#### Vol. 302, No. 2, 2019

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

### Makhlouf Derridj

Vol. 302 (2019), No. 2, 511–543
##### Abstract

Following our preceding papers devoted to the case of general Hörmander’s operators $P$ with analytic-Gevrey coefficients on an open set $\Omega$ in ${ℝ}^{n}$, for which we established local relations of domination by powers of $P$ and derived from it local ${s}^{\prime }$-Gevrey regularity of local $s$-Gevrey vectors of $P$ (with, furthermore, suitable relations between $s,{s}^{\prime }$ and the coefficient of the Sobolev estimate satisfied by $P$), this article deals with the case of Hörmander’s operators of first kind (or of degenerate elliptic kind). We establish, in this case, precise local relations of domination by powers of $P$ which give, when applied to the ${s}^{\prime }$-Gevrey regularity of $s$-Gevrey vectors of $P$, in ${\Omega }_{0}$, with ${\overline{Omega}}_{0}\subset \Omega$, an optimal relation between $s,{s}^{\prime }$ and the type of ${\overline{Omega}}_{0}$ with respect to the system $X$ of vector fields whose sum of squares is the leading part of $P$.

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