Hypersingular integral equation

Remarks

More generally, an integral equation on a domain or surface of dimension $m$ is called hypersingular when its kernel grows like $\abs{x - t}^{-m-1}$ or faster near the diagonal.

Example: the first kind equation $\frac{1}{\pi}\, \mathrm{f.p.}\!\int_{-1}^{1} \frac{\varphi(t)}{(t - x)^2}\, dt = -1$ has the solution $\varphi(t) = \sqrt{1 - t^2}$. Check: the finite part is the $x$-derivative of $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}}{t - x}\, dt = -x$, which is $-1$. Here $\mathrm{PV}$ is the Cauchy principal value. Equations of this form, with the condition $\varphi(\pm 1) = 0$, describe the opening of a straight crack under uniform pressure.

Non-example: $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt = f(x)$ has a Cauchy kernel of order $1$, so it is a singular but not a hypersingular equation.

The finite part is the derivative of a principal value. So, when $\varphi(\pm 1) = 0$, an integration by parts turns a hypersingular equation of the first kind for $\varphi$ into a Cauchy singular equation for $\varphi'$.