Hypoellipticity and loss of derivatives with Appendix Analyticity and   loss of derivatives
  For each value of k, two complex vector fields satisfying the bracket condition are exhibited the sum of whose squares is hypoelliptic but not subelliptic - in fact the operator loses k-1 derivatives in Sobolev norms. In the Appendix it is proven that this operator is analytic hypoelliptic. 
Joseph J. Kohn, Makhlouf Derridj and David S. Tartakoff