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 $\gamma \in <0,1]$, uniformly for all $ϕ\in W^{\infty,2}(\Ri^d)$, 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 \[ c \|(I+H)^αϕ\|_2\geq \|Δ^{αγ} ϕ\|_2 \] is valid for all $\varphi \in D(H^{\alpha })$ 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 $\alpha \ge 0$. 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'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 $H_0$.

24 pages

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