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 \[ c (��, (I+H_0)��)\geq \|��^{��/2} ��\|_2^2 \] for some $c>0$ and $��\in<0,1]$, uniformly for all $��\in W^{\infty,2}(\Ri^d)$, where $��$ denotes the usual Laplacian. Then we prove that $D(H^��) \subseteq D(��^{����})$ for all $��\in [0,2^{-1} (m + 1 + ��^{-1})>$. Hence there is a $c>0$ such that the norm estimate \[ c \|(I+H)^����\|_2\geq \|��^{����} ��\|_2 \] is valid for all $��\in D(H^��)$ 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 $��\geq0$. 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��rmander's criterion for hypoellipticity, then $H_0$ satisfies the subellipticity condition for $��=r^{-1}$ where $r$ is the rank of the set of vector fields. Consequently $D(H^n) \subseteq D(��^{n/r})$ for all $n \in \Ni$, where $H$ is the closure of $H_0$.

24 pages