The purpose of the paper is to study the behavior at infinity of Fourier-Laplace transforms of distributions or more generally plurisubharmonic functions \(u\) in \({\mathbb C}^n\), \(u\in PSH({\mathbb C}^n)\). If \(H\) is a supporting function in \({\mathbb R}^n\), that is, \(H\) is convex and positively homogeneous of degree 1, then \(P_H\) will denote the set of \(u\in PSH({\mathbb C}^n)\) such that \[ u(\zeta)\leq H(\Im\zeta),\quad\zeta\in{\mathbb C}^n;\quad u(\zeta)=H(\Im\zeta),\quad\zeta\in{\mathbf CR}^n . \] The set \(L_\infty(u)\) of limits of \(T_tu=u(t\cdot)/t\) as \(t\to +\infty\) is a compact \(T\) invariant subset of the set \(P_H\). In the paper some basic properties of \(P_H\) are proved when \(H\) has various smoothness properties, in particular, when the first derivatives are Lipschitz continuous in \({\mathbb R}^n \backslash\{0\}\), then \[ \frac 1{C_{2n}r^{2n}}\int_{B(\zeta,r)}|u(w)-H(\Im w)|d\lambda(w) \leq Mr(r/(r+|\Im\zeta|))^k , \] \[ \frac 1{C_{2n}r^{2n}}\int_{B(0,r)}|u(\zeta+w)-H(\Im\zeta) - \langle H'(\Im\zeta),\Im w\rangle|d\lambda(w) \leq 2Mr(r/(r+|\Im\zeta|))^k , \] if \(\zeta\in{\mathbb CR}^n\). Here \(B(\zeta,r)\) is the ball\(\subset{\mathbb C}^n\) with center \(\zeta\) and radius \(r\), and \(C_{2n}r^{2n}\) denotes its volume. When \(\theta\in{\mathbb R}^n\), \(\varphi\in{\mathbb R}\), and \(\zeta\in{\mathbb CR}^n\), we have with a constant \(C\) depending only on the dimension, if \(|\theta|