Extension of domain, principle of

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)