Maximal Subellipticity

Maximally subelliptic partial differential equations (PDEs) are a far-reaching generalization of elliptic PDEs. Elliptic PDEs hold a special place: sharp results are known for general linear and even fully nonlinear elliptic PDEs. Over the past half-century, important results for elliptic PDEs have been generalized to maximally subelliptic PDEs. This text presents this theory and generalizes the sharp, interior regularity theory for general linear and fully nonlinear elliptic PDEs to the maximally subelliptic setting. Discusses sharp regularity theory for linear and fully nonlinear maximally subelliptic PDEs. Covers Gaussian bounds for the heat equation for maximally subelliptic PDEs. Presents function spaces adapted to maximally subelliptic PDEs. Contents 1 Introduction 2 Ellipticity 3 Vector fields and Carnot–Carathéodory geometry 4 Pseudo-differential operators 5 Singular integrals 6 Besov and Triebel–Lizorkin spaces 7 Zygmund–Hölder spaces 8 Linear maximally subelliptic operators 9 Nonlinear maximally subelliptic equations A Canonical coordinates Bibliography Symbol Index Index