Geodesic
The Laplace Transform of Generalized Functions
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 357-374

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In this paper we develop a theory of Laplace transforms for generalized functions. Some fundamental results in this field were given by Schwartz in (3) for the n-dimensional bilateral case from the point of view of topological vector spaces, and in (4) in a form amenable to operational use. Our presentation characterizes a one-dimensional theory of Laplace transforms with a half-plane of convergence (indicating an extension of the usual classical transform) and with the property that Laplace transforms are analytic functions satisfying the fundamental convolution-multiplication theorem. Section 1 is devoted to defining the Laplace transform of generalized functions and also to showing how the property of a half-plane of convergence is intrinsic to this definition. 
Benedetto, John. The Laplace Transform of Generalized Functions. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 357-374. doi: 10.4153/CJM-1966-038-5
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