Calderón–Zygmund Variable Kernels and Their Commutators

In this chapter we consider singular integrals with kernels depending on a parameter, and prove theorems about singular integrals and commutators of \(L^{\infty }\) functions and singular integral operators (Theorems 11.2 and 11.3), generalizing Theorems 10.2 and 10.3 in Chap. 10. The main idea of proof is to reduce the variable kernel case to the constant kernel case. This is done by expanding the kernel into a series of spherical harmonics (Theorem 4.41), each term defining a constant kernel operator treated in Chap. 10.