On Gevrey vectors of L. Hörmander’s operators

Abstract:

We study the regularity of Gevrey vectors of L. Hörmander’s operators: \begin{equation*} P=\sum _{j=1}^{m} X_j^2+X_0+c, \end{equation*} where $X_0$, $X_1$, …, $X_m$ are real vector fields in an open set $\Omega \subset \mathbb {R}^n$ and $c$ is a smooth function. More precisely, we prove the following: If the coefficients of $P$ are in the Gevrey class $G^k(\Omega )$, $k\in \mathbb N$, $k\geq 1$, and $P$ satisfies the following estimate with $p/q$ rational, $0<p\leq q$: \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 $\Omega _0\subset \overline {\Omega _0}\subset \Omega$, then $G^k(P, \Omega _0)\subset G^{k\frac {q}{p}}(\Omega _0)$. 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 $k\geq 1$ not necessarily integer, for Hörmander’s operators on a torus.