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

Alexander V. Abanin; Ryuichi Ishimura; Le Hai Khoi

doi:10.1016/j.crma.2010.01.015

Complex Analysis

Surjectivity criteria for convolution operators in

A^{- \infty}

[Critères de surjectivité pour des opérateurs de convolution dans

A^{- \infty}

]

Présenté par : Jean-Pierre Demailly

Alexander V. Abanin ¹ ; Ryuichi Ishimura ² ; Le Hai Khoi ³

¹ Southern Institute of Mathematics (SIM), Vladikavkaz 362027, and Southern Federal University (SFU), Rostov-on-Don 344090, The Russian Federation
² Graduate School of Science, Course of Mathematics and Informatics, Chiba University, Chiba 263-8522, Japan
³ Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University (NTU), 637371 Singapore

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 253-256.

Résumé
Abstract

Le but de cet article est d'établir des critères de surjectivité pour des opérateurs de convolution, opérant de $A^{- \infty} (Ω + K)$ dans $A^{- \infty} (Ω)$ (Ω et K étant, respectivement, un domaine convexe borné et un compact convexe dans $C^{n} (n > 1)$ ). Ils seront obtenus en les reliant au problème de division. Une représentation explicite des solutions des équations de convolution correspondantes sera également donnée sous forme de série de Dirichlet.

The goal of this Note is to prove criteria for surjectivity of convolution operators acting from $A^{- \infty} (Ω + K)$ into $A^{- \infty} (Ω)$ (Ω and K being a bounded convex domain and a convex compact set in $C^{n} (n > 1)$ , respectively). This is obtained in a connection with the division problem. The explicit representation of solutions of the corresponding convolution equations in a form of Dirichlet series is also given.

Reçu le : 2009-09-30
Accepté le : 2010-01-15
Publié le : 2010-02-21

DOI : 10.1016/j.crma.2010.01.015

Affiliations des auteurs :

Alexander V. Abanin ¹ ; Ryuichi Ishimura ² ; Le Hai Khoi ³

@article{CRMATH_2010__348_5-6_253_0,
     author = {Alexander V. Abanin and Ryuichi Ishimura and Le Hai Khoi},
     title = {Surjectivity criteria for convolution operators in $ {A}^{-\infty }$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {253--256},
     publisher = {Elsevier},
     volume = {348},
     number = {5-6},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.015},
     language = {en},
}

TY  - JOUR
AU  - Alexander V. Abanin
AU  - Ryuichi Ishimura
AU  - Le Hai Khoi
TI  - Surjectivity criteria for convolution operators in $ {A}^{-\infty }$
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 253
EP  - 256
VL  - 348
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2010.01.015
LA  - en
ID  - CRMATH_2010__348_5-6_253_0
ER  -

%0 Journal Article
%A Alexander V. Abanin
%A Ryuichi Ishimura
%A Le Hai Khoi
%T Surjectivity criteria for convolution operators in $ {A}^{-\infty }$
%J Comptes Rendus. Mathématique
%D 2010
%P 253-256
%V 348
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2010.01.015
%G en
%F CRMATH_2010__348_5-6_253_0

Alexander V. Abanin; Ryuichi Ishimura; Le Hai Khoi. Surjectivity criteria for convolution operators in $ {A}^{-\infty }$. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 253-256. doi : 10.1016/j.crma.2010.01.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.015/

Bibliographie
Cité par

[1] A.V. Abanin; Le Hai Khoi On the duality between $A^{- \infty} (D)$ and $A_{D}^{- \infty}$ for convex domains, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 863-866

[2] 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, in press

[3] 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., in press

[4] R. Ishimura; Y. Okada The existence and the continuation of holomorphic solutions for convolution equations in tube domains, Bull. Soc. Math. France, Volume 122 (1994), pp. 413-433

[5] R. Ishimura; J. Okada Sur la condition (S) de Kawai et la propriété de croissance régulière d'une fonction sous-harmonique et d'une fonction entière, Kyushu J. Math., Volume 48 (1994), pp. 257-263

[6] T. Kawai On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., Volume 17 (1970), pp. 467-517

[7] A.S. Krivosheev A criterion for the solvability of nonhomogeneous convolution equations in convex domains of $C^{n}$ , Math. USSR-Izv., Volume 36 (1991), pp. 497-517

[8] A. Martineau Équations différentielles d'ordre infini, Bull. Soc. Math. France, Volume 95 (1967), pp. 109-154

[9] S. Momm A division problem in the space of entire functions of exponential type, Ark. Mat., Volume 32 (1994), pp. 213-236

[10] R. Sigurdsson Convolution equations in domains of $C^{n}$ , Ark. Mat., Volume 29 (1991), pp. 285-305

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)

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

A.V. Abanin; Le Hai Khoi

C. R. Math (2011)