On single-layer potentials, pseudo-gradients and a jump theorem for an isotropic $$\alpha $$ -stable stochastic process

Abstract

The aim of this paper is a behavior investigation of pseudo-gradients with respect to the spatial variable of a single-layer potential if the spatial point tends to some point in the carrier surface of the potential. The potentials connect to the generator of an isotropic \(\alpha \)-stable stochastic process with the power \(\alpha \in (1,2]\). This generator is the fractional Laplacian of the order \(\alpha \). Pseudo-gradient or fractional gradient is the pseudo-differential operator of the gradient type. Its order \(\beta \) is a positive number less than \(\alpha \). The jump theorem is known in the case of \(\beta =\alpha -1\). We present here the corresponding results in the cases of \(\beta <\alpha -1\) and \(\beta >\alpha -1\). In the first case there are no jumps, and in the second case there are no finite limits of the fractional gradients of the single-layer potential, when the spatial argument tends to some point placed on the potential carrier.