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 \[ c (ϕ, (I+H_0)ϕ)\geq \|Δ^{γ/2} ϕ\|_2^2 \] for some $c>0$ and $γ\in<0,1]$, uniformly for all $ϕ\in W^{\infty,2}(\Ri^d)$, where $Δ$ denotes the usual Laplacian. Then we prove that $D(H^α) \subseteq D(Δ^{αγ})$ for all $α\in [0,2^{-1} (m + 1 + γ^{-1})>$. Hence there is a $c>0$ such that the norm estimate \[ c \|(I+H)^αϕ\|_2\geq \|Δ^{αγ} ϕ\|_2 \] is valid for all $ϕ\in D(H^α)$ 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 $α\geq0$. 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 Hörmander&#39;s criterion for hypoellipticity, then $H_0$ satisfies the subellipticity condition for $γ=r^{-1}$ where $r$ is the rank of the set of vector fields. Consequently $D(H^n) \subseteq D(Δ^{n/r})$ for all $n \in \Ni$, where $H$ is the closure of $H_0$.