Tauberian and Abelian theorems for rapidly decaying distributions and their applications to stable laws

We establish some assertions of Tauberian and Abelian types which enable us to find connections between the asymptotic properties of the Laplace transform at infinity and the asymptotics of the corresponding densities of rapidly decaying distributions (at infinity or in some neighborhood of zero). As applications of our Tauberian type theorems we present asymptotics for the density $f^{(\alpha,\rho)}(x)$ of “extreme” stable laws with parameters $(\alpha,\rho)$ for $\rho=\pm1$ and $x$ lying in the domain of rapid decay of $f^{(\alpha,\rho)}(x)$. This asymptotics had been found in [1–5] by a more complicated method.