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}

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2009-09-30
Accepted: 2010-01-15
Published online: 2010-02-21

DOI: 10.1016/j.crma.2010.01.015

Author's affiliations:

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

@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/

References
Cited by

[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

Cited by Sources:

Comments - Policy