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Spaces of generalized functions known as Boehmians contain all Schwartz distributions as well as all ultradistributions of Beurling and Roumieu type. Even though these spaces are quite large, it is still possible to investigate local behavior. In this note, ultradifferential operators are introduced in the context of Boehmians. As an application, some Abelian type theorems are established for the unilateral Laplace transform. [ABSTRACT FROM AUTHOR]

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 ( α,ρ)( x) of “extreme” stable laws with parameters ( α, ρ) for ρ = ±1 and x lying in the domain of rapid decay of f ( α,ρ)( x). This asymptotics had been found in [1–5] by a more complicated method. [ABSTRACT FROM AUTHOR]