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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.

##### MSC:
 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
##### Keywords:
temperate microfunction solutions; Cauchy problem
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