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1
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Regularity of certain sets in ℂⁿ

100%
Dieu N.
Annales Polonici Mathematici
|
2003
|
tom 82
|
nr 3
219-232
EN
A subset K of ℂⁿ is said to be regular in the sense of pluripotential theory if the pluricomplex Green function (or Siciak extremal function) $V_K$ is continuous in ℂⁿ. We show that K is regular if the intersections of K with sufficiently many complex lines are regular (as subsets of ℂ). A complete characterization of regularity for Reinhardt sets is also given.
2
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A unicity theorem for plurisubharmonic functions

100%
Dieu N.
Annales Polonici Mathematici
|
2011
|
tom 100
|
nr 2
159-165
EN
We give sufficient conditions for unicity of plurisubharmonic functions in Cegrell classes.
3
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Weighted Bernstein-Markov property in ℂⁿ

64%
Dieu N., Hiep P.
Annales Polonici Mathematici
|
2012
|
tom 105
|
nr 2
101-123
EN
We study the weighted Bernstein-Markov property for subsets in ℂⁿ which might not be bounded. An application concerning approximation of the weighted Green function using Bergman kernels is also given.
4
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Complete pluripolar graphs in $ℂ^N$

64%
Dieu N., Manh P.
Annales Polonici Mathematici
|
2014
|
tom 112
|
nr 1
85-100
EN
Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that $Γ_g(F):={(z, g(z)): z ∈ F}$ is complete pluripolar in $ℂ^{N+1}$. Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that $Γ_g(D)$ is complete pluripolar in $ℂ^{N+1}$. These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86] and Levenberg, Martin and Poletsky [Analytic disks and pluripolar sets, Indiana Univ. Math. J. 41 (1992), 515-532].
5
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A new class of pluripolar sets

64%
Dieu N., Long T.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 3
229-245
EN
Let D be a domain in ℂⁿ. We introduce a class of pluripolar sets in D which is essentially contained in the class of complete pluripolar sets. An application of this new class to the problem of approximation of holomorphic functions is also given.
6
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Some properties of Reinhardt domains

51%
Hai L., Dieu N., Tuyen N.
Annales Polonici Mathematici
|
2003
|
tom 82
|
nr 3
203-217
EN
We first establish the equivalence between hyperconvexity of a fat bounded Reinhardt domain and the existence of a Stein neighbourhood basis of its closure. Next, we give a necessary and sufficient condition on a bounded Reinhardt domain D so that every holomorphic mapping from the punctured disk $Δ_*$ into D can be extended holomorphically to a map from Δ into D.
7
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Remarks on pluripolar hulls

51%
Hai L., Dieu N., Long T.
Annales Polonici Mathematici
|
2004
|
tom 84
|
nr 3
225-236
EN
The aim of the paper is to establish some results on pluripolar hulls and to define pluripolar hulls of certain graphs.
8
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B-regularity of certain domains in ℂⁿ

51%
Dieu N., Dung N., Hung D.
Annales Polonici Mathematici
|
2005
|
tom 86
|
nr 2
137-152
EN
We study the B-regularity of some classes of domains in ℂⁿ. The results include a complete characterization of B-regularity in the class of Reinhardt domains, we also give some sufficient conditions for Hartogs domains to be B-regular. The last result yields sufficient conditions for preservation of B-regularity under holomorphic mappings.
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