On the microlocal decomposition of ultradistributions and ultradifferentiable functions

A. Eida

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 26 (2001) no. 1.

Résumé

The microlocal decomposition for ultradistributions and ultradifferentiable functions is derived by Bengel-Schapira's method and these classes of functions are microlocalized as subsheaves $\FC_M^*$, $\FC_M^{d,*}$ of the sheaf $\FC_M$ of Sato's microfunctions on a real analytic manifold $M$. Moreover, the exactness of the sequences $$\sexact{\FA_M}{\FDB^*_M}{\pi_*\FC^*_M}$$ and $$\sexact{\FA_M}{\FDF^*_M}{\pi_*\FC_M^{d,*}}$$ is shown and some fundamental properties on $\FC_M^*$, $\FC_M^{d,*}$ are described. Here $\FDB^*_M$ is a sheaf of ultradistributions and $\FDF^*_M$ is a sheaf of ultradifferentiable functions. We give some solvability conditions applicable to partial differential equations by operating Aoki's classes of microdifferential operators of infinite order on these sheaves.