Ultradifferentiable Fundamental Kernels of Linear Partial Differential Operators on Non-quasianalytic Classes of Roumieu Type

Abstract

Let $P$ be a linear partial differential operator with coeffcients in the Roumieu class ${\mathcal{E}}_{\{\omega \}}(\Omega )$. We prove that if $P$ and its transposed operator ${}^{t}P$ are $\{\omega \}$-hypoelliptic in $\Omega$ and surjective on the space ${\mathcal{E}}_{\{\omega \}}(\Omega )$, then $P$ has a global two-sided ultradifferentiable fundamental kernel in $\Omega$, thus extending to the Roumieu classes the well-known analogous result of B. Malgrange in the $C^{\infty }$ class. This result is new even for Gevrey classes.