#### Vol. 1, No. 3, 2019

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

### Makhlouf Derridj

Vol. 1 (2019), No. 3, 321–345
##### Abstract

Given a general Hörmander’s operator $P={\sum }_{j=1}^{m}{X}_{j}^{2}+Y+b$ in an open set $\Omega \subset {ℝ}^{n}$, where $Y,{X}_{1},\dots ,{X}_{m}$ are smooth real vector fields in $\Omega$, $b\in {C}^{\infty }\left(\Omega \right)$, and given also an open, relatively compact set ${\Omega }_{0}$ with ${\overline{\Omega }}_{0}\subset \Omega$, and $s\in ℝ$, $s\ge 1$, such that the coefficients of $P$ are in ${G}^{s}\left({\Omega }_{0}\right)$ and $P$ satisfies a $\frac{1}{p}$-Sobolev estimate in ${\Omega }_{0}$, our aim is to establish local estimates reflecting local domination of ordinary derivatives by powers of $P$, in ${\Omega }_{0}$. As an application, we give a direct proof of the ${G}^{2ps}\left({\Omega }_{0}\right)$-regularity of any ${G}^{s}\left({\Omega }_{0}\right)$-vector of $P$.

##### Keywords
Gevrey vectors, Degenerate elliptic-parabolic differential operators.
##### Mathematical Subject Classification 2010
Primary: 35B65, 35G99, 35J70, 35K65