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Asymptotic Maximum Principles for Subharmonic and Plurisubharmonic Functions

Published online by Cambridge University Press:  20 November 2018

P. M. Gauthier ,
R. Grothmann  and
W. Hengartner
P. M. Gauthier
Affiliation:
Université de Montréal, Montréal, Québec
R. Grothmann
Affiliation:
Kath, Universitàt Eichstätt, Federal Republic of Germany
W. Hengartner
Affiliation:
Université Laval, Québec, Québec
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Let Ω be a bounded open set in Rn. An immediate consequence of the maximum principle is that if s is a function continuous on and subharmonic on Ω, then

(1)

Of course (1) is no longer true if Ω is not bounded. For example in C ∼ R2 consider the functions

However, if we restrict the growth of s, then (1) may still hold even if the open set Ω is no longer bounded and such is the theme of Phragmèn-Lindelöf type theorems. If we assume even more, namely, that s is upper-bounded, then we can again infer (1) for unbounded open sets Ω. We shall return to this point later.

In the present note, we wish to prove (1) for an arbitrary subharmonic function s on an open subset Ω of Rn. In particular, we do not assume that s is bounded or even of restricted growth. Rather, we impose restrictions on the (possibly unbounded) set Ω.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 2 , 01 April 1988 , pp. 477 - 486
Copyright
Copyright © Canadian Mathematical Society 1988

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