Weakly singular kernel

Summary

A weakly singular kernel is a kernel that may become infinite on the diagonal $x = s$. It may grow there only like a power $\abs{x - s}^{-\alpha}$ with exponent $\alpha$ below the dimension. Such a singularity is integrable, so the integral operator still makes sense on continuous functions and is compact there. So the Fredholm theory for continuous kernels carries over. Kernels of this type arise in potential theory and in Abel's integral equation.