Gevrey regularity of Gevrey vectors of second-order partial differential operators with non-negative characteristic form

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Given a second-order partial differential operator $$P = P^\mathrm{o} + X + a$$ on an open set $$\Omega $$ in $${\mathbb {R}}^n$$ , where $$P^\mathrm{o}$$ is the principal part and X is a real vector field, with non-negative real characteristic form, we study the $$s'$$ -Gevrey regularity on an open subset $$\omega $$ of $$\Omega $$ , of s-Gevrey vectors of P on $$\omega $$ . For that we associate to any subset $$A \subset \Omega $$ an integer (finite or $$+\infty $$ ) named the type of A with respect to $$P^\mathrm{o}$$ and denoted $$\tau (A; P^\mathrm{o}$$ ) (see in the next sections, precise definitions, facts and remarks about it). Denoting the space of s-Gevrey vectors of P in $$\omega $$ by $$G^s(\omega ,P)$$ , we prove that $$G^{s}(\omega ; P) \subset G^{s'} (\omega )$$ , with $$s' = \tau (\omega ; P)\cdot s$$ under the assumption that the coefficients of P are in $$G^{s} (\omega )$$ . Moreover, $$s'$$ is optimal.