Uniform subellipticity

Abstract

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 $H_0$.