Let $P$ be a linear partial differential operator with coefficients in the Roumieu class ${\cal E}_{\{\omega\}}(\Omega)$. We prove that if $P$ and its transpose operator $^tP$ are $\{\omega\}$-hypoelliptic in $\Omega$ and surjective on the space ${\cal E}_{\{\omega\}}(\Omega)$, 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 and hence, answering in positive to a question posed by T. GRAMCHEV and L. RODINO. As the Roumieu classes contain the Gevrey classes, this result extends also for Gevrey classes. We point out that this result is new even for Gevrey classes.
Ultradifferentiable fundamental kernels of linear partial differential operators on non-quasianalytic classes of Roumieu type
ALBANESE, Angela Anna;
2007-01-01
Abstract
Let $P$ be a linear partial differential operator with coefficients in the Roumieu class ${\cal E}_{\{\omega\}}(\Omega)$. We prove that if $P$ and its transpose operator $^tP$ are $\{\omega\}$-hypoelliptic in $\Omega$ and surjective on the space ${\cal E}_{\{\omega\}}(\Omega)$, 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 and hence, answering in positive to a question posed by T. GRAMCHEV and L. RODINO. As the Roumieu classes contain the Gevrey classes, this result extends also for Gevrey classes. We point out that this result is new even for Gevrey classes.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
