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On The Mapping Properties of Certain Exceptional Sets in R2

Published online by Cambridge University Press:  20 November 2018

H. L. Jackson
H. L. Jackson*
Affiliation:
McMaster University, Hamilton, Ontario
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It is known that minimally thin and semithin sets in R2 are preserved by conformai mappings (see [3]) but it is not known whether or not analogous results hold true for ordinary thin and semithin sets respectively. In an unpublished work, Brelot has shown that ordinary thin sets at the origin are preserved by mappings of the form f(z) = z∝ (∝ > 0) where one always considers the principal branch of the mapping. We shall prove this result along with an analogous one for ordinary semithin sets and will see that the implications established by Jackson (see [4, Theorem 4]), for ordinary and minimally thin sets and by Brelot (see [2, p. 152]) for semithin sets in a half plane hold true for any wedge shaped region with vertex at the origin.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 44 - 49
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Brelot, M., Points irréguliers et transformations continues en théorie du potentiel, J. Math. Pures Appl. 19 (1940), 319–337.Google Scholar
2. Brelot, M., On topologies and boundaries in potential theory, Lecture Notes in Math., vol. 175 (Springer).Google Scholar
3. Brelot, M. and D∞b, J. L., Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble) IS (1963), 395–415.Google Scholar
4. Jackson, H. L., Some results on thin sets in a half plane, Ann. Inst. Fourier (Grenoble) 20 (1970), 201–218.Google Scholar
5. Jackson, H. L., On minimally thin sets in a Stolz domain, Can. Math. Bull. 15 (1972), 219–223.Google Scholar