Given a general Hörmander’s operator
$P={\sum}_{j=1}^{m}{X}_{j}^{2}+Y+b$ in an open
set
$\Omega \subset {\mathbb{R}}^{n}$, where
$Y,{X}_{1},\dots ,{X}_{m}$ are smooth real
vector fields in
$\Omega $,
$b\in {C}^{\infty}\left(\Omega \right)$, and given also an open,
relatively compact set
${\Omega}_{0}$
with
${\overline{\Omega}}_{0}\subset \Omega $,
and
$s\in \mathbb{R}$,
$s\ge 1$, such that the
coefficients of
$P$
are in
${G}^{s}\left({\Omega}_{0}\right)$ and
$P$ satisfies a
$\frac{1}{p}$Sobolev
estimate in
${\Omega}_{0}$,
our aim is to establish local estimates reflecting local domination of ordinary derivatives by
powers of
$P$,
in
${\Omega}_{0}$.
As an application, we give a direct proof of the
${G}^{2ps}\left({\Omega}_{0}\right)$regularity of
any
${G}^{s}\left({\Omega}_{0}\right)$vector
of
$P$.
