Surjective linear partial differential operators with variable coefficients on non-quasianalytic classes of Roumieu type

Let $P$ be a linear partial differential operator with variable coefficients in the Roumieu class $\{\cal E}_{\{\omega\}}(\Omega)$. We prove that if $P$ is $\{\omega\}$-hypoelliptic and has an $\{\omega\}$-fundamental kernel in $\Omega$, then $P$ is a surjective operator on the space $\{\cal E}_{\{\omega\}}(\Omega)$. Applications are given to elliptic second order partial differential operators and to Mizohata operator. The surjectivity of Mizohata operator on Gevrey classes had been proved by Cattabriga and Zanghirati in 1990 with a different method.