Multidimensional Tuberian comparison theorems for generalized functions in cones

This article deals with the proofs of some multidimensional Tauberian comparison theorems for generalized functions with supports in homogeneous cones, in particular, for measures and functions whose Laplace transforms have nonnegative imaginary parts. “Admissible” generalized functions, which can be regarded as multidimensional analogues of the so-called $R$-$O$-functions of Karamata, serve as comparison functions in these theorems. For circular and $n$-faced cones a criterion is obtained for admissibility which generalizes the well-known Keldysh Tauberian condition to the multidimensional case.
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