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''Carleman's principle''
 
''Carleman's principle''
  
The [[Harmonic measure|harmonic measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370501.png" /> of an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370502.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370503.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370504.png" /> can only increase when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370505.png" /> is extended across arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370507.png" />. More precisely, let the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370508.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e0370509.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705010.png" />-plane consist of a finite number of Jordan curves, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705011.png" /> be a part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705012.png" /> consisting of a finite number of arcs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705013.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705014.png" /> be an extension of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705015.png" /> across the complementary arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705016.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705018.png" /> is a part of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705020.png" />. Then for the harmonic measures one has the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705022.png" />, where equality only holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705023.png" />. The principle of extension of domain also holds for harmonic measure on domains in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705025.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705027.png" />.
+
The [[Harmonic measure|harmonic measure]] $  \omega ( z, \alpha , D) $
 +
of an arc $  \alpha $
 +
of the boundary $  \Gamma $
 +
of a domain $  D $
 +
can only increase when $  D $
 +
is extended across arcs $  \beta \subset  \Gamma $,  
 +
$  \alpha \cup \beta = \Gamma $.  
 +
More precisely, let the boundary $  \Gamma $
 +
of a domain $  D $
 +
in the complex $  z $-
 +
plane consist of a finite number of Jordan curves, let $  \alpha $
 +
be a part of $  \Gamma $
 +
consisting of a finite number of arcs of $  \Gamma $,  
 +
and let $  D  ^  \prime  $
 +
be an extension of the domain $  D $
 +
across the complementary arcs $  \beta = \Gamma \setminus  \alpha $,  
 +
that is, $  D \subset  D  ^  \prime  $
 +
and $  \alpha $
 +
is a part of the boundary $  \Gamma  ^  \prime  $
 +
of $  D  ^  \prime  $.  
 +
Then for the harmonic measures one has the inequality $  \omega ( z, \alpha , D) \leq  \omega ( z, \alpha  , D  ^  \prime  ) $,  
 +
$  z \in D $,  
 +
where equality only holds if $  D  ^  \prime  = D $.  
 +
The principle of extension of domain also holds for harmonic measure on domains in the Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
or $  \mathbf C  ^ {n} $,  
 +
$  n \geq  1 $.
  
The principle of extension of domain finds important applications in various problems concerned with estimating harmonic measure. For example, T. Carleman [[#References|[1]]] used the principle of extension of domain to obtain the solution of the Carleman–Milloux problem: Let the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705028.png" /> of a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705029.png" /> consist of a finite number of Jordan arcs, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705030.png" /> be a point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705031.png" />, or let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705032.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705033.png" /> be the disc of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705034.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705035.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705036.png" /> be the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705038.png" />. It is required to find a lower bound for the harmonic measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705039.png" /> depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705042.png" />. The solution is given by
+
The principle of extension of domain finds important applications in various problems concerned with estimating harmonic measure. For example, T. Carleman [[#References|[1]]] used the principle of extension of domain to obtain the solution of the Carleman–Milloux problem: Let the boundary $  \Gamma $
 +
of a simply-connected domain $  D $
 +
consist of a finite number of Jordan arcs, let $  \zeta $
 +
be a point on $  \Gamma $,  
 +
or let $  \zeta \notin \overline{D}\; $,  
 +
let $  \Delta = \{ {z } : {| z - \zeta | < R } \} $
 +
be the disc of radius $  R $
 +
with centre $  \zeta $,  
 +
and let $  \alpha $
 +
be the part of $  \Gamma $
 +
in $  \Delta _ {R} = \Delta \cap D $.  
 +
It is required to find a lower bound for the harmonic measure $  \omega ( z, \alpha , \Delta _ {R} ) $
 +
depending only on $  R $
 +
and $  | z - \zeta | $,  
 +
$  z \in \Delta _ {R} $.  
 +
The solution is given by
  
<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/e/e037/e037050/e03705043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\omega ( z, \alpha , \Delta _ {R} )  \geq  \
 +
{
 +
\frac{2} \pi
 +
}  \mathop{\rm arc}  \mathop{\rm tan} \
 +
\left (
 +
{
 +
\frac{2}{\theta ( R) }
 +
}  \mathop{\rm ln} \
 +
{
 +
\frac{R}{| z - \zeta | }
 +
}
 +
\right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705044.png" /> is the sum of the lengths of arcs of the intersection
+
where $  R \theta ( R) $
 +
is the sum of the lengths of arcs of the intersection
  
<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/e/e037/e037050/e03705045.png" /></td> </tr></table>
+
$$
 +
\{ {z } : {| z - \zeta | = R } \}
 +
\cap D.
 +
$$
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037050/e03705046.png" />, it follows that
+
Since $  \theta ( R) \leq  2 \pi $,  
 +
it follows that
  
<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/e/e037/e037050/e03705047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\omega ( z, \alpha , \Delta _ {R} )  \geq  \
 +
{
 +
\frac{2} \pi
 +
}  \mathop{\rm arc}  \mathop{\rm tan} \
 +
\left ( {
 +
\frac{1} \pi
 +
}  \mathop{\rm ln} \
 +
{
 +
\frac{R}{| z - \zeta | }
 +
}
 +
\right ) ,\ \
 +
z \in \Delta _ {R} .
 +
$$
  
 
There exist generalizations of the Carleman–Milloux problem and refinements of formulas (1), (2) (see [[#References|[3]]]). The principle of extension of domain also allows one to prove the Lindelöf theorems (cf. [[Lindelöf theorem|Lindelöf theorem]]). Various applications of the principle of extension of domain and of formulas of the type (1), (2) were given by H. Milloux (see [[#References|[2]]], and also [[#References|[3]]], [[#References|[4]]]).
 
There exist generalizations of the Carleman–Milloux problem and refinements of formulas (1), (2) (see [[#References|[3]]]). The principle of extension of domain also allows one to prove the Lindelöf theorems (cf. [[Lindelöf theorem|Lindelöf theorem]]). Various applications of the principle of extension of domain and of formulas of the type (1), (2) were given by H. Milloux (see [[#References|[2]]], and also [[#References|[3]]], [[#References|[4]]]).

Latest revision as of 19:38, 5 June 2020


Carleman's principle

The harmonic measure $ \omega ( z, \alpha , D) $ of an arc $ \alpha $ of the boundary $ \Gamma $ of a domain $ D $ can only increase when $ D $ is extended across arcs $ \beta \subset \Gamma $, $ \alpha \cup \beta = \Gamma $. More precisely, let the boundary $ \Gamma $ of a domain $ D $ in the complex $ z $- plane consist of a finite number of Jordan curves, let $ \alpha $ be a part of $ \Gamma $ consisting of a finite number of arcs of $ \Gamma $, and let $ D ^ \prime $ be an extension of the domain $ D $ across the complementary arcs $ \beta = \Gamma \setminus \alpha $, that is, $ D \subset D ^ \prime $ and $ \alpha $ is a part of the boundary $ \Gamma ^ \prime $ of $ D ^ \prime $. Then for the harmonic measures one has the inequality $ \omega ( z, \alpha , D) \leq \omega ( z, \alpha , D ^ \prime ) $, $ z \in D $, where equality only holds if $ D ^ \prime = D $. The principle of extension of domain also holds for harmonic measure on domains in the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, or $ \mathbf C ^ {n} $, $ n \geq 1 $.

The principle of extension of domain finds important applications in various problems concerned with estimating harmonic measure. For example, T. Carleman [1] used the principle of extension of domain to obtain the solution of the Carleman–Milloux problem: Let the boundary $ \Gamma $ of a simply-connected domain $ D $ consist of a finite number of Jordan arcs, let $ \zeta $ be a point on $ \Gamma $, or let $ \zeta \notin \overline{D}\; $, let $ \Delta = \{ {z } : {| z - \zeta | < R } \} $ be the disc of radius $ R $ with centre $ \zeta $, and let $ \alpha $ be the part of $ \Gamma $ in $ \Delta _ {R} = \Delta \cap D $. It is required to find a lower bound for the harmonic measure $ \omega ( z, \alpha , \Delta _ {R} ) $ depending only on $ R $ and $ | z - \zeta | $, $ z \in \Delta _ {R} $. The solution is given by

$$ \tag{1 } \omega ( z, \alpha , \Delta _ {R} ) \geq \ { \frac{2} \pi } \mathop{\rm arc} \mathop{\rm tan} \ \left ( { \frac{2}{\theta ( R) } } \mathop{\rm ln} \ { \frac{R}{| z - \zeta | } } \right ) , $$

where $ R \theta ( R) $ is the sum of the lengths of arcs of the intersection

$$ \{ {z } : {| z - \zeta | = R } \} \cap D. $$

Since $ \theta ( R) \leq 2 \pi $, it follows that

$$ \tag{2 } \omega ( z, \alpha , \Delta _ {R} ) \geq \ { \frac{2} \pi } \mathop{\rm arc} \mathop{\rm tan} \ \left ( { \frac{1} \pi } \mathop{\rm ln} \ { \frac{R}{| z - \zeta | } } \right ) ,\ \ z \in \Delta _ {R} . $$

There exist generalizations of the Carleman–Milloux problem and refinements of formulas (1), (2) (see [3]). The principle of extension of domain also allows one to prove the Lindelöf theorems (cf. Lindelöf theorem). Various applications of the principle of extension of domain and of formulas of the type (1), (2) were given by H. Milloux (see [2], and also [3], [4]).

References

[1] T. Carleman, "Sur les fonctions inverses des fonctions entières" Ark. Mat. Ast. Fys. , 15 : 10 (1921)
[2] H. Milloux, "Le théorème de M. Picard, suites des fonctions holomorphes, fonctions méromorphes et fonctions entières" J. Math. Pures Appl. , 3 (1924)
[3] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[4] M.A. Evgrafov, "Analytic functions" , Saunders , Philadelphia (1966) (Translated from Russian)
How to Cite This Entry:
Extension of domain, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_domain,_principle_of&oldid=15054
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article