# Journal of Operator Theory

Volume 62, Issue 1, Summer 2009 pp. 125-149.

Uniform subellipticity**Authors**: A.F.M. ter Elst (1) and Derek W. Robinson (2)

**Author institution:**(1) Department of Mathematics, University of Auckland, Auckland, New Zealand

(2) Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia

**Summary:**$\newcommand{\Ri}{\mathcal{R}}$ We prove that uniform subellipticity of a positive symmetric second order partial differential operator on $L_2(\Ri^d)$ is self-improving in the sense that it automatically extends to higher powers of the operator. The range of extension is governed by the degree of smoothness of the coefficients of the operator. Secondly, if the operator is of the form $\sum\limits^N_{i=1}X_i^* \, X_i$, where the $X_i$ are vector fields on $\Ri^d$ with coefficients in $C_{\mathrm b}^\infty(\Ri^d)$ satisfying a uniform version of Hörmander's criterion for hypoellipticity, then we prove that it is uniformly subelliptic of order~$r^{-1}$, where $r$ is the rank of the set of vector fields.

**Keywords:**Subelliptic operator, Hormander sums of squares, double commutators.

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