Boundary behaviour of hypersingular potentials

In this article we study the boundary behaviour of surface potentials on the plane $mathbb{R}^{n-1}$ ($ngeq 2$) with hypersingular kernels. We show the existence of pointwise and ``weak'' one-sided limits for such hypersingular integrals. The pointwise result will be also expressed in terms of Hadamard finite-part integrals. Moreover, we determine a two-sided jump relation which yields necessary and sufficient conditions for the validity of a Lyapunov-Tauber property. The paper ends with some applications to harmonic, elastic and Stokes potentials.