Hartogs functions
Definition.
Let be an open set and let be the smallest class of functions on to that contains all of the functions where is holomorphic on and suchthat is closed with respect to the following conditions:
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If , then.
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If then for all .
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If and,then .
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If andthe 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 closureof is compact), then .
These functions are called the Hartogs functions.
It is known that if then the upper semi-continuous Hartogs functionsare 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.