The paper is a summary of a manuscript of the author and a collaborator, Ragnar Sigurdsson, where they studied the asymptotic Fourier-Laplace transform behaviour of a function \(f \in {\mathcal E}' (\mathbb{R}^ n)\). As a summary, the paper contains a lot of results, densely presented, closely connected with the differential operators with constant coefficients and convolution equations, especially with the plurisubharmonic functions in \(\mathbb{C}^ n\) \((\text{PSH} (\mathbb{C}^ n))\).
Doing a severe sorting of the presented results, we can enumerate:
1) If \(F\), the Fourier-Laplace transform of \(f \in {\mathcal E}' (\mathbb{R}^ n)\) is any entire analytic function such that \[ \bigl | F (\zeta) \bigr | \leq e^{C + A | \zeta |},\;\zeta \in \mathbb{C}^ n, \] then \(u = \log | F |\) belongs to the set \(\text{PSH} (\mathbb{C}^ n)\), and \(u(\zeta) \leq C + A | \zeta |\).
2) The indicator function \(j_ u\) of \(u\), defined in the paper, is a PSH function, homogeneous of degree 1.
3) If \(u \in \text{PSH} (\mathbb{C}^ n)\) satisfies the inequality \[ u(\zeta) \leq C_ \varepsilon + A | \text{Im} \zeta | + \varepsilon | \zeta |,\;\zeta \in \mathbb{C}^ n,\;\varepsilon > 0, \] then the indicator function \(j_ u\) vanishes in \(\mathbb{R}^ n\) and \(j_ u (\zeta) \leq A | \text{Im} \zeta |\), \(\zeta \in \mathbb{C}^ n\).
There are also many other results exposed in the paper, but very difficult to present in such a review, due to their lengths and their references to other papers of the author. Many of them are compared with results of other authors.