Geodesic
Tauberian theorems for generalized functions with supports in~cones
Sbornik. Mathematics, Tome 36 (1980) no. 1, pp. 75-86

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In this article the authors prove several multidimensional theorems of Tauberian type, connecting the behavior at infinity of generalized functions with support in a cone with the behavior of their Fourier–Laplace transforms in a neighborhood of zero. As corollaries they deduce a strengthened version of V. S. Vladimirov's Tauberian theorem and an analog of the theorem of Lindelöf for the edge of a tube domain over a cone. Bibliography: 5 titles. 
@article{SM_1980_36_1_a4,
     author = {Yu. N. Drozhzhinov and B. I. Zavialov},
     title = {Tauberian theorems for generalized functions with supports in~cones},
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     volume = {36},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1980_36_1_a4/}
}
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Yu. N. Drozhzhinov; B. I. Zavialov. Tauberian theorems for generalized functions with supports in~cones. Sbornik. Mathematics, Tome 36 (1980) no. 1, pp. 75-86. http://geodesic.mathdoc.fr/item/SM_1980_36_1_a4/