Singular integral

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator

${\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,}$

whose kernel function ${\displaystyle K:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }$ is singular along the diagonal ${\displaystyle x=y}$. Specifically, the singularity is such that ${\displaystyle |K(x,y)|}$ is of size ${\displaystyle |x-y|^{-n}}$ asymptotically as ${\displaystyle |x-y|\to 0}$. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over ${\displaystyle |y-x|\to \epsilon }$ as ${\displaystyle \epsilon \to 0}$, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp spaces, for example ${\displaystyle L^{p}(\mathbb {R} ^{n})}$.