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A Criterion for a Set and Its Image under Quasiconformal Mapping to be of α(0 < α ≦ 2)-Dimensional Measure Zero

Published online by Cambridge University Press:  22 January 2016

Kazuo Ikoma
Kazuo Ikoma*
Affiliation:
Yamagata University
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Let w = w(z) be any K-quasiconformai mapping (in the sense of Pfluger-Ahlfors) of a domain D in the z-plane into the w-plane. Since w = w(z) is a measurable mapping (vid. Bers [1]), it transforms any set of Hausdorffs 2-dimensional measure zero in D into such another one. However, A. Mori [5] showed that for 0 < α ≤ 2, any set of -dimensional measure zero in a Jordan domain D is transformed by w = w(z)into a set of α-dimensional measure zero.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 22 , June 1963 , pp. 203 - 209
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

[1] Bers, L.: On a theorem of Mori and the definition of quasiconformality, Trans. Amer. Math. Soc, 84, 78–84 (1957).Google Scholar
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[3] Ikoma, K.: On a property of the boundary correspondence under quasiconf ormai mappings, Nagoya Math. Journ., 16, 185–188 (1960).Google Scholar
[4] Kuroda, T.: A criterion for a set to be of 1-dimensional measure zero, Jap. Journ. Math., 29, 48–51 (1959).Google Scholar
[5] Mori, A.: On quasi-conformality and pseudo-analyticity, Trans. Amer. Math. Soc, 84, 56–77, (1957).Google Scholar
[6] Teichmiiller, O.: Untersuchungen Über konforme und quasikonforme Abbildung, Deutsche Math., 3, 621–678 (1938).Google Scholar