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

Abanin, Alexander V.; Ishimura, Ryuichi; Khoi, Le Hai

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

Abanin, Alexander V.¹ ; Ishimura, Ryuichi ² ; Khoi, Le Hai 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, Tome 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 :

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

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Abanin, Alexander V.; Ishimura, Ryuichi; Khoi, Le Hai. Surjectivity criteria for convolution operators in $ {A}^{-\infty }$. Comptes Rendus. Mathématique, Tome 348 (2010) no. 5-6, pp. 253-256. doi : 10.1016/j.crma.2010.01.015. http://www.numdam.org/articles/10.1016/j.crma.2010.01.015/

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