Calderón–Zygmund operator

A Calderón–Zygmund operator $T$ on $\mathbb R^n$ is a bounded linear map $L^2\to L^2$ admitting a Calderón–Zygmund kernel $K$ such that

Tf(x)=\int K(x,y)f(y)\,dy

for every compactly supported $L^2$ function $f$ and almost every $x$ outside its support. The integral there is absolutely convergent.

What is included in the definition

The $L^2$ bound is an assumption in addition to the kernel estimates. The definition does not insist that the full operator be exactly a principal-value integral; local terms invisible off the support may be present. This convention includes a multiple of the identity, whose off-diagonal kernel is zero.

References

Tao, Fourier analysis lecture notes 4, §2.