On the Behaviour of the Distributional Stieltjes Transformation at the Origin

Abstract

The behaviour of the distributional Stieltjes transformation at the origin 0 is investigated for distributions of ${\mathcal{S}}_{+}^{\prime }$ having appropriate quasiasymptotic behaviour at $0^{+}$. These new results follow by known ones for the asymptotic behaviour at $\infty$. A Tauberian-type result for the behaviour at 0 is also obtained.