The Calder´on-Zygmund Theory II: Maximal Hypoellipticity

This chapter remains in the single-parameter case and turns to the case when the metric is a Carnot–Carathéodory (or sub-Riemannian) metric. It defines a class of singular integral operators adapted to this metric. The chapter has two major themes. The first is a more general reprise of the trichotomy described in Chapter 1 (Theorem 2.0.29). The second theme is a generalization of the fact that Euclidean singular integral operators are closely related to elliptic partial differential equations. The chapter also introduces a quantitative version of the classical Frobenius theorem from differential geometry. This “quantitative Frobenius theorem” can be thought of as yielding “scaling maps” which are well adapted to the Carnot–Carathéodory geometry, and is of central use throughout the rest of the monograph.