# Subelliptic Operators

## Summary

If *P* is an elliptic operator of order *m* in a *C*∞ manifold *X* then *Pu*∈*H* _{(s)} ^{loc} ; implies *u*∈*H* _{(s+m)} ^{loc} (Theorem 18.1.29). This result can be microlocalized (Theorem 18.1.31): If *Pu*∈*H* _{(s)} ^{loc} at a point in the cotangent bundle where *P* is non-characteristic then *u*∈*H* _{(s+m)} ^{loc} 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* _{(s)} ^{loc} ; implies *u* _{(s+m-δ)} ^{loc} 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.

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## Notes

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