Contents §1. Subanalytic Sets §1.1. Review on subanalytic sets §1.2. Subanalytic subsets of R 2 §2. Tempered Holomorphic Functions §2.1. The subanalytic site §2.2. Definition and main properties of O t X sa §2.3. Pull-back of tempered holomorphic functions §3. Existence Theorem §3.1. Some classical results §3.2. Existence theorem for tempered holomorphic functions §4. Tempered Holomorphic Solutions §4.1. Classical results on D-modules §4.2. Existence theorem for holonomic D X -modules §4.3. The

more »

... case of a single operator §4.4. R-constructibility for tempered holomorphic solutions References Communicated by M. Kashiwara. Abstract Let X be a complex curve, X sa the subanalytic site associated to X, M a holonomic D X -module. Let O t X sa be the sheaf on X sa of tempered holomorphic functions and S ol(M) (resp. S ol t (M)) the complex of holomorphic (resp. tempered holomorphic) solutions of M. We prove that the natural morphism is an isomorphism. As a consequence, we prove that S ol t (M) is R-constructible in the sense of sheaves on X sa . Such a result is conjectured by M. Kashiwara and P. Schapira in [15] in any dimension. is an isomorphism. Second, we prove that the complex S ol t (M) is R-constructible in the sense of [15] . In that paper the authors conjectured such a result in any dimension. Our results being on a complex curve, it is natural to look for extensions of them in higher dimensions. In [22] , C. Sabbah conjectured and widely developed the higher dimensional version of Hukuhara-Turrittin's Theorem. Recently Y. André announced the proof of Sabbah's conjecture. Such results would be at the base of a possible extensions of our results.