Tajima, Shinichi Analyse microlocale sur les variétés de Cauchy-Riemann et problème de Lewy pour les solutions hyperfonctions. (French) Zbl 0583.32035 J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 569-583 (1984). This paper deals with the Lewy problem, i.e., the problem of holomorphic extensions of solutions of the tangential Cauchy-Riemann equation on a real analytic submanifold in a complex manifold and its generalization, from the algebraic analytic view point. The author reformulates the problem by using the language of \({\mathcal D}\)-modules and gives a necessary and sufficient condition for the Lewy problem for a real analytic hypersurface. Next, he considers a general system \({\mathcal M}\) on a complex manifold X satisfying \({\mathcal E}xt^ k_{{\mathcal D}_ X}({\mathcal M},{\mathcal O}_ X)=0\) and gives two equivalent conditions in order that any tangential holomorphic function on a real analytic hypersurface N is extended to a holomorphic solution of \({\mathcal M}\). Lastly, he gives some considerations when N is a subvariety of codimension more than two. Reviewer: M.Muro MSC: 32D15 Continuation of analytic objects in several complex variables 32V40 Real submanifolds in complex manifolds 32C36 Local cohomology of analytic spaces 32A45 Hyperfunctions Keywords:Lewy problem; holomorphic extensions; real analytic submanifold in a complex manifold PDFBibTeX XMLCite \textit{S. Tajima}, J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 569--583 (1984; Zbl 0583.32035)