Let be complex-valued vector fields in and assume that they satisfy the bracket condition (i.e. that their Lie algebra spans all vector fields), and , where is adjoint of . The operator is subelliptic at point if there exists a neighborhood of , a real number , and a constant , such that
for all . The author studies whether is hypoelliptic and whether it satisfies the subelliptic estimate (1).
If spans the complex tangent space at the origin, then a subelliptic estimate
is satisfied, with .
For there exist complex vector fields and on a neighborhood of the origin in such that the two vectorfields and their commutators of order span the complexified tangent space at the origin, and when the subelliptic estimate (1) does not hold. Moreover, when , the operator loses derivatives in the sup norms and derivatives in the Sobolev norms.
If and are the vector fields given in Theorem 2 then the operator is hypoelliptic. More precisely, if is a distribution solution of with and if is an open set such that , then . Theorem 2 shows that the loss of derivatives is .
The author introduces subelliptic multipliers to establish subelliptic estimates for the -Neumann problem. To prove these theorems he uses subelliptic multipliers.