Calderón–Zygmund kernel

Summary

A Calderón–Zygmund kernel is a function $K(x, y)$ on $\R^n \times \R^n$ off the diagonal. It is at most of size $\abs{x - y}^{-n}$, and it is smooth in a quantitative way away from the diagonal. These are the kernels of the singular integral operators of Calderón–Zygmund theory, such as the Hilbert and Riesz transforms. The size and smoothness conditions are what turn an $L^2$ bound into $L^p$ bounds for $1 < p < \infty$.