Calderón–Zygmund kernel

A Calderón–Zygmund kernel on $\mathbb R^n$ is a measurable function $K(x,y)$ , defined for $x\ne y$ , such that for some $C>0$ and $0<\alpha\le1$ ,

|K(x,y)|\le C|x-y|^{-n},

and, whenever $|x-x'|\le|x-y|/2$ ,

|K(x,y)-K(x',y)|+|K(y,x)-K(y,x')| \le C\frac{|x-x'|^\alpha}{|x-y|^{n+\alpha}}.

The second condition is a scale-dependent Hölder bound in each variable.

Kernel versus operator

These estimates concern points away from the diagonal. They neither define a value on the diagonal nor by themselves guarantee a bounded singular-integral operator. In particular, an operator may contain a multiple of the identity that cannot be read from its off-diagonal kernel.

References

Tao, Fourier analysis lecture notes 4, §2.