On conformal maps of infinitely connected Dirichlet regions

Author:
V. C. Williams

Journal:
Trans. Amer. Math. Soc. **155** (1971), 427-453

MSC:
Primary 30.40

DOI:
https://doi.org/10.1090/S0002-9947-1971-0280698-8

MathSciNet review:
0280698

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Abstract: Let $D$ be a plane region of arbitrary connectivity $( > 1)$ for which the Dirichlet problem is solvable. There exists a conformal map of $D$ onto a region bounded by two level loci of $H$, a nontrivial harmonic measure. $H$ is essentially the difference of two logarithmic potentials. The two measures involved are mutually singular probability measures. Further properties of these measures, and of $H$, are derived. The special case in which $D$ is of connectivity 2 is the classical theorem which states that an annular region is conformally equivalent to a region bounded by two circles. The case in which $D$ is of finite connectivity was treated by J. L. Walsh in 1956. A similar generalization of the Riemann mapping theorem is also established. Finally, converses of the above results are also valid.

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Keywords:
Infinitely connected Dirichlet region,
harmonic measure,
logarithmic potential,
subharmonic function

Article copyright:
© Copyright 1971
American Mathematical Society