  Let $P(\partial_0,\partial_1,...,\partial_n)$ be a PDO on $\symR^{1+n}$ with
constant coefficients. It is proved that (i) the real parts of the
$\lambda$-roots of the polynomial $P(\lambda,i\xi_1,...,i\xi_n)$ are bounded
from above when $(\xi_1,...,\xi_n)$ ranges over $\symR^n$ if and only if (ii)
$P$ has a fundamental solution with support in $H_+=\{(x_0,x_1,\allowbreak...,
x_n)\in \symR^{1+n}:x_0\ge0\}$ having some special properties expressed in
terms of the L. Schwartz space $\calO^{\prime}_C$ of rapidly decreasing
distributions. Moreover, it is proved that the fundamental solution with
support in $H_+$ having these special properties is unique.
