary values  $  \mathop{\rm ln}  | f ^ { \prime } ( e ^ {i \theta } ) | $,  
 +
which exist almost-everywhere on $  E $.
  
A complete solution of the problem of characterizing domains of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055130/k05513036.png" /> (domains of Smirnov type) has so far (1989) not been obtained (see [[#References|[2]]], [[#References|[3]]]).
+
M.V. Keldysh and M.A. Lavrent'ev [[#References|[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 [[#References|[2]]], [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.J. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.J. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:14, 5 June 2020


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
How to Cite This Entry:
Keldysh-Lavrent'ev example. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Keldysh-Lavrent%27ev_example&oldid=16860
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article