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On the microlocal decomposition of ultradistributions and ultradifferentiable functions. (English) Zbl 1008.58018

Studying the problem of microlocal decomposition of generalized functions started with decompositions of distributions. M. Sato introduced the microlocal point of view. The author studies the same problem introducing microlocalized sheaves for ultradistributions and ultradifferentiable functions by Bengel-Schapira’s methods. His main result is the following: Let \(M\) be a real analytic manifold. The sheaf \(C_M^*\) (\(C_M^{d,*}\) respectively) is supple, where \(C_M^*\) and \(C_M^{d,*}\) are subsheaves of the sheaf \(C_M\) of Sato’s microfunctions on \(M\). At the end he gives solvability conditions applicable to partial differential equations.

MSC:

58J15 Relations of PDEs on manifolds with hyperfunctions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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