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单词 HartogsFunctions
释义

Hartogs functions


Definition.

Let Gn be an open set and let Gbe the smallest class of functionsMathworldPlanetmath on G to {-} that contains all of the functionszlog|f(z)| where f is holomorphic on G and suchthat G is closed with respect to the following conditions:

  • If φ1,φ2G, thenφ1+φ2G.

  • If φG thenaφ1G for all a0.

  • If {φk}G andφ1φ2,then limkφkG.

  • If {φk}G andthe sequence is uniformly bounded above on compact sets, thensupkφkG.

  • If φG andφ^(w):=lim supwzφ(w),then φ^G

  • If φ|UU for allUG where U is relatively compact (the closureof U is compact), then φG.

These functions are called the Hartogs functions.

It is known that if n=1 then the upper semi-continuous Hartogs functionsare precisely the subharmonic functions on G.

Theorem (H. Bremerman).

All plurisubharmonic functionsMathworldPlanetmath are Hartogs functions ifG is a domain of holomorphy.

References

  • 1 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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更新时间:2025/5/4 17:49:10