Hypoellipticity and loss of derivatives

Let {X1, . . . ,Xp} be complex-valued vector fields in Rn and assume that they satisfy the bracket condition (i.e. that their Lie algebra spans all vector fields). Our object is to study the operator E = ∑ X∗ i Xi, where X ∗ i is the L2 adjoint of Xi. A result of Hörmander is that when the Xi are real then E is hypoelliptic and furthemore it is subelliptic (the restriction of a destribution u to an open set U is “smoother” then the restriction of Eu to U). When the Xi are complex-valued if the bracket condition of order one is satisfied (i.e. if the {Xi, [Xi,Xj ]} span), then we prove that the operator E is still subelliptic. This is no longer true if brackets of higher order are needed to span. For each k ≥ 1 we give an example of two complex-valued vector fields, X1 and X2, such that the bracket condition of order k+1 is satisfied and we prove that the operator E = X∗ 1X1 + X ∗ 2X2 is hypoelliptic but that it is not subelliptic. In fact it “loses” k derivatives in the sense that, for each m, there exists a distribution u whose restriction to an open set U has the property that the DαEu are bounded on U whenever |α| ≤ m and for some β, with |β| = m − k + 1, the restriction of Dβu to U is not locally bounded.