Subelliptic Operators

Summary

If P is an elliptic operator of order m in a C∞ manifold X then Pu∈H ^loc_(s) ; implies u∈H ^loc_(s+m) (Theorem 18.1.29). This result can be microlocalized (Theorem 18.1.31): If Pu∈H ^loc_(s) at a point in the cotangent bundle where P is non-characteristic then u∈H ^loc_(s+m) there. This is the strongest possible result on (micro-)hypoellipticity.

The purpose of this chapter is to give a complete study of the next simplest case where Pu∈H ^loc_(s) ; implies u ^loc_(s+m-δ) for some fixed δ∈(0,1). One calls P subelliptic with loss of δ derivatives then. The condition δ< 1 guarantees that subellipticity is only a condition on the principal symbol.

In Section 26.4 we have already seen that condition(Ψ) is necessary for hypoellipticity. In Section 27.1 another necessary condition on the principal symbol of a subelliptic operator is obtained by a scaling argument. These results together suggest the necessary and sufficient condition for sub-ellipticity stated as Theorem 27.1.11. However, to prove the necessity completely we also need a symplectic study of the Taylor expansion of the symbol given in Section 27.2. The general proof of sufficiency is long so we give a short proof for operators satisfying condition (P) in Section 27.3. Section 27.4 is devoted to a detailed discussion of the local properties of a general subelliptic symbol. The proof of Theorem 27.1.11 is then completed in Sections 27.5 and 27.6 by means of a localization argument in several steps.