Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-18T06:11:43.557Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on n-harmonic majorants

Published online by Cambridge University Press:  17 April 2009

Hong Oh Kim  and
Chang Ock Lee
Hong Oh Kim
Affiliation:
Department of Applied Mathematics, Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryang, Seoul, Korea.
Chang Ock Lee
Affiliation:
Department of Applied Mathematics, Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryang, Seoul, Korea.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.

We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:

If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 3 , December 1986 , pp. 461 - 472
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Duren, P.L., Theory of HP spaces, (Academic Press, New York, NY. 1970).Google Scholar
[2]Flett, T.M., “The dual of an inequality of Hardy and Littlewood and some related inequalities”, J. Math. Anal. Appl. 38 (1972), 746–765.CrossRefGoogle Scholar
[3]Garnett, J.B., Bounded analytic functions, (Academic Press, New York, NY 1981).Google Scholar
[4]Hardy, G.H. and Littlewood, J.E., “Some properties of fractional intergrals, II”, Math. Z. 34 (1932), 403–439.CrossRefGoogle Scholar
[5]Hardy, G.H. and Littlewood, J.E., “Theorems concerning mean values of analytic or harmonic functions”, Quart. J. Math. 12 (1941), 221–256.CrossRefGoogle Scholar
[6]Kim, H.O., “Derivatives of Blaschke products”, Pacific J. Math., 114 (1984), 175–189.CrossRefGoogle Scholar
[7]Kim, H.O., “on a theorem of Hardy and Littlewood on the polydisc”, Proc. Amer. Math. Soc. (to appear).Google Scholar
[8]Rudin, W., Function therory in polydiscs, (W.A. Benjamin, Inc., New York, NY, 1969).Google Scholar
[9]Yamashita, S., “Dirichlet-finite functions and harmonic majorants”, Illinois J. Math., 25 (1981), 626–631.CrossRefGoogle Scholar