Comparison Tauberian theorems and hyperbolic operators with constant coefficients

As comparison Tauberian theorems one usually means theorems which by a prescribed asymptotic behavior of the ratio of some integral transforms of two (generalized) functions make a conclusion on asymptotic behavior of other integral transformations of these functions. In the work we prove the comparison Tauberian function for the generalized functions whose Laplace transform have a bounded argument. In particular, examples of these functions are the kernels and the fundamental solutions of differential operators with constant coefficients hyperbolic w.r.t. a cone.