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

A.V. Abanin; Le Hai Khoi

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

A.V. Abanin ^1,² ; Le Hai Khoi 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, Volume 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 :

A.V. Abanin ^{1, 2} ; Le Hai Khoi ³

@article{CRMATH_2011__349_21-22_1155_0,
     author = {A.V. Abanin and Le Hai Khoi},
     title = {Cauchy{\textendash}Fantappi\`e transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1155--1158},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.013},
     language = {en},
}

TY  - JOUR
AU  - A.V. Abanin
AU  - Le Hai Khoi
TI  - Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1155
EP  - 1158
VL  - 349
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.013
LA  - en
ID  - CRMATH_2011__349_21-22_1155_0
ER  -

%0 Journal Article
%A A.V. Abanin
%A Le Hai Khoi
%T Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains
%J Comptes Rendus. Mathématique
%D 2011
%P 1155-1158
%V 349
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2011.10.013
%G en
%F CRMATH_2011__349_21-22_1155_0

A.V. Abanin; Le Hai Khoi. Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1155-1158. doi : 10.1016/j.crma.2011.10.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.013/

Bibliographie
Cité par

[1] A.V. Abanin, Le Hai Khoi, Pre-dual of the function algebra $A^{- \infty} (D)$ and representation of functions in Dirichlet series, Complex Anal. Oper. Theory (, available online on Feb 18, 2010). | DOI

[2] A.V. Abanin; Le Hai Khoi Dual of the function algebra $A^{- \infty} (D)$ and representation of functions in Dirichlet series, Proc. Amer. Math. Soc., Volume 138 (2010), pp. 3623-3635

[3] M. Andersson; M. Passare; R. Sigurdsson Complex Convexity and Analytic Functionals, Birkhäuser, 2004

[4] D. Barrett Duality between $A^{\infty}$ and $A^{- \infty}$ on domains with non-degenerate corners, Contemp. Math. AMS, Volume 185 (1995), pp. 77-87

[5] K.D. Bierstedt; R. Meise Summers, J. Math. Anal. Appl., Volume 277 (2003), pp. 651-669

[6] J. Bonet; P. Domański Sampling sets and sufficient sets for $A^{- \infty}$ , J. Math. Anal. Appl., Volume 277 (2003), pp. 651-669

[7] Y.J. Choi; Le Hai Khoi; K.T. Kim On an explicit construction of weakly sufficient sets for the function algebra $A^{- \infty}$ , Compl. Var. Elliptic Equation, Volume 54 (2009), pp. 879-897

[8] B.A. Derzhavets Spaces of functions, analytic in some linearly convex domains of $C^{n}$ , having prescribed behavior near the boundary, Izv. Vyssh. Uchebn. Zaved. Mat., Volume 29 (1985) no. 6, pp. 10-13 (in Russian). English translation in Soviet Math. (Iz. VUZ), 29, 6, 1985, pp. 11-14

[9] O.V. Epifanov Variations of weakly sufficient sets in spaces of analytic functions, Izv. Vyssh. Uchebn. Zaved. Mat., Volume 30 (1986) no. 7, pp. 50-56 (in Russian). English translation in Soviet Math. (Iz. VUZ), 30, 7, 1986, pp. 67-74

[10] C.A. Horowitz; B. Korenblum; B. Pinchuk Sampling sequences for $A^{- \infty}$ , Michigan Math. J., Volume 44 (1997), pp. 389-398

[11] Le Hai Khoi Sets of uniqueness, weakly sufficient sets and sampling sets for $A^{- \infty} (B)$ , Bull. Korean Math. Soc., Volume 47 (2010), pp. 933-950

[12] C.O. Kiselman A study of the Bergman projection in certain Hartogs domains, Proc. Symp. Pure Math., Volume 52 (1991), pp. 219-232

[13] D.M. Schneider Sufficient sets for some spaces of entire functions, Trans. Amer. Math. Soc., Volume 197 (1974), pp. 161-180

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

On the duality between $A^{- \infty} (D)$ and $A_{D}^{- \infty}$ for convex domains

Alexander V. Abanin; Le Hai Khoi

C. R. Math (2009)

Surjectivity criteria for convolution operators in $A^{- \infty}$

Alexander V. Abanin; Ryuichi Ishimura; Le Hai Khoi

C. R. Math (2010)