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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On conformal maps of infinitely connected Dirichlet regions
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by V. C. Williams PDF
Trans. Amer. Math. Soc. 155 (1971), 427-453 Request permission

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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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