A Remark on Sums of Squares of Complex Vector Fields

Let {Zj} be a finite collection of vector fields, with smooth complex-valued coefficients, defined in an open subset U of Euclidean space. Let Z∗ j be the formal adjoint of Zj , with respect to the Hilbert space structure L2 associated to some measure with a smooth nonvanishing density. Consider the operator L = ∑ j Z ∗ jZj, which we shall refer to as a sum of squares. L is said to be hypoelliptic in U if for any open subset V ⊂ U and any distribution u ∈ D′(V ) such that L(u) ∈ C∞(V ), necessarily u ∈ C∞(V ). Assume throughout this paragraph only that all vector fields are real. Then a well-known sufficient condition for hypoellipticity is the bracket condition of Hörmander, that the Lie algebra generated by {Zj} should span the tangent space to U at each of its points. This condition ensures, and is equivalent to, the condition that L is subelliptic in the sense that for any relatively compact open subset V ⋐ U , there exist ε > 0 and C <∞ such that for all u ∈ C2 0 (V ),