Let
X
1
,
X
2
,
…
,
X
q
X_1,X_2,\ldots ,X_q
be a system of real smooth vector fields, satisfying Hörmander’s condition in some bounded domain
Ω
⊂
R
n
\Omega \subset \mathbb {R}^n
(
n
>
q
n>q
). We consider the differential operator
L
=
∑
i
=
1
q
a
i
j
(
x
)
X
i
X
j
,
\begin{equation*} \mathcal {L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*}
where the coefficients
a
i
j
(
x
)
a_{ij}(x)
are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:
μ
|
ξ
|
2
≤
∑
i
,
j
=
1
q
a
i
j
(
x
)
ξ
i
ξ
j
≤
μ
−
1
|
ξ
|
2
\begin{equation*} \mu |\xi |^2\leq \sum _{i,j=1}^qa_{ij}(x)\xi _i\xi _j\leq \mu ^{-1}|\xi |^2 \end{equation*}
for a.e.
x
∈
Ω
x\in \Omega
, every
ξ
∈
R
q
\xi \in \mathbb {R}^q
, some constant
μ
\mu
. Moreover, we assume that the coefficients
a
i
j
a_{ij}
belong to the space VMO (“Vanishing Mean Oscillation”), defined with respect to the subelliptic metric induced by the vector fields
X
1
,
X
2
,
…
,
X
q
X_1,X_2,\ldots ,X_q
. We prove the following local
L
p
\mathcal {L}^p
-estimate:
‖
X
i
X
j
f
‖
L
p
(
Ω
′
)
≤
c
{
‖
L
f
‖
L
p
(
Ω
)
+
‖
f
‖
L
p
(
Ω
)
}
\begin{equation*} \left \|X_iX_jf\right \|_{\mathcal {L}^p(\Omega ’)}\leq c\left \{\left \|\mathcal {L}f\right \|_{\mathcal {L}^p(\Omega )}+\left \|f\right \|_{\mathcal {L}^p(\Omega )}\right \} \end{equation*}
for every
Ω
′
⊂⊂
Ω
\Omega ’\subset \subset \Omega
,
1
>
p
>
∞
1>p>\infty
. We also prove the local Hölder continuity for solutions to
L
f
=
g
\mathcal {L}f=g
for any
g
∈
L
p
g\in \mathcal {L}^p
with
p
p
large enough. Finally, we prove
L
p
\mathcal {L}^p
-estimates for higher order derivatives of
f
f
, whenever
g
g
and the coefficients
a
i
j
a_{ij}
are more regular.