Maximally subelliptic partial differential equations (PDE) are a far-reaching generalization of elliptic PDE. Elliptic PDE hold a special place: sharp results are known for general linear and even fully nonlinear elliptic PDE. Over the past half-century, important results for elliptic PDE have been generalized to maximally subelliptic PDE. This text presents this theory and generalizes the sharp, interior regularity theory for general linear and fully nonlinear elliptic PDE to the maximally subelliptic setting.
