On Gevrey vectors of L. Hörmander’s operators

We study the regularity of Gevrey vectors of L. Hörmander’s operators: P = ∑ j = 1 m X j 2 + X 0 + c , \begin{equation*} P=\sum _{j=1}^{m} X_j^2+X_0+c, \end{equation*} where X 0 X_0 , X 1 X_1 , …, X m X_m are real vector fields in an open set Ω ⊂ R n \Omega \subset \mathbb {R}^n and c c is a smooth function. More precisely, we prove the following: If the coefficients of P P are in the Gevrey class G k ( Ω ) G^k(\Omega ) , k ∈ N k\in \mathbb N , k ≥ 1 k\geq 1 , and P P satisfies the following estimate with p / q p/q rational, 0 > p ≤ q 0>p\leq q : | | v | | p / q 2 ≤ C ( | ( P v , v ) | + | | v | | 2 ) , ∀ v ∈ D ( Ω 0 ) , \begin{eqnarray} ||v ||^2_{p/q}\leq C(|(Pv,v)|+||v ||^2), \; \forall v \in \mathcal D(\Omega _0), \end{eqnarray} for some open subset Ω 0 ⊂ Ω 0 ¯ ⊂ Ω \Omega _0\subset \overline {\Omega _0}\subset \Omega , then G k ( P , Ω 0 ) ⊂ G k q p ( Ω 0 ) G^k(P, \Omega _0)\subset G^{k\frac {q}{p}}(\Omega _0) . This provides in particular a local version of a recent result of N. Braun Rodrigues, G Chinni, P. D. Cordaro, and M. R. Jahnke, giving a global such result, with k ≥ 1 k\geq 1 not necessarily integer, for Hörmander’s operators on a torus.