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.