Hartogs functions
Definition.
Let G⊂ℂn be an open set and let ℱG
be the smallest class of functions on G to ℝ∪{-∞} that contains all of the functions
z↦log|f(z)| where f is holomorphic on G and such
that ℱG is closed with respect to the following conditions:
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•
If φ1,φ2∈ℱG, then φ1+φ2∈ℱG.
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•
If φ∈ℱG then aφ1∈ℱG for all a≥0.
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If {φk}∈ℱG and φ1≥φ2≥…, then lim.
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If and the sequence is uniformly bounded above on compact sets, then .
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If and , then
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If for all where is relatively compact (the closure of is compact), then .
These functions are called the Hartogs functions.
It is known that if then the upper semi-continuous Hartogs functions are precisely the subharmonic functions on .
Theorem (H. Bremerman).
All plurisubharmonic functions are Hartogs functions if
is a domain of holomorphy.
References
- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title | Hartogs functions |
---|---|
Canonical name | HartogsFunctions |
Date of creation | 2013-03-22 14:29:27 |
Last modified on | 2013-03-22 14:29:27 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 7 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 32U05 |