Local estimates for Hörmander's operators with Gevrey coefficients and application to the regularity of their Gevrey vectors

Abstract

Given a general Hörmander’s operator $P={\sum }_{j=1}^m{X}_j^2+Y+b$ in an open set $\mathrm{\Omega }\subset {ℝ}^n$, where $Y,X_1,\dots ,X_m$ are smooth real vector fields in $\mathrm{\Omega }$, $b\in C^{\infty }\left(\mathrm{\Omega }\right)$, and given also an open, relatively compact set ${\mathrm{\Omega }}_0$ with ${\overline{\mathrm{\Omega }}}_0\subset \mathrm{\Omega }$, and $s\in ℝ$, $s\ge 1$, such that the coefficients of $P$ are in $G^s\left({\mathrm{\Omega }}_0\right)$ and $P$ satisfies a $\frac{1}{p}$-Sobolev estimate in ${\mathrm{\Omega }}_0$, our aim is to establish local estimates reflecting local domination of ordinary derivatives by powers of $P$, in ${\mathrm{\Omega }}_0$. As an application, we give a direct proof of the $G^{2ps}\left({\mathrm{\Omega }}_0\right)$-regularity of any $G^s\left({\mathrm{\Omega }}_0\right)$-vector of $P$.