Uniform Subellipticity

We prove that uniform subellipticity of a positive symmetric second-order partial differential operator on L₂(R^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 N operator. Secondly, if the operator is of the form X_i X_i, where the Xi are ^N∑ _i=1, vector fields on Rd with coefficients in C^∞_b (R^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.