Brian Street Livres

The book presents a groundbreaking theory of multi-parameter singular integrals linked to Carnot-Carathéodory balls, beginning with a comprehensive overview of Calderón-Zygmund singular integrals and their relevance to linear partial differential equations. It explores multi-parameter Carnot-Carathéodory geometry, utilizing a quantitative version of Frobenius's theorem. Through various examples, it illustrates the natural emergence of these integrals in different contexts. The concluding chapter establishes a unifying general theory, appealing to graduate students and researchers in singular integrals and related disciplines.

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.