Inversion formulas for the Laplace transform. I: The octant (Q1082588)

The author derived, in the first part, some inversion formulas for the Laplace transforms of tempered distributions with support in an n- dimensional octant. In the second part of this paper, it is shown that these inversion formulas are useful for studying the inverse of the Laplace transformation of generalized functions with support in a light cone of arbitrary dimension. Denoting the space of tempered distributions with support on the positive half-axis by \({\mathcal S}'(\bar R_+)\), the author defines the Laplace transform in accordance with [\textit{V. S. Vladimirov}, Methods of the theory of functions of many complex variables, (Russian) (1964; Zbl 0125.319), p. 280] of a generalized function \(f\in {\mathcal S}'(\bar R_+)\) as follows: A function F(z), that is holomorphic in the upper half-plane, is the Laplace transform \(F(z)=(f(t),e^{izt})\) of a generalized function \(f\in {\mathcal S}'(\bar R_+)\) if and only if it satisfies the estimate \[ | F(x+iy)| \leq c(1+| z|^ n)y^{-p} \] for certain positive c, n and p. Then the author proves two theorems and concludes the paper by remarking that a formula occurring in theorem 2 is a modification of the inversion formula in [\textit{Yu. M. Zinov'ev}, Teor. Mat. Fiz. 38, 153- 162 (1979; Zbl 0409.46044)] for the Laplace transform of generalized functions with support on the half-axis [\(\mu\),\(\infty)\) where \(\mu >0\).