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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.
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Abstract

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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

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