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

Given a second-order partial differential operator P= Po+ X+ a on an open set Ω in Rn, where Po 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 ω of Ω , of s-Gevrey vectors of P on ω. For that we associate to any subset A⊂ Ω an integer (finite or + ∞) named the type of A with respect to Po and denoted τ(A; Po) (see in the next sections, precise definitions, facts and remarks about it). Denoting the space of s-Gevrey vectors of P in ω by Gs(ω, P) , we prove that Gs(ω;P)⊂Gs′(ω), with s′= τ(ω; P) · s under the assumption that the coefficients of P are in Gs(ω). Moreover, s′ is optimal. © 2020, Springer Nature Switzerland AG.