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0. Introduction. In the present report, we consider the convolution operator u∗ defined by a hyperfunction u(x) with compact support, operating on holomorphic functions in tube domains invariant by any real translation. In all what follows we impose the natural condition called (S) introduced by Kawai [Kaw]. This condition is in fact equivalent to the classical property of completely regular growth in pure-imaginary directions ([IOk]). First we have under this condition (S), the existence of solutions in any open tube domain. And conversely, we also have that the existence of solutions in some special tube domain implies the condition (S), meaning that (S) is a sufficient and almost necessary condition for the existence. Secondly, we will define the characteristic set Car(u∗) of u∗ and prove that all solutions of the homogeneous equation u ∗ f = 0 are extended holomorphically to any non-characteristic direction. Finally, by these results, we will conclude under the condition (S) that the micro-support of the complex defined by the operator u∗ is included in the characteristic set Car(u∗). Most of the results are based on our paper [IO] and on [IOk] and we refer to them for the details of proofs. The authors take this occasion to express their thanks to Polish Academy of Sciences and Institute of Mathematics for inviting one of the authors and for all hospitality during his stay in Warszawa, especially to Professor B. Ziemian and also to secretaries of Banach Center.

The micro-support of the complex defined by a convolution operator in tube domains