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On the Extension and Wedge Product of Positive Currents
Stockholm University, Faculty of Science, Department of Mathematics. (Complex Analysis)
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This dissertation is concerned with extensions and wedge products of positive currents. Our study can be considered as a generalization for classical works done earlier in this field.

Paper I deals with the extension of positive currents across different types of sets. For closed complete pluripolar obstacles, we show the existence of such extensions. To do so, further Hausdorff dimension conditions are required. Moreover, we study the case when these obstacles are zero sets of strictly k-convex functions.

In Paper II, we discuss the wedge product of positive pluriharmonic (resp. plurisubharmonic) current of bidimension (p,p) with the Monge-Ampère operator of plurisubharmonic function. In the first part of the paper, we define this product when the locus points of the plurisubharmonic function are located in a (2p-2)-dimensional closed set (resp. (2p-4)-dimensional sets), in the sense of Hartogs. The second part treats the case when these locus points are contained in a compact complete pluripolar sets and p≥2 (resp. p≥3).

Paper III studies the extendability of negative S-plurisubharmonic current of bidimension (p,p) across a (2p-2)-dimensional closed set. Using only the positivity of S, we show that such extensions exist in the case when these obstacles are complete pluripolar, as well as zero sets of C2-plurisubharmoinc functions.

Place, publisher, year, edition, pages
Department of Mathematics, Stockholm University , 2012. , 20 p.
Keyword [en]
Closed positive currents, plurisubharmonic currents, plurisubharmonic functions, pluripolar sets, k-convex functions, wedge product of currents
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-71035ISBN: 978-91-7447-447-3 (print)OAI: oai:DiVA.org:su-71035DiVA: diva2:486254
Public defence
2012-03-02, lecture room 14, house 5, Kräftriket, Roslagsvägen 101, Stockholm, 14:00 (English)
Opponent
Supervisors
Note
At the time of doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Accepted. Paper 2: Manuscript. Paper 3: Manuscript.Available from: 2012-02-09 Created: 2012-01-25 Last updated: 2012-02-10Bibliographically approved
List of papers
1. Extension of Positive currents with Special Properties of Monge-Ampere Operators
Open this publication in new window or tab >>Extension of Positive currents with Special Properties of Monge-Ampere Operators
2013 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 113, no 1, 108-127 p.Article in journal (Refereed) Published
Abstract [en]

This paper deals with the extension of positive currents across different types of sets. For closed complete pluripolar obstacles, we show the existence of such extensions. To do so, further Hausdorff dimension conditions are required. Moreover, we study the case when these obstacles are zero sets of strictly $k$-convex functions.

Keyword
Closed positive currents, pluripolar sets, plurisubharmonic functions, plurisubharmonic currents, strictly k-convex functions
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-70839 (URN)000325836900007 ()
Available from: 2012-01-24 Created: 2012-01-24 Last updated: 2017-12-08Bibliographically approved
2. The inductive wedge product of positive currents
Open this publication in new window or tab >>The inductive wedge product of positive currents
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we discuss the wedge product of positive pluriharmonic (resp. plurisubharmonic) current of bidimension $(p,p)$ with the Monge-Ampère operator of plurisubharmonic function. In the first part of the paper, we define this product when the locus points of the plurisubharmonic function are located in a (2p-2)-dimensional closed set (resp. (2p-4)-dimensional sets), in the sense of Hartogs. The second part treats the case when these locus points are contained in a compact complete pluripolar sets and p≥ 2 (resp. p≥3).

Keyword
Wedge product of currents, plurisubharmonic currents, plurisubharmonic functions
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-71026 (URN)
Available from: 2012-01-25 Created: 2012-01-25 Last updated: 2012-02-01Bibliographically approved
3. The extendability of S-plurisubharmonic currents:  = Sur le prolongement des courants S-plurisousharmoniques
Open this publication in new window or tab >>The extendability of S-plurisubharmonic currents:  = Sur le prolongement des courants S-plurisousharmoniques
2012 (English)In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 350, no 23-24, 1023-1026 p.Article in journal (Other academic) Published
Abstract [en]

This paper studies the extendability of negative S-plurisubharmonic current of bidimension (p,p) across a (2p-2)-dimensional closed set. Using only the positivity of S, we show that such extensions exist in the case when these obstacles are complete pluripolar, as well as zero sets of C2-plurisubharmoinc functions.

Abstract [fr]

Dans cette Note, nous étudions le prolongement aux ensembles fermés des courants S-plurisousharmoniques négatifs

Keyword
Plurisubharmonic currents
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-71028 (URN)10.1016/j.crma.2012.10.032 (DOI)
Available from: 2012-01-25 Created: 2012-01-25 Last updated: 2017-12-08Bibliographically approved

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