Singular integral operator ($Tf(x)=\mathrm{PV}\int K(x,t)f(t)dt$)

A singular integral operator has a kernel whose ordinary integral can fail at points on the integration domain. A specified limiting prescription, such as a Cauchy principal value, can still define the operator by cancellation. For the Hilbert transform kernel $1/(t-x)$, subtracting $f(x)$ from a locally Hölder-continuous density leaves $[f(t)-f(x)]/(t-x)$ locally integrable; the constant part is interpreted by symmetric deletion. The limiting prescription and the class of allowed densities are part of the operator's definition, not optional details.

An integral equation with a kernel singular on the integration contour needs a prescribed limiting interpretation such as a Cauchy principal value. For the Cauchy kernel, the Sokhotski–Plemelj theorem converts a linear scalar equation into a Riemann-Hilbert problem for its two analytic boundary values. Normalization, endpoint regularity and the index of that jump determine homogeneous freedom and solvability.