Local estimates for Hörmander's operators with Gevrey coefficients and application to the regularity of their Gevrey vectors
General Hörmander's operators of the form $ P = \sum_{j=1} ^m X_j ^2 + Y + b $ in an open set $ \Omega \subset \mathbb{R}^n$, where $Y, X_1,\dots, X_m $ are smooth vector fields in $ \Omega $ and $b \in C^\infty ( \Omega )$, are considered. More precisely, the author studies the existence of local estimates giving local domination of the ordinary derivatives by powers of $P$, when the coefficients of $P$ are in Gevrey classes $ G^s (\Omega_0)$, $ \overline{\Omega_0} \subset \Omega$, and when $P$ satisfies a ``$\frac{1}{p}$-Sobolev estimate''. A related results given in the reference [\textit{M. Derridj}, ``On Gevrey vectors of L. Hörmander's operators'', Preprint (to appear)] from the list of references. \par Such estimates are used to obtain $G^{2ps}(\Omega_0)$-regularity for $G^{s}(\Omega_0)$-vectors of $P$, with any $s\geq 1$. The proof is therefore considered as a direct proof of the Gevrey type regularity since it does not use the method of addition of an extra variable. \par The exposition starts with a nice introduction pointing out the main results and surveys in the field. Section 2, which contains the main definitions, is followed by the section with preliminary lemmas and propositions, with the main result, Theorem 4.2 given in Section 4. Finally, an application of the main result to Gevrey regularity for Gevrey vectors is explained in the last Section 5.
degenerate elliptic-parabolic differential operators
scientific article; zbMATH DE number 7027458
scientific article