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Inverse image for the functor \(\mu\text{hom}\). (English) Zbl 0751.58041

The author studies the microlocal inverse image of sheaves on \(C^ \infty\)-manifolds. Let \(f: X\to Y\) be a morphism from the real \(C^ \infty\)-manifold \(X\) to \(Y\) and let \(F\), \(K\) be sheaves on \(X\). Then the microlocalization of the functors \(f^{-1}\) and \(f^ !\) are defined and denoted by \(f^{-1}_{\mu,p}\) and \(f^ !_{\mu,p}\), respectively. The author obtained the theorem which asserts that the natural morphism: \[ \mu\text{hom}(f^{-1}_{\mu,p}K,f^ !_{\mu,p}F\otimes\omega_ Y^{\otimes-1})_{p_ Y}\to \mu\text{hom}(K,F\otimes\omega_ Y^{\otimes-1})_{p_ X} \] is an isomorphism under some natural conditions. Here, \(\omega_ X\) denotes the dualizing complex on \(X\) and \(\mu\text{hom}\) the microlocalization bifunctor introduced by Kashiwara and Schapira. This theorem can be interpreted as a statement of the microlocal well-posedness for the Cauchy problem.
Reviewer: M.Muro (Yanagido)

MSC:

58J99 Partial differential equations on manifolds; differential operators
18C99 Categories and theories
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