Calderón–Zygmund Kernels and Their Commutators

This chapter and Chap. 11 are the heart of the subject. TheCalderón–Zygmund kernel Calderón–Zygmund theory of singular integrals continues to be one of the most influential works in modern history of analysis. The first main result (Theorem 10.2) asserts the existence of singular integral operators andBMO (bounded mean oscillation) the function second main result (Theorem 10.3) concerns commutators of bounded mean oscillation functions (BMO) and singular integral operators. It should be emphasized that singular integral operatorsBounded mean oscillation (BMO) with non-smooth kernels provide a powerful tool to deal with smoothness of solutions of partial differential equations, with minimal assumptionsAssumption of regularity on the coefficients (see [26, 28, 107]). The results discussed here are adapted from Coifman–Rochberg–Weiss [39] and Bramanti–Cerutti [17].