Multidimensional Abelian and Tauberian comparison theorems

Theorems in which a specified asymptotic behavior of the quotient of two (generalized) functions leads to a conclusion about the asymptotic behavior of the quotient of integral transforms of them are called Abelian comparison theorems. The theorems converse to them are called Tauberian comparison theorems. This article concerns some Abelian and Tauberian comparison theorems for generalized functions with supports in pointed cones. The Laplace transform is used as an integral transform. It is shown that additional “Abelian” conditions are needed for the validity of Abelian theorems in the multidimensional case.
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