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Micro-support and Cauchy problem for temperate solutions of regular \({\mathcal D}\)-modules. (English) Zbl 1038.35006
Summary: Let \(X\) be a complex manifold, \(V\) a smooth involutive submanifold of \(T^*X\), \({\mathcal M}\) a microdifferential system regular along \(V\), and \(F\) an \(\mathbb{R}\)-constructible sheaf on \(X\). We study the complex of temperate microfunction solutions of \({\mathcal M}\) associated with \(F\), that is, the complex \(R{\mathcal H}om_{{\mathcal D}_X} ({\mathcal M},{\mathcal T}\mu hom(F,{\mathcal O}_X))\). We give a bound to its micro-support and solve the Cauchy problem under a suitable hyperbolicity assumption.

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
58J10 Differential complexes
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