Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains

Abanin, A.V.; Khoi, Le Hai

doi:10.1016/j.crma.2011.10.013

Complex Analysis

Cauchy–Fantappiè transformation and mutual dualities between

A^{- \infty} (Ω)

and

A^{\infty} (\tilde{Ω})

for lineally convex domains
[La transformation de Cauchy–Fantappiè et les dualités mutuelles entre

A^{- \infty} (Ω)

A^{\infty} (\tilde{Ω})

pour des domaines linéellement convexes]

Présenté par : Jean-Pierre Demailly

Abanin, A.V.^1,² ; Khoi, Le Hai ³

¹ Southern Institute of Mathematics (SIM), Vladikavkaz 362027, The Russian Federation
² Southern Federal University (SFU), Rostov-on-Don 344090, The Russian Federation
³ Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University (NTU), 637371 Singapore

Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1155-1158.

Résumé
Abstract

Dans cette Note nous présentons les résultats suivants : (i) la description par la transformation de Cauchy–Fantappiè des fonctionnelles analytiques, des dualités mutuelles entre le $(DFS)$ -espace $A^{- \infty} (Ω)$ des fonctions holomorphes dans le domaine linéellement convexe Ω de $C^{n} (n ⩾ 2)$ avec croissance polynomiale près de la frontière de ∂Ω, et le $(FS)$ -espace $A^{\infty} (\tilde{Ω})$ des fonctions holomorphes dans lʼintérieur de lʼensemble conjugué $\tilde{Ω}$ qui sont dans $C^{\infty} (\tilde{Ω})$ ; (ii) lʼexistence dʼensembles suffisants dénombrables dans $A^{- \infty} (Ω)$ et $A^{\infty} (\tilde{Ω})$ ; (iii) la possibilité (respectivement, lʼimpossibilité) de représentation des fonctions de $A^{\infty} (\tilde{Ω})$ (respectivement, $A^{- \infty} (Ω)$ ) sous la forme de séries de fractions partielles.

In this Note we present the following results: (i) a description, via the Cauchy–Fantappiè transformation of analytic functionals, of the mutual dualities between the $(DFS)$ -space $A^{- \infty} (Ω)$ of holomorphic functions in a bounded lineally convex domain Ω of $C^{n} (n ⩾ 2)$ with polynomial growth near the boundary ∂Ω, and the $(FS)$ -space $A^{\infty} (\tilde{Ω})$ of holomorphic functions in the interior of the conjugate set $\tilde{Ω}$ that are in $C^{\infty} (\tilde{Ω})$ ; (ii) the existence of countable sufficient sets in $A^{- \infty} (Ω)$ and $A^{\infty} (\tilde{Ω})$ ; (iii) a possibility (respectively, the failure) of representing functions from $A^{\infty} (\tilde{Ω})$ (respectively, $A^{- \infty} (Ω)$ ) in the form of series of partial fractions.

Reçu le : 2011-08-22
Accepté le : 2011-10-18
Publié le : 2011-11-01

DOI : 10.1016/j.crma.2011.10.013

Affiliations des auteurs :

Abanin, A.V. ^{1, 2} ; Khoi, Le Hai ³

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Abanin, A.V.; Khoi, Le Hai. Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains. Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1155-1158. doi : 10.1016/j.crma.2011.10.013. http://www.numdam.org/articles/10.1016/j.crma.2011.10.013/

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