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Envelopes of holomorphy for bounded holomorphic functionsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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1992 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Umeå universitet , 1992. , p. 7
##### Series

Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 2
##### Keywords [en]

holomorphicfunction, boundedholomorphic function, domain of holo¬ morphy, envelope of holomorphy, Gleason’s problem, convex set, plurisubharmonic function, pluripolar set, poly normally convex set
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:umu:diva-141155ISBN: 91-7174-677-3 (print)OAI: oai:DiVA.org:umu-141155DiVA, id: diva2:1152792
#####

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##### Projects

digitalisering@umu.seAvailable from: 2017-10-26 Created: 2017-10-26 Last updated: 2018-06-09Bibliographically approved

Some problems concerning holomorphic continuation of the class of bounded holomorphic functions from bounded domains in C^{n} that are domains of holomorphy are solved. A bounded domain of holomorphy Ω in C^{2} with nonschlicht H°°-envelope of holomorphy is constructed and it is shown that there is a point in D for which Gleason’s Problem for H°°(Ω) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C^{2} for which the volume of the H°°-envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves.

If H°°(Ω) is the Banach algebra of bounded holomorphic functions on a bounded domain Ω in C^{n} and if *p* is a point in Ω, then the following problem is known as Gleason’s Problem for H^{oo}(Ω) :

Is the maximal ideal in H°°(Ω) consisting of functions vanishing at *p* generated

by (*z*_{1} -*p*_{1}) , ... , (z* _{n}* -

A sufficient condition for solving Gleason’s Problem for 77°° (Ω) for all points in Ω is given. In particular, this condition is fulfilled by a convex domain Ω with Lip_{1}+e boundary (0 < e < 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C^{2}. One of the ideas in the methods of proof is integration along specific polygonal lines.

Certain properties of some open sets defined by global plurisubharmonic functions in C^{n} are studied. More precisely, the sets D_{u} = {z e C^{n} : *u*(*z*) < 0} and E_{h} = {{*z _{,}w*) e C

A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions.

isbn
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