  We establish two global subellipticity properties of positive symmetric
second-order partial differential operators on $L_2(\Ri^d)$. First, if $m \in
\Ni$ then we consider operators $H_0$ with coefficients in
$W^{m+1,\infty}(\Ri^d)$ and domain $D(H_0)=W^{\infty,2}(\Ri^d)$ satisfying the
subellipticity property \[ c (\phi, (I+H_0)\phi)\geq \|\Delta^{\gamma/2}
\phi\|_2^2 \] for some $c>0$ and $\gamma\in<0,1]$, uniformly for all $\phi\in
W^{\infty,2}(\Ri^d)$, where $\Delta$ denotes the usual Laplacian. Then we prove
that $D(H^\alpha) \subseteq D(\Delta^{\alpha \gamma})$ for all $\alpha \in
[0,2^{-1} (m + 1 + \gamma^{-1})>$. Hence there is a $c>0$ such that the norm
estimate \[ c \|(I+H)^\alpha \phi\|_2\geq \|\Delta^{\alpha \gamma} \phi\|_2 \]
is valid for all $\phi\in D(H^\alpha)$ where $H$ denotes the self-adjoint
closure of $H_0$. In particular, if the coefficients of $H_0$ are in
$C_b^\infty(\Ri^d)$ then the conclusion is valid for all $\alpha\geq0$.
  Secondly, we prove that if \[ H_0=\sum^N_{i=1}X_i^* X_i, \] where the $X_i$
are vector fields on $\Ri^d$ with coefficients in $C_b^\infty(\Ri^d)$
satisfying a uniform version of H\"ormander's criterion for hypoellipticity,
then $H_0$ satisfies the subellipticity condition for $\gamma=r^{-1}$ where $r$
is the rank of the set of vector fields. Consequently $D(H^n) \subseteq
D(\Delta^{n/r})$ for all $n \in \Ni$, where $H$ is the closure of $H_0$.

Analysis of PDEs