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This book develops a new theory of multi-parameter singular integrals associated with Carnot-Carathéodory balls. Brian Street first details the classical theory of Calderón-Zygmund singular integrals and applications to linear partial differential equations. He then outlines the theory of multi-parameter Carnot-Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. Street then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. Multi-parameter Singular Integrals will interest graduate students and researchers working in singular integrals and related fields.

Singular integrals. --- Transformations (Mathematics) --- CaldernКygmund singular integrals. --- CaldernКygmund. --- CarnotЃarathodory balls. --- CarnotЃarathodory geometry. --- CarnotЃarathodory metric. --- Euclidean singular integral operators. --- Frobenius theorem. --- Frobenius. --- LittlewoodАaley theory. --- Schwartz space. --- Sobolev spaces. --- convolution. --- elliptic partial differential equations. --- elliptic partial differential operators. --- flag kernels. --- invariant operators. --- linear partial differential equation. --- non-homogeneous kernels. --- pseudodifferential operators. --- singular integral operator. --- singular integral operators. --- singular integrals. --- strengthened cancellation.