View source for Keldysh-Lavrent'ev example

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]]).

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

<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>

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" />.

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" />,

<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>

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" />.

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]]]).

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Piranian,   "Two monotonic, singular, uniformly almost smooth functions"  ''Duke Math. J.'' , '''33'''  (1966)  pp. 254–262</TD></TR></table>