Dedicated to Yum-Tong Siu for his 60th birthday. Abstract Let {X1 , . . . , Xp } be complex-valued vector fields in Rn and assume that they satisfy the bracket condition (. that their Lie algebra spans all vector fields). Our object is to study the operator E = Xi∗ Xi , where Xi∗ is the L2 adjoint of Xi . A result of H¨rmander is that when the Xi are real then E is o hypoelliptic and furthemore it is subelliptic (the restriction of a destribution u to an open set U is “smoother” then the restriction.
