Singular integral operator

Summary

A singular integral operator is an integral operator $Tf(x) = \int K(x, y)f(y) \, dy$ whose kernel has a non-integrable singularity on the diagonal $x = y$, of size $\abs{x - y}^{-n}$ in $\R^n$. The integral exists only as a principal value, thanks to cancellation in the kernel. The Hilbert transform and the Riesz transforms are the basic examples. Calderón–Zygmund theory shows that such operators are bounded on $L^p$ for $1 < p < \infty$, which is central in harmonic analysis and elliptic PDE.