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Description:
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We study the ∂ -Neumann
operator and the Kobayashi metric . We observe that under certain
conditions , a higher -dimensional domain fibered over Ω can
inherit noncompactness of the d -bar -Neumann
operator from the base domain Ω . Thus we have a domain
which has noncompact d -bar -Neumann operator but
does not necessarily have the standard conditions which usually
are satisfied with noncompact d -bar -Neumann operator .
We define the property K which is related to the Kobayashi metric and gives
information about holomorphic structure of fat subdomains . We
find an equivalence between compactness of the d -bar -Neumann operator and the property K in any convex domain .
We also find a local property of the Kobayashi metric [Theorem IV .1] , in
which the domain is not necessary pseudoconvex .
We find a more
general condition than finite type for the local regularity of the
d -bar -Neumann operator with the vector -field
method . By this generalization , it is possible for an analytic
disk to be on the part of boundary where we have local
regularity of the d -bar -Neumann operator . By Theorem V .2 , we show that an isolated infinite -type point in the
boundary of the domain is not an obstruction for the local
regularity of the d -bar -Neumann operator . |