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Extreme Points in the Hardy Class H1 of a Riemann Surface

Published online by Cambridge University Press:  20 November 2018

Walter Pranger
Walter Pranger*
Affiliation:
DePaul University, Chicago, Illinois
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The theorems presented here extend known results on the set of extreme points of the unit ball of the Hardy class H1 of a disk to the situation of an arbitrary Riemann surface. Several new results are obtained. The initial motivation for this work was provided by the theorem of de Leeuw and Rudin [2, p. 471] characterizing the extreme points in the case ol a disk. Careful scrutiny of the proof of that theorem yields one necessary and one sufficient condition for being an extreme point in H1 of an arbitrary surface (Theorems 1 and 4 below). The material presented here on compact bordered surfaces is closely related to the beautiful results of Gamelin and Voichick [4] and the results of Forelli [3].

For a subharmonic function u, which has a harmonic majorant on the Riemann surface R, Mu will denote the least harmonic majorant of u.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 6 , December 1971 , pp. 969 - 976
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Ahlfors, L., Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv. 24 (1950), 100–134.Google Scholar
2. Leeuw, K. de and Rudin, W., Extreme points and extremium problems in H1 , Pacific J. Math. 8 (1958), 467–485.Google Scholar
3. Forelli, F., Extreme points in H1(R), Can. J. Math. 19 (1967), 312–320.Google Scholar
4. Gamelin, T. and Voichick, M., Extreme points in spaces of analytic functions, Can. J. Math. 20 (1968), 919–928.Google Scholar
5. Hahn, K. T. and Mitchell, J., Hp spaces on bounded symmetric domains, Trans. Amer. Math. Soc. 146 (1969), 521–531.Google Scholar
6. Heins, M., Hardy classes on Riemann surfaces, Lecture Notes in Mathematics, No. 98 (Springer, Berlin, 1969).Google Scholar
7. Hoffman, K., Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, New Jersey, 1965).Google Scholar
8. Hörmander, L., An introduction to complex analysis in several variables (D. Van Nostrand, Princeton, New Jersey 1966).Google Scholar
9. Royden, H., The Riemann-Roch theorem, Comment. Math. Helv. 34 (1960), 37–51.Google Scholar
10. Rudin, W., Function theory in polydisks (Benjamin, New York, 1969).Google Scholar