## 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

### Keywords:

microlocal analysis; microhyperbolicity
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### References:

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