Conformal mapping, boundary properties of a

Properties of functions mapping one domain in the complex plane conformally onto another that show up near the boundary of the mapped domain and on the boundary itself. Among such properties are: the possibility of continuously extending a function $ w = f ( z) $ mapping a given domain $ G _ {1} $ conformally onto a domain $ G _ {2} $ to some point $ \zeta $ of the boundary $ \Gamma _ {1} $ of $ G _ {1} $ or to the entire boundary $ \Gamma _ {1} $ of this domain; the nature of the discontinuity in the case where such an extension is not possible; conformality of the extended mapping at the boundary points $ \zeta \in \Gamma _ {1} $; the differentiability or smoothness properties of the extended function on $ \Gamma _ {1} $ and on the closed domain $ \overline{G}\; _ {1} = G _ {1} \cup \Gamma _ {1} $; or the membership of the derivative of the mapping function to various classes of analytic functions in $ G _ {1} $, etc. These properties are studied in their dependence on the properties of the boundaries of $ G _ {1} $ and $ G _ {2} $. From among the most general boundary properties of conformal mappings one can distinguish: For any simply-connected domains $ G _ {1} $ and $ G _ {2} $ and any univalent conformal mapping $ w = f ( z) $ of $ G _ {1} $ onto $ G _ {2} $, this mapping sets up a one-to-one correspondence between the prime ends (cf. Limit elements) of these domains in the sense that the class of all equivalent paths lying in $ G _ {1} $ and defining some prime end $ \zeta $ of $ G _ {1} $ is taken by this mapping into the class of all equivalent paths lying in $ G _ {2} $ and defining some prime end $ \omega $ of $ G _ {2} $( the inverse mapping $ z = f ^ { - 1 } ( w) $, $ w \in G _ {2} $, takes the class of equivalent paths defining $ \omega $ into the class of equivalent paths defining $ \zeta $). Furthermore, $ f $ determines, in a special topology, a homeomorphism of the domain $ G $ with its prime ends adjoined (regarded together with the points $ z \in G _ {1} $ as points of a topological space) onto the domain $ G _ {2} $ with prime ends adjoined. One usually considers the case when one of the domains $ G _ {1} $, $ G _ {2} $ is the unit disc $ D = \{ {z } : {| z | < 1 } \} $( more rarely, the half-plane or a sector); the general case reduces to this particular case.

Let $ w = f ( z) $ be a univalent conformal mapping of the disc $ D $ with boundary $ C = \{ {z } : {| z | = 1 } \} $ onto a bounded domain $ G $ with boundary $ \Gamma $, let $ z = \phi ( w) $ be its inverse: $ \phi ( f ( z) ) = z $ for $ z \in D $. Then one has the following results.

1) In order that $ w = f ( z) $ be continuously extendable to a point $ \zeta \in C $ it is necessary and sufficient that the prime end of $ G $ corresponding to $ \zeta $ under this mapping be a prime end of the first kind (that is, it consists of a single point). In order that $ z = \phi ( w) $ be continuously extendable to a point $ \omega \in \Gamma $ it is necessary and sufficient that $ \omega $ be part of just one prime end (more precisely, be part of just one support of a prime end of $ G $). If $ \Gamma $ is a closed Jordan curve, then $ f $ is continuously extendable onto $ C $, and $ \phi $ onto $ \Gamma $, so that the extended functions realize a one-to-one bicontinuous mapping (a homeomorphism) of the closed domains $ \overline{D}\; $, $ \overline{G}\; $ onto each other.

In what follows $ \Gamma $ denotes a Jordan curve and it is supposed that the functions $ f $ and $ \phi $ are extended onto $ C $ and $ \Gamma $, respectively.

2) If $ \Gamma $ is a closed rectifiable Jordan curve, then the boundary functions $ f ( \zeta ) $, $ \zeta \in C $, and $ \phi ( \omega ) $, $ \omega \in \Gamma $, are absolutely continuous. Thus, $ \omega = f ( \zeta ) $, $ \zeta \in C $, and $ \zeta = \phi ( \omega ) $, $ \omega \in \Gamma $, take boundary sets of measure zero to boundary sets of measure zero. The function $ f ( z) $ has a finite non-zero derivative relative to the closed disc $ \overline{D}\; $ at almost-every point $ \zeta \in \Gamma $, while $ \phi ( w) $ has a finite non-zero derivative at almost-every point $ \omega \in \Gamma $. Consequently, these mappings are conformal (that is, have the property of constant dilation and preservation of angles) at almost-every boundary point of their respective domains. The function $ f ^ { \prime } ( z) $ belongs to the Hardy class $ H ^ {1} $.

3) Let $ \Gamma $ be a closed rectifiable Jordan curve with the following property: For any distinct points $ \omega _ {1} , \omega _ {2} \in \Gamma $, the ratio of the length of the smaller of the arcs into which these points divide $ \Gamma $ to the distance $ | \omega - \omega _ {2} | $ between these points is bounded from above by some quantity $ d $ that does not depend on $ \omega _ {1} $ or $ \omega _ {2} $. Then $ f ( z) $ satisfies the Hölder condition of order $ 2 ( 1 + d ) ^ {-} 2 $ on $ D $.

4) Let $ \Gamma $ be a smooth closed Jordan curve. A point $ \omega _ {2} \in \Gamma $ is fixed and for $ - \infty < s < \infty $ an arc of length $ | s | $ is laid-off along $ \Gamma $ in the positive (when $ s > 0 $) or negative (when $ s < 0 $) direction of traversal of $ G $. Let $ \omega ( s) $ be the end of the arc laid-off, and let $ \tau ( s) $ be the angle between the positive direction of the real axis and the positive direction of the tangent at $ \omega ( s) $( the value of $ \tau ( s) $ is chosen so that the function $ \tau ( s) $ is continuous). If there exists for some $ p = 0 , 1 \dots $ a derivative $ \tau ^ {(} p) ( s) $ satisfying a Hölder condition of some positive order $ \alpha < 1 $, then the function $ f ^ { ( p+ 1) } ( z) $ is continuous and satisfies the Hölder condition of the same order $ \alpha $ on the closed disc $ \overline{D}\; $, moreover $ f ^ { \prime } ( z) \neq 0 $ on $ \overline{D}\; $ and $ \phi ^ {(} p+ 1) ( w) $ is continuous and satisfies the Hölder condition of order $ \alpha $ on $ G $ with, moreover, $ \phi ^ \prime ( w) \neq 0 $ on $ \overline{G}\; $.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)
[2]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6
[3]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[5]	S.E. Warschawski, "On differentiability at the boundary in conformal mapping" Proc. Amer. Math. Soc. , 12 : 4 (1961) pp. 614–620
[6]	O.D. Kellogg, "Harmonic functions and Green's integral" Trans. Amer. Math. Soc. , 13 : 1 (1912) pp. 109–132
[7]	E.P. Dolzhenko, "Smoothness of harmonic and analytic functions at boundary points of a domain" Izv. Akad. Nauk SSSR Ser. Mat. , 29 (1965) pp. 1069–1084 (In Russian)