Local Tauberian theorems in spaces of distributions related to cones, and their applications

In this article we introduce and study special spaces of distributions related to a given cone. These spaces occupy an intermediate position between the space of temperate distributions and the class of distributions concentrated on a cone. The properties of these spaces are investigated. In particular, we prove that they are convolution algebras. Quasi-asymptotic properties of distributions belonging to these spaces are thoroughly studied. To this end we prove several complex Tauberian and Abelian theorems in which the role of the integral transformation is played by the Laplace transformation. This transformation establishes an isomorphism between these spaces and the classes of functions holomorphic in special wedge-shaped domains. These results are applied to the study of the asymptotic behaviour of functions holomorphic in wedge-shaped domains at boundary points. A local theorem on non-compensation of singularities of holomorphic functions is proved.