On the microlocal decomposition of ultradistributions and ultradifferentiable functions

A. Eida

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.